NAME

       PDL::LinearAlgebra::Special - Special matrices for PDL

SYNOPSIS

        use PDL::LinearAlgebra::Special;

        $a = mhilb(5,5);

DESCRIPTION

       This module provides some constructors of well known matrices.

FUNCTIONS

   mhilb
       Construct Hilbert matrix from specifications list or template ndarray

        PDL(Hilbert)  = mpart(PDL(template) | ARRAY(specification))

        my $hilb   = mhilb(float,5,5);

   mtri
       Return zeroed matrix with upper or lower triangular part from another matrix.  Return trapezoid matrix if
       entry matrix is not square.  Supports threading.  Uses tricpy or tricpy.

        PDL = mtri(PDL, SCALAR)
        SCALAR : UPPER = 0 | LOWER = 1, default = 0

        my $a = random(10,10);
        my $b = mtri($a, 0);

   mvander
       Return (primal) Vandermonde matrix from vector.

       mvander(M,P) is a rectangular version of mvander(P) with M Columns.

   mpart
       Return antisymmetric and symmetric part of a real or complex square matrix.

        ( PDL(antisymmetric), PDL(symmetric) )  = mpart(PDL, SCALAR(conj))
        conj : if true Return AntiHermitian, Hermitian part.

        my $a = random(10,10);
        my ( $antisymmetric, $symmetric )  = mpart($a);

   mhankel
       Return Hankel matrix also known as persymmetric matrix.  Handles complex data.

        mhankel(c,r), where c and r are vectors, returns matrix whose first column
        is c and whose last row is r. The last element of c prevails.
        mhankel(c) returns matrix with element below skew diagonal (anti-diagonal) equals
        to zero. If c is a scalar number, make it from sequence beginning at one.

       The elements are:

               H (i,j) = c (i+j),  i+j+1 <= m;
               H (i,j) = r (i+j-m+1),  otherwise
               where m is the size of the vector.

       If c is a scalar number, it's determinant can be computed by:

                               floor(n/2)    n
               Det(H(n)) = (-1)      *      n

   mtoeplitz
       Return toeplitz matrix.  Handles complex data.

        mtoeplitz(c,r), where c and r are vectors, returns matrix whose first column
        is c and whose last row is r. The last element of c prevails.
        mtoeplitz(c) returns symmetric matrix.

   mpascal
       Return Pascal matrix (from Pascal's triangle) of order N.

        mpascal(N,uplo).
        uplo:
               0 => upper triangular (Cholesky factor),
               1 => lower triangular (Cholesky factor),
               2 => symmetric.

       This matrix is obtained by writing Pascal's triangle (whose elements are binomial coefficients from index
       and/or index sum) as a matrix and truncating appropriately.  The symmetric Pascal is positive definite,
       its inverse has integer entries.

       Their determinants are all equal to one and:

               S = L * U
               where S, L, U are symmetric, lower and upper pascal matrix respectively.

   mcompanion
       Return a matrix with characteristic polynomial equal to p if p is monic.  If p is not monic the
       characteristic polynomial of A is equal to p/c where c is the coefficient of largest degree in p (here p
       is in descending order).

        mcompanion(PDL(p),SCALAR(charpol)).
        charpol:
               0 => first row is -P(1:n-1)/P(0),
               1 => last column is -P(1:n-1)/P(0),

AUTHOR

       Copyright (C) Grégory Vanuxem 2005-2007.

       This library is free software; you can redistribute it and/or modify it under the terms of the artistic
       license as specified in the Artistic file.

perl v5.38.2                                       2024-03-31                   PDL::LinearAlgebra::Special(3pm)