armadillo
Armadillo
C++ linear algebra library
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[top] API Documentation for Armadillo 7.400


Preamble

 
  • Please cite the following tech report if you use Armadillo in your research and/or software. Citations are useful for the continued development and maintenance of the library.

    Conrad Sanderson and Ryan Curtin. Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software, Vol. 1, pp. 26, 2016.


Overview
Matrix, Vector, Cube and Field Classes
Member Functions & Variables
Generated Vectors/Matrices/Cubes
    eye generate identity matrix
    linspace generate vector with linearly spaced elements
    logspace generate vector with logarithmically spaced elements
    ones generate object filled with ones
    randi generate object with random integer values in specified interval
    randu / randn generate object with random values (uniform and normal distributions)
    randg generate object with random values (gamma distribution)
    regspace generate vector with regularly spaced elements
    speye generate sparse identity matrix
    spones generate sparse matrix with non-zero elements set to one
    sprandu / sprandn generate sparse matrix with non-zero elements set to random values
    toeplitz generate Toeplitz matrix
    zeros generate object filled with zeros

Functions of Vectors/Matrices/Cubes
    abs obtain magnitude of each element
    accu accumulate (sum) all elements
    all check whether all elements are non-zero, or satisfy a relational condition
    any check whether any element is non-zero, or satisfies a relational condition
    approx_equal approximate equality
    as_scalar convert 1x1 matrix to pure scalar
    clamp obtain clamped elements according to given limits
    cond condition number of matrix
    conj obtain complex conjugate of each element
    conv_to convert between matrix types
    cross cross product
    cumsum cumulative sum
    cumprod cumulative product
    det / log_det determinant
    diagmat generate diagonal matrix from given matrix or vector
    diagvec extract specified diagonal
    diff differences between adjacent elements
    dot/cdot/norm_dot dot product
    eps obtain distance of each element to next largest floating point representation
    expmat matrix exponential
    expmat_sym matrix exponential (symmetric)
    find find indices of non-zero elements, or elements satisfying a relational condition
    find_finite find indices of finite elements
    find_nonfinite find indices of non-finite elements
    find_unique find indices of unique elements
    fliplr / flipud reverse order of columns or rows
    imag / real extract imaginary/real part
    ind2sub convert linear index to subscripts
    index_min / index_max indices of extremum values
    inplace_trans in-place transpose
    is_finite check whether all elements are finite
    join_rows / join_cols concatenation of matrices
    join_slices concatenation of cubes
    kron Kronecker tensor product
    logmat matrix logarithm
    logmat_sympd matrix logarithm (symmetric)
    min / max return extremum values
    nonzeros return non-zero values
    norm various norms of vectors and matrices
    normalise normalise vectors to unit p-norm
    prod product of elements
    rank rank of matrix
    rcond reciprocal of condition number
    repmat replicate matrix in block-like fashion
    reshape change size while keeping elements
    resize change size while keeping elements and preserving layout
    shift shift elements
    shuffle randomly shuffle elements
    size obtain dimensions of given object
    sort sort elements
    sort_index vector describing sorted order of elements
    sqrtmat square root of matrix
    sqrtmat_sympd square root of matrix (symmetric)
    sum sum of elements
    sub2ind convert subscripts to linear index
    symmatu / symmatl generate symmetric matrix from given matrix
    trace sum of diagonal elements
    trans transpose of matrix
    trapz trapezoidal numerical integration
    trimatu / trimatl generate triangular matrix from given matrix
    unique return unique elements
    vectorise convert matrix to vector
    misc functions miscellaneous element-wise functions: exp, log, pow, sqrt, round, sign, ...
    trig functions trigonometric element-wise functions: cos, sin, ...

Decompositions, Factorisations, Inverses and Equation Solvers (Dense Matrices)
    chol Cholesky decomposition
    eig_sym eigen decomposition of dense symmetric/hermitian matrix
    eig_gen eigen decomposition of dense general square matrix
    eig_pair eigen decomposition for pair of general dense square matrices
    inv inverse of general square matrix
    inv_sympd inverse of symmetric positive definite matrix
    lu   lower-upper decomposition
    null orthonormal basis of null space
    orth orthonormal basis of range space
    pinv pseudo-inverse
    qr   QR decomposition
    qr_econ economical QR decomposition
    qz   generalised Schur decomposition
    schur Schur decomposition
    solve solve systems of linear equations
    svd singular value decomposition
    svd_econ economical singular value decomposition
    syl Sylvester equation solver

Decompositions, Factorisations and Equation Solvers (Sparse Matrices)
    eigs_sym limited number of eigenvalues & eigenvectors of sparse symmetric real matrix
    eigs_gen limited number of eigenvalues & eigenvectors of sparse general square matrix
    spsolve solve sparse systems of linear equations
    svds limited number of singular values & singular vectors of sparse matrix

Signal & Image Processing
Statistics & Clustering
    stats functions mean, median, standard deviation, variance
    cov covariance
    cor correlation
    hist histogram of counts
    histc histogram of counts with user specified edges
    princomp principal component analysis
    running_stat running statistics of one dimensional process/signal
    running_stat_vec running statistics of multi-dimensional process/signal
    kmeans cluster data into disjoint sets
    gmm_diag model data as a Gaussian Mixture Model (GMM)

Miscellaneous




Matrix, Vector, Cube and Field Classes



Mat<type>
mat
cx_mat
  • The root matrix class is Mat<type>, where type is one of:
    • float, double, std::complex<float>, std::complex<double>, short, int, long, and unsigned versions of short, int, long

  • For convenience the following typedefs have been defined:
      mat  =  Mat<double>
      fmat  =  Mat<float>
      cx_mat  =  Mat<cx_double>
      cx_fmat  =  Mat<cx_float>
      umat  =  Mat<uword>
      imat  =  Mat<sword>

  • In this documentation the mat type is used for convenience; it is possible to use other types instead, eg. fmat

  • Functions which use LAPACK or ATLAS (generally matrix decompositions) are only valid for the following types: mat, fmat, cx_mat, cx_fmat

  • Elements are stored with column-major ordering (ie. column by column)

  • Constructors:
      mat()
      mat(n_rows, n_cols)
      mat(n_rows, n_cols, fill_type)
      mat(size(X))
      mat(size(X), fill_type)
      mat(mat)
      mat(sp_mat)
      mat(vec)
      mat(rowvec)
      mat(initializer_list)
      mat(string)
      mat(std::vector)   (treated as a column vector)
      cx_mat(mat,mat)   (for constructing a complex matrix out of two real matrices)

  • When specifying the size with n_rows and n_cols, by default the memory is uninitialised; memory can be initialised by specifying the fill_type, which is one of: fill::zeros, fill::ones, fill::eye, fill::randu, fill::randn, fill::none, with the following meanings:
      fill::zeros = set all elements to 0
      fill::ones = set all elements to 1
      fill::eye = set the elements along the main diagonal to 1 and off-diagonal elements to 0
      fill::randu = set each element to a random value from a uniform distribution in the [0,1] interval
      fill::randn = set each element to a random value from a normal/Gaussian distribution with zero mean and unit variance
      fill::none = do not modify the elements

  • The string format for the constructor is elements separated by spaces, and rows denoted by semicolons. For example, the 2x2 identity matrix can be created using "1 0; 0 1".
    Caveat: string based initialisation is slower than directly setting the elements or using element initialisation.

  • Advanced constructors:

      mat(ptr_aux_mem, n_rows, n_cols, copy_aux_mem = true, strict = false)

        Create a matrix using data from writable auxiliary (external) memory, where ptr_aux_mem is a pointer to the memory. By default the matrix allocates its own memory and copies data from the auxiliary memory (for safety). However, if copy_aux_mem is set to false, the matrix will instead directly use the auxiliary memory (ie. no copying); this is faster, but can be dangerous unless you know what you are doing!

        The strict parameter comes into effect only when copy_aux_mem is set to false (ie. the matrix is directly using auxiliary memory)
        • when strict is set to false, the matrix will use the auxiliary memory until a size change
        • when strict is set to true, the matrix will be bound to the auxiliary memory for its lifetime; the number of elements in the matrix can't be changed
        • the default setting of strict in versions 6.000+ is false
        • the default setting of strict in versions 5.600 and earlier is true

      mat(const ptr_aux_mem, n_rows, n_cols)

        Create a matrix by copying data from read-only auxiliary memory, where ptr_aux_mem is a pointer to the memory

      mat::fixed<n_rows, n_cols>

        Create a fixed size matrix, with the size specified via template arguments. Memory for the matrix is allocated at compile time. This is generally faster than dynamic memory allocation, but the size of the matrix can't be changed afterwards (directly or indirectly).

        For convenience, there are several pre-defined typedefs for each matrix type (where the types are: umat, imat, fmat, mat, cx_fmat, cx_mat). The typedefs specify a square matrix size, ranging from 2x2 to 9x9. The typedefs were defined by simply appending a two digit form of the size to the matrix type -- for example, mat33 is equivalent to mat::fixed<3,3>, while cx_mat44 is equivalent to cx_mat::fixed<4,4>.

      mat::fixed<n_rows, n_cols>(const ptr_aux_mem)

        Create a fixed size matrix, with the size specified via template arguments; data is copied from auxiliary memory, where ptr_aux_mem is a pointer to the memory


  • Examples:
      mat A(5, 5, fill::randu);
      double x = A(1,2);
      
      mat B = A + A;
      mat C = A * B;
      mat D = A % B;
      
      cx_mat X(A,B);
      
      B.zeros();
      B.set_size(10,10);
      B.ones(5,6);
      
      B.print("B:");
      
      mat::fixed<5,6> F;
      
      double aux_mem[24];
      mat H(&aux_mem[0], 4, 6, false);  // use auxiliary memory
      

  • Caveat: For mathematical correctness, scalars are treated as 1x1 matrices during initialisation. As such, the code below will not generate a 5x5 matrix with every element equal to 123.0:
      mat A(5,5);  A = 123.0;
      
    Use the following code instead:
      mat A(5,5);  A.fill(123.0);
      

  • See also:



Col<type>
vec
cx_vec
  • Classes for column vectors (matrices with one column)

  • The Col<type> class is derived from the Mat<type> class and inherits most of the member functions

  • For convenience the following typedefs have been defined:
      vec  =  colvec  =  Col<double>
      fvec  =  fcolvec  =  Col<float>
      cx_vec  =  cx_colvec  =  Col<cx_double>
      cx_fvec  =  cx_fcolvec  =  Col<cx_float>
      uvec  =  ucolvec  =  Col<uword>
      ivec  =  icolvec  =  Col<sword>

  • In this documentation, the vec and colvec types have the same meaning and are used interchangeably

  • In this documentation, the types vec or colvec are used for convenience; it is possible to use other types instead, eg. fvec, fcolvec

  • Functions which take Mat as input can generally also take Col as input. Main exceptions are functions which require square matrices

  • Constructors
      vec()
      vec(n_elem)
      vec(n_elem, fill_type)
      vec(size(X))
      vec(size(X), fill_type)
      vec(vec)
      vec(mat)   (a std::logic_error exception is thrown if the given matrix has more than one column)
      vec(initializer_list)
      vec(string)   (elements separated by spaces)
      vec(std::vector)
      cx_vec(vec,vec)   (for constructing a complex vector out of two real vectors)

  • When specifying the size with n_elem, by default the memory is uninitialised; memory can be initialised by specifying the fill_type, as per the Mat class

  • Advanced constructors:

      vec(ptr_aux_mem, number_of_elements, copy_aux_mem = true, strict = false)

        Create a column vector using data from writable auxiliary (external) memory, where ptr_aux_mem is a pointer to the memory. By default the vector allocates its own memory and copies data from the auxiliary memory (for safety). However, if copy_aux_mem is set to false, the vector will instead directly use the auxiliary memory (ie. no copying); this is faster, but can be dangerous unless you know what you are doing!

        The strict parameter comes into effect only when copy_aux_mem is set to false (ie. the vector is directly using auxiliary memory)
        • when strict is set to false, the vector will use the auxiliary memory until a size change
        • when strict is set to true, the vector will be bound to the auxiliary memory for its lifetime; the number of elements in the vector can't be changed
        • the default setting of strict in versions 6.000+ is false
        • the default setting of strict in versions 5.600 and earlier is true

      vec(const ptr_aux_mem, number_of_elements)

        Create a column vector by copying data from read-only auxiliary memory, where ptr_aux_mem is a pointer to the memory

      vec::fixed<number_of_elements>

        Create a fixed size column vector, with the size specified via the template argument. Memory for the vector is allocated at compile time. This is generally faster than dynamic memory allocation, but the size of the vector can't be changed afterwards (directly or indirectly).

        For convenience, there are several pre-defined typedefs for each vector type (where the types are: uvec, ivec, fvec, vec, cx_fvec, cx_vec as well as the corresponding colvec versions). The pre-defined typedefs specify vector sizes ranging from 2 to 9. The typedefs were defined by simply appending a single digit form of the size to the vector type -- for example, vec3 is equivalent to vec::fixed<3>, while cx_vec4 is equivalent to cx_vec::fixed<4>.

      vec::fixed<number_of_elements>(const ptr_aux_mem)

        Create a fixed size column vector, with the size specified via the template argument; data is copied from auxiliary memory, where ptr_aux_mem is a pointer to the memory


  • Examples:
      vec x(10);
      vec y = zeros<vec>(10);
      
      mat A = randu<mat>(10,10);
      vec z = A.col(5); // extract a column vector
      

  • Caveat: For mathematical correctness, scalars are treated as 1x1 matrices during initialisation. As such, the code below will not generate a column vector with every element equal to 123.0:
      vec a(5);  a = 123.0;
      
    Use the following code instead:
      vec a(5);  a.fill(123.0);
      

  • See also:



Row<type>
rowvec
cx_rowvec
  • Classes for row vectors (matrices with one row)

  • The template Row<type> class is derived from the Mat<type> class and inherits most of the member functions

  • For convenience the following typedefs have been defined:
      rowvec  =  Row<double>
      frowvec  =  Row<float>
      cx_rowvec  =  Row<cx_double>
      cx_frowvec  =  Row<cx_float>
      urowvec  =  Row<uword>
      irowvec  =  Row<sword>

  • In this documentation, the rowvec type is used for convenience; it is possible to use other types instead, eg. frowvec

  • Functions which take Mat as input can generally also take Row as input. Main exceptions are functions which require square matrices

  • Constructors
      rowvec()
      rowvec(n_elem)
      rowvec(n_elem, fill_type)
      rowvec(size(X))
      rowvec(size(X), fill_type)
      rowvec(rowvec)
      rowvec(mat)   (a std::logic_error exception is thrown if the given matrix has more than one row)
      rowvec(initializer_list)
      rowvec(string)   (elements separated by spaces)
      rowvec(std::vector)
      cx_rowvec(rowvec,rowvec)   (for constructing a complex row vector out of two real row vectors)

  • When specifying the size with n_elem, by default the memory is uninitialised; memory can be initialised by specifying the fill_type, as per the Mat class

  • Advanced constructors:

      rowvec(ptr_aux_mem, number_of_elements, copy_aux_mem = true, strict = false)

        Create a row vector using data from writable auxiliary (external) memory, where ptr_aux_mem is a pointer to the memory. By default the vector allocates its own memory and copies data from the auxiliary memory (for safety). However, if copy_aux_mem is set to false, the vector will instead directly use the auxiliary memory (ie. no copying); this is faster, but can be dangerous unless you know what you are doing!

        The strict parameter comes into effect only when copy_aux_mem is set to false (ie. the vector is directly using auxiliary memory)
        • when strict is set to false, the vector will use the auxiliary memory until a size change
        • when strict is set to true, the vector will be bound to the auxiliary memory for its lifetime; the number of elements in the vector can't be changed
        • the default setting of strict in versions 6.000+ is false
        • the default setting of strict in versions 5.600 and earlier is true

      rowvec(const ptr_aux_mem, number_of_elements)

        Create a row vector by copying data from read-only auxiliary memory, where ptr_aux_mem is a pointer to the memory

      rowvec::fixed<number_of_elements>

        Create a fixed size row vector, with the size specified via the template argument. Memory for the vector is allocated at compile time. This is generally faster than dynamic memory allocation, but the size of the vector can't be changed afterwards (directly or indirectly).

        For convenience, there are several pre-defined typedefs for each vector type (where the types are: urowvec, irowvec, frowvec, rowvec, cx_frowvec, cx_rowvec). The pre-defined typedefs specify vector sizes ranging from 2 to 9. The typedefs were defined by simply appending a single digit form of the size to the vector type -- for example, rowvec3 is equivalent to rowvec::fixed<3>, while cx_rowvec4 is equivalent to cx_rowvec::fixed<4>.

      rowvec::fixed<number_of_elements>(const ptr_aux_mem)

        Create a fixed size row vector, with the size specified via the template argument; data is copied from auxiliary memory, where ptr_aux_mem is a pointer to the memory


  • Examples:
      rowvec x(10);
      rowvec y = zeros<rowvec>(10);
      
      mat    A = randu<mat>(10,10);
      rowvec z = A.row(5); // extract a row vector
      

  • Caveat: For mathematical correctness, scalars are treated as 1x1 matrices during initialisation. As such, the code below will not generate a row vector with every element equal to 123.0:
      rowvec r(5);  r = 123.0;
      
    Use the following code instead:
      rowvec r(5);  r.fill(123.0);
      

  • See also:



Cube<type>
cube
cx_cube
  • Classes for cubes, also known as "3D matrices" or 3rd order tensors

  • The cube class is Cube<type>, where type is one of:
    • float, double, std::complex<float>, std::complex<double>, short, int, long and unsigned versions of short, int, long

  • For convenience the following typedefs have been defined:
      cube  =  Cube<double>
      fcube  =  Cube<float>
      cx_cube  =  Cube<cx_double>
      cx_fcube  =  Cube<cx_float>
      ucube  =  Cube<uword>
      icube  =  Cube<sword>

  • In this documentation the cube type is used for convenience; it is possible to use other types instead, eg. fcube

  • Cube data is stored as a set of slices (matrices) stored contiguously within memory. Within each slice, elements are stored with column-major ordering (ie. column by column)

  • Each slice can be interpreted as a matrix, hence functions which take Mat as input can generally also take cube slices as input

  • Constructors:
      cube()
      cube(n_rows, n_cols, n_slices)
      cube(n_rows, n_cols, n_slices, fill_type)
      cube(size(X))
      cube(size(X), fill_type)
      cube(cube)
      cx_cube(cube, cube)   (for constructing a complex cube out of two real cubes)

  • When specifying the cube size with n_rows, n_cols and n_slices, by default the memory is uninitialised; memory can be initialised by specifying the fill_type, as per the Mat class (except for fill::eye)

  • Advanced constructors:

      cube::fixed<n_rows, n_cols, n_slices>

        Create a fixed size cube, with the size specified via template arguments. Memory for the cube is allocated at compile time. This is generally faster than dynamic memory allocation, but the size of the cube can't be changed afterwards (directly or indirectly).

      cube(ptr_aux_mem, n_rows, n_cols, n_slices, copy_aux_mem = true, strict = false)

        Create a cube using data from writable auxiliary (external) memory, where ptr_aux_mem is a pointer to the memory. By default the cube allocates its own memory and copies data from the auxiliary memory (for safety). However, if copy_aux_mem is set to false, the cube will instead directly use the auxiliary memory (ie. no copying); this is faster, but can be dangerous unless you know what you are doing!

        The strict parameter comes into effect only when copy_aux_mem is set to false (ie. the cube is directly using auxiliary memory)
        • when strict is set to false, the cube will use the auxiliary memory until a size change
        • when strict is set to true, the cube will be bound to the auxiliary memory for its lifetime; the number of elements in the cube can't be changed
        • the default setting of strict in versions 6.000+ is false
        • the default setting of strict in versions 5.600 and earlier is true

      cube(const ptr_aux_mem, n_rows, n_cols, n_slices)

        Create a cube by copying data from read-only auxiliary memory, where ptr_aux_mem is a pointer to the memory


  • Examples:
      cube x(1,2,3);
      cube y = randu<cube>(4,5,6);
      
      mat A = y.slice(1);  // extract a slice from the cube
                           // (each slice is a matrix)
      
      mat B = randu<mat>(4,5);
      y.slice(2) = B;     // set a slice in the cube
      
      cube q = y + y;     // cube addition
      cube r = y % y;     // element-wise cube multiplication
      
      cube::fixed<4,5,6> f;
      f.ones();
      

  • Caveats:

    • The size of individual slices can't be changed. For example, the following will not work:
        cube c(5,6,7);
        c.slice(0) = randu<mat>(10,20); // wrong size
        
    • For mathematical correctness, scalars are treated as 1x1x1 cubes during initialisation. As such, the code below will not generate a cube with every element equal to 123.0:
        cube c(5,6,7);  c = 123.0;
        
      Use the following code instead:
        cube c(5,6,7);  c.fill(123.0);
        

  • See also:



field<object_type>
  • Class for storing arbitrary objects in matrix-like or cube-like layouts

  • Somewhat similar to a matrix or cube, but instead of each element being a scalar, each element can be a vector, or matrix, or cube

  • Each element can have an arbitrary size (eg. in a field of matrices, each matrix can have a different size)

  • Constructors, where object_type is another class, eg. vec, mat, std::string, etc:
      field<object_type>()
      field<object_type>(n_elem)
      field<object_type>(n_rows, n_cols)
      field<object_type>(n_rows, n_cols, n_slices)
      field<object_type>(size(X))
      field<object_type>(field<object_type>)

  • Caveat: to store a set of matrices of the same size, the Cube class is more efficient

  • Examples:
      mat A = randn(2,3);
      mat B = randn(4,5);
      
      field<mat> F(2,1);
      F(0,0) = A;
      F(1,0) = B; 
      
      F.print("F:");
      
      F.save("mat_field");
      

  • See also:



SpMat<type>
sp_mat
sp_cx_mat
  • The root sparse matrix class is SpMat<type>, where type is one of:
    • float, double, std::complex<float>, std::complex<double>, short, int, long and unsigned versions of short, int, long

  • For convenience the following typedefs have been defined:
      sp_mat  =  SpMat<double>
      sp_fmat  =  SpMat<float>
      sp_cx_mat  =  SpMat<cx_double>
      sp_cx_fmat  =  SpMat<cx_float>
      sp_umat  =  SpMat<uword>
      sp_imat  =  SpMat<sword>

  • In this documentation the sp_mat type is used for convenience; it is possible to use other types instead, eg. sp_fmat

  • Constructors:
      sp_mat()
      sp_mat(n_rows, n_cols)
      sp_mat(size(X))
      sp_mat(sp_mat)
      sp_mat(mat)
      sp_mat(string)
      sp_cx_mat(sp_mat,sp_mat)   (for constructing a complex matrix out of two real matrices)

  • Elements are stored in the compressed sparse column (CSC) format

  • All elements are treated as zero by default (ie. the matrix is initialised to contain zeros)

  • This class behaves in a similar manner to the Mat class, however, member functions which set all elements to non-zero values (and hence do not make sense for sparse matrices) have been deliberately omitted; examples of omitted functions: .fill(), .ones(), += scalar, etc.

