SIMPLE SOLUTIONS

# PDL::LINEARALGEBRA - reference manual online

Linear Algebra utils for PDL.

Chapter
2015-12-19
```LinearAlgebra(3pm)             User Contributed Perl Documentation             LinearAlgebra(3pm)

NAME
PDL::LinearAlgebra - Linear Algebra utils for PDL

SYNOPSIS
use PDL::LinearAlgebra;

\$a = random (100,100);
(\$U, \$s, \$V) = mdsvd(\$a);

DESCRIPTION
This module provides a convenient interface to PDL::LinearAlgebra::Real and
PDL::LinearAlgebra::Complex. Its primary purpose is educational.  You have to know that
routines defined here are not optimized, particularly in term of memory. Since Blas and
Lapack use a column major ordering scheme some routines here need to transpose matrices
before calling fortran routines and transpose back (see the documentation of each
routine). If you need optimized code use directly  PDL::LinearAlgebra::Real and
PDL::LinearAlgebra::Complex. It's planned to "port" this module to PDL::Matrix such that
transpositions will not be necessary, the major problem is that two new modules need to be
created PDL::Matrix::Real and PDL::Matrix::Complex.

FUNCTIONS
setlaerror
Sets action type when an error is encountered, returns previous type. Available values are
NO, WARN and BARF (predefined constants).  If, for example, in computation of the inverse,
singularity is detected, the routine can silently return values from computation (see
manuals), warn about singularity or barf. BARF is the default value.

# h : x -> g(f(x))

\$a = sequence(5,5);
\$err = setlaerror(NO);
(\$b, \$info)= f(\$a);
setlaerror(\$err);
\$info ? barf "can't compute h" : return g(\$b);

getlaerror
Gets action type when an error is encountered.

0 => NO,
1 => WARN,
2 => BARF

t
PDL = t(PDL, SCALAR(conj))
conj : Conjugate Transpose = 1 | Transpose = 0, default = 1;

Convenient function for transposing real or complex 2D array(s).  For PDL::Complex, if
conj is true returns conjugate transposed array(s) and doesn't support dataflow.  Supports

issym
PDL = issym(PDL, SCALAR|PDL(tol),SCALAR(hermitian))
tol : tolerance value, default: 1e-8 for double else 1e-5
hermitian : Hermitian = 1 | Symmetric = 0, default = 1;

Checks symmetricity/Hermitianicity of matrix.  Supports threading.

diag
Returns i-th diagonal if matrix in entry or matrix with i-th diagonal with entry. I-th
diagonal returned flows data back&forth.  Can be used as lvalue subs if your perl supports

PDL = diag(PDL, SCALAR(i), SCALAR(vector)))
i      : i-th diagonal, default = 0
vector : create diagonal matrices by threading over row vectors, default = 0

my \$a = random(5,5);
my \$diag  = diag(\$a,2);
# If your perl support lvaluable subroutines.
\$a->diag(-2) .= pdl(1,2,3);
# Construct a (5,5,5) PDL (5 matrices) with
# diagonals from row vectors of \$a
\$a->diag(0,1)

tritosym
Returns symmetric or Hermitian matrix from lower or upper triangular matrix.  Supports
inplace and threading.  Uses tricpy or ctricpy from Lapack.

PDL = tritosym(PDL, SCALAR(uplo), SCALAR(conj))
uplo : UPPER = 0 | LOWER = 1, default = 0
conj : Hermitian = 1 | Symmetric = 0, default = 1;

# Assume \$a is symmetric triangular
my \$a = random(10,10);
my \$b = tritosym(\$a);

positivise
Returns entry pdl with changed sign by row so that average of positive sign > 0.  In other
words threads among dimension 1 and row  =  -row if sum(sign(row)) < 0.  Works inplace.

my \$a = random(10,10);
\$a -= 0.5;
\$a->xchg(0,1)->inplace->positivise;

mcrossprod
Computes the cross-product of two matrix: A' x  B.  If only one matrix is given, takes B
to be the same as A.  Supports threading.  Uses crossprod or ccrossprod.

PDL = mcrossprod(PDL(A), (PDL(B))

my \$a = random(10,10);
my \$crossproduct = mcrossprod(\$a);

mrank
Computes the rank of a matrix, using a singular value decomposition.  from Lapack.

