PDL::LinearAlgebra::Special(3) Special matrices for PDL

## SYNOPSIS

use PDL::LinearAlgebra::Mtype;
\$a = mhilb(5,5);

## DESCRIPTION

This module provides some constructors of well known matrices.

## mhilb

Construct Hilbert matrix from specifications list or template piddle

``` 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. 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
```

## mtoeplitz

``` 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, 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.
```

## 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),
```