  • Batch insertion constructors:
    • form 1: sp_mat(locations, values, sort_locations = true)
    • form 2: sp_mat(locations, values, n_rows, n_cols, sort_locations = true, check_for_zeros = true)
    • form 3: sp_mat(add_values, locations, values, n_rows, n_cols, sort_locations = true, check_for_zeros = true)
    • form 4: sp_mat(rowind, colptr, values, n_rows, n_cols)

      • Using batch insertion constructors is generally much faster than consecutively inserting values using element access operators

      • For forms 1, 2, 3, locations is a dense matrix of type umat, with a size of 2 x N, where N is the number of values to be inserted; the location of the i-th element is specified by the contents of the i-th column of the locations matrix, where the row is in locations(0,i), and the column is in locations(1,i)

      • For form 4, rowind is a dense column vector of type uvec containing the row indices of the values to be inserted, and colptr is a dense column vector of type uvec (with length n_cols + 1) containing indices of values corresponding to the start of new columns; the vectors correspond to the arrays used by the compressed sparse column format; this form is useful for copying data from other CSC sparse matrix containers

      • For all forms, values is a dense column vector containing the values to be inserted; it must have the same element type as the sparse matrix. For forms 1 and 2, the value in values[i] will be inserted at the location specified by the i-th column of the locations matrix.

      • For form 3, add_values is either true or false; when set to true, identical locations are allowed, and the values at identical locations are added

      • The size of the constructed matrix is either automatically determined from the maximal locations in the locations matrix (form 1), or manually specified via n_rows and n_cols (forms 2, 3, 4)

      • If sort_locations is set to false, the locations matrix is assumed to contain locations that are already sorted according to column-major ordering

      • If check_for_zeros is set to false, the values vector is assumed to contain no zero values

  • Caveats:

  • Examples:
      sp_mat A(5,6);
      sp_mat B(6,5);
      
      A(0,0) = 1;
      A(1,0) = 2;
      
      B(0,0) = 3;
      B(0,1) = 4;
      
      sp_mat C = 2*B;
      
      sp_mat D = A*C;
      
      
      // batch insertion of two values at (5, 6) and (9, 9)
      umat locations;
      locations << 5 << 9 << endr
                << 6 << 9 << endr;
      
      vec values;
      values << 1.5 << 3.2 << endr;
      
      sp_mat X(locations, values);
      

  • See also:



operators:  +    *  /  %  ==  !=  <=  >=  <  >
  • Overloaded operators for Mat, Col, Row and Cube classes

  • Meanings:

      +    
      Addition of two objects

      Subtraction of one object from another or negation of an object
           
      /
      Element-wise division of an object by another object or a scalar
      *
      Matrix multiplication of two objects; not applicable to the Cube class unless multiplying a cube by a scalar
           
      %
      Schur product: element-wise multiplication of two objects
           
      ==
      Element-wise equality evaluation of two objects; generates a matrix of type umat with entries that indicate whether at a given position the two elements from the two objects are equal (1) or not equal (0)
      !=
      Element-wise non-equality evaluation of two objects
           
      >=
      As for ==, but the check is for "greater than or equal to"
      <=
      As for ==, but the check is for "less than or equal to"
           
      >
      As for ==, but the check is for "greater than"
      <
      As for ==, but the check is for "less than"

  • Caveat: operators involving an equality comparison (ie. ==, !=, >=, <=) are not recommended for matrices of type mat or fmat, due to the necessarily limited precision of the underlying element types; you may wish to use approx_equal() instead

  • A std::logic_error exception is thrown if incompatible object sizes are used

  • If the +, and % operators are chained, Armadillo will try to avoid the generation of temporaries; no temporaries are generated if all given objects are of the same type and size

  • If the * operator is chained, Armadillo will try to find an efficient ordering of the matrix multiplications

  • Examples:
      mat A = randu<mat>(5,10);
      mat B = randu<mat>(5,10);
      mat C = randu<mat>(10,5);
      
      mat P = A + B;
      mat Q = A - B;
      mat R = -B;
      mat S = A / 123.0;
      mat T = A % B;
      mat U = A * C;
      
      // V is constructed without temporaries
      mat V = A + B + A + B;
      
      imat AA = "1 2 3; 4 5 6; 7 8 9;";
      imat BB = "3 2 1; 6 5 4; 9 8 7;";
      
      // compare elements
      umat ZZ = (AA >= BB);
      

  • See also:





Member Functions & Variables



attributes
    .n_rows     number of rows; present in Mat, Col, Row, Cube, field and SpMat
    .n_cols     number of columns; present in Mat, Col, Row, Cube, field and SpMat
    .n_elem     total number of elements; present in Mat, Col, Row, Cube, field and SpMat
    .n_slices     number of slices; present in Cube
    .n_nonzero     number of non-zero elements; present in SpMat


element/object access via (), [] and .at()
  • Provide access to individual elements or objects stored in a container object (ie. Mat, Col, Row, Cube, field)

      (n)  
      For vec and rowvec, access the n-th element. For mat, cube and field, access the n-th element/object under the assumption of a flat layout, with column-major ordering of data (ie. column by column). A std::logic_error exception is thrown if the requested element is out of bounds. The bounds check can be optionally disabled at compile-time to get more speed.
           
      .at(n)  or  [n] 
      As for (n), but without a bounds check. Not recommended for use unless your code has been thoroughly debugged.
           
      (i,j)
      For mat and 2D field classes, access the element/object stored at the i-th row and j-th column. A std::logic_error exception is thrown if the requested element is out of bounds. The bounds check can be optionally disabled at compile-time to get more speed.
           
      .at(i,j)
      As for (i,j), but without a bounds check. Not recommended for use unless your code has been thoroughly debugged.
           
      (i,j,k)
      For cube and 3D field classes, access the element/object stored at the i-th row, j-th column and k-th slice. A std::logic_error exception is thrown if the requested element is out of bounds. The bounds check can be optionally disabled at compile-time to get more speed.
           
      .at(i,j,k)
      As for (i,j,k), but without a bounds check. Not recommended for use unless your code has been thoroughly debugged.

  • The bounds checks used by the (n), (i,j) and (i,j,k) access forms can be disabled by defining the ARMA_NO_DEBUG macro before including the armadillo header file (eg. #define ARMA_NO_DEBUG). Disabling the bounds checks is not recommended until your code has been thoroughly debugged -- it's better to write correct code first, and then maximise its speed.

  • The indices of elements are specified via the uword type, which is a typedef for an unsigned integer type. When using loops to access elements, it's best to use uword instead of int. For example: for(uword i=0; i<X.n_elem; ++i) { X(i) = ... }

  • Caveat: for sparse matrices, using element access operators to insert values via loops can be inefficient; you may wish to use batch insertion constructors instead

  • Examples:
      mat A = randu<mat>(10,10);
      A(9,9) = 123.0;
      double x = A.at(9,9);
      double y = A[99];
      
      vec p = randu<vec>(10,1);
      p(9) = 123.0;
      double z = p[9];
      

  • See also:



element initialisation
  • When using the C++11 standard, elements in Mat, Col, Row can be set via initialiser lists

  • When using the old C++98 standard, elements can be set via the << operator; special element endr indicates "end of row" (conceptually similar to std::endl)

  • Caveat: using the << operator is slower than using initialiser lists

  • Examples:
      // C++11
      
      vec v = { 1, 2, 3 };
      
      mat A = { {1, 3, 5},
                {2, 4, 6} };
      
      
      // C++98
      
      mat B;
      
      B << 1 << 3 << 5 << endr
        << 2 << 4 << 6 << endr;
      

  • See also:



.zeros()
  (member function of Mat, Col, Row, SpMat, Cube)
.zeros( n_elem )
  (member function of Col and Row)
.zeros( n_rows, n_cols )
  (member function of Mat and SpMat)
.zeros( n_rows, n_cols, n_slices )
  (member function of Cube)
.zeros( size(X) )
  (member function of Mat, Col, Row, Cube, SpMat)
  • Set the elements of an object to zero, optionally first changing the size to specified dimensions

  • Examples:
      mat A(5,10);  A.zeros();   // or:  mat A(5,10,fill::zeros);
      
      mat B;  B.zeros(10,20);
      
      mat C;  C.zeros( size(B) );
      

  • See also:



.ones()
  (member function of Mat, Col, Row, Cube)
.ones( n_elem )
  (member function of Col and Row)
.ones( n_rows, n_cols )
  (member function of Mat)
.ones( n_rows, n_cols, n_slices )
  (member function of Cube)
.ones( size(X) )
  (member function of Mat, Col, Row, Cube)
  • Set all the elements of an object to one, optionally first changing the size to specified dimensions

  • Examples:
      mat A(5,10);  A.ones();   // or:  mat A(5,10,fill::ones);
      
      mat B;  B.ones(10,20);
      
      mat C;  C.ones( size(B) );
      

  • See also:



.eye()
.eye( n_rows, n_cols )
.eye( size(X) )
  • Member functions of Mat and SpMat

  • Set the elements along the main diagonal to one and off-diagonal elements to zero, optionally first changing the size to specified dimensions

  • An identity matrix is generated when n_rows = n_cols

  • Examples:
      mat A(5,5);  A.eye();  // or:  mat A(5,5,fill::eye);
      
      mat B;  B.eye(5,5);
      
      mat C;  C.eye( size(B) );
      

  • See also:



.randu()
  (member function of Mat, Col, Row, Cube)
.randu( n_elem )
  (member function of Col and Row)
.randu( n_rows, n_cols )
  (member function of Mat)
.randu( n_rows, n_cols, n_slices )
  (member function of Cube)
.randu( size(X) )
  (member function of Mat, Col, Row, Cube)

.randn()
  (member function of Mat, Col, Row, Cube)
.randn( n_elem )
  (member function of Col and Row)
.randn( n_rows, n_cols )
  (member function of Mat)
.randn( n_rows, n_cols, n_slices )
  (member function of Cube)
.randn( size(X) )
  (member function of Mat, Col, Row, Cube)
  • Set all the elements to random values, optionally first changing the size to specified dimensions

  • .randu() uses a uniform distribution in the [0,1] interval

  • .randn() uses a normal/Gaussian distribution with zero mean and unit variance

  • To change the RNG seed, use arma_rng::set_seed(value) or arma_rng::set_seed_random() functions

  • Examples:
      mat A(5,10);  A.randu();   // or:  mat A(5,10,fill::randu);
      
      mat B;  B.randu(10,20);
      
      mat C;  C.randu( size(B) );
      
      arma_rng::set_seed_random();  // set the seed to a random value
      

  • See also:



.fill( value )


.imbue( functor )
.imbue( lambda_function )   (C++11 only)
  • Member functions of Mat, Col, Row and Cube

  • Imbue (fill) with values provided by a functor or lambda function

  • For matrices, filling is done column-by-column (ie. column 0 is filled, then column 1, ...)

  • For cubes, filling is done slice-by-slice, with each slice treated as a matrix

  • Examples:
      // C++11 only example
      // need to include <random>
      
      std::mt19937 engine;  // Mersenne twister random number engine
      
      std::uniform_real_distribution<double> distr(0.0, 1.0);
        
      mat A(4,5);
        
      A.imbue( [&]() { return distr(engine); } );
      

  • See also:



.replace( old_value, new_value )
  • Member function of Mat, Col, Row, Cube and SpMat

  • For all elements equal to old_value, set them to new_value

  • The type of old_value and new_value must match the type of elements used by the container object (eg. for mat the type is double)

  • Caveats:
    • floating point numbers (float and double) are approximations due to their necessarily limited precision
    • for sparse matrices (SpMat), replacement is not done when old_value = 0

  • Examples:
      mat A(5,6,fill::randu);
      
      A.diag().fill(datum::nan);
      
      A.replace(datum::nan, 0);  // replace each NaN with 0
      

  • See also:



.transform( functor )
.transform( lambda_function )   (C++11 only)


.for_each( functor )
.for_each( lambda_function )   (C++11 only)
  • Member functions of Mat, Col, Row, Cube and field

  • For each element, pass its reference to a functor or lambda function

  • For matrices, the processing is done column-by-column

  • For cubes, processing is done slice-by-slice, with each slice treated as a matrix

  • Examples:
      // C++11 only examples
      
      mat A = ones<mat>(4,5);
      
      // add 123 to each element
      A.for_each( [](mat::elem_type& val) { val += 123.0; } );  // NOTE: the '&' is crucial!
      
      
      field<mat> F(2,3);
      
      // set the size of all matrices in field F
      F.for_each( [](mat& X) { X.zeros(4,5); } );  // NOTE: the '&' is crucial!
      

  • See also:



.set_size( n_elem )
  (member function of Col, Row, field)
.set_size( n_rows, n_cols )
  (member function of Mat, SpMat, field)
.set_size( n_rows, n_cols, n_slices )
  (member function of Cube and field)
.set_size( size(X) )
  (member function of Mat, Col, Row, Cube, SpMat, field)
  • Change the size of an object, without explicitly preserving data and without initialising the elements

  • If you need to initialise the elements to zero while changing the size, use .zeros() instead

  • If you need to explicitly preserve data while changing the size, use .reshape() or .resize() instead;
    caveat: .reshape() and .resize() are considerably slower than .set_size()

  • Examples:
      mat A;  A.set_size(5,10);       // or:  mat A(5,10);
      
      mat B;  B.set_size( size(A) );  // or:  mat B( size(A) );
      
      vec v;  v.set_size(100);        // or:  vec v(100);
      

  • See also:



.reshape( n_rows, n_cols )
  (member function of Mat and SpMat)
.reshape( n_rows, n_cols, n_slices )
  (member function of Cube)
.reshape( size(X) )
  (member function of Mat, Cube, SpMat)
  • Recreate the object according to given size specifications, with the elements taken from the previous version of the object in a column-wise manner; the elements in the generated object are placed column-wise (ie. the first column is filled up before filling the second column)

  • The layout of the elements in the recreated object will be different to the layout in the previous version of the object

  • If the total number of elements in the previous version of the object is less than the specified size, the extra elements in the recreated object are set to zero

  • If the total number of elements in the previous version of the object is greater than the specified size, only a subset of the elements is taken

  • Caveats:
    • do not use .reshape() if you simply want to change the size without preserving data; use .set_size() instead, which is much faster
    • to grow/shrink the object while preserving the elements as well as the layout of the elements, use .resize() instead
    • to create a vector representation of a matrix (ie. concatenate all the columns or rows), use vectorise() instead

  • Examples:
      mat A = randu<mat>(4,5);
      
      A.reshape(5,4);
      

  • See also:



.resize( n_elem )
  (member function of Col, Row)
.resize( n_rows, n_cols )
  (member function of Mat and SpMat)
.resize( n_rows, n_cols, n_slices )
  (member function of Cube)
.resize( size(X) )
  (member function of Mat, Col, Row, Cube, SpMat)


.copy_size( A )
  • Set the size to be the same as object A

  • Object A must be of the same root type as the object being modified (eg. you can't set the size of a matrix by providing a cube)

  • Examples:
      mat A = randu<mat>(5,6);
      mat B;
      B.copy_size(A);
      
      cout << B.n_rows << endl;
      cout << B.n_cols << endl;
      

  • See also:



.reset()


submatrix views
  • A collection of member functions of Mat, Col and Row classes that provide read/write access to submatrix views

  • contiguous views for matrix X:

      X.col( col_number )
      X.row( row_number )

      X.cols( first_col, last_col )
      X.rows( first_row, last_row )

      X.submat( first_row, first_col, last_row, last_col )

      X( span(first_row, last_row), span(first_col, last_col) )

      Xfirst_row, first_col, size(n_rowsn_cols) )
      Xfirst_row, first_col, size(Y) )    (Y is a mat)

      X( span(first_row, last_row), col_number )
      X( row_number, span(first_col, last_col) )

      X.head_cols( number_of_cols )
      X.head_rows( number_of_rows )

      X.tail_cols( number_of_cols )
      X.tail_rows( number_of_rows )

      X.unsafe_col( col_number )

  • contiguous views for vector V:

      V( span(first_index, last_index) )
      V.subvec( first_index, last_index )

      V.subvec( first_index, size(W) )    (W is a vector)

      V.head( number_of_elements )
      V.tail( number_of_elements )
  •           
  • non-contiguous views for matrix or vector X:

      X.elem( vector_of_indices )
      X( vector_of_indices )

      X.cols( vector_of_column_indices )
      X.rows( vector_of_row_indices )

      X.submat( vector_of_row_indices, vector_of_column_indices )
      X( vector_of_row_indices, vector_of_column_indices )


  • related views (documented separately)

  • Instances of span(start,end) can be replaced by span::all to indicate the entire range

  • For functions requiring one or more vector of indices, eg. X.submat(vector_of_row_indices, vector_of_column_indices), each vector of indices must be of type uvec

  • In the function X.elem(vector_of_indices), elements specified in vector_of_indices are accessed. X is interpreted as one long vector, with column-by-column ordering of the elements of X. The vector_of_indices must evaluate to a vector of type uvec (eg. generated by the find() function). The aggregate set of the specified elements is treated as a column vector (ie. the output of X.elem() is always a column vector).

  • The function .unsafe_col() is provided for speed reasons and should be used only if you know what you are doing. It creates a seemingly independent Col vector object (eg. vec), but uses memory from the existing matrix object. As such, the created vector is not alias safe, and does not take into account that the underlying matrix memory could be freed (eg. due to any operation involving a size change of the matrix).

  • Examples:
      mat A = zeros<mat>(5,10);
      
      A.submat( 0,1, 2,3 )      = randu<mat>(3,3);
      A( span(0,2), span(1,3) ) = randu<mat>(3,3);
      A( 0,1, size(3,3) )       = randu<mat>(3,3);
      
      mat B = A.submat( 0,1, 2,3 );
      mat C = A( span(0,2), span(1,3) );
      mat D = A( 0,1, size(3,3) );
      
      A.col(1)        = randu<mat>(5,1);
      A(span::all, 1) = randu<mat>(5,1);
      
      mat X = randu<mat>(5,5);
      
      // get all elements of X that are greater than 0.5
      vec q = X.elem( find(X > 0.5) );
      
      // add 123 to all elements of X greater than 0.5
      X.elem( find(X > 0.5) ) += 123.0;
      
      // set four specific elements of X to 1
      uvec indices;
      indices << 2 << 3 << 6 << 8;
      
      X.elem(indices) = ones<vec>(4);
      
      // add 123 to the last 5 elements of vector a
      vec a(10, fill::randu);
      a.tail(5) += 123.0;
      
      // add 123 to the first 3 elements of column 2 of X
      X.col(2).head(3) += 123;
      

  • See also:



subcube views and slices
  • A collection of member functions of the Cube class that provide subcube views

  • contiguous views for cube Q:

      Q.slice( slice_number )
      Q.slices( first_slice, last_slice )

      Q.subcube( first_row, first_col, first_slice, last_row, last_col, last_slice )

      Q( span(first_row, last_row), span(first_col, last_col), span(first_slice, last_slice) )

      Qfirst_row, first_col, first_slice, size(n_rows, n_cols, n_slices) )
      Qfirst_row, first_col, first_slice, size(R) )      (R is a cube)

      Q.head_slices( number_of_slices )
      Q.tail_slices( number_of_slices )

      Q.tube( row, col )
      Q.tube( first_row, first_col, last_row, last_col )
      Q.tube( span(first_row, last_row), span(first_col, last_col) )
      Q.tube( first_row, first_col, size(n_rows, n_cols) )
  •           
  • non-contiguous views for cube Q:

      Q.elem( vector_of_indices )
      Q( vector_of_indices )

  • related views (documented separately)

  • Instances of span(a,b) can be replaced by:
    • span() or span::all, to indicate the entire range
    • span(a), to indicate a particular row, column or slice

  • An individual slice, accessed via .slice(), is an instance of the Mat class (a reference to a matrix is provided)

  • All .tube() forms are variants of .subcube(), using first_slice = 0 and last_slice = Q.n_slices-1

  • The .tube(row,col) form uses row = first_row = last_row, and col = first_col = last_col

  • In the function Q.elem(vector_of_indices), elements specified in vector_of_indices are accessed. Q is interpreted as one long vector, with slice-by-slice and column-by-column ordering of the elements of Q. The vector_of_indices must evaluate to a vector of type uvec (eg. generated by the find() function). The aggregate set of the specified elements is treated as a column vector (ie. the output of Q.elem() is always a column vector).

  • Examples:
      cube A = randu<cube>(2,3,4);
      
      mat  B = A.slice(1); // each slice is a matrix
      
      A.slice(0) = randu<mat>(2,3);
      A.slice(0)(1,2) = 99.0;
      
      A.subcube(0,0,1,  1,1,2)             = randu<cube>(2,2,2);
      A( span(0,1), span(0,1), span(1,2) ) = randu<cube>(2,2,2);
      A( 0,0,1, size(2,2,2) )              = randu<cube>(2,2,2);
      
      // add 123 to all elements of A greater than 0.5
      A.elem( find(A > 0.5) ) += 123.0;
      
      cube C = A.head_slices(2);  // get first two slices
      
      A.head_slices(2) += 123.0;
      

  • See also:



subfield views
  • A collection of member functions of the field class that provide subfield views

  • For a 2D field F, the subfields are accessed as:

    • F.row( row_number )
      F.col( col_number )

      F.rows( first_row, last_row )
      F.cols( first_col, last_col )

      F.subfield( first_row, first_col, last_row, last_col )

      F( span(first_row, last_row), span(first_col, last_col) )

      Ffirst_row, first_col, size(G) )    (G is a 2D field)
      Ffirst_row, first_col, size(n_rows, n_cols) )

  • For a 3D field F, the subfields are accessed as:

    • F.slice( slice_number )

      F.slices( first_slice, last_slice )

      F.subfield( first_row, first_col, first_slice, last_row, last_col, last_slice )

      F( span(first_row, last_row), span(first_col, last_col), span(first_slice, last_slice) )

      Ffirst_row, first_col, first_slice, size(G) )    (G is a 3D field)
      Ffirst_row, first_col, first_slice, size(n_rows, n_cols, n_slices) )

  • Instances of span(a,b) can be replaced by:
    • span() or span::all, to indicate the entire range
    • span(a), to indicate a particular row or column

  • See also:



.diag()
.diag( k )
  • Member function of Mat and SpMat

  • Read/write access to a diagonal in a matrix

  • The argument k is optional; by default the main diagonal is accessed (k=0)

  • For k > 0, the k-th super-diagonal is accessed (top-right corner)

  • For k < 0, the k-th sub-diagonal is accessed (bottom-left corner)

  • The diagonal is interpreted as a dense column vector within expressions

  • Examples:
      mat X = randu<mat>(5,5);
      
      vec a = X.diag();
      vec b = X.diag(1);
      vec c = X.diag(-2);
      
      X.diag()  = randu<vec>(5);
      X.diag() += 6;
      X.diag().ones();
      

  • See also:



.each_col()
.each_row()

.each_col( vector_of_indices )
.each_row( vector_of_indices )

.each_col( lambda_function )   (C++11 only)
.each_row( lambda_function )   (C++11 only)
  • Member functions of Mat

  • Repeat a vector operation on each column or row of a matrix

  • Supported operations for .each_col() / .each_row() and .each_col(vector_of_indices) / .each_row(vector_of_indices) forms:

      + addition      += in-place addition
      - subtraction      -= in-place subtraction
      % element-wise multiplication      %= in-place element-wise multiplication
      / element-wise division      /= in-place element-wise division
      = assignment (copy)         

  • The argument vector_of_indices is optional; by default all columns or rows are used

  • If the argument vector_of_indices is specified, it must evaluate to a vector of type uvec; the vector contains a list of indices of the columns or rows to be used

  • If the lambda_function is specified, the function must accept a reference to a Col or Row object with the same element type as the underlying matrix

  • Examples:
      mat X = ones<mat>(6,5);
      vec v = linspace<vec>(10,15,6);
      
      X.each_col() += v;         // in-place addition of v to each column vector of X
      
      mat Y = X.each_col() + v;  // generate Y by adding v to each column vector of X
      
      // subtract v from columns 0 through to 3 in X
      X.cols(0,3).each_col() -= v;
      
      
      uvec indices(2);
      indices(0) = 2;
      indices(1) = 4;
      
      X.each_col(indices) = v;   // copy v to columns 2 and 4 in X
      
      
      X.each_col( [](vec& a){ a.print(); } );     // lambda function with non-const vector
      
      const mat& XX = X;
      XX.each_col( [](const vec& b){ b.print(); } );  // lambda function with const vector
      

  • See also:



.each_slice()
.each_slice( vector_of_indices )
.each_slice( lambda_function )   (C++11 only)
  • Member function of Cube

  • Repeat a matrix operation on each slice of a cube

  • Supported operations for .each_slice() and .each_slice(vector_of_indices) forms:

      + addition      += in-place addition
      - subtraction      -= in-place subtraction
      % element-wise multiplication      %= in-place element-wise multiplication
      / element-wise division      /= in-place element-wise division
      = assignment (copy)         

  • The argument vector_of_indices is optional; by default all slices are used

  • If the argument vector_of_indices is specified, it must evaluate to a vector of type uvec; the vector contains a list of indices of the slices to be used

  • If the lambda_function is specified, the function must accept a reference to a Mat object with the same element type as the underlying cube

  • Examples:
      cube C(4,5,6, fill::randu);
      
      mat M = repmat(linspace<vec>(1,4,4), 1, 5);
      
      C.each_slice() += M;          // in-place addition of M to each slice of C
      
      cube D = C.each_slice() + M;  // generate D by adding M to each slice of C
      
      
      uvec indices(2);
      indices(0) = 2;
      indices(1) = 4;
      
      C.each_slice(indices) = M;    // copy M to slices 2 and 4 in C
      
      
      C.each_slice( [](mat& X){ X.print(); } );     // lambda function with non-const matrix
      
      const cube& CC = C;
      CC.each_slice( [](const mat& X){ X.print(); } );  // lambda function with const matrix
      

  • See also:



.set_imag( X )
.set_real( X )
  • Set the imaginary/real part of an object

  • X must have the same size as the recipient object

  • Examples:
         mat A = randu<mat>(4,5);
         mat B = randu<mat>(4,5);
      
      cx_mat C = zeros<cx_mat>(4,5);
      
      C.set_real(A);
      C.set_imag(B);
      

  • Caveat: to directly construct a complex matrix out of two real matrices, the following code is faster:
         mat A = randu<mat>(4,5);
         mat B = randu<mat>(4,5);
      
      cx_mat C = cx_mat(A,B);
      

  • See also:



.insert_rows( row_number, X )
.insert_rows( row_number, number_of_rows )
.insert_rows( row_number, number_of_rows, set_to_zero )
  (member functions of Mat and Col)
 
.insert_cols( col_number, X )
.insert_cols( col_number, number_of_cols )
.insert_cols( col_number, number_of_cols, set_to_zero )
  (member functions of Mat and Row)
 
.insert_slices( slice_number, X )
.insert_slices( slice_number, number_of_slices )
.insert_slices( slice_number, number_of_slices, set_to_zero )
  (member functions of Cube)
  • Functions with the X argument: insert a copy of X at the specified row/column/slice
    • if inserting rows, X must have the same number of columns as the recipient object
    • if inserting columns, X must have the same number of rows as the recipient object
    • if inserting slices, X must have the same number of rows and columns as the recipient object (ie. all slices must have the same size)

  • Functions with the number_of_... argument: expand the object by creating new rows/columns/slices. By default, the new rows/columns/slices are set to zero. If set_to_zero is false, the memory used by the new rows/columns/slices will not be initialised.

  • Examples:
      mat A = randu<mat>(5,10);
      mat B = ones<mat>(5,2);
      
      // at column 2, insert a copy of B;
      // A will now have 12 columns
      A.insert_cols(2, B);
      
      // at column 1, insert 5 zeroed columns;
      // B will now have 7 columns
      B.insert_cols(1, 5);
      

  • See also:



.shed_row( row_number )
.shed_rows( first_row, last_row )
  (member functions of Mat, Col and SpMat)
 
.shed_col( column_number )
.shed_cols( first_column, last_column )
  (member functions of Mat, Row and SpMat)
 
.shed_slice( slice_number )
.shed_slices( first_slice, last_slice )
  (member functions of Cube)


.swap_rows( row1, row2 )
.swap_cols( col1, col2 )
  • Member functions of Mat, Col, Row and SpMat

  • Swap the contents of specified rows or columns

  • Examples:
      mat X = randu<mat>(5,5);
      X.swap_rows(0,4);
      

  • See also:



.swap( X )
  • Member function of Mat, Col, Row and Cube

  • Swap contents with object X

  • Examples:
      mat A = zeros<mat>(4,5);
      mat B =  ones<mat>(6,7);
      
      A.swap(B);
      

  • See also:



.memptr()
  • Member function of Mat, Col, Row and Cube

  • Obtain a raw pointer to the memory used for storing elements

  • The function can be used for interfacing with libraries such as FFTW

  • Data for matrices is stored in a column-by-column order

  • Data for cubes is stored in a slice-by-slice (matrix-by-matrix) order

  • Caveat: the pointer becomes invalid after any operation involving a size change or aliasing

  • Caveat: this function is not recommended for use unless you know what you are doing!

  • Examples:
            mat A = randu<mat>(5,5);
      const mat B = randu<mat>(5,5);
      
            double* A_mem = A.memptr();
      const double* B_mem = B.memptr();
      

  • See also:



.colptr( col_number )


iterators (matrices & vectors)
  • STL-style iterators and associated member functions of Mat, Col, Row and SpMat

  • Member functions:

      .begin()  
      iterator referring to the first element
      .end()  
      iterator referring to the past-the-end element
       
      .begin_row( row_number )  
      iterator referring to the first element of the specified row
      .end_row( row_number )  
      iterator referring to the past-the-end element of the specified row
       
      .begin_col( col_number )  
      iterator referring to the first element of the specified column
      .end_col( col_number )  
      iterator referring to the past-the-end element of the specified column

  • Iterator types:

      mat::iterator
      vec::iterator
      rowvec::iterator
      sp_mat::iterator
       
      random access iterators, for read/write access to elements (which are stored column by column)
         
       
      mat::const_iterator
      vec::const_iterator
      rowvec::const_iterator
      sp_mat::const_iterator
       
      random access iterators, for read-only access to elements (which are stored column by column)
         
       
      mat::col_iterator
      vec::col_iterator
      rowvec::col_iterator
       
      random access iterators, for read/write access to the elements of a specific column
         
       
      mat::const_col_iterator
      vec::const_col_iterator
      rowvec::const_col_iterator
       
      random access iterators, for read-only access to the elements of a specific column
         
       
      mat::row_iterator
      sp_mat::row_iterator
       
      rudimentary forward iterator, for read/write access to the elements of a specific row
         
       
      mat::const_row_iterator
      sp_mat::const_row_iterator
       
      rudimentary forward iterator, for read-only access to the elements of a specific row
         
       
      vec::row_iterator
      rowvec::row_iterator
       
      random access iterators, for read/write access to the elements of a specific row
         
       
      vec::const_row_iterator
      rowvec::const_row_iterator
       
      random access iterators, for read-only access to the elements of a specific row

  • Examples:
      mat X = randu<mat>(5,5);
      
      
      mat::iterator a = X.begin();
      mat::iterator b = X.end();
      
      for(mat::iterator i=a; i!=b; ++i)
        {
        cout << *i << endl;
        }
      
      
      mat::col_iterator c = X.begin_col(1);  // start of column 1
      mat::col_iterator d = X.end_col(3);    // end of column 3
      
      for(mat::col_iterator i=c; i!=d; ++i)
        {
        cout << *i << endl;
        (*i) = 123.0;
        }
      

  • See also:



iterators (cubes)
  • STL-style iterators and associated member functions of Cube

  • Member functions:

      .begin()  
      iterator referring to the first element
      .end()  
      iterator referring to the past-the-end element
       
      .begin_slice( slice_number )  
      iterator referring to the first element of the specified slice
      .end_slice( slice_number )  
      iterator referring to the past-the-end element of the specified slice

  • Iterator types:

      cube::iterator  
      random access iterator, for read/write access to elements; the elements are ordered slice by slice; the elements within each slice are ordered column by column
         
       
      cube::const_iterator  
      random access iterators, for read-only access to elements
         
       
      cube::slice_iterator  
      random access iterator, for read/write access to the elements of a particular slice; the elements are ordered column by column
         
       
      cube::const_slice_iterator  
      random access iterators, for read-only access to the elements of a particular slice

  • Examples:
      cube X = randu<cube>(2,3,4);
      
      
      cube::iterator a = X.begin();
      cube::iterator b = X.end();
      
      for(cube::iterator i=a; i!=b; ++i)
        {
        cout << *i << endl;
        }
      
      
      cube::slice_iterator c = X.begin_slice(1);  // start of slice 1
      cube::slice_iterator d = X.end_slice(2);    // end of slice 2
      
      for(cube::slice_iterator i=c; i!=d; ++i)
        {
        cout << *i << endl;
        (*i) = 123.0;
        }
      

  • See also:



STL-style container functions
  • Member functions that mimic the containers in the C++ Standard Template Library:

    .clear()  
    causes an object to have no elements
    .empty()  
    returns true if the object has no elements; returns false if the object has one or more elements
    .size()  
    returns the total number of elements

  • Examples:
      mat A = randu<mat>(5,5);
      cout << A.size() << endl;
      
      A.clear();
      cout << A.empty() << endl;
      

  • See also:



.t()
.st()
  • Member functions of any matrix or vector expression

  • For real (non-complex) matrix:
    • .t() provides a transposed copy of the matrix
    • .st() not applicable

  • For complex matrix:
    • .t() provides a Hermitian transpose (ie. the conjugate of the elements is taken during the transpose)
    • .st() provides a transposed copy without taking the conjugate of the elements

  • Examples:
      mat A = randu<mat>(4,5);
      mat B = A.t();
      

  • See also:



.i()
  • Member function of any matrix expression

  • Provides an inverse of the matrix expression

  • If the matrix expression is not square sized, a std::logic_error exception is thrown

  • If the matrix expression appears to be singular, the output matrix is reset and a std::runtime_error exception is thrown

  • Caveats:
    • if matrix A is know to be symmetric positive definite, it's faster to use inv_sympd() instead
    • to solve a system of linear equations, such as Z = inv(X)*Y, it is faster and more accurate to use solve() instead

  • Examples:
      mat A = randu<mat>(4,4);
      mat B = randu<mat>(4,1);
      
      mat X = A.i();
      
      mat Y = (A+A).i();
      
  • See also:



.min()
.max()


.index_min()
.index_max()


.eval()
  • Member function of any matrix or vector expression

  • Explicitly forces the evaluation of a delayed expression and outputs a matrix

  • This function should be used sparingly and only in cases where it is absolutely necessary; indiscriminate use can cause slow downs

  • Examples:
      cx_mat A( randu<mat>(4,4), randu<mat>(4,4) );
      
      real(A).eval().save("A_real.dat", raw_ascii);
      imag(A).eval().save("A_imag.dat", raw_ascii);
      

  • See also:



.in_range( i )
  (member of Mat, Col, Row, Cube, SpMat, field)
.in_range( span(start, end) )
  (member of Mat, Col, Row, Cube, SpMat, field)
 
.in_range( row, col )
  (member of Mat, Col, Row, SpMat, field)
.in_range( span(start_row, end_row), span(start_col, end_col) )
  (member of Mat, Col, Row, SpMat, field)
 
.in_range( row, col, slice )
  (member of Cube and field)
.in_range( span(start_row, end_row), span(start_col, end_col), span(start_slice, end_slice) )
  (member of Cube and field)
 
.in_range( first_row, first_col, size(X) )   (X is a matrix or field)
  (member of Mat, Col, Row, SpMat, field)
.in_range( first_row, first_col, size(n_rows, n_cols) )
  (member of Mat, Col, Row, SpMat, field)
 
.in_range( first_row, first_col, first_slice, size(Q) )   (Q is a cube or field)
  (member of Cube and field)
.in_range( first_row, first_col, first_slice, size(n_rows, n_cols n_slices) )
  (member of Cube and field)
  • Returns true if the given location or span is currently valid

  • Returns false if the object is empty, the location is out of bounds, or the span is out of bounds

  • Instances of span(a,b) can be replaced by:
    • span() or span::all, to indicate the entire range
    • span(a), to indicate a particular row, column or slice

  • Examples:
      mat A = randu<mat>(4,5);
      
      cout << A.in_range(0,0) << endl;  // true
      cout << A.in_range(3,4) << endl;  // true
      cout << A.in_range(4,5) << endl;  // false
      

  • See also:



.is_empty()
  • Returns true if the object has no elements

  • Returns false if the object has one or more elements

  • Examples:
      mat A = randu<mat>(5,5);
      cout << A.is_empty() << endl;
      
      A.reset();
      cout << A.is_empty() << endl;
      

  • See also:



.is_square()
  • Member function of Mat and SpMat

  • Returns true if the matrix is square, ie. number of rows is equal to the number of columns

  • Returns false if the matrix is not square

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = randu<mat>(6,7);
      
      cout << A.is_square() << endl;
      cout << B.is_square() << endl;
      

  • See also:



.is_vec()
.is_colvec()
.is_rowvec()
  • Member functions of Mat and SpMat

  • .is_vec():
    • returns true if the matrix can be interpreted as a vector (either column or row vector)
    • returns false if the matrix does not have exactly one column or one row

  • .is_colvec():
    • returns true if the matrix can be interpreted as a column vector
    • returns false if the matrix does not have exactly one column

  • .is_rowvec():
    • returns true if the matrix can be interpreted as a row vector
    • returns false if the matrix does not have exactly one row

  • Caveat: do not assume that the vector has elements if these functions return true; it is possible to have an empty vector (eg. 0x1)

  • Examples:
      mat A = randu<mat>(1,5);
      mat B = randu<mat>(5,1);
      mat C = randu<mat>(5,5);
      
      cout << A.is_vec() << endl;
      cout << B.is_vec() << endl;
      cout << C.is_vec() << endl;
      

  • See also:



.is_sorted()
.is_sorted( sort_direction )
.is_sorted( sort_direction, dim )
  • Member function of Mat, Row and Col

  • If the object is a vector, return a bool indicating whether the elements are sorted

  • If the object is a matrix, return a bool indicating whether the elements are sorted in each column (dim=0), or each row (dim=1)

  • The sort_direction argument is optional; sort_direction is either "ascend" or "descend"; by default "ascend" is used

  • The dim argument is optional; by default dim=0 is used

  • For matrices and vectors with complex numbers, order is checked via absolute values

  • Examples:
      vec a = randu<vec>(10);
      vec b = sort(a);
      
      bool check1 = a.is_sorted();
      bool check2 = b.is_sorted();
      
      
      mat A = randu<mat>(10,10);
      
      // check whether each column is sorted in descending manner
      cout << A.is_sorted("descend") << endl;
      
      // check whether each row is sorted in ascending manner
      cout << A.is_sorted("ascend", 1) << endl;
      

  • See also:



.is_finite()
  • Member function of Mat, Col, Row, Cube, SpMat

  • Returns true if all elements of the object are finite

  • Returns false if at least one of the elements of the object is non-finite (±infinity or NaN)

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = randu<mat>(5,5);
      
      B(1,1) = datum::inf;
      
      cout << A.is_finite() << endl;
      cout << B.is_finite() << endl;
      

  • See also:



.has_inf()
  • Member function of Mat, Col, Row, Cube, SpMat

  • Returns true if at least one of the elements of the object is ±infinity

  • Returns false otherwise

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = randu<mat>(5,5);
      
      B(1,1) = datum::inf;
      
      cout << A.has_inf() << endl;
      cout << B.has_inf() << endl;
      

  • See also:



.has_nan()
  • Member function of Mat, Col, Row, Cube, SpMat

  • Returns true if at least one of the elements of the object is NaN (not-a-number)

  • Returns false otherwise

  • Caveat: NaN is not equal to anything, even itself

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = randu<mat>(5,5);
      
      B(1,1) = datum::nan;
      
      cout << A.has_nan() << endl;
      cout << B.has_nan() << endl;
      

  • See also:



.print()
.print( header )

.print( stream )
.print( stream, header )
  • Member functions of Mat, Col, Row, SpMat, Cube and field

  • Print the contents of an object to the std::cout stream (default), or a user specified stream, with an optional header string

  • Objects can also be printed using the << stream operator

  • Elements of a field can only be printed if there is an associated operator<< function defined

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = randu<mat>(6,6);
      
      A.print();
      
      // print a transposed version of A
      A.t().print();
      
      // "B:" is the optional header line
      B.print("B:");
      
      cout << A << endl;
      
      cout << "B:" << endl;
      cout << B << endl;
      

  • See also:



.raw_print()
.raw_print( header )

.raw_print( stream )
.raw_print( stream, header )
  • Member functions of Mat, Col, Row, SpMat and Cube

  • Similar to the .print() member function, with the difference that no formatting of the output is done; the stream's parameters such as precision, cell width, etc. can be set manually

  • If the cell width is set to zero, a space is printed between the elements

  • Examples:
      mat A = randu<mat>(5,5);
      
      cout.precision(11);
      cout.setf(ios::fixed);
      
      A.raw_print(cout, "A:");
      

  • See also:



saving/loading matrices & cubes

.save( name )
.save( name, file_type )

.save( stream )
.save( stream, file_type )
       .load( name )
.load( name, file_type )

.load( stream )
.load( stream, file_type )

  • Member functions of Mat, Col, Row and Cube

  • Store/retrieve data in a file or stream
    • caveat: the stream must be opened in binary mode

  • The default file_type for .save() is arma_binary (see below)

  • The default file_type for .load() is auto_detect (see below)

  • On success, .save() and .load() will return a bool set to true

  • .save() will return a bool set to false if the saving process fails

  • .load() will return a bool set to false if the loading process fails; additionally, the object will be reset so it has no elements

  • file_type can be one of the following:

      auto_detect
      Used by .load() only: try to automatically detect the file type as one of the formats described below. This is the default operation.

      raw_ascii
      Numerical data stored in raw ASCII format, without a header. The numbers are separated by whitespace. The number of columns must be the same in each row. Cubes are loaded as one slice. Data which was saved in Matlab/Octave using the -ascii option can be read in Armadillo, except for complex numbers. Complex numbers are stored in standard C++ notation, which is a tuple surrounded by brackets: eg. (1.23,4.56) indicates 1.24 + 4.56i.

      raw_binary
      Numerical data stored in machine dependent raw binary format, without a header. Matrices are loaded to have one column, while cubes are loaded to have one slice with one column. The .reshape() function can be used to alter the size of the loaded matrix/cube without losing data.

      arma_ascii
      Numerical data stored in human readable text format, with a simple header to speed up loading. The header indicates the type of matrix as well as the number of rows and columns. For cubes, the header additionally specifies the number of slices.

      arma_binary
      Numerical data stored in machine dependent binary format, with a simple header to speed up loading. The header indicates the type of matrix as well as the number of rows and columns. For cubes, the header additionally specifies the number of slices. arma_binary is the default file_type for .save()

      csv_ascii
      Numerical data stored in comma separated value (CSV) text format, without a header. Applicable to Mat only.

      hdf5_binary
      Numerical data stored in portable HDF5 binary format.
      Caveat: support for HDF5 must be enabled within Armadillo's configuration; the hdf5.h header file must be available on your system and you will need to link with the hdf5 library (eg. -lhdf5)

      hdf5_binary_trans
      As per hdf5_binary, but save/load the data with columns transposed to rows (and vice versa)

      pgm_binary
      Image data stored in Portable Gray Map (PGM) format. Applicable to Mat only. Saving int, float or double matrices is a lossy operation, as each element is copied and converted to an 8 bit representation. As such the matrix should have values in the [0,255] interval, otherwise the resulting image may not display correctly.

      ppm_binary
      Image data stored in Portable Pixel Map (PPM) format. Applicable to Cube only. Saving int, float or double matrices is a lossy operation, as each element is copied and converted to an 8 bit representation. As such the cube/field should have values in the [0,255] interval, otherwise the resulting image may not display correctly.


  • Examples:
      mat A = randu<mat>(5,5);
      
      A.save("A1.mat");  // default save format is arma_binary
      
      A.save("A2.mat", arma_ascii);  // force saving in arma_ascii format
      
      mat B;
      B.load("A1.mat");  // automatically detect format type
      
      mat C;
      C.load("A2.mat", arma_ascii);  // force loading in arma_ascii format
      
      
      // example of testing for success
      mat D;
      bool status = D.load("A2.mat");
      
      if(status == true)
        {
        cout << "loaded okay" << endl;
        }
      else
        {
        cout << "problem with loading" << endl;
        }
      

  • See also:



saving/loading fields

.save( name )
.save( name, file_type )

.save( stream )
.save( stream, file_type )
       .load( name )
.load( name, file_type )

.load( stream )
.load( stream, file_type )

  • Store/retrieve data in a file or stream
    • caveat: the stream must be opened in binary mode

  • On success, .save() and .load() will return a bool set to true

  • .save() will return a bool set to false if the saving process fails

  • .load() will return a bool set to false if the loading process fails; additionally, the field will be reset so it has no elements

  • Fields with objects of type std::string are saved and loaded as raw text files. The text files do not have a header. Each string is separated by a whitespace. load() will only accept text files that have the same number of strings on each line. The strings can have variable lengths.

  • Other than storing string fields as text files, the following file formats are supported:

      auto_detect

    • .load(): try to automatically detect the field format type as one of the formats described below; this is the default operation.

    • arma_binary

    • Objects are stored in machine dependent binary format
    • Default type for fields of type Mat, Col, Row or Cube
    • Only applicable to fields of type Mat, Col, Row or Cube

    • ppm_binary

    • Image data stored in Portable Pixmap Map (PPM) format.
    • Only applicable to fields of type Mat, Col or Row
    • .load(): loads the specified image and stores the red, green and blue components as three separate matrices; the resulting field is comprised of the three matrices, with the red, green and blue components in the first, second and third matrix, respectively
    • .save(): saves a field with exactly three matrices of equal size as an image; it is assumed that the red, green and blue components are stored in the first, second and third matrix, respectively; saving int, float or double matrices is a lossy operation, as each matrix element is copied and converted to an 8 bit representation

  • See also:





Generated Vectors/Matrices



eye( n_rows, n_cols )
eye( size(X) )
  • Generate a matrix with the elements along the main diagonal set to one and off-diagonal elements set to zero

  • An identity matrix is generated when n_rows = n_cols

  • Usage:
    • matrix_type X = eye<matrix_type>( n_rows, n_cols )
    • matrix_type Y = eye<matrix_type>( size(X) )

  • Examples:
      mat A = eye<mat>(5,5);  // or:  mat A(5,5,fill::eye);
      
      mat B = 123.0 * eye<mat>(5,5);
      
      mat C = eye<mat>( size(B) );
      

  • See also:



linspace( start, end )
linspace( start, end, N )
  • Generate a vector with N elements; the values of the elements are linearly spaced from start to (and including) end

  • The argument N is optional; by default N = 100

  • Usage:
    • vector_type v = linspace<vector_type>(start, end, N)

  • Caveat: if N ≤ 1, the generated vector will have a size of one, with the single element equal to end

  • Examples:
      vec a = linspace<vec>(0, 5, 6);
      
      vec b = linspace<vec>(5, 0, 6);
      

  • See also:



logspace( A, B )
logspace( A, B, N )
  • Generate a vector with N elements; the values of the elements are logarithmically spaced from 10A to (and including) 10B

  • The argument N is optional; by default N = 50

  • Usage:
    • vector_type v = logspace<vector_type>(A, B, N)

  • Examples:
      vec a = logspace<vec>(0, 5, 6);
      
      vec b = logspace<vec>(5, 0, 6);
      

  • See also:



ones( n_elem )
ones( n_rows, n_cols )
ones( n_rows, n_cols, n_slices )
ones( size(X) )
  • Generate a vector, matrix or cube with all elements set to one

  • Usage:
    • vector_type v = ones<vector_type>( n_elem )
    • matrix_type X = ones<matrix_type>( n_rows, n_cols )
    • matrix_type Y = ones<matrix_type>( size(X) )
    • cube_type Q = ones<cube_type>( n_rows, n_cols, n_slices )
    • cube_type R = ones<cube_type>( size(Q) )

  • Examples:
      vec  v = ones<vec>(10);
      uvec u = ones<uvec>(11);
      mat  A = ones<mat>(5,6);
      cube Q = ones<cube>(5,6,7);
      
      mat  B = 123.0 * ones<mat>(5,6);
      

  • See also:



randi( n_elem )
randi( n_elem, distr_param(a,b) )

randi( n_rows, n_cols )
randi( n_rows, n_cols, distr_param(a,b) )

randi( n_rows, n_cols, n_slices )
randi( n_rows, n_cols, n_slices, distr_param(a,b) )

randi( size(X) )
randi( size(X), distr_param(a,b) )
  • Generate a vector, matrix or cube with the elements set to random integer values in the [a,b] interval

  • The default distribution parameters are a=0 and b=maximum_int

  • Usage:
    • vector_type v = randi<vector_type>( n_elem )
    • vector_type v = randi<vector_type>( n_elem, distr_param(a,b) )

    • matrix_type X = randi<matrix_type>( n_rows, n_cols )
    • matrix_type X = randi<matrix_type>( n_rows, n_cols, distr_param(a,b) )
    • matrix_type Y = randi<matrix_type>( size(X) )
    • matrix_type Y = randi<matrix_type>( size(X), distr_param(a,b) )

    • cube_type Q = randi<cube_type>( n_rows, n_cols, n_slices )
    • cube_type Q = randi<cube_type>( n_rows, n_cols, n_slices, distr_param(a,b) )
    • cube_type R = randi<cube_type>( size(Q) )
    • cube_type R = randi<cube_type>( size(Q), distr_param(a,b) )

  • To change the RNG seed, use arma_rng::set_seed(value) or arma_rng::set_seed_random() functions

  • Caveat: to generate a continuous distribution with floating point values (ie. float or double), use randu() or randn() instead

  • Examples:
      imat A = randi<imat>(5, 6);
      
      imat A = randi<imat>(6, 7, distr_param(-10, +20));
      
      arma_rng::set_seed_random();  // set the seed to a random value
      
  • See also:



randu( n_elem )
randu( n_rows, n_cols )
randu( n_rows, n_cols, n_slices )
randu( size(X) )

randn( n_elem )
randn( n_rows, n_cols )
randn( n_rows, n_cols, n_slices )
randn( size(X) )
  • Generate a vector, matrix or cube with the elements set to random floating point values

  • randu() uses a uniform distribution in the [0,1] interval

  • randn() uses a normal/Gaussian distribution with zero mean and unit variance

  • Usage:
    • vector_type v = randu<vector_type>( n_elem )
    • matrix_type X = randu<matrix_type>( n_rows, n_cols )
    • matrix_type Y = randu<matrix_type>( size(X) )
    • cube_type Q = randu<cube_type>( n_rows, n_cols, n_slices )
    • cube_type R = randu<cube_type>( size(Q) )

  • To change the RNG seed, use arma_rng::set_seed(value) or arma_rng::set_seed_random() functions

  • Caveat: to generate a matrix with random integer values instead of floating point values, use randi() instead

  • Examples:
      vec  v = randu<vec>(5);
      mat  A = randu<mat>(5,6);
      cube Q = randu<cube>(5,6,7);
      
      arma_rng::set_seed_random();  // set the seed to a random value
      
  • See also:



randg( n_elem )
randg( n_elem, distr_param(a,b) )

randg( n_rows, n_cols )
randg( n_rows, n_cols, distr_param(a,b) )

randg( n_rows, n_cols, n_slices )
randg( n_rows, n_cols, n_slices, distr_param(a,b) )

randg( size(X) )
randg( size(X), distr_param(a,b) )
  • Generate a vector, matrix or cube with the elements set to random values from a gamma distribution:
        x a-1 exp( -x / b )
      p(x|a,b) = 
        b a Γ(a)

  • The default distribution parameters are a=1 and b=1

  • Usage:
    • vector_type v = randg<vector_type>( n_elem )
    • vector_type v = randg<vector_type>( n_elem, distr_param(a,b) )

    • matrix_type X = randg<matrix_type>( n_rows, n_cols )
    • matrix_type X = randg<matrix_type>( n_rows, n_cols, distr_param(a,b) )
    • matrix_type Y = randg<matrix_type>( size(X) )
    • matrix_type Y = randg<matrix_type>( size(X), distr_param(a,b) )

    • cube_type Q = randg<cube_type>( n_rows, n_cols, n_slices )
    • cube_type Q = randg<cube_type>( n_rows, n_cols, n_slices, distr_param(a,b) )
    • cube_type R = randg<cube_type>( size(Q) )
    • cube_type R = randg<cube_type>( size(Q), distr_param(a,b) )

  • Caveat: randg() is available only when using a C++11 compiler

  • To change the RNG seed, use arma_rng::set_seed(value) or arma_rng::set_seed_random() functions

  • Examples:
      vec v = randg<vec>(100, distr_param(2,1));
      
      mat X = randg<mat>(10, 10, distr_param(2,1));
      
  • See also:



regspace( start, end )
regspace( start, delta, end )
  • Generate a vector with regularly spaced elements:
    [  (start + 0*delta),  (start + 1*delta),  (start + 2*delta),  ,  (start + M*delta)  ]
    where M = floor((end-start)/delta), so that (start + M*delta) ≤ end

  • Similar in operation to the Matlab/Octave colon operator, ie. start:end  and  start:delta:end

  • If delta is not specified:
    • delta = +1, if start ≤ end
    • delta = −1, if start > end   (caveat: this is different to Matlab/Octave)

  • An empty vector is generated when one of the following conditions is true:
    • start < end, and delta < 0
    • start > end, and delta > 0
    • delta = 0

  • Usage:
    • vector_type v = regspace<vector_type>(start, end)
    • vector_type v = regspace<vector_type>(start, delta, end)

  • Examples:
       vec a = regspace< vec>(0,  9);       // 0,  1, ...,   9
      uvec b = regspace<uvec>(2,  2,  10);  // 2,  4, ...,  10
      ivec c = regspace<ivec>(0, -1, -10);  // 0, -1, ..., -10
      

  • Caveat: do not use regspace() to specify ranges for contiguous submatrix views; use span() instead

  • See also:



speye( n_rows, n_cols )
speye( size(X) )
  • Generate a sparse matrix with the elements along the main diagonal set to one and off-diagonal elements set to zero

  • An identity matrix is generated when n_rows = n_cols

  • Usage:
    • sparse_matrix_type X = speye<sparse_matrix_type>( n_rows, n_cols )
    • sparse_matrix_type Y = speye<sparse_matrix_type>( size(X) )

  • Examples:
      sp_mat A = speye<sp_mat>(5,5);
      

  • See also:



spones( A )
  • Generate a sparse matrix with the same structure as sparse matrix A, but with the non-zero elements set to one

  • Examples:
      sp_mat A = sprandu<sp_mat>(100, 200, 0.1);
      
      sp_mat B = spones(A);
      
  • See also:



sprandu( n_rows, n_cols, density )
sprandn( n_rows, n_cols, density )

sprandu( size(X), density )
sprandn( size(X), density )
  • Generate a sparse matrix with the non-zero elements set to random values

  • The density argument specifies the percentage of non-zero elements; it must be in the [0,1] interval

  • sprandu() uses a uniform distribution in the [0,1] interval

  • sprandn() uses a normal/Gaussian distribution with zero mean and unit variance

  • Usage:
    • sparse_matrix_type X = sprandu<sparse_matrix_type>( n_rows, n_cols, density )
    • sparse_matrix_type Y = sprandu<sparse_matrix_type>( size(X), density )

  • To change the RNG seed, use arma_rng::set_seed(value) or arma_rng::set_seed_random() functions

  • Examples:
      sp_mat A = sprandu<sp_mat>(100, 200, 0.1);
      
  • See also:



toeplitz( A )
toeplitz( A, B )
circ_toeplitz( A )


zeros( n_elem )
zeros( n_rows, n_cols )
zeros( n_rows, n_cols, n_slices )
zeros( size(X) )
  • Generate a vector, matrix or cube with the elements set to zero

  • Usage:
    • vector_type v = zeros<vector_type>( n_elem )
    • matrix_type X = zeros<matrix_type>( n_rows, n_cols )
    • matrix_type Y = zeros<matrix_type>( size(X) )
    • cube_type Q = zeros<cube_type>( n_rows, n_cols, n_slices )
    • cube_type R = zeros<cube_type>( size(Q) )

  • Examples:
      vec  v = zeros<vec>(10);
      uvec u = zeros<uvec>(11);
      mat  A = zeros<mat>(5,6);
      cube Q = zeros<cube>(5,6,7);
      

  • See also:





Functions of Vectors/Matrices/Cubes



abs( X )
  • Obtain the magnitude of each element

  • Usage for non-complex matrices and cubes:
    • matrix_type Y = abs(X)
    • cube_type Y = abs(X)
    • X and Y must have the same matrix_type / cube_type

  • Usage for complex matrices and cubes:
    • non_complex_matrix_type Y = abs(X)
    • non_complex_cube_type Y = abs(X)
    • X must be a have complex matrix / cube type, eg. cx_mat or cx_cube
    • The type of Y must be related to the type of X, eg. if X has the type cx_mat, then the type of Y must be mat

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = abs(A); 
      
      cx_mat X = randu<cx_mat>(5,5);
      mat    Y = abs(X);
      
  • See also:



accu( X )
  • Accumulate (sum) all elements of a vector, matrix or cube

  • Examples:
      mat A(5, 6, fill::randu);
      mat B(5, 6, fill::randu);
      
      double x = accu(A);
      
      double y = accu(A % B);
      
      // accu(A % B) is a "multiply-and-accumulate" operation
      // as operator % performs element-wise multiplication
      

  • See also:



all( V )
all( X )
all( X, dim )
  • For vector V, return true if all elements of the vector are non-zero or satisfy a relational condition

  • For matrix X and
    • dim=0, return a row vector (of type urowvec or umat), with each element (0 or 1) indicating whether the corresponding column of X has all non-zero elements
    • dim=1, return a column vector (of type ucolvec or umat), with each element (0 or 1) indicating whether the corresponding row of X has all non-zero elements

  • The dim argument is optional; by default dim=0 is used

  • Relational operators can be used instead of V or X, eg. A > 0.5

  • Examples:
      vec V = randu<vec>(10);
      mat X = randu<mat>(5,5);
      
      
      // status1 will be set to true if vector V has all non-zero elements
      bool status1 = all(V);
      
      // status2 will be set to true if vector V has all elements greater than 0.5
      bool status2 = all(V > 0.5);
      
      // status3 will be set to true if matrix X has all elements greater than 0.6;
      // note the use of vectorise()
      bool status3 = all(vectorise(X) > 0.6);
      
      // generate a row vector indicating which columns of X have all elements greater than 0.7
      umat A = all(X > 0.7);
      
      

  • See also:



any( V )
any( X )
any( X, dim )
  • For vector V, return true if any element of the vector is non-zero or satisfies a relational condition

  • For matrix X and
    • dim=0, return a row vector (of type urowvec or umat), with each element (0 or 1) indicating whether the corresponding column of X has any non-zero elements
    • dim=1, return a column vector (of type ucolvec or umat), with each element (0 or 1) indicating whether the corresponding row of X has any non-zero elements

  • The dim argument is optional; by default dim=0 is used

  • Relational operators can be used instead of V or X, eg. A > 0.9

  • Examples:
      vec V = randu<vec>(10);
      mat X = randu<mat>(5,5);
      
      
      // status1 will be set to true if vector V has any non-zero elements
      bool status1 = any(V);
      
      // status2 will be set to true if vector V has any elements greater than 0.5
      bool status2 = any(V > 0.5);
      
      // status3 will be set to true if matrix X has any elements greater than 0.6;
      // note the use of vectorise()
      bool status3 = any(vectorise(X) > 0.6);
      
      // generate a row vector indicating which columns of X have elements greater than 0.7
      umat A = any(X > 0.7);
      
      

  • See also:



approx_equal( A, B, method, tol )
approx_equal( A, B, method, abs_tol, rel_tol )
  • Return true if all corresponding elements in A and B are approximately equal

  • Return false if any of the corresponding elements in A and B are not approximately equal, or if A and B have different dimensions

  • The argument method controls how the approximate equality is determined; it is one of:
      "absdiff" ↦ scalars x and y are considered equal if |x − y| ≤ tol
      "reldiff" ↦ scalars x and y are considered equal if |x − y| / max( |x|, |y| ) ≤ tol
      "both" ↦ scalars x and y are considered equal if |x − y| ≤ abs_tol  or  |x − y| / max( |x|, |y| ) ≤ rel_tol

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = A + 0.001;
      
      bool same1 = approx_equal(A, B, "absdiff", 0.002);
      
      
      mat C = 1000 * randu<mat>(5,5);
      mat D = C + 1;
      
      bool same2 = approx_equal(C, D, "reldiff", 0.1);
      
      bool same3 = approx_equal(C, D, "both", 2, 0.1);
      
  • See also:



as_scalar( expression )
  • Evaluate an expression that results in a 1x1 matrix, followed by converting the 1x1 matrix to a pure scalar

  • If a binary or trinary expression is given (ie. 2 or 3 terms), the function will try to exploit the fact that the result is a 1x1 matrix by using optimised expression evaluations

  • Examples:
      rowvec r = randu<rowvec>(5);
      colvec q = randu<colvec>(5);
      mat    X = randu<mat>(5,5);
      
      // examples of some expressions
      // for which optimised implementations exist
      
      double a = as_scalar(r*q);
      double b = as_scalar(r*X*q);
      double c = as_scalar(r*diagmat(X)*q);
      double d = as_scalar(r*inv(diagmat(X))*q);
      

  • See also:



clamp( X, min_val, max_val )
  • Create a copy of X with each element clamped to be between min_val and max_val

  • Examples:
      mat A = randu<mat>(5,5);
      
      mat B = clamp(A, 0.2, 0.8); 
      
      mat C = clamp(A, A.min(), 0.8); 
      
      mat D = clamp(A, 0.2, A.max()); 
      
  • See also:



cond( A )
  • Return the condition number of matrix A (the ratio of the largest singular value to the smallest)

  • Large condition numbers suggest that matrix A is nearly singular

  • The computation is based on singular value decomposition; if the decomposition fails, a std::runtime_error exception is thrown

  • Caveat: the rcond() function is faster for providing an estimate of the reciprocal of the condition number

  • Examples:
      mat    A = randu<mat>(5,5);
      double c = cond(A);
      

  • See also:



conj( X )
  • Obtain the complex conjugate of each element in a complex matrix or cube

  • Examples:
      cx_mat X = randu<cx_mat>(5,5);
      cx_mat Y = conj(X);
      

  • See also:



conv_to< type >::from( X )
  • Convert from one matrix type to another (eg. mat to imat), or one cube type to another (eg. cube to icube)

  • Conversion between std::vector and Armadillo matrices/vectors is also possible

  • Conversion of a mat object into colvec, rowvec or std::vector is possible if the object can be interpreted as a vector

  • Examples:
      mat  A = randu<mat>(5,5);
      fmat B = conv_to<fmat>::from(A);
      
      typedef std::vector<double> stdvec;
      
      stdvec x(3);
      x[0] = 0.0; x[1] = 1.0;  x[2] = 2.0;
      
      colvec y = conv_to< colvec >::from(x);
      stdvec z = conv_to< stdvec >::from(y); 
      

  • See also:



cross( A, B )


cumsum( V )
cumsum( X )
cumsum( X, dim )
  • For vector V, return a vector of the same orientation, containing the cumulative sum of elements

  • For matrix X, return a matrix containing the cumulative sum of elements in each column (dim=0), or each row (dim=1)

  • The dim argument is optional; by default dim=0 is used

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = cumsum(A);
      mat C = cumsum(A, 1);
      
      vec x = randu<vec>(10);
      vec y = cumsum(x);
      

  • See also:



cumprod( V )
cumprod( X )
cumprod( X, dim )
  • For vector V, return a vector of the same orientation, containing the cumulative product of elements

  • For matrix X, return a matrix containing the cumulative product of elements in each column (dim=0), or each row (dim=1)

  • The dim argument is optional; by default dim=0 is used

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = cumprod(A);
      mat C = cumprod(A, 1);
      
      vec x = randu<vec>(10);
      vec y = cumprod(x);
      

  • See also:



det( A )
log_det( val, sign, A )
  • Determinant of square matrix A

  • If A is not square sized, a std::logic_error exception is thrown

  • log_det(): find the log determinant, such that the determinant is equal to exp(val)*sign

  • Caveat: for large matrices log_det() is more precise than det()

  • Examples:
      mat    A = randu<mat>(5,5);
      double x = det(A);
      
      double val;
      double sign;
      
      log_det(val, sign, A);
      

  • See also:



diagmat( V )
diagmat( V, k )
diagmat( X )
diagmat( X, k )
  • Generate a diagonal matrix from vector V or matrix X

  • Given vector V, generate a square matrix with the k-th diagonal containing a copy of the vector; all other elements are set to zero

  • Given matrix X, generate a matrix with the k-th diagonal containing a copy of the k-th diagonal of X; all other elements are set to zero

  • The argument k is optional; by default the main diagonal is used (k=0)

  • For k > 0, the k-th super-diagonal is used (top-right corner)

  • For k < 0, the k-th sub-diagonal is used (bottom-left corner)

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = diagmat(A);
      mat C = diagmat(A,1);
      
      vec v = randu<vec>(5);
      mat D = diagmat(v);
      mat E = diagmat(v,1);
      

  • See also:



diagvec( A )
diagvec( A, k )
  • Extract the k-th diagonal from matrix A

  • The argument k is optional; by default the main diagonal is extracted (k=0)

  • For k > 0, the k-th super-diagonal is extracted (top-right corner)

  • For k < 0, the k-th sub-diagonal is extracted (bottom-left corner)

  • The extracted diagonal is interpreted as a column vector

  • Examples:
      mat A = randu<mat>(5,5);
      
      vec d = diagvec(A);
      

  • See also:



diff( V )
diff( V, k )

diff( X )
diff( X, k )
diff( X, k, dim )
  • For vector V, return a vector of the same orientation, containing the differences between consecutive elements

  • For matrix X, return a matrix containing the differences between consecutive elements in each column (dim=0), or each row (dim=1)

  • The optional argument k indicates that the differences are calculated recursively k times; by default k=1 is used

  • The resulting number of differences is n - k, where n is the number of elements; if n ≤ k, the number of differences is zero (ie. an empty vector/matrix is returned)

  • The argument dim is optional; by default dim=0

  • Examples:
      vec a = linspace<vec>(1,10,10);
      
      vec b = diff(a);
      

  • See also:



dot( A, B )
cdot( A, B )
norm_dot( A, B )
  • dot(A,B): dot product of A and B, treating A and B as vectors

  • cdot(A,B): as per dot(A,B), but the complex conjugate of A is used

  • norm_dot(A,B): normalised dot product; equivalent to dot(A,B) / (∥A∥•∥B∥)

  • Caveat: norm() is more robust for calculating the norm, as it handles underflows and overflows

  • Examples:
      vec a = randu<vec>(10);
      vec b = randu<vec>(10);
      
      double x = dot(a,b);
      

  • See also:



eps( X )


B = expmat(A)
expmat(B, A)


B = expmat_sym(A)
expmat_sym(B, A)
  • Matrix exponential of symmetric/hermitian matrix A

  • If A is not square sized, a std::logic_error exception is thrown

  • If the matrix exponential cannot be found:
    • B = expmat_sym(A) resets B and throws a std::runtime_error exception
    • expmat_sym(B,A) resets B and returns a bool set to false (exception is not thrown)

  • Caveat: there is currently no check whether A is symmetric

  • Caveat: the matrix exponential operation is generally not the same as applying the exp() function to each element

  • Examples:
      mat A = randu<mat>(5,5);
      
      mat B = A*A.t();   // make symmetric matrix
      
      mat C = expmat_sym(B);
      

  • See also:



find( X )
find( X, k )
find( X, k, s )
  • Return a column vector containing the indices of elements of X that are non-zero or satisfy a relational condition

  • The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)

  • X is interpreted as a vector, with column-by-column ordering of the elements of X

  • Relational operators can be used instead of X, eg. A > 0.5

  • If k=0 (default), return the indices of all non-zero elements, otherwise return at most k of their indices

  • If s="first" (default), return at most the first k indices of the non-zero elements

  • If s="last", return at most the last k indices of the non-zero elements

  • Examples:
      mat  A  = randu<mat>(5,5);
      mat  B  = randu<mat>(5,5);
      
      uvec q1 = find(A > B);
      uvec q2 = find(A > 0.5);
      uvec q3 = find(A > 0.5, 3, "last");
      
      // change elements of A greater than 0.5 to 1
      A.elem( find(A > 0.5) ).ones();
      

  • See also:



find_finite( X )
  • Return a column vector containing the indices of elements of X that are finite (ie. not ±Inf and not NaN)

  • The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)

  • X is interpreted as a vector, with column-by-column ordering of the elements of X

  • Examples:
      mat A = randu<mat>(5,5);
      
      A(1,1) = datum::inf;
      
      // accumulate only finite elements
      double val = accu( A.elem( find_finite(A) ) );
      

  • See also:



find_nonfinite( X )
  • Return a column vector containing the indices of elements of X that are non-finite (ie. ±Inf or NaN)

  • The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)

  • X is interpreted as a vector, with column-by-column ordering of the elements of X

  • Examples:
      mat A = randu<mat>(5,5);
      
      A(1,1) = datum::inf;
      
      // change non-finite elements to zero
      A.elem( find_nonfinite(A) ).zeros();
      

  • See also:



find_unique( X )
find_unique( X, ascending_indices )
  • Return a column vector containing the indices of unique elements of X

  • The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)

  • X is interpreted as a vector, with column-by-column ordering of the elements of X

  • The ascending_indices argument is optional; it is one of:
      true = the returned indices are sorted to be ascending (default setting)
      false = the returned indices are in arbitrary order (faster operation)

  • Examples:
      mat A;
      
      A << 2 << 2 << 4 << endr
        << 4 << 6 << 6 << endr;
      
      uvec indices = find_unique(A);
      

  • See also:



fliplr( X )
flipud( X )
  • fliplr(): generate a copy of matrix X, with the order of the columns reversed

  • flipud(): generate a copy of matrix X, with the order of the rows reversed

  • Examples:
      mat A = randu<mat>(5,5);
      
      mat B = fliplr(A);
      mat C = flipud(A);
      

  • See also:



imag( X )
real( X )
  • Extract the imaginary/real part of a complex matrix or cube