SCALAR = mrank(PDL, SCALAR(TOL))
TOL:   tolerance value, default : mnorm(dims(PDL),'inf') * mnorm(PDL) * EPS

my \$a = random(10,10);
my \$b = mrank(\$a, 1e-5);

mnorm
Computes norm of real or complex matrix Supports threading.

PDL(norm) = mnorm(PDL, SCALAR(ord));
ord :
0|'inf' : Infinity norm
1|'one' : One norm
2|'two' : norm 2 (default)
3|'fro' : frobenius norm

my \$a = random(10,10);
my \$norm = mnorm(\$a);

mdet
Computes determinant of a general square matrix using LU factorization.  Supports
threading.  Uses getrf or cgetrf from Lapack.

PDL(determinant) = mdet(PDL);

my \$a = random(10,10);
my \$det = mdet(\$a);

mposdet
Compute determinant of a symmetric or Hermitian positive definite square matrix using
Cholesky factorization.  Supports threading.  Uses potrf or cpotrf from Lapack.

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

my \$a = random(10,10);
my \$det = mposdet(\$a);

mcond
Computes the condition number (two-norm) of a general matrix.

The condition number in two-n is defined:

norm (a) * norm (inv (a)).

Uses a singular value decomposition.  Supports threading.

PDL = mcond(PDL)

my \$a = random(10,10);
my \$cond = mcond(\$a);

mrcond
Estimates the reciprocal condition number of a general square matrix using LU
factorization in either the 1-norm or the infinity-norm.

The reciprocal condition number is defined:

1/(norm (a) * norm (inv (a)))

Supports threading.  Works on transposed array(s)

PDL = mrcond(PDL, SCALAR(ord))
ord :
0 : Infinity norm (default)
1 : One norm

my \$a = random(10,10);
my \$rcond = mrcond(\$a,1);

morth
Returns an orthonormal basis of the range space of matrix A.

PDL = morth(PDL(A), SCALAR(tol))
tol : tolerance for determining rank, default: 1e-8 for double else 1e-5

my \$a = sequence(10,10);
my \$ortho = morth(\$a, 1e-8);

mnull
Returns an orthonormal basis of the null space of matrix A.  Works on transposed array.

PDL = mnull(PDL(A), SCALAR(tol))
tol : tolerance for determining rank, default: 1e-8 for double else 1e-5

my \$a = sequence(10,10);
my \$null = mnull(\$a, 1e-8);

minv
Computes inverse of a general square matrix using LU factorization. Supports inplace and
threading.  Uses getrf and getri or cgetrf and cgetri from Lapack and returns "inverse,
info" in array context.

PDL(inv)  = minv(PDL)

my \$a = random(10,10);
my \$inv = minv(\$a);

mtriinv
Computes inverse of a triangular matrix. Supports inplace and threading.  Uses trtri or
ctrtri from Lapack.  Returns "inverse, info" in array context.

(PDL, PDL(info))) = mtriinv(PDL, SCALAR(uplo), SCALAR|PDL(diag))
uplo : UPPER = 0 | LOWER = 1, default = 0
diag : UNITARY DIAGONAL = 1, default = 0

# Assume \$a is upper triangular
my \$a = random(10,10);
my \$inv = mtriinv(\$a);

msyminv
Computes inverse of a symmetric square matrix using the Bunch-Kaufman diagonal pivoting
method.  Supports inplace and threading.  Uses sytrf and sytri or csytrf and csytri from
Lapack and returns "inverse, info" in array context.

(PDL, (PDL(info))) = msyminv(PDL, SCALAR|PDL(uplo))
uplo : UPPER = 0 | LOWER = 1, default = 0

# Assume \$a is symmetric
my \$a = random(10,10);
my \$inv = msyminv(\$a);

mposinv
Computes inverse of a symmetric positive definite square matrix using Cholesky
factorization.  Supports inplace and threading.  Uses potrf and potri or cpotrf and cpotri
from Lapack and returns "inverse, info" in array context.

(PDL, (PDL(info))) = mposinv(PDL, SCALAR|PDL(uplo))
uplo : UPPER = 0 | LOWER = 1, default = 0

# Assume \$a is symmetric positive definite
my \$a = random(10,10);
\$a = \$a->crossprod(\$a);
my \$inv = mposinv(\$a);

mpinv
Computes pseudo-inverse (Moore-Penrose) of a general matrix.  Works on transposed array.

PDL(pseudo-inv)  = mpinv(PDL, SCALAR(tol))
TOL:   tolerance value, default : mnorm(dims(PDL),'inf') * mnorm(PDL) * EPS

my \$a = random(5,10);
my \$inv = mpinv(\$a);

mlu
Computes LU factorization.  Uses getrf or cgetrf from Lapack and returns L, U, pivot and
info.  Works on transposed array.

(PDL(l), PDL(u), PDL(pivot), PDL(info)) = mlu(PDL)

my \$a = random(10,10);
(\$l, \$u, \$pivot, \$info) = mlu(\$a);

mchol
Computes Cholesky decomposition of a symmetric matrix also knows as symmetric square root.
If inplace flag is set, overwrite  the leading upper or lower triangular part of A else
returns triangular matrix. Returns "cholesky, info" in array context.  Supports threading.
Uses potrf or cpotrf from Lapack.

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

my \$a = random(10,10);
\$a = crossprod(\$a, \$a);
my \$u  = mchol(\$a);

mhessen
Reduces a square matrix to Hessenberg form H and orthogonal matrix Q.

It reduces a general matrix A to upper Hessenberg form H by an orthogonal similarity
transformation:

Q' x A x Q = H

or

A = Q x H x Q'

Uses gehrd and orghr or cgehrd and cunghr from Lapack and returns "H" in scalar context
else "H" and "Q".  Works on transposed array.

(PDL(h), (PDL(q))) = mhessen(PDL)

my \$a = random(10,10);
(\$h, \$q) = mhessen(\$a);

mschur
Computes Schur form, works inplace.

A = Z x T x Z'

Supports threading for unordered eigenvalues.  Uses gees or cgees from Lapack and returns
schur(T) in scalar context.  Works on tranposed array(s).

( PDL(schur), (PDL(eigenvalues), (PDL(left schur vectors), PDL(right schur vectors), \$sdim), ↲
\$info) ) = mschur(PDL(A), SCALAR(schur vector),SCALAR(left eigenvector), SCALAR(right eigenvector),SCALAR(select_func), SCALAR(backtransform), SCALAR(norm))
schur vector        : Schur vectors returned, none = 0 | all = 1 | selected = 2, default = 0
left eigenvector    : Left eigenvectors returned, none = 0 | all = 1 | selected = 2, default ↲
= 0
right eigenvector   : Right eigenvectors returned, none = 0 | all = 1 | selected = 2, defaul ↲
t = 0
select_func         : Select_func is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue is selected if PerlInt select_func(PDL::Complex(w)) is t ↲
rue;
Note that a selected complex eigenvalue may no longer
satisfy select_func(PDL::Complex(w)) = 1 after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned).
All eigenvalues/vectors are selected if select_func is undefined.
backtransform       : Whether or not backtransforms eigenvectors to those of A.
Only supported if schur vectors are computed, default = 1.
norm                : Whether or not computed eigenvectors are normalized to have Euclidean  ↲
norm equal to
1 and largest component real, default = 1

Returned values     :
Schur form T (SCALAR CONTEXT),
eigenvalues,
Schur vectors (Z) if requested,
left eigenvectors if requested
right eigenvectors if requested
sdim: Number of eigenvalues selected if select_func is defined.
info: Info output from gees/cgees.

my \$a = random(10,10);
my \$schur  = mschur(\$a);
sub select{
my \$m = shift;
# select "discrete time" eigenspace
return \$m->Cabs < 1 ? 1 : 0;
}
my (\$schur,\$eigen, \$svectors,\$evectors)  = mschur(\$a,1,1,0,\&select);

mschurx
Computes Schur form, works inplace.  Uses geesx or cgeesx from Lapack and returns schur(T)
in scalar context.  Works on transposed array.

( PDL(schur) (,PDL(eigenvalues))  (, PDL(schur vectors), HASH(result)) ) = mschurx(PDL, SCAL ↲
AR(schur vector), SCALAR(left eigenvector), SCALAR(right eigenvector),SCALAR(select_func), SCALAR(sense), SCALAR(backtransform), SCALAR(norm))
schur vector        : Schur vectors returned, none = 0 | all = 1 | selected = 2, default = 0
left eigenvector    : Left eigