PDL::LinearAlgebra::Special - Special matrices for PDL
use PDL::LinearAlgebra::Mtype; $a = mhilb(5,5);
This module provides some constructors of well known matrices.
Contruct Hilbert matrix from specifications list or template piddle
PDL(Hilbert) = mpart(PDL(template) | ARRAY(specification))
my $hilb = mhilb(float,5,5);
PDL = mtri(PDL, SCALAR) SCALAR : UPPER = 0 | LOWER = 1, default = 0
my $a = random(10,10); my $b = mtri($a, 0);
Return (primal) Vandermonde matrix from vector.
mvander(M,P) is a rectangular version of mvander(P) with M Columns.
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);
Return Hankel matrix also known as persymmetric matrix. For complex, needs object of type PDL::Complex.
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 whith 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
Return toeplitz matrix. For complex need object of type PDL::Complex.
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.
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, it's 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.
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),
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.