  • Examples:
      cx_mat C = randu<cx_mat>(5,5);
      
      mat    A = imag(C);
      mat    B = real(C);
      

  • Caveat: versions 4.4, 4.5 and 4.6 of the GCC C++ compiler have a bug when using the -std=c++0x compiler option (ie. experimental support for C++11); to work around this bug, preface Armadillo's imag() and real() with the arma namespace qualification, eg. arma::imag(C)

  • See also:



uvec sub = ind2sub( size(X), index )
umat sub = ind2sub( size(X), vector_of_indices )
  • Convert a linear index, or a vector of indices, to subscript notation

  • The argument size(X) can be replaced with size(n_rows, n_cols) or size(n_rows, n_cols, n_slices)

  • std::logic_error exception is thrown if an index is out of range

  • When only one index is given, the subscripts are returned in a vector of type uvec

  • When a vector of indices (of type uvec) is given, the corresponding subscripts are returned in each column of an m x n matrix of type umat; m=2 for matrix subscripts, while m=3 for cube subscripts

  • Examples:
      mat M(4, 5, fill::randu);
      
      uvec s = ind2sub( size(M), 6 );
      
      cout << "row: " << s(0) << endl;
      cout << "col: " << s(1) << endl;
      
      
      uvec indices = find(M > 0.5);
      umat t       = ind2sub( size(M), indices );
      
      
      cube Q(2,3,4);
      
      uvec u = ind2sub( size(Q), 8 );
      
      cout << "row:   " << u(0) << endl;
      cout << "col:   " << u(1) << endl;
      cout << "slice: " << u(2) << endl;
      

  • See also:



index_min( V )
index_min( M )
index_min( M, dim )
       index_max( V )
index_max( M )
index_max( M, dim )
  • For vector V, return the linear index of the extremum value; the returned index is of type uword

  • For matrix M and:
    • dim=0, return a row vector (of type urowvec or umat), with each column containing the index of the extremum value in the corresponding column of M
    • dim=1, return a column vector (of type uvec or umat), with each row containing the index of the extremum value in the corresponding row of M

  • For each column and row, the index starts at zero

  • The dim argument is optional; by default dim=0 is used

  • For objects with complex numbers, absolute values are used for comparison

  • Examples:
      vec   v = randu<vec>(10);
      uword i = index_max(v);
      
      double max_val_in_v = v(i);
      
      
      mat     M  = randu<mat>(5,6);
      
      urowvec ii = index_max(M);
      ucolvec jj = index_max(M,1);
      
      double max_val_in_col_2 = M( ii(2), 2 );
      
      double max_val_in_row_4 = M( 4, jj(4) );
      

  • See also:



inplace_trans( X )
inplace_trans( X, method )

inplace_strans( X )
inplace_strans( X, method )
  • In-place / in-situ transpose of matrix X

  • For real (non-complex) matrix:
    • inplace_trans() performs a normal transpose
    • inplace_strans() not applicable

  • For complex matrix:
    • inplace_trans() performs a Hermitian transpose (ie. the conjugate of the elements is taken during the transpose)
    • inplace_strans() provides a transposed copy without taking the conjugate of the elements

  • The argument method is optional

  • By default, a greedy transposition algorithm is used; a low-memory algorithm can be used instead by explicitly setting method to "lowmem"

  • The low-memory algorithm is considerably slower than the greedy algorithm; using the low-memory algorithm is only recommended for cases where X takes up more than half of available memory (ie. very large X)

  • Examples:
      mat X = randu<mat>(4,5);
      mat Y = randu<mat>(20000,30000);
      
      inplace_trans(X);            // use greedy algorithm by default
      
      inplace_trans(Y, "lowmem");  // use low-memory (and slow) algorithm
      

  • See also:



is_finite( X )
  • Returns true if all elements in X are finite

  • Returns false if at least one element in X is non-finite (±infinity or NaN)

  • X can be a scalar (eg. double), vector, matrix or cube

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = randu<mat>(5,5);
      
      B(1,1) = datum::inf;
      
      cout << is_finite(A) << endl;
      cout << is_finite(B) << endl;
      
      cout << is_finite( 0.123456789 ) << endl;
      cout << is_finite( datum::nan  ) << endl;
      cout << is_finite( datum::inf  ) << endl;
      

  • See also:



join_rows( A, B )
join_horiz( A, B )

join_cols( A, B )
join_vert( A, B )
  • join_rows() and join_horiz(): horizontal concatenation; for two matrices A and B, join each row of A with the corresponding row of B; matrices A and B must have the same number of rows

  • join_cols() and join_vert(): vertical concatenation; for two matrices A and B, join each column of A with the corresponding column of B; matrices A and B must have the same number of columns

  • Examples:
      mat A = randu<mat>(4,5);
      mat B = randu<mat>(4,6);
      mat C = randu<mat>(6,5);
      
      mat X = join_rows(A,B);
      mat Y = join_cols(A,C);
      

  • See also:



join_slices( cube C, cube D )
join_slices( mat M, mat N )

join_slices( mat M, cube C )
join_slices( cube C, mat M )
  • for two cubes C and D: join the slices of C with the slices of D; cubes C and D must have the same number of rows and columns (ie. all slices must have the same size)

  • for two matrices M and N: treat M and N as cube slices and join them to form a cube with 2 slices; matrices M and N must have the same number of rows and columns

  • for matrix M and cube C: treat M as a cube slice and join it with the slices of C; matrix M and cube C must have the same number of rows and columns

  • Examples:
      cube C(5, 10, 3, fill::randu);
      cube D(5, 10, 4, fill::randu);
      
      cube E = join_slices(C,D);
      
      mat M(10, 20, fill::randu);
      mat N(10, 20, fill::randu);
      
      cube Q = join_slices(M,N);
      
      cube R = join_slices(Q,M);
      
      cube S = join_slices(M,Q);
      

  • See also:



kron( A, B )
  • Kronecker tensor product

  • Given matrix A (with n rows and p columns) and matrix B (with m rows and q columns), generate a matrix (with nm rows and pq columns) that denotes the tensor product of A and B

  • Examples:
      mat A = randu<mat>(4,5);
      mat B = randu<mat>(5,4);
      
      mat K = kron(A,B);
      

  • See also:



B = logmat(A)
logmat(B, A)
  • Complex matrix logarithm of general square matrix A

  • If A is not square sized, a std::logic_error exception is thrown

  • If the matrix logarithm cannot be found:
    • B = logmat(A) resets B and throws a std::runtime_error exception
    • logmat(B,A) resets B and returns a bool set to false (exception is not thrown)

  • Caveat: if matrix A is symmetric positive definite, using logmat_sympd() is faster

  • Caveat: the matrix logarithm operation is generally not the same as applying the log() function to each element

  • Examples:
         mat A = randu<mat>(5,5);
      
      cx_mat B = logmat(A);
      

  • See also:



B = logmat_sympd(A)
logmat_sympd(B, A)


min( V )
min( M )
min( M, dim )
min( Q )
min( Q, dim )
min( A, B )
       max( V )
max( M )
max( M, dim )
max( Q )
max( Q, dim )
max( A, B )
  • For vector V, return the extremum value

  • For matrix M, return the extremum value for each column (dim=0), or each row (dim=1)

  • For cube Q, return the extremum values of elements along dimension dim, where dim ∈ { 0, 1, 2 }

  • The dim argument is optional; by default dim=0 is used

  • For two matrices/cubes A and B, return a matrix/cube containing element-wise extremum values

  • For objects with complex numbers, absolute values are used for comparison

  • Examples:
      colvec v = randu<colvec>(10,1);
      double x = max(v);
      
      mat    M = randu<mat>(10,10);
      
      rowvec a = max(M);
      rowvec b = max(M,0); 
      colvec c = max(M,1);
      
      // element-wise maximum
      mat X = randu<mat>(5,6);
      mat Y = randu<mat>(5,6);
      mat Z = arma::max(X,Y);  // use arma:: prefix to distinguish from std::max()
      

  • See also:



nonzeros(X)
  • Return a column vector containing the non-zero values of X

  • X can be a sparse or dense matrix

  • Examples:
      sp_mat A = sprandu<sp_mat>(100, 100, 0.1);
         vec a = nonzeros(A);
      
      mat B(100, 100, fill::eye);
      vec b = nonzeros(B);
      

  • See also:



norm( X )
norm( X, p )
  • Compute the p-norm of X, where X can be a vector or matrix

  • For vectors, p is an integer ≥1, or one of: "-inf", "inf", "fro"

  • For matrices, p is one of: 1, 2, "inf", "fro"; the calculated norm is the induced norm (not entrywise norm)

  • "-inf" is the minimum norm, "inf" is the maximum norm, while "fro" is the Frobenius norm

  • The argument p is optional; by default p=2 is used

  • For vector norm with p=2 and matrix norm with p="fro", a robust algorithm is used to reduce the likelihood of underflows and overflows

  • To obtain the zero norm or Hamming norm (ie. the number of non-zero elements), use this expression: accu(X != 0)

  • Examples:
      vec    q = randu<vec>(5);
      
      double x = norm(q, 2);
      double y = norm(q, "inf");
      

  • See also:



normalise( V )
normalise( V, p )

normalise( X )
normalise( X, p )
normalise( X, p, dim )
  • For vector V, return its normalised version (ie. having unit p-norm)

  • For matrix X, return its normalised version, where each column (dim=0) or row (dim=1) has been normalised to have unit p-norm

  • The p argument is optional; by default p=2 is used

  • The dim argument is optional; by default dim=0 is used

  • Examples:
      vec A = randu<vec>(10);
      vec B = normalise(A);
      vec C = normalise(A, 1);
      
      mat X = randu<mat>(5,6);
      mat Y = normalise(X);
      mat Z = normalise(X, 2, 1);
      

  • See also:



prod( V )
prod( M )
prod( M, dim )
  • For vector V, return the product of all elements

  • For matrix M, return the product of elements in each column (dim=0), or each row (dim=1)

  • The dim argument is optional; by default dim=0 is used

  • Examples:
      colvec v = randu<colvec>(10,1);
      double x = prod(v);
      
      mat    M = randu<mat>(10,10);
      
      rowvec a = prod(M);
      rowvec b = prod(M,0);
      colvec c = prod(M,1);
      

  • See also:



rank( X )
rank( X, tolerance )
  • Return the rank of matrix X

  • Any singular values less than tolerance are treated as zero

  • The tolerance argument is optional; by default tolerance is max(m,n)*max_sv*datum::eps, where:
    • m = number of rows and n = number of columns in X
    • max_sv = maximal singular value of X
    • datum::eps = difference between 1 and the least value greater than 1 that is representable

  • The computation is based on singular value decomposition; if the decomposition fails, a std::runtime_error exception is thrown

  • Caveat: to confirm whether a matrix is singular, use rcond() or cond()

  • Examples:
      mat   A = randu<mat>(4,5);
      
      uword r = rank(A);
      

  • See also:



rcond( A )
  • Return an estimate of the reciprocal of the condition number of square matrix A

  • Values close to 1 suggest that A is well-conditioned

  • Values close to 0 suggest that A is badly conditioned

  • If A is not square sized, a std::logic_error exception is thrown

  • Examples:
      mat    A = randu<mat>(5,5);
      double r = rcond(A);
      

  • See also:



repmat( A, num_copies_per_row, num_copies_per_col )
  • Generate a matrix by replicating matrix A in a block-like fashion

  • The generated matrix has the following size:
      rows = num_copies_per_row*A.n_rows
      cols = num_copies_per_col*A.n_cols

  • Examples:
      mat A = randu<mat>(2, 3);
      
      mat B = repmat(A, 4, 5);
      
  • See also:



reshape( mat, n_rows, n_cols )
reshape( mat, size(X) )

reshape( cube, n_rows, n_cols, n_slices )
reshape( cube, size(X) )
  • Generate a matrix or cube sized according to given size specifications, whose elements are taken from the given matrix/cube in a column-wise manner; the elements in the generated object are placed column-wise (ie. the first column is filled up before filling the second column)

  • The layout of the elements in the generated object will be different to the layout in the given object

  • If the total number of elements in the given object is less than the specified size, the remaining elements in the generated object are set to zero

  • If the total number of elements in the given object is greater than the specified size, only a subset of elements is taken from the given object

  • Caveats:
    • do not use reshape() if you simply want to change the size without preserving data; use .set_size() instead, which is much faster
    • to grow/shrink a matrix while preserving the elements as well as the layout of the elements, use resize() instead
    • to create a vector representation of a matrix (ie. concatenate all the columns or rows), use vectorise() instead

  • Examples:
      mat A = randu<mat>(10, 5);
      mat B = reshape(A, 5, 10);
      

  • See also:



resize( mat, n_rows, n_cols )
resize( mat, size(X) )

resize( cube, n_rows, n_cols, n_slices )
resize( cube, size(X) )
  • Generate a matrix or cube sized according to given size specifications, whose elements as well as the layout of the elements are taken from the given matrix/cube

  • Caveat: do not use resize() if you simply want to change the size without preserving data; use .set_size() instead, which is much faster

  • Examples:
      mat A = randu<mat>(4, 5);
      mat B = resize(A, 7, 6);
      

  • See also:



shift( V, N )
shift( X, N )
shift( X, N, dim )
  • For vector V, generate a copy of the vector with the elements shifted by N positions in a circular manner

  • For matrix X, generate a copy of the matrix with the elements shifted by N positions in each column (dim=0), or each row (dim=1)

  • N can be positive of negative

  • The dim argument is optional; by default dim=0 is used

  • Examples:
      mat A = randu<mat>(4,5);
      mat B = shift(A, -1);
      mat C = shift(A, +1);
      

  • See also:



shuffle( V )
shuffle( X )
shuffle( X, dim )
  • For vector V, generate a copy of the vector with the elements shuffled

  • For matrix X, generate a copy of the matrix with the elements shuffled in each column (dim=0), or each row (dim=1)

  • The dim argument is optional; by default dim=0 is used

  • Examples:
      mat A = randu<mat>(4,5);
      mat B = shuffle(A);
      

  • See also:



size( X )
size( n_rows, n_cols )
size( n_rows, n_cols, n_slices )
  • Obtain the dimensions of object X, or explicitly specify the dimensions

  • The dimensions can be used in conjunction with:

  • The dimensions support simple arithmetic operations; they can also be printed and compared for equality/inequality

  • Examples:
      mat A(5,6);
      
      mat B(size(A), fill::zeros);
      
      mat C; C.randu(size(A));
      
      mat D = ones<mat>(size(A));
      
      mat E(10,20, fill::ones);
      E(3,4,size(C)) = C;    // access submatrix of E
      
      mat F( size(A) + size(E) );
      
      mat G( size(A) * 2 );
      
      cout << "size of A: " << size(A) << endl;
      
      bool is_same_size = (size(A) == size(E));
      

  • See also:



sort( V )
sort( V, sort_direction )

sort( X )
sort( X, sort_direction )
sort( X, sort_direction, dim )
  • For vector V, return a vector which is a sorted version of the input vector

  • For matrix X, return a matrix with the elements of the input matrix sorted in each column (dim=0), or each row (dim=1)

  • The dim argument is optional; by default dim=0 is used

  • The sort_direction argument is optional; sort_direction is either "ascend" or "descend"; by default "ascend" is used

  • For matrices and vectors with complex numbers, sorting is via absolute values

  • Examples:
      mat A = randu<mat>(10,10);
      mat B = sort(A);
      
  • See also:



sort_index( X )
sort_index( X, sort_direction )

stable_sort_index( X )
stable_sort_index( X, sort_direction )
  • Return a vector which describes the sorted order of the elements of X (ie. it contains the indices of the elements of X)

  • The output vector must have the type uvec (ie. the indices are stored as unsigned integers of type uword)

  • X is interpreted as a vector, with column-by-column ordering of the elements of X

  • The sort_direction argument is optional; sort_direction is either "ascend" or "descend"; by default "ascend" is used

  • The stable_sort_index() variant preserves the relative order of elements with equivalent values

  • For matrices and vectors with complex numbers, sorting is via absolute values

  • Examples:
      vec  q       = randu<vec>(10);
      uvec indices = sort_index(q);
      

  • See also:



B = sqrtmat(A)
sqrtmat(B,A)


B = sqrtmat_sympd(A)
sqrtmat_sympd(B,A)


sum( V )
sum( M )
sum( M, dim )
sum( Q )
sum( Q, dim )
  • For vector V, return the sum of all elements

  • For matrix M, return the sum of elements in each column (dim=0), or each row (dim=1)

  • For cube Q, return the sums of elements along dimension dim, where dim ∈ { 0, 1, 2 }; for example, dim=0 indicates the sum of elements in each column within each slice

  • The dim argument is optional; by default dim=0 is used

  • Caveat: to get a sum of all the elements regardless of the object type (ie. vector, or matrix, or cube), use accu() instead

  • Examples:
      colvec v = randu<colvec>(10,1);
      double x = sum(v);
      
      mat    M = randu<mat>(10,10);
      
      rowvec a = sum(M);
      rowvec b = sum(M,0);
      colvec c = sum(M,1);
      
      double y = accu(M);   // find the overall sum regardless of object type
      

  • See also:



uword index = sub2ind( size(M), row, col )(M is a matrix)
uvec indices = sub2ind( size(M), matrix_of_subscripts )
    
uword index = sub2ind( size(Q), row, col, slice )(Q is a cube)
uvec indices = sub2ind( size(Q), matrix_of_subscripts )
  • Convert subscripts to a linear index

  • The argument size(X) can be replaced with size(n_rows, n_cols) or size(n_rows, n_cols, n_slices)

  • For the matrix_of_subscripts argument, the subscripts must be stored in each column of an m x n matrix of type umat; m=2 for matrix subscripts, while m=3 for cube subscripts

  • std::logic_error exception is thrown if a subscript is out of range

  • Examples:
      mat  M(4,5);
      cube Q(4,5,6);
      
      uword i = sub2ind( size(M), 2, 3 );
      uword j = sub2ind( size(Q), 2, 3, 4 );
      

  • See also:



symmatu( A )
symmatu( A, do_conj )

symmatl( A )
symmatl( A, do_conj )
  • symmatu(A): generate symmetric matrix from square matrix A, by reflecting the upper triangle to the lower triangle

  • symmatl(A): generate symmetric matrix from square matrix A, by reflecting the lower triangle to the upper triangle

  • If A is a complex matrix, the reflection uses the complex conjugate of the elements; to disable the complex conjugate, set do_conj to false

  • If A is non-square, a std::logic_error exception is thrown

  • Examples:
      mat A = randu<mat>(5,5);
      
      mat B = symmatu(A);
      mat C = symmatl(A);
      

  • See also:



trace( X )
  • Sum of the elements on the main diagonal of matrix X

  • If X is an expression, the function will try to use optimised expression evaluations to calculate only the diagonal elements

  • Examples:
      mat    A = randu<mat>(5,5);
      
      double x = trace(A);
      

  • See also:



trans( A )
strans( A )
  • For real (non-complex) matrix:
    • trans() provides a transposed copy of the matrix
    • strans() not applicable

  • For complex matrix:
    • trans() provides a Hermitian transpose (ie. the conjugate of the elements is taken during the transpose)
    • strans() provides a transposed copy without taking the conjugate of the elements

  • Examples:
      mat A = randu<mat>(5,10);
      
      mat B = trans(A);
      mat C = A.t();    // equivalent to trans(A), but more compact
      

  • See also:



trapz( X, Y )
trapz( X, Y, dim )

trapz( Y )
trapz( Y, dim )


trimatu( A )
trimatl( A )


unique( A )
  • Return the unique elements of A, sorted in ascending order

  • If A is a vector, the output is also a vector with the same orientation (row or column) as A; if A is a matrix, the output is always a column vector

  • Examples:
      mat X;
      X << 1 << 2 << endr
        << 2 << 3 << endr;
      
      mat Y = unique(X);
      

  • See also:



vectorise( A )
vectorise( A, dim )
  • Generate a column vector (dim=0) or row vector (dim=1) from matrix A

  • The argument dim is optional; by default dim=0 is used

  • For dim=0, the elements are copied from X column-wise; equivalent to concatenating all the columns of A

  • For dim=1, the elements are copied from X row-wise; equivalent to concatenating all the rows of A

  • Concatenating columns is faster than concatenating rows

  • Examples:
      mat A = randu<mat>(4, 5);
      
      vec v = vectorise(A);
      

  • See also:



miscellaneous element-wise functions:
    exp    exp2    exp10    trunc_exp
    log    log2    log10    trunc_log
    pow    square   sqrt     
    floor    ceil    round    trunc
    erf    erfc         
    lgamma            
    sign             
  • Apply a function to each element

  • Usage:
    • matrix_type B = fn(A)
    • cube_type B = fn(A)
    • A and B must have the same matrix_type/cube_type
    • fn(A) is one of:

      exp(A)    base-e exponential: e x
      exp2(A)    base-2 exponential: 2 x
      exp10(A)    base-10 exponential: 10 x
      trunc_exp(A)   base-e exponential, truncated to avoid infinity   (only for float and double elements)
      log(A)    natural log: loge x
      log2(A)    base-2 log: log2 x
      log10(A)    base-10 log: log10 x
      trunc_log(A)   natural log, truncated to avoid ±infinity   (only for float and double elements)
      pow(A, p)    raise to the power of p: x p
      square(A)    square: x 2
      sqrt(A)    square root: x ½
      floor(A)   largest integral value that is not greater than the input value
      ceil(A)   smallest integral value that is not less than the input value
      round(A)   round to nearest integer, with halfway cases rounded away from zero
      trunc(A)   round to nearest integer, towards zero
      erf(A)   error function
      erfc(A)   complementary error function
      lgamma(A)   natural log of the gamma function
      sign(A)   signum function; for each element a in A, the corresponding element b in B is:
        ⎧ −1 if a < 0
        b = ⎨  0 if a = 0
        ⎩  +1 if a > 0
      if a is complex and non-zero, then b = a / abs(a)

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = exp(A);
      

  • See also:



trigonometric element-wise functions (cos, sin, tan, ...)
  • Apply a trigonometric function to each element

  • Usage:
    • matrix_type Y = trig_fn(X)
    • cube_type Y = trig_fn(X)
    • X and Y must have the same matrix_type/cube_type
    • trig_fn is one of:
      • cos family: cos, acos, cosh, acosh
      • sin family: sin, asin, sinh, asinh
      • tan family: tan, atan, tanh, atanh

  • Examples:
      mat X = randu<mat>(5,5);
      mat Y = cos(X);
      
  • See also:





Decompositions, Factorisations, Inverses and Equation Solvers (Dense Matrices)



R = chol( X )
R = chol( X, layout )

chol( R, X )
chol( R, X, layout )
  • Cholesky decomposition of matrix X

  • Matrix X must be symmetric and positive-definite

  • By default, R is upper triangular, such that R.t()*R = X

  • The argument layout is optional; layout is either "upper" or "lower", which specifies whether R is upper or lower triangular

  • If the decomposition fails:
    • R = chol(X) resets R and throws a std::runtime_error exception
    • chol(R,X) resets R and returns a bool set to false (exception is not thrown)

  • Examples:
      mat X = randu<mat>(5,5);
      mat Y = X.t()*X;
      
      mat R1 = chol(Y);
      mat R2 = chol(Y, "lower");
      

  • See also:



vec eigval = eig_sym( X )

eig_sym( eigval, X )

eig_sym( eigval, eigvec, X )
eig_sym( eigval, eigvec, X, method )
  • Eigen decomposition of dense symmetric/hermitian matrix X

  • The eigenvalues and corresponding eigenvectors are stored in eigval and eigvec, respectively

  • The eigenvalues are in ascending order

  • If X is not square sized, a std::logic_error exception is thrown

  • The method argument is optional; method is either "dc" or "std"
    • "dc" indicates divide-and-conquer method (default setting)
    • "std" indicates standard method
    • the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices

  • If the decomposition fails:
    • eigval = eig_sym(X) resets eigval and throws a std::runtime_error exception
    • eig_sym(eigval,X) resets eigval and returns a bool set to false (exception is not thrown)
    • eig_sym(eigval,eigvec,X) resets eigval & eigvec and returns a bool set to false (exception is not thrown)

  • Caveat: there is currently no check whether X is symmetric

  • Examples:
      // for matrices with real elements
      
      mat A = randu<mat>(50,50);
      mat B = A.t()*A;  // generate a symmetric matrix
      
      vec eigval;
      mat eigvec;
      
      eig_sym(eigval, eigvec, B);
      
      
      // for matrices with complex elements
      
      cx_mat C = randu<cx_mat>(50,50);
      cx_mat D = C.t()*C;
      
         vec eigval2;
      cx_mat eigvec2;
      
      eig_sym(eigval2, eigvec2, D);
      

  • See also:



cx_vec eigval = eig_gen( X )

eig_gen( eigval, X )

eig_gen( eigval, eigvec, X )
  • Eigen decomposition of dense general (non-symmetric/non-hermitian) square matrix X

  • The eigenvalues and corresponding eigenvectors are stored in eigval and eigvec, respectively

  • If X is not square sized, a std::logic_error exception is thrown

  • If the decomposition fails:
    • eigval = eig_gen(X) resets eigval and throws a std::runtime_error exception
    • eig_gen(eigval,X) resets eigval and returns a bool set to false (exception is not thrown)
    • eig_gen(eigval,eigvec,X) resets eigval & eigvec and returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(10,10);
      
      cx_vec eigval;
      cx_mat eigvec;
      
      eig_gen(eigval, eigvec, A);
      

  • See also:



cx_vec eigval = eig_pair( A, B )

eig_pair( eigval, A, B )

eig_pair( eigval, eigvec, A, B )
  • Eigen decomposition for pair of general dense square matrices A and B of the same size, such that A*eigvec = B*eigvec*diagmat(eigval)

  • The eigenvalues and corresponding eigenvectors are stored in eigval and eigvec, respectively

  • If A or B is not square sized, a std::logic_error exception is thrown

  • If the decomposition fails:
    • eigval = eig_pair(A,B) resets eigval and throws a std::runtime_error exception
    • eig_pair(eigval,A,B) resets eigval and returns a bool set to false (exception is not thrown)
    • eig_pair(eigval,eigvec,A,B) resets eigval & eigvec and returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(10,10);
      mat B = randu<mat>(10,10);
      
      cx_vec eigval;
      cx_mat eigvec;
      
      eig_pair(eigval, eigvec, A, B);
      

  • See also:



B = inv( A )
inv( B, A )
  • Inverse of general square matrix A

  • If A is not square sized, a std::logic_error exception is thrown

  • If A appears to be singular:
    • B = inv(A) resets B and throws a std::runtime_error exception
    • inv(B,A) resets B and returns a bool set to false (exception is not thrown)

  • Caveats:
    • if matrix A is know to be symmetric positive definite, using inv_sympd() is faster
    • if matrix A is know to be diagonal, use inv( diagmat(A) )
    • to solve a system of linear equations, such as Z = inv(X)*Y, using solve() is faster and more accurate

  • Examples:
      mat A = randu<mat>(5,5);
      
      mat B = inv(A);
      

  • See also:



B = inv_sympd( A )
inv_sympd( B, A )


lu( L, U, P, X )
lu( L, U, X )
  • Lower-upper decomposition (with partial pivoting) of matrix X

  • The first form provides a lower-triangular matrix L, an upper-triangular matrix U, and a permutation matrix P, such that P.t()*L*U = X

  • The second form provides permuted L and U, such that L*U = X; note that in this case L is generally not lower-triangular

  • If the decomposition fails:
    • lu(L,U,P,X) resets L, U, P and returns a bool set to false (exception is not thrown)
    • lu(L,U,X) resets L, U and returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(5,5);
      
      mat L, U, P;
      
      lu(L, U, P, A);
      
      mat B = P.t()*L*U;
      

  • See also:



B = null( A )
B = null( A, tolerance )

null( B, A )
null( B, A, tolerance )
  • Find the orthonormal basis of the null space of matrix A

  • The dimension of the range space is the number of singular values of A not greater than tolerance

  • The tolerance argument is optional; by default tolerance is max(m,n)*max_sv*datum::eps, where:
    • m = number of rows and n = number of columns in A
    • max_sv = maximal singular value of A
    • datum::eps = difference between 1 and the least value greater than 1 that is representable

  • The computation is based on singular value decomposition; if the decomposition fails:
    • B = null(A) resets B and throws a std::runtime_error exception
    • null(B,A) resets B and returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(5,6);
      
      A.row(0).zeros();
      A.col(0).zeros();
      
      mat B = null(A);
      

  • See also:



B = orth( A )
B = orth( A, tolerance )

orth( B, A )
orth( B, A, tolerance )
  • Find the orthonormal basis of the range space of matrix A, so that B.t()*B ≈ eye(r,r), where r = rank(A)

  • The dimension of the range space is the number of singular values of A greater than tolerance

  • The tolerance argument is optional; by default tolerance is max(m,n)*max_sv*datum::eps, where:
    • m = number of rows and n = number of columns in A
    • max_sv = maximal singular value of A
    • datum::eps = difference between 1 and the least value greater than 1 that is representable

  • The computation is based on singular value decomposition; if the decomposition fails:
    • B = orth(A) resets B and throws a std::runtime_error exception
    • orth(B,A) resets B and returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(5,6);
      
      mat B = orth(A);
      

  • See also:



B = pinv( A )
B = pinv( A, tolerance )
B = pinv( A, tolerance, method )

pinv( B, A )
pinv( B, A, tolerance )
pinv( B, A, tolerance, method )
  • Moore-Penrose pseudo-inverse of matrix A

  • The computation is based on singular value decomposition

  • The tolerance argument is optional

  • The default tolerance is max(m,n)*max_sv*datum::eps, where:
    • m = number of rows and n = number of columns in A
    • max_sv = maximal singular value of A
    • datum::eps = difference between 1 and the least value greater than 1 that is representable

  • Any singular values less than tolerance are treated as zero

  • The method argument is optional; method is either "dc" or "std"
    • "dc" indicates divide-and-conquer method (default setting)
    • "std" indicates standard method
    • the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices

  • If the decomposition fails:
    • B = pinv(A) resets B and throws a std::runtime_error exception
    • pinv(B,A) resets B and returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(4,5);
      
      mat B = pinv(A);        // use default tolerance
      
      mat C = pinv(A, 0.01);  // set tolerance to 0.01
      

  • See also:



qr( Q, R, X )


qr_econ( Q, R, X )
  • Economical decomposition of X (with size m x n) into an orthogonal matrix Q and a right triangular matrix R, such that Q*R = X

  • If m > n, only the first n rows of R and the first n columns of Q are calculated (ie. the zero rows of R and the corresponding columns of Q are omitted)

  • If the decomposition fails, Q and R are reset and the function returns a bool set to false (exception is not thrown)

  • Examples:
      mat X = randu<mat>(6,5);
      mat Q, R;
      
      qr_econ(Q,R,X);
      

  • See also:



qz( AA, BB, Q, Z, A, B )
  • Generalised Schur decomposition for pair of general square matrices A and B of the same size,
    such that A = Q.t()*AA*Z.t() and B = Q.t()*BB*Z.t()

  • The left and right Schur vectors are stored in Q and Z, respectively

  • In the complex-valued problem, the generalised eigenvalues are found in diagvec(AA) / diagvec(BB)

  • If A or B is not square sized, a std::logic_error exception is thrown

  • If the decomposition fails, AA, BB, Q and Z are reset, and the function returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(10,10);
      mat B = randu<mat>(10,10);
      
      mat AA;
      mat BB;
      mat Q;
      mat Z; 
      
      qz(AA, BB, Q, Z, A, B);
      

  • See also:



S = schur( X )

schur( S, X )

schur( U, S, X )
  • Schur decomposition of square matrix X, such that X = U*S*U.t()

  • U is a unitary matrix containing the Schur vectors

  • S is an upper triangular matrix, called the Schur form of X

  • If X is not square sized, a std::logic_error exception is thrown

  • If the decomposition fails:
    • S = schur(X) resets S and throws a std::runtime_error exception
    • schur(S,X) resets S and returns a bool set to false (exception is not thrown)
    • schur(U,S,X) resets U & S and returns a bool set to false (exception is not thrown)

  • Caveat: in general, Schur decomposition is not unique

  • Examples:
      mat X(20,20, fill::randu);
      
      mat U;
      mat S;
      
      schur(U, S, X);
      

  • See also:



X = solve( A, B )
X = solve( A, B, settings )

solve( X, A, B )
solve( X, A, B, settings )
  • Solve a dense system of linear equations, A*X = B, where X is unknown; similar functionality to the \ operator in Matlab/Octave, ie. X = A \ B

  • A can be square (critically determined system), or non-square (under/over-determined system)

  • B can be a vector or matrix

  • The number of rows in A and B must be the same

  • The settings argument is optional; it is one of the following, or a combination thereof:

    solve_opts::fast   fast mode: do not apply iterative refinement and/or equilibration
    solve_opts::equilibrate   equilibrate the system before solving   (matrix A must be square)
    solve_opts::no_approx   do not find approximate solutions for rank deficient systems

    the above settings can be combined using the + operator; for example: solve_opts::fast + solve_opts::no_approx

  • Caveat: using solve_opts::fast will speed up finding the solution, but for poorly conditioned systems the solution may have lower quality

  • If A is known to be a triangular matrix, the solution can be computed faster by explicitly indicating that A is triangular through trimatu() or trimatl();
    indicating a triangular matrix also implies that solve_opts::fast is enabled

  • If no solution is found:
    • X = solve(A,B) resets X and throws a std::runtime_error exception
    • solve(X,A,B) resets X and returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(5,5);
      vec b = randu<vec>(5);
      mat B = randu<mat>(5,5);
      
      vec x1 = solve(A, b);
      
      vec x2;
      bool status = solve(x2, A, b);
      
      mat X1 = solve(A, B);
      
      mat X2 = solve(A, B, solve_opts::fast);  // enable fast mode
      
      mat X3 = solve(trimatu(A), B);  // indicate that A is triangular
      

  • See also:



vec s = svd( X )

svd( vec s, X )

svd( mat U, vec s, mat V, mat X )
svd( mat U, vec s, mat V, mat X, method )

svd( cx_mat U, vec s, cx_mat V, cx_mat X )
svd( cx_mat U, vec s, cx_mat V, cx_mat X, method )
  • Singular value decomposition of dense matrix X

  • If X is square, it can be reconstructed using X = U*diagmat(s)*V.t()

  • The singular values are in descending order

  • The method argument is optional; method is either "dc" or "std"
    • "dc" indicates divide-and-conquer method (default setting)
    • "std" indicates standard method
    • the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices

  • If the decomposition fails, the output objects are reset and:
    • s = svd(X) resets s and throws a std::runtime_error exception
    • svd(s,X) resets s and returns a bool set to false (exception is not thrown)
    • svd(U,s,V,X) resets U, s, V and returns a bool set to false (exception is not thrown)

  • Examples:
      mat X = randu<mat>(5,5);
      
      mat U;
      vec s;
      mat V;
      
      svd(U,s,V,X);
      

  • See also:



svd_econ( mat U, vec s, mat V, mat X )
svd_econ( mat U, vec s, mat V, mat X, mode )
svd_econ( mat U, vec s, mat V, mat X, mode, method )

svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X )
svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X, mode )
svd_econ( cx_mat U, vec s, cx_mat V, cx_mat X, mode, method )
  • Economical singular value decomposition of dense matrix X

  • The singular values are in descending order

  • The mode argument is optional; mode is one of:
      "both" = compute both left and right singular vectors (default operation)
      "left" = compute only left singular vectors
      "right" = compute only right singular vectors

  • The method argument is optional; method is either "dc" or "std"
    • "dc" indicates divide-and-conquer method (default setting)
    • "std" indicates standard method
    • the divide-and-conquer method provides slightly different results than the standard method, but is considerably faster for large matrices

  • If the decomposition fails, U, s, V are reset and a bool set to false is returned (exception is not thrown)

  • Examples:
      mat X = randu<mat>(4,5);
      
      mat U;
      vec s;
      mat V;
      
      svd_econ(U, s, V, X);
      

  • See also:



X = syl( A, B, C )
syl( X, A, B, C )
  • Solve the Sylvester equation, ie. AX + XB + C = 0, where X is unknown

  • Matrices A, B and C must be square sized

  • If no solution is found:
    • syl(A,B,C) resets X and throws a std::runtime_error exception
    • syl(X,A,B,C) resets X and returns a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(5,5);
      mat B = randu<mat>(5,5);
      mat C = randu<mat>(5,5);
      
      mat X1 = syl(A, B, C);
      
      mat X2;
      syl(X2, A, B, C);
      

  • See also:





Decompositions, Factorisations and Equation Solvers (Sparse Matrices)



vec eigval = eigs_sym( X, k )
vec eigval = eigs_sym( X, k, form )
vec eigval = eigs_sym( X, k, form, tol )

eigs_sym( eigval, X, k )
eigs_sym( eigval, X, k, form )
eigs_sym( eigval, X, k, form, tol )

eigs_sym( eigval, eigvec, X, k )
eigs_sym( eigval, eigvec, X, k, form )
eigs_sym( eigval, eigvec, X, k, form, tol )
  • Obtain a limited number of eigenvalues and eigenvectors of sparse symmetric real matrix X

  • k specifies the number of eigenvalues and eigenvectors

  • The argument form is optional; form is one of:
      "lm" = obtain eigenvalues with largest magnitude (default operation)
      "sm" = obtain eigenvalues with smallest magnitude
      "la" = obtain eigenvalues with largest algebraic value
      "sa" = obtain eigenvalues with smallest algebraic value

  • The argument tol is optional; it specifies the tolerance for convergence

  • The eigenvalues and corresponding eigenvectors are stored in eigval and eigvec, respectively

  • If X is not square sized, a std::logic_error exception is thrown

  • If the decomposition fails:
    • eigval = eigs_sym(X,k) resets eigval and throws a std::runtime_error exception
    • eigs_sym(eigval,X,k) resets eigval and returns a bool set to false (exception is not thrown)
    • eigs_sym(eigval,eigvec,X,k) resets eigval & eigvec and returns a bool set to false (exception is not thrown)

  • Caveat: there is currently no check whether X is symmetric

  • Examples:
      // generate sparse matrix
      sp_mat A = sprandu<sp_mat>(1000, 1000, 0.1);
      sp_mat B = A.t()*A;
      
      vec eigval;
      mat eigvec;
      
      eigs_sym(eigval, eigvec, B, 5);  // find 5 eigenvalues/eigenvectors
      

  • See also:



cx_vec eigval = eigs_gen( X, k )
cx_vec eigval = eigs_gen( X, k, form )
cx_vec eigval = eigs_gen( X, k, form, tol )

eigs_gen( eigval, X, k )
eigs_gen( eigval, X, k, form )
eigs_gen( eigval, X, k, form, tol )

eigs_gen( eigval, eigvec, X, k )
eigs_gen( eigval, eigvec, X, k, form )
eigs_gen( eigval, eigvec, X, k, form, tol )
  • Obtain a limited number of eigenvalues and eigenvectors of sparse general (non-symmetric/non-hermitian) square matrix X

  • k specifies the number of eigenvalues and eigenvectors

  • The argument form is optional; form is one of:
      "lm" = obtain eigenvalues with largest magnitude (default operation)
      "sm" = obtain eigenvalues with smallest magnitude
      "lr" = obtain eigenvalues with largest real part
      "sr" = obtain eigenvalues with smallest real part
      "li" = obtain eigenvalues with largest imaginary part
      "si" = obtain eigenvalues with smallest imaginary part

  • The argument tol is optional; it specifies the tolerance for convergence

  • The eigenvalues and corresponding eigenvectors are stored in eigval and eigvec, respectively

  • If X is not square sized, a std::logic_error exception is thrown

  • If the decomposition fails:
    • eigval = eigs_gen(X,k) resets eigval and throws a std::runtime_error exception
    • eigs_gen(eigval,X,k) resets eigval and returns a bool set to false (exception is not thrown)
    • eigs_gen(eigval,eigvec,X,k) resets eigval & eigvec and returns a bool set to false (exception is not thrown)

  • Examples:
      // generate sparse matrix
      sp_mat A = sprandu<sp_mat>(1000, 1000, 0.1);  
      
      cx_vec eigval;
      cx_mat eigvec;
      
      eigs_gen(eigval, eigvec, A, 5);  // find 5 eigenvalues/eigenvectors
      

  • See also:



X = spsolve( A, B )
X = spsolve( A, B, solver )
X = spsolve( A, B, solver, settings )

spsolve( X, A, B )
spsolve( X, A, B, solver )
spsolve( X, A, B, solver, settings )
  • Solve a sparse system of linear equations, A*X = B, where A is a sparse matrix, B is a dense matrix or vector, and X is unknown

  • The number of rows in A and B must be the same

  • If no solution is found:
    • X = spsolve(A, B) resets X and throws a std::runtime_error exception
    • spsolve(X, A, B) resets X and returns a bool set to false (no exception is thrown)

  • The solver argument is optional; solver is either "superlu" or "lapack"; by default "superlu" is used
    • For "superlu", ARMA_USE_SUPERLU must be enabled in config.hpp
    • For "lapack", sparse matrix A is converted to a dense matrix before using the LAPACK solver; this considerably increases memory usage

  • The settings argument is optional

    • when solver is "superlu", settings is an instance of the superlu_opts structure:
      struct superlu_opts
        {
        bool             equilibrate;  // default: false
        bool             symmetric;    // default: false
        double           pivot_thresh; // default: 1.0
        permutation_type permutation;  // default: superlu_opts::COLAMD
        refine_type      refine;       // default: superlu_opts::REF_DOUBLE
        };
      
    • equilibrate is either true or false; it indicates whether to equilibrate the system (scale the rows and columns of A to have unit norm)

    • symmetric is either true or false; it indicates whether to use SuperLU symmetric mode, which gives preference to diagonal pivots

    • pivot_threshold is in the range [0.0, 1.0], used for determining whether a diagonal entry is an acceptable pivot (details in SuperLU documentation)

    • permutation specifies the type of column permutation; it is one of:

      superlu_opts::NATURAL  natural ordering
      superlu_opts::MMD_ATA  minimum degree ordering on structure of A.t() * A
      superlu_opts::MMD_AT_PLUS_A  minimum degree ordering on structure of A.t() + A
      superlu_opts::COLAMD  approximate minimum degree column ordering

    • refine specifies the type of iterative refinement; it is one of:

      superlu_opts::REF_NONE   no refinement
      superlu_opts::REF_SINGLE   iterative refinement in single precision
      superlu_opts::REF_DOUBLE   iterative refinement in double precision
      superlu_opts::REF_EXTRA   iterative refinement in extra precision

  • Examples:
      sp_mat A = sprandu<sp_mat>(1000, 1000, 0.1);
      
      vec b = randu<vec>(1000);
      mat B = randu<mat>(1000, 5);
      
      vec x = spsolve(A, b);  // solve one system
      mat X = spsolve(A, B);  // solve several systems
      
      bool status = spsolve(x, A, b);  // use default solver
      if(status == false)  { cout << "no solution" << endl; }
      
      spsolve(x, A, b, "lapack");   // use LAPACK  solver
      spsolve(x, A, b, "superlu");  // use SuperLU solver
      
      superlu_opts settings;
      
      settings.permutation = superlu_opts::NATURAL;
      settings.refine      = superlu_opts::REF_NONE;
      
      spsolve(x, A, b, "superlu", settings);
      

  • See also:



vec s = svds( X, k )
vec s = svds( X, k, tol )

svds( vec s, X, k )
svds( vec s, X, k, tol )

svds( mat U, vec s, mat V, sp_mat X, k )
svds( mat U, vec s, mat V, sp_mat X, k, tol )

svds( cx_mat U, vec s, cx_mat V, sp_cx_mat X, k )
svds( cx_mat U, vec s, cx_mat V, sp_cx_mat X, k, tol )
  • Obtain a limited number of singular values and singular vectors of sparse matrix X

  • k specifies the number of singular values and singular vectors

  • The argument tol is optional; it specifies the tolerance for convergence

  • The singular values are calculated via sparse eigen decomposition of:
    ⎡ zeros(X.n_rows, X.n_rows)  X ⎤
    ⎣ X.t() zeros(X.n_cols, X.n_cols) ⎦

  • The singular values are in descending order

  • If the decomposition fails, the output objects are reset and:
    • s = svds(X,k) resets s and throws a std::runtime_error exception
    • svds(s,X,k) resets s and returns a bool set to false (exception is not thrown)
    • svds(U,s,V,X,k) resets U, s, V and returns a bool set to false (exception is not thrown)

  • Caveat: svds() is intended only for finding a few singular values from a large sparse matrix; to find all singular values, use svd() instead

  • Examples:
      sp_mat X = sprandu<sp_mat>(100, 200, 0.1);
      
      mat U;
      vec s;
      mat V;
      
      svds(U, s, V, X, 10);
      

  • See also:





Signal & Image Processing



conv( A, B )
conv( A, B, shape )
  • 1D convolution of vectors A and B

  • The orientation of the result vector is the same as the orientation of A (ie. either column or row vector)

  • The shape argument is optional; it is one of:
        "full" = return the full convolution (default setting), with the size equal to A.n_elem + B.n_elem - 1
        "same" = return the central part of the convolution, with the same size as vector A

  • The convolution operation is also equivalent to FIR filtering

  • Examples:
      vec A(256, fill::randu);
      
      vec B(16, fill::randu);
      
      vec C = conv(A, B);
      
      vec D = conv(A, B, "same");
      

  • See also:



conv2( A, B )
conv2( A, B, shape )
  • 2D convolution of matrices A and B

  • The shape argument is optional; it is one of:
        "full" = return the full convolution (default setting), with the size equal to size(A) + size(B) - 1
        "same" = return the central part of the convolution, with the same size as matrix A

  • The implementation of 2D convolution in this version is preliminary; it is not yet fully optimised

  • Examples:
      mat A(256, 256, fill::randu);
      
      mat B(16, 16, fill::randu);
      
      mat C = conv2(A, B);
      
      mat D = conv2(A, B, "same");
      

  • See also:



cx_mat Y =  fft( X )
cx_mat Y =  fft( X, n )

cx_mat Z = ifft( cx_mat Y )
cx_mat Z = ifft( cx_mat Y, n )
  • fft(): fast Fourier transform of a vector or matrix (real or complex)

  • ifft(): inverse fast Fourier transform of a vector or matrix (complex only)

  • If given a matrix, the transform is done on each column vector of the matrix

  • The optional n argument specifies the transform length:
    • if n is larger than the length of the input vector, a zero-padded version of the vector is used
    • if n is smaller than the length of the input vector, only the first n elements of the vector are used

  • If n is not specified, the transform length is the same as the length of the input vector

  • Caveat: the transform is fastest when the transform length is a power of 2, eg. 64, 128, 256, 512, 1024, ...

  • The implementation of the transform in this version is preliminary; it is not yet fully optimised

  • Examples:
         vec X = randu<vec>(100);
      cx_vec Y = fft(X, 128);
      

  • See also:



cx_mat Y =  fft2( X )
cx_mat Y =  fft2( X, n_rows, n_cols )

cx_mat Z = ifft2( cx_mat Y )
cx_mat Z = ifft2( cx_mat Y, n_rows, n_cols )
  • fft2(): 2D fast Fourier transform of a matrix (real or complex)

  • ifft2(): 2D inverse fast Fourier transform of a matrix (complex only)

  • The optional arguments n_rows and n_cols specify the size of the transform; a truncated and/or zero-padded version of the input matrix is used

  • Caveat: the transform is fastest when both n_rows and n_cols are a power of 2, eg. 64, 128, 256, 512, 1024, ...

  • The implementation of the transform in this version is preliminary; it is not yet fully optimised

  • Examples:
         mat A = randu<mat>(100,100);
      cx_mat B = fft2(A);
      cx_mat C = fft2(A, 128, 128);
      

  • See also:



interp1( X, Y, XI, YI )
interp1( X, Y, XI, YI, method )
interp1( X, Y, XI, YI, method, extrapolation_value )
  • 1D data interpolation

  • Given a 1D function specified in vectors X and Y (where X specifies locations and Y specifies the corresponding values),
    generate vector YI which contains interpolated values at locations XI

  • The method argument is optional; it is one of:
      "nearest" = interpolate using single nearest neighbour
      "linear" = linear interpolation between two nearest neighbours (default setting)
      "*nearest" = as per "nearest", but faster by assuming that X and XI are monotonically increasing
      "*linear" = as per "linear", but faster by assuming that X and XI are monotonically increasing

  • If a location in XI is outside the domain of X, the corresponding value in YI is set to extrapolation_value

  • The extrapolation_value argument is optional; by default it is datum::nan (not-a-number)

  • Examples:
      vec x = linspace<vec>(0, 3, 20);
      vec y = square(x);
      
      vec xx = linspace<vec>(0, 3, 100);
      
      vec yy;
      
      interp1(x, y, xx, yy);  // use linear interpolation by default
      
      interp1(x, y, xx, yy, "*linear");  // faster than "linear"
      
      interp1(x, y, xx, yy, "nearest");
      

  • See also:





Statistics & Clustering



mean, median, stddev, var
    mean( V )
    mean( M )
    mean( M, dim )
    mean( Q )
    mean( Q, dim )

        ⎫ 
    ⎪ 
    ⎬  mean (average value)
    ⎪ 
    ⎭ 
    median( V )
    median( M )
    median( M, dim )

        ⎫ 
    ⎬  median
    ⎭ 
    stddev( V )
    stddev( V, norm_type )
    stddev( M )
    stddev( M, norm_type )
    stddev( M, norm_type, dim )

        ⎫ 
    ⎪ 
    ⎬  standard deviation
    ⎪ 
    ⎭ 
    var( V )
    var( V, norm_type )
    var( M )
    var( M, norm_type )
    var( M, norm_type, dim )

        ⎫ 
    ⎪ 
    ⎬  variance
    ⎪ 
    ⎭ 
  • For vector V, return the statistic calculated using all the elements of the vector

  • For matrix M, find the statistic for each column (dim=0), or each row (dim=1)

  • For cube Q, find the statistics of elements along dimension dim, where dim ∈ { 0, 1, 2 }

  • The dim argument is optional; by default dim=0 is used

  • The norm_type argument is optional; by default norm_type=0 is used

  • For the var() and stddev() functions:
    • the default norm_type=0 performs normalisation using N-1 (where N is the number of samples), providing the best unbiased estimator
    • using norm_type=1 performs normalisation using N, which provides the second moment around the mean

  • Examples:
      mat A    = randu<mat>(5,5);
      mat B    = mean(A);
      mat C    = var(A);
      double m = mean(mean(A));
      
      vec    v = randu<vec>(5);
      double x = var(v);
      

  • See also:



cov( X, Y )
cov( X, Y, norm_type )

cov( X )
cov( X, norm_type )
  • For two matrix arguments X and Y, if each row of X and Y is an observation and each column is a variable, the (i,j)-th entry of cov(X,Y) is the covariance between the i-th variable in X and the j-th variable in Y

  • For vector arguments, the type of vector is ignored and each element in the vector is treated as an observation

  • For matrices, X and Y must have the same dimensions

  • For vectors, X and Y must have the same number of elements

  • cov(X) is equivalent to cov(X, X)

  • The default norm_type=0 performs normalisation using N-1 (where N is the number of observations), providing the best unbiased estimation of the covariance matrix (if the observations are from a normal distribution). Using norm_type=1 causes normalisation to be done using N, which provides the second moment matrix of the observations about their mean

  • Examples:
      mat X = randu<mat>(4,5);
      mat Y = randu<mat>(4,5);
      
      mat C = cov(X,Y);
      mat D = cov(X,Y, 1);
      

  • See also:



cor( X, Y )
cor( X, Y, norm_type )

cor( X )
cor( X, norm_type )
  • For two matrix arguments X and Y, if each row of X and Y is an observation and each column is a variable, the (i,j)-th entry of cor(X,Y) is the correlation coefficient between the i-th variable in X and the j-th variable in Y

  • For vector arguments, the type of vector is ignored and each element in the vector is treated as an observation

  • For matrices, X and Y must have the same dimensions

  • For vectors, X and Y must have the same number of elements

  • cor(X) is equivalent to cor(X, X)

  • The default norm_type=0 performs normalisation of the correlation matrix using N-1 (where N is the number of observations). Using norm_type=1 causes normalisation to be done using N

  • Examples:
      mat X = randu<mat>(4,5);
      mat Y = randu<mat>(4,5);
      
      mat R = cor(X,Y);
      mat S = cor(X,Y, 1);
      

  • See also:



hist( V )
hist( V, n_bins )
hist( V, centers )

hist( X, centers )
hist( X, centers, dim )
  • For vector V, produce an unsigned vector of the same orientation as V (ie. either uvec or urowvec) that represents a histogram of counts

  • For matrix X, produce a umat matrix containing either column histogram counts (for dim=0, default), or row histogram counts (for dim=1)

  • The bin centers can be automatically determined from the data, with the number of bins specified via n_bins (default is 10); the range of the bins is determined by the range of the data

  • The bin centers can also be explicitly specified via the centers vector; the vector must contain monotonically increasing values (eg. 0.1, 0.2, 0.3, ...)

  • Examples:
      vec  v  = randn<vec>(1000); // Gaussian distribution
      
      uvec h1 = hist(v, 11);
      uvec h2 = hist(v, linspace<vec>(-2,2,11));
      

  • See also:



histc( V, edges )
histc( X, edges )
histc( X, edges, dim )
  • For vector V, produce an unsigned vector of the same orientation as V (ie. either uvec or urowvec) that contains the counts of the number of values that fall between the elements in the edges vector

  • For matrix X, produce a umat matrix containing either column histogram counts (for dim=0, default), or row histogram counts (for dim=1)

  • The edges vector must contain monotonically increasing values (eg. 0.1, 0.2, 0.3, ...)

  • Examples:
      vec  v = randn<vec>(1000);  // Gaussian distribution
      
      uvec h = histc(v, linspace<vec>(-2,2,11));
      

  • See also:



mat coeff = princomp( mat X )
cx_mat coeff = princomp( cx_mat X )

princomp( mat coeff, mat X )
princomp( cx_mat coeff, cx_mat X )

princomp( mat coeff, mat score, mat X )
princomp( cx_mat coeff, cx_mat score, cx_mat X )

princomp( mat coeff, mat score, vec latent, mat X )
princomp( cx_mat coeff, cx_mat score, vec latent, cx_mat X )

princomp( mat coeff, mat score, vec latent, vec tsquared, mat X )
princomp( cx_mat coeff, cx_mat score, vec latent, cx_vec tsquared, cx_mat X )
  • Principal component analysis of matrix X

  • Each row of X is an observation and each column is a variable

  • output objects:
    • coeff: principal component coefficients
    • score: projected data
    • latent: eigenvalues of the covariance matrix of X
    • tsquared: Hotteling's statistic for each sample

  • The computation is based on singular value decomposition

  • If the decomposition fails:
    • coeff = princomp(X) resets coeff and throws a std::runtime_error exception
    • remaining forms of princomp() reset all output matrices and return a bool set to false (exception is not thrown)

  • Examples:
      mat A = randu<mat>(5,4);
      
      mat coeff;
      mat score;
      vec latent;
      vec tsquared;
      
      princomp(coeff, score, latent, tsquared, A);
      

  • See also:



running_stat<type>
  • Class for keeping the running statistics of a continuously sampled one dimensional process/signal

  • Useful if the storage of individual samples (scalars) is not required, or the number of samples is not known beforehand

  • type is one of: float, double, cx_float, cx_double

  • For an instance of running_stat named as X, the member functions are:

      X(scalar)  
      update the statistics so far using the given scalar
      X.min()  
      get the minimum value so far
      X.max()  
      get the maximum value so far
      X.mean()  
      get the mean or average value so far
      X.var()  and  X.var(norm_type)  
      get the variance so far
      X.stddev()  and  X.stddev(norm_type)  
      get the standard deviation so far
      X.reset()  
      reset all statistics and set the number of samples to zero
      X.count()  
      get the number of samples so far

  • The norm_type argument is optional; by default norm_type=0 is used

  • For the .var() and .stddev() functions, the default norm_type=0 performs normalisation using N-1 (where N is the number of samples so far), providing the best unbiased estimator; using norm_type=1 causes normalisation to be done using N, which provides the second moment around the mean

  • The return type of .count() depends on the underlying form of type: it is either float or double

  • Examples:
      running_stat<double> stats;
      
      for(uword i=0; i<10000; ++i)
        {
        double sample = randn();
        stats(sample);
        }
      
      cout << "mean = " << stats.mean() << endl;
      cout << "var  = " << stats.var()  << endl;
      cout << "min  = " << stats.min()  << endl;
      cout << "max  = " << stats.max()  << endl;
      

  • See also:



running_stat_vec<vec_type>
running_stat_vec<vec_type>(calc_cov)
  • Class for keeping the running statistics of a continuously sampled multi-dimensional process/signal

  • Useful if the storage of individual samples (vectors) is not required, or if the number of samples is not known beforehand

  • This class is similar to running_stat, with the difference that vectors are processed instead of scalars

  • vec_type is the vector type of the samples; for example: vec, cx_vec, rowvec, ...

  • For an instance of running_stat_vec named as X, the member functions are:

      X(vector)  
      update the statistics so far using the given vector
      X.min()  
      get the vector of minimum values so far
      X.max()  
      get the vector of maximum values so far
      X.mean()  
      get the mean vector so far
      X.var()  and  X.var(norm_type)  
      get the vector of variances so far
      X.stddev()  and  X.stddev(norm_type)  
      get the vector of standard deviations so far
      X.cov()  and  X.cov(norm_type)  
      get the covariance matrix so far; valid if calc_cov=true during construction of running_stat_vec
      X.reset()  
      reset all statistics and set the number of samples to zero
      X.count()  
      get the number of samples so far

  • The calc_cov argument is optional; by default calc_cov=false, indicating that the covariance matrix will not be calculated; to enable the covariance matrix, use calc_cov=true during construction; for example: running_stat_vec<vec> X(true);

  • The norm_type argument is optional; by default norm_type=0 is used

  • For the .var() and .stddev() functions, the default norm_type=0 performs normalisation using N-1 (where N is the number of samples so far), providing the best unbiased estimator; using norm_type=1 causes normalisation to be done using N, which provides the second moment around the mean

  • The return type of .count() depends on the underlying form of vec_type: it is either float or double

  • Examples:
      running_stat_vec<vec> stats;
      
      vec sample;
      
      for(uword i=0; i<10000; ++i)
        {
        sample = randu<vec>(5);
        stats(sample);
        }
      
      cout << "mean = " << endl << stats.mean() << endl;
      cout << "var  = " << endl << stats.var()  << endl;
      cout << "max  = " << endl << stats.max()  << endl;
      
      //
      //
      
      running_stat_vec<rowvec> more_stats(true);
      
      for(uword i=0; i<20; ++i)
        {
        sample = randu<rowvec>(3);
        
        sample(1) -= sample(0);
        sample(2) += sample(1);
        
        more_stats(sample);
        }
      
      cout << "covariance matrix = " << endl;
      cout << more_stats.cov() << endl;
      
      rowvec sd = more_stats.stddev();
      
      cout << "correlations = " << endl;
      cout << more_stats.cov() / (sd.t() * sd);
      

  • See also:



kmeans( means, data, k, seed_mode, n_iter, print_mode )
  • Cluster given data into k disjoint sets

  • The means parameter is the output matrix for storing the resulting centroids of the sets, with each centroid stored as a column vector

  • The data parameter is the input data matrix, with each sample stored as a column vector

  • The k parameter indicates the number of centroids

  • The seed_mode parameter specifies how the initial centroids are seeded; it is one of:
      keep_existing   use the centroids specified in the means matrix as the starting point
      static_subset   use a subset of the data vectors (repeatable)
      random_subset   use a subset of the data vectors (random)
      static_spread   use a maximally spread subset of data vectors (repeatable)
      random_spread   use a maximally spread subset of data vectors (random start)

  • The n_iter parameter specifies the number of clustering iterations; about 10 is typically sufficient

  • The print_mode parameter is either true or false, indicating whether progress is printed during clustering

  • If the clustering fails, the means matrix is reset and a bool set to false is returned

  • The clustering will run faster on multi-core machines when OpenMP is enabled in your compiler (eg. -fopenmp in GCC)

  • Caveats:

    • the number of data vectors (number of columns in the data matrix) should be much larger than k  (number of centroids)

    • seeding the initial centroids with static_spread and random_spread can be much more time consuming than with static_subset and random_subset

  • Examples:
      uword d = 5;       // dimensionality
      uword N = 10000;   // number of vectors
      
      mat data(d, N, fill::randu);
      
      mat means;
      
      bool status = kmeans(means, data, 2, random_subset, 10, true);
      
      if(status == false)
        {
        cout << "clustering failed" << endl;
        }
      
      means.print("means:");
      

  • See also:



gmm_diag
  • Class for modelling data as a Gaussian Mixture Model (GMM), under the assumption of diagonal covariance matrices

  • Can also be used for clustering and vector quantisation (VQ)

  • Provides associated parameter estimation algorithms: k-means clustering and Expectation-Maximisation (EM)

  • The k-means and EM algorithms are parallelised (multi-threaded) when compiling with OpenMP enabled (eg. the -fopenmp option in gcc)

  • Data is modelled as:
      n_gaus-1 
    p(x) =  hg  N( x | mg Cg )
      g=0 
    where:
    • n_gaus is the number of Gaussians
    • N( x | mg Cg ) represents a Gaussian (normal) distribution
    • for each Gaussian g:
      • hg is the heft (weight), with hg ≥ 0 and ∑hg = 1
      • mg is the mean (centroid) vector with dimensionality n_dims
      • Cg is the diagonal covariance matrix

  • For an instance of gmm_diag named as M, the member functions and variables are:

      M.log_p(V)  
      return a scalar representing the log-likelihood of vector V
      M.log_p(V, g)  
      return a scalar representing the log-likelihood of vector V, according to Gaussian g
       
       
       
      M.log_p(X)  
      return a row vector (of type rowvec) containing log-likelihoods of each column vector in matrix X
      M.log_p(X, g)  
      return a row vector (of type rowvec) containing log-likelihoods of each column vector in matrix X, according to Gaussian g
       
       
       
      M.avg_log_p(X)  
      return a scalar representing the average log-likelihood of all column vectors in matrix X
      M.avg_log_p(X, g)  
      return a scalar representing the average log-likelihood of all column vectors in matrix X, according to Gaussian g
       
       
       
      M.assign(V, dist_mode)  
      return the index of the closest mean (or Gaussian) to vector V;
      dist_mode is one of:
      eucl_dist   Euclidean distance (takes only means into account)
      prob_dist   probabilistic "distance", defined as the inverse likelihood (takes into account means, covariances and hefts)
      M.assign(X, dist_mode)  
      return a row vector containing the indices of the closest means (or Gaussians) to each column vector in matrix X
       
       
       
      M.raw_hist(X, dist_mode)  
      return a row vector (of type urowvec) representing the raw histogram of counts; each entry is the number of counts corresponding to a Gaussian; each count is the number times the corresponding Gaussian was the closest to each column vector in matrix X; parameter dist_mode is either eucl_dist or prob_dist (as per the .assign() function above)
      M.norm_hist(X, dist_mode)  
      return a row vector (of type rowvec) containing normalised counts; the vector sums to one
       
       
       
      M.generate()  
      return a column vector representing a random sample generated according to the model's parameters
      M.generate(N)  
      return a matrix containing N column vectors, with each vector representing a random sample generated according to the model's parameters
       
       
       
      M.save(filename)  
      save the model to the given filename
      M.load(filename)  
      load the model from the given filename
       
       
       
      M.n_gaus()  
      return the number of means/Gaussians in the model
      M.n_dims()  
      return the dimensionality of the means/Gaussians in the model
       
       
       
      M.reset(n_dims, n_gaus)  
      set the model to have dimensionality n_dims, with n_gaus number of Gaussians;
      all the means are set to zero, all diagonal covariances are set to one, and all the hefts (weights) are set to be uniform
       
       
       
      M.means  
      read-only matrix containing the means (centroids), stored as column vectors
      M.dcovs  
      read-only matrix containing the diagonal covariance matrices, stored as column vectors
      M.hefts  
      read-only row vector containing the hefts (weights)
       
       
       
      M.set_means(X)  
      set the means to be as specified in matrix X;
      the number of means and their dimensionality must match the existing model
      M.set_dcovs(X)  
      set the diagonal covariances matrices to be as specified in matrix X;
      the number of diagonal covariance matrices and their dimensionality must match the existing model
      M.set_hefts(V)  
      set the hefts (weights) of the model to be as specified in row vector V;
      the number of hefts must match the existing model
       
       
       
      M.set_params(meansdcovshefts)  
      set all the parameters in one hit;
      the number of Gaussians and dimensionality can be different from the existing model
       
       
       
      M.learn(data, n_gaus, dist_mode, seed_mode, km_iter, em_iter, var_floor, print_mode)
       
       
      learn the model parameters via the k-means and/or EM algorithms,
      and return a bool variable, with true indicating success, and false indicating failure

  • Parameters for the .learn() member function:
           
      data   matrix containing training samples; each sample is stored as a column vector
           
      n_gaus   set the number of Gaussians to n_gaus
           
      dist_mode   specifies the distance used during the seeding of initial means and k-means clustering:
      eucl_dist   Euclidean distance
      maha_dist   Mahalanobis distance, which uses a global diagonal covariance matrix estimated from the training samples
           
      seed_mode   specifies how the initial means are seeded prior to running k-means and/or EM algorithms:
      keep_existing   keep the existing model (do not modify the means, covariances and hefts)
      static_subset   a subset of the training samples (repeatable)
      random_subset   a subset of the training samples (random)
      static_spread   a maximally spread subset of training samples (repeatable)
      random_spread   a maximally spread subset of training samples (random start)
           
      km_iter   the number of iterations of the k-means algorithm
           
      em_iter   the number of iterations of the EM algorithm
           
      var_floor   the variance floor (smallest allowed value) for the diagonal covariances
           
      print_mode   either true or false; enable or disable printing of progress during the k-means and EM algorithms


  • Notes:

    • for probabilistic applications, better model parameters are typically learned with dist_mode set to maha_dist

    • for vector quantisation applications, model parameters should be learned with dist_mode set to eucl_dist, and the number of EM iterations set to zero

    • in general, a sufficient number of k-means and EM iterations is typically about 10

    • the number of training samples should be much larger than the number of Gaussians

    • seeding the initial means with static_spread and random_spread can be much more time consuming than with static_subset and random_subset

    • the k-means and EM algorithms will run faster on multi-core machines when OpenMP is enabled in your compiler (eg. -fopenmp in GCC)

  • Examples:
      // create synthetic data with 2 Gaussians
      
      uword d = 5;       // dimensionality
      uword N = 10000;   // number of vectors
      
      mat data(d, N, fill::zeros);
      
      vec mean0 = linspace<vec>(1,d,d);
      vec mean1 = mean0 + 2;
      
      uword i = 0;
      
      while(i < N)
        {
        if(i < N)  { data.col(i) = mean0 + randn<vec>(d); ++i; }
        if(i < N)  { data.col(i) = mean0 + randn<vec>(d); ++i; }
        if(i < N)  { data.col(i) = mean1 + randn<vec>(d); ++i; }
        }
      
      
      // model the data as a GMM with 2 Gaussians
      
      gmm_diag model;
      
      bool status = model.learn(data, 2, maha_dist, random_subset, 10, 5, 1e-10, true);
      
      if(status == false)
        {
        cout << "learning failed" << endl;
        }
      
      model.means.print("means:");
      
      double  scalar_likelihood = model.log_p( data.col(0)    );
      rowvec     set_likelihood = model.log_p( data.cols(0,9) );
      
      double overall_likelihood = model.avg_log_p(data);
      
      uword   gaus_id  = model.assign( data.col(0),    eucl_dist );
      urowvec gaus_ids = model.assign( data.cols(0,9), prob_dist );
      
      urowvec hist1 = model.raw_hist (data, prob_dist);
       rowvec hist2 = model.norm_hist(data, eucl_dist);
      
      model.save("my_model.gmm");
      

  • See also:





Miscellaneous



constants (pi, inf, speed of light, ...)
    datum::pi   π, the ratio of any circle's circumference to its diameter
    datum::inf   ∞, infinity
    datum::nan   “not a number” (NaN); caveat: NaN is not equal to anything, even itself
         
    datum::e   base of the natural logarithm
    datum::sqrt2   square root of 2
    datum::eps   the difference between 1 and the least value greater than 1 that is representable (type and machine dependent)
         
    datum::log_min   log of minimum non-zero value (type and machine dependent)
    datum::log_max   log of maximum value (type and machine dependent)
    datum::euler   Euler's constant, aka Euler-Mascheroni constant
         
    datum::gratio   golden ratio
    datum::m_u   atomic mass constant (in kg)
    datum::N_A   Avogadro constant
         
    datum::k   Boltzmann constant (in joules per kelvin)
    datum::k_evk   Boltzmann constant (in eV/K)
    datum::a_0   Bohr radius (in meters)
         
    datum::mu_B   Bohr magneton
    datum::Z_0   characteristic impedance of vacuum (in ohms)
    datum::G_0   conductance quantum (in siemens)
         
    datum::k_e   Coulomb's constant (in meters per farad)
    datum::eps_0   electric constant (in farads per meter)
    datum::m_e   electron mass (in kg)
         
    datum::eV   electron volt (in joules)
    datum::ec   elementary charge (in coulombs)
    datum::F   Faraday constant (in coulombs)
         
    datum::alpha   fine-structure constant
    datum::alpha_inv   inverse fine-structure constant
    datum::K_J   Josephson constant
         
    datum::mu_0   magnetic constant (in henries per meter)
    datum::phi_0   magnetic flux quantum (in webers)
    datum::R   molar gas constant (in joules per mole kelvin)
         
    datum::G   Newtonian constant of gravitation (in newton square meters per kilogram squared)
    datum::h   Planck constant (in joule seconds)
    datum::h_bar   Planck constant over 2 pi, aka reduced Planck constant (in joule seconds)
         
    datum::m_p   proton mass (in kg)
    datum::R_inf   Rydberg constant (in reciprocal meters)
    datum::c_0   speed of light in vacuum (in meters per second)
         
    datum::sigma   Stefan-Boltzmann constant
    datum::R_k   von Klitzing constant (in ohms)
    datum::b   Wien wavelength displacement law constant

  • The constants are stored in the Datum<type> class, where type is either float or double;
    for convenience, Datum<double> is typedefed as datum, and Datum<float> is typedefed as fdatum

  • Caveat: datum::nan is not equal to anything, even itself; to check whether a given number x is finite, use is_finite(x)

  • The physical constants were mainly taken from NIST 2014 CODATA values, and some from WolframAlpha (as of 2009-06-23)

  • Examples:
      cout << "2.0 * pi = " << 2.0 * datum::pi << endl;
      
      cout << "speed of light = " << datum::c_0 << endl;
      
      cout << "log_max for floats = ";
      cout << fdatum::log_max << endl;
      
      cout << "log_max for doubles = ";
      cout << datum::log_max << endl;
      
  • See also:



wall_clock
  • Simple wall clock timer class for measuring the number of elapsed seconds

  • Examples:
      wall_clock timer;
      
      mat A = randu<mat>(100,100);
      mat B = randu<mat>(100,100);
      mat C;
      
      timer.tic();
      
      for(uword i=0; i<100000; ++i)
        {
        C = A + B + A + B;
        }
      
      double n = timer.toc();
      
      cout << "number of seconds: " << n << endl;
      



logging of warnings and errors

set_stream_err1( user_stream )
set_stream_err2( user_stream )

std::ostream& x = get_stream_err1()
std::ostream& x = get_stream_err2()
  • By default, warnings and messages associated with exceptions are printed to the std::cout stream
    • the printing can be disabled by defining ARMA_DONT_PRINT_ERRORS before including the armadillo header
    • detailed information about errors is always reported via the base std::exception class

  • set_stream_err1(): change the stream for messages associated with std::logic_error exceptions (eg. out of bounds accesses)

  • set_stream_err2(): change the stream for warnings and messages associated with std::runtime_error and std::bad_alloc exceptions (eg. failed decompositions, out of memory)

  • get_stream_err1(): get a reference to the stream for messages associated with std::logic_error exceptions

  • get_stream_err2(): get a reference to the stream for warnings and messages associated with std::runtime_error exceptions

  • Examples:
      // print "hello" to the current err1 stream
      get_stream_err1() << "hello" << endl;
      
      // change the err2 stream to be a file
      ofstream f("my_log.txt");
      set_stream_err2(f);
      
      // trying to invert a singular matrix
      // will print a message to the err2 stream
      // and throw an exception
      mat X = zeros<mat>(5,5);
      mat Y = inv(X);
      
      // disable messages being printed to the err2 stream
      std::ostream nullstream(0);
      set_stream_err2(nullstream);
      

  • Caveat: set_stream_err1() and set_stream_err2() will not change the stream used by .print()

  • See also:



uword, sword
  • uword is a typedef for an unsigned integer type; it is used for matrix indices as well as all internal counters and loops

  • sword is a typedef for a signed integer type

  • The minimum width of both uword and sword is either 32 or 64 bits:
    • when using the old C++98 / C++03 standards, the default width is 32 bits
    • when using the new C++11 / C++14 standards, the default width is 64 bits on 64-bit platforms

  • The width can also be forcefully set to 64 bits by enabling ARMA_64BIT_WORD via editing include/armadillo_bits/config.hpp

  • See also:



cx_double, cx_float


Examples of Matlab/Octave syntax and conceptually corresponding Armadillo syntax

    Matlab/Octave   Armadillo   Notes
             
    A(1, 1)   A(0, 0)   indexing in Armadillo starts at 0
    A(k, k)   A(k-1, k-1)    
             
    size(A,1)   A.n_rows   read only
    size(A,2)   A.n_cols    
    size(Q,3)   Q.n_slices   Q is a cube (3D array)
    numel(A)   A.n_elem    
             
    A(:, k)   A.col(k)   this is a conceptual example only; exact conversion from Matlab/Octave to Armadillo syntax will require taking into account that indexing starts at 0
    A(k, :)   A.row(k)    
    A(:, p:q)   A.cols(p, q)    
    A(p:q, :)   A.rows(p, q)    
    A(p:q, r:s)   A( span(p,q), span(r,s) )   A( span(first_row, last_row), span(first_col, last_col) )
             
    Q(:, :, k)   Q.slice(k)   Q is a cube (3D array)
    Q(:, :, t:u)   Q.slices(t, u)    
    Q(p:q, r:s, t:u)   Q( span(p,q), span(r,s), span(t,u) )    
             
    A'   A.t() or trans(A)   matrix transpose / Hermitian transpose
    (for complex matrices, the conjugate of each element is taken)
             
    A = zeros(size(A))   A.zeros()    
    A = ones(size(A))   A.ones()    
    A = zeros(k)   A = zeros<mat>(k,k)    
    A = ones(k)   A = ones<mat>(k,k)    
             
    C = complex(A,B)   cx_mat C = cx_mat(A,B)    
             
    A .* B   A % B   element-wise multiplication
    A ./ B   A / B   element-wise division
    A \ B   solve(A,B)   conceptually similar to inv(A)*B, but more efficient
    A = A + 1;   A++    
    A = A - 1;   A--    
             
    A = [ 1 2; 3 4; ]   << 1 << 2 << endr
       << 3 << 4 << endr;
      element initialisation, with special element endr indicating end of row
             
    X = A(:)   X = vectorise(A)    
    X = [ A  B ]   X = join_horiz(A,B)    
    X = [ A; B ]   X = join_vert(A,B)    
             
    A   cout << A << endl;
    or
    A.print("A =");
       
             
    save ‑ascii 'A.dat' A   A.save("A.dat", raw_ascii);   Matlab/Octave matrices saved as ascii are readable by Armadillo (and vice-versa)
    load ‑ascii 'A.dat'   A.load("A.dat", raw_ascii);    
             
    A = randn(2,3);
    B = randn(4,5);
    F = { A; B }
      mat A = randn(2,3);
    mat B = randn(4,5);
    field<mat> F(2,1);
    F(0,0) = A;
    F(1,0) = B;
      fields store arbitrary objects, such as matrices



example program
    #include <iostream>
    #include <armadillo>
    
    using namespace std;
    using namespace arma;
    
    int main()
      {
      mat A = randu<mat>(4,5);
      mat B = randu<mat>(4,5);
      
      cout << A*B.t() << endl;
      
      return 0;
      }
    
  • If you save the above program as example.cpp, under Linux and Mac OS X it can be compiled using:
    g++ example.cpp -o example -O2 -larmadillo

  • As Armadillo is a template library, we strongly recommend to enable optimisation when compiling programs (eg. when compiling with GCC or clang, use the -O2 or -O3 options)

  • See also the example program that comes with the Armadillo archive



config.hpp
  • Armadillo can be configured via editing the file include/armadillo_bits/config.hpp. Specific functionality can be enabled or disabled by uncommenting or commenting out a particular #define, listed below.

    ARMA_USE_LAPACK   Enable use of LAPACK, or a high-speed replacement for LAPACK (eg. Intel MKL, AMD ACML or the Accelerate framework). Armadillo requires LAPACK for functions such as svd(), inv(), eig_sym(), solve(), etc.
         
    ARMA_DONT_USE_LAPACK   Disable use of LAPACK. Overrides ARMA_USE_LAPACK
         
    ARMA_USE_BLAS   Enable use of BLAS, or a high-speed replacement for BLAS (eg. OpenBLAS, Intel MKL, AMD ACML or the Accelerate framework). BLAS is used for matrix multiplication. Without BLAS, Armadillo will use a built-in matrix multiplication routine, which might be slower for large matrices.
         
    ARMA_DONT_USE_BLAS   Disable use of BLAS. Overrides ARMA_USE_BLAS
         
    ARMA_USE_NEWARP   Enable use of the built-in reimplementation of ARPACK (Armadillo 7.x). This is used for the eigen decomposition of real (non-complex) sparse matrices, ie. eigs_gen(), eigs_sym() and svds(). Requires ARMA_USE_LAPACK to be enabled.
         
    ARMA_DONT_USE_NEWARP   Disable use of the built-in reimplementation of ARPACK. Overrides ARMA_USE_NEWARP
         
    ARMA_USE_ARPACK   Enable use of ARPACK, or a high-speed replacement for ARPACK. Armadillo requires ARPACK for the eigen decomposition of complex sparse matrices, ie. eigs_gen(), eigs_sym() and svds()
         
    ARMA_DONT_USE_ARPACK   Disable use of ARPACK. Overrides ARMA_USE_ARPACK
         
    ARMA_USE_SUPERLU   Enable use of SuperLU, which is used by spsolve() for finding the solutions of sparse systems; you will need to link with the superlu library, for example -lsuperlu
    Caveat: Armadillo 6.x requires SuperLU 4.3, while Armadillo 7.x requires SuperLU 5.2
         
    ARMA_DONT_USE_SUPERLU   Disable use of SuperLU. Overrides ARMA_USE_SUPERLU
         
    ARMA_USE_HDF5   Enable the ability to save and load matrices stored in the HDF5 format; the hdf5.h header file must be available on your system and you will need to link with the hdf5 library (eg. -lhdf5)
         
    ARMA_DONT_USE_HDF5   Disable the use of the HDF5 library. Overrides ARMA_USE_HDF5
         
    ARMA_DONT_USE_WRAPPER   Disable going through the run-time Armadillo wrapper library (libarmadillo.so) when calling LAPACK, BLAS, ARPACK, SuperLU and HDF5 functions. You will need to directly link with LAPACK, BLAS, etc (eg. -llapack -lblas)
         
    ARMA_USE_CXX11   Use C++11 features, such as initialiser lists; automatically enabled when using a compiler in C++11 mode, for example, g++ -std=c++11
         
    ARMA_DONT_USE_CXX11   Disable use of C++11 features. Overrides ARMA_USE_CXX11
         
    ARMA_BLAS_CAPITALS   Use capitalised (uppercase) BLAS and LAPACK function names (eg. DGEMM vs dgemm)
         
    ARMA_BLAS_UNDERSCORE   Append an underscore to BLAS and LAPACK function names (eg. dgemm_ vs dgemm). Enabled by default.
         
    ARMA_BLAS_LONG   Use "long" instead of "int" when calling BLAS and LAPACK functions
         
    ARMA_BLAS_LONG_LONG   Use "long long" instead of "int" when calling BLAS and LAPACK functions
         
    ARMA_USE_TBB_ALLOC   Use Intel TBB scalable_malloc() and scalable_free() instead of standard malloc() and free() for managing matrix memory
         
    ARMA_USE_MKL_ALLOC   Use Intel MKL mkl_malloc() and mkl_free() instead of standard malloc() and free() for managing matrix memory
         
    ARMA_64BIT_WORD   Use 64 bit integers. Automatically enabled when using a C++11 compiler on 64-bit platforms. Useful if you require matrices/vectors capable of holding more than 4 billion elements. Your machine and compiler must have support for 64 bit integers (eg. via "long" or "long long"). This can also be enabled by adding #define ARMA_64BIT_WORD before each instance of #include <armadillo>
         
    ARMA_NO_DEBUG   Disable all run-time checks, such as bounds checking. This will result in faster code, but you first need to make sure that your code runs correctly! We strongly recommend to have the run-time checks enabled during development, as this greatly aids in finding mistakes in your code, and hence speeds up development. We recommend that run-time checks be disabled only for the shipped version of your program (ie. final release build).
         
    ARMA_EXTRA_DEBUG   Print out the trace of internal functions used for evaluating expressions. Not recommended for normal use. This is mainly useful for debugging the library.
         
    ARMA_MAT_PREALLOC   The number of pre-allocated elements used by matrices and vectors. Must be always enabled and set to an integer that is at least 1. By default set to 16. If you mainly use lots of very small vectors (eg. ≤ 4 elements), change the number to the size of your vectors.
         
    ARMA_DEFAULT_OSTREAM   The default stream used for printing error messages and by .print(). Must be always enabled. By default defined to std::cout
         
    ARMA_PRINT_ERRORS   Print errors and warnings encountered during program execution
         
    ARMA_DONT_PRINT_ERRORS   Do not print errors or warnings. Overrides ARMA_PRINT_ERRORS
         

  • See also:



History of API Additions, Changes and Deprecations

  • API Version Policy
    • Armadillo's version number is A.B.C, where A is a major version, B is a minor version, and C is the patch level (indicating bug fixes)

    • Within each major version (eg. 4.x), minor versions are backwards compatible with earlier minor versions. For example, code written for version 4.000 will work with version 4.100, 4.120, 4.200, etc. However, as each minor version may have more features (ie. API extensions) than earlier versions, code specifically written for version 4.100 doesn't necessarily work with 4.000

    • We don't like changes to existing APIs and strongly prefer not to break any user software. However, to allow evolution, we reserve the right to alter the APIs in future major versions of Armadillo while remaining backwards compatible in as many cases as possible (eg. version 5.x may have slightly different APIs than 4.x). In a rare instance the user API may need to be tweaked if a bug fix absolutely requires it

    • This policy is applicable to the APIs described in this documentation; it is not applicable to internal functions (ie. the underlying internal implementation details may change across consecutive minor versions)


  • List of additions and changes for each version:

    • Version 7.400:

    • Version 7.300:

    • Version 7.200:

    • Version 7.100:

    • Version 6.700:
      • added trapz() for numerical integration
      • added logmat() for calculating the matrix logarithm
      • added regspace() for generating vectors with regularly spaced elements
      • added logspace() for generating vectors with logarithmically spaced elements
      • added approx_equal() for determining approximate equality
      • expanded .save() and .load() with hdf5_binary_trans file type, to save/load data with columns transposed to rows

    • Version 6.600:
      • expanded sum(), mean(), min(), max() to handle cubes
      • expanded Cube class to handle arbitrarily sized empty cubes (eg. 0x5x2)
      • added shift() for circular shifts of elements
      • added sqrtmat() for finding the square root of a matrix

    • Version 6.500:
      • added conv2() for 2D convolution
      • added stand-alone kmeans() function for clustering data
      • added trunc()
      • extended conv() to optionally provide central convolution
      • faster handling of multiply-and-accumulate by accu() when using Intel MKL, ATLAS or OpenBLAS

    • Version 6.400:

    • Version 6.300:

    • Version 6.200:
      • expanded diagmat() to handle non-square matrices and arbitrary diagonals
      • expanded trace() to handle non-square matrices

    • Version 6.100:
      • faster norm() and normalise() when using Intel MKL, ATLAS or OpenBLAS
      • added Schur decomposition: schur()
      • stricter handling of matrix objects by hist() and histc()
      • advanced constructors for using auxiliary memory by Mat, Col, Row and Cube now have the default of strict = false
      • Cube class now delays allocation of .slice() related structures until needed
      • expanded join_slices() to handle joining cubes with matrices

    • Version 5.600:

    • Version 5.500:
      • expanded object constructors and generators to handle size() based specification of dimensions

    • Version 5.400:
      • added find_unique() for finding indices of unique values
      • added diff() for calculating differences between consecutive elements
      • added cumprod() for calculating cumulative product
      • added null() for finding the orthonormal basis of null space
      • expanded interp1() to handle repeated locations
      • expanded unique() to handle complex numbers
      • faster flipud()
      • faster row-wise cumsum()

    • Version 5.300:

    • Version 5.200:
      • added orth() for finding the orthonormal basis of the range space of a matrix
      • expanded element initialisation to handle nested initialiser lists (C++11)

    • Version 5.100:

    • Version 5.000:

    • Version 4.650:
      • added randg() for generating random values from gamma distributions (C++11 only)
      • added .head_rows() and .tail_rows() to submatrix views
      • added .head_cols() and .tail_cols() to submatrix views
      • expanded eigs_sym() to optionally calculate eigenvalues with smallest/largest algebraic values

    • Version 4.600:
      • added .head() and .tail() to submatrix views
      • faster matrix transposes within compound expressions
      • faster in-place matrix multiplication
      • faster accu() and norm() when compiling with -O3 -ffast-math -march=native (gcc and clang)

    • Version 4.550:
      • added matrix exponential function: expmat()
      • faster .log_p() and .avg_log_p() functions in the gmm_diag class when compiling with OpenMP enabled
      • faster handling of in-place addition/subtraction of expressions with an outer product

    • Version 4.500:
      • faster handling of complex vectors by norm()
      • expanded chol() to optionally specify output matrix as upper or lower triangular
      • better handling of non-finite values when saving matrices as text files

    • Version 4.450:
      • faster handling of matrix transposes within compound expressions
      • expanded symmatu()/symmatl() to optionally disable taking the complex conjugate of elements
      • expanded sort_index() to handle complex vectors
      • expanded the gmm_diag class with functions to generate random samples

    • Version 4.400:
      • faster handling of subvectors by dot()
      • faster handling of aliasing by submatrix views
      • added clamp() for clamping values to be between lower and upper limits
      • added gmm_diag class for statistical modelling of data using Gaussian Mixture Models
      • expanded batch insertion constructors for sparse matrices to add values at repeated locations

    • Version 4.320:
      • expanded eigs_sym() and eigs_gen() to use an optional tolerance parameter
      • expanded eig_sym() to automatically fall back to standard decomposition method if divide-and-conquer fails
      • cmake-based installer enables use of C++11 random number generator when using gcc 4.8.3+ in C++11 mode

    • Version 4.300:

    • Version 4.200:
      • faster transpose of sparse matrices
      • more efficient handling of aliasing during matrix multiplication
      • faster inverse of matrices marked as diagonal

    • Version 4.100:
      • added normalise() for normalising vectors to unit p-norm
      • extended the field class to handle 3D layout
      • extended eigs_sym() and eigs_gen() to obtain eigenvalues of various forms (eg. largest or smallest magnitude)
      • automatic SIMD vectorisation of elementary expressions (eg. matrix addition) when using Clang 3.4+ with -O3 optimisation
      • faster handling of sparse submatrix views

    • Version 4.000:

    • Version 3.930:

    • Version 3.920:
      • faster .zeros()
      • faster round(), exp2() and log2() when using C++11
      • added signum function: sign()
      • added move constructors when using C++11
      • added 2D fast Fourier transform: fft2()
      • added .tube() for easier extraction of vectors and subcubes from cubes
      • added specification of a fill type during construction of Mat, Col, Row and Cube classes, eg. mat X(4, 5, fill::zeros)

    • Version 3.910:
      • faster multiplication of a matrix with a transpose of itself, ie. X*X.t() and X.t()*X
      • added vectorise() for reshaping matrices into vectors
      • added all() and any() for indicating presence of elements satisfying a relational condition

    • Version 3.900:
      • added automatic SSE2 vectorisation of elementary expressions (eg. matrix addition) when using GCC 4.7+ with -O3 optimisation
      • faster median()
      • faster handling of compound expressions with transposes of submatrix rows
      • faster handling of compound expressions with transposes of complex vectors
      • added support for saving & loading of cubes in HDF5 format

    • Version 3.820:
      • faster as_scalar() for compound expressions
      • faster transpose of small vectors
      • faster matrix-vector product for small vectors
      • faster multiplication of small fixed size matrices

    • Version 3.810:

    • Version 3.800:
    • Version 3.6:
      • faster handling of compound expressions with submatrices and subcubes
      • faster trace()
      • added support for loading matrices as text files with NaN and Inf elements
      • added stable_sort_index(), which preserves the relative order of elements with equivalent values
      • added handling of sparse matrices by mean(), var(), norm(), abs(), square(), sqrt()
      • added saving and loading of sparse matrices in arma_binary format

    • Version 3.4:
    • Version 3.2:
      • added unique(), for finding unique elements of a matrix
      • added .eval(), for forcing the evaluation of delayed expressions
      • faster eigen decomposition via optional use of divide-and-conquer algorithm
      • faster transpose of vectors and compound expressions
      • faster handling of diagonal views
      • faster handling of tiny fixed size vectors (≤ 4 elements)

    • Version 3.0:
    • Version 2.4:
      • added shorter forms of transposes: .t() and .st()
      • added .resize() and resize()
      • added optional use of 64 bit indices (allowing matrices to have more than 4 billion elements), enabled via ARMA_64BIT_WORD in include/armadillo_bits/config.hpp
      • added experimental support for C++11 initialiser lists, enabled via ARMA_USE_CXX11 in include/armadillo_bits/config.hpp
      • refactored code to eliminate warnings when using the Clang C++ compiler
      • umat, uvec, .min() and .max() have been changed to use the uword type instead of the u32 type; by default the uword and u32 types are equivalent (ie. unsigned integer type with a minimum width 32 bits); however, when the use of 64 bit indices is enabled via ARMA_64BIT_WORD in include/armadillo_bits/config.hpp, the uword type then has a minimum width of 64 bits

    • Version 2.2:
    • Version 2.0:
    • Version 1.2:
      • added .min() & .max() member functions of Mat and Cube
      • added floor() and ceil()
      • added representation of “not a number”: math::nan()
      • added representation of infinity: math::inf()
      • added standalone is_finite()
      • .in_range() expanded to use span() arguments
      • fixed size matrices and vectors can use auxiliary (external) memory
      • submatrices and subfields can be accessed via X( span(a,b)span(c,d) )
      • subcubes can be accessed via X( span(a,b)span(c,d)span(e,f) )
      • the two argument version of span can be replaced by span::all or span(), to indicate an entire range
      • for cubes, the two argument version of span can be replaced by a single argument version, span(a), to indicate a single column, row or slice
      • arbitrary "flat" subcubes can be interpreted as matrices; for example:
          cube Q = randu<cube>(5,3,4);
          mat  A = Q( span(1), span(1,2), span::all );
          // A has a size of 2x4
          
          vec v = ones<vec>(4);
          Q( span(1), span(1), span::all ) = v;
          
      • added interpretation of matrices as triangular through trimatu() / trimatl()
      • added explicit handling of triangular matrices by solve() and inv()
      • extended syntax for submatrices, including access to elements whose indices are specified in a vector
      • added ability to change the stream used for logging of errors and warnings
      • added ability to save/load matrices in raw binary format
      • added cumulative sum function: cumsum()

    • Changed in 1.0 (compared to earlier 0.x development versions):
      • the 3 argument version of lu(), eg. lu(L,U,X), provides L and U which should be the same as produced by Octave 3.2 (this was not the case in versions prior to 0.9.90)

      • rand() has been replaced by randu(); this has been done to avoid confusion with std::rand(), which generates random numbers in a different interval

      • In versions earlier than 0.9.0, some multiplication operations directly converted result matrices with a size of 1x1 into scalars. This is no longer the case. If you know the result of an expression will be a 1x1 matrix and wish to treat it as a pure scalar, use the as_scalar() wrapping function

      • Almost all functions have been placed in the delayed operations framework (for speed purposes). This may affect code which assumed that the output of some functions was a pure matrix. The solution is easy, as explained below.

        In general, Armadillo queues operations before executing them. As such, the direct output of an operation or function cannot be assumed to be a directly accessible matrix. The queued operations are executed when the output needs to be stored in a matrix, eg. mat B = trans(A) or mat B(trans(A)). If you need to force the execution of the delayed operations, place the operation or function inside the corresponding Mat constructor. For example, if your code assumed that the output of some functions was a pure matrix, eg. chol(m).diag(), change the code to mat(chol(m)).diag(). Similarly, if you need to pass the result of an operation such as A+B to one of your own functions, use my_function( mat(A+B) ).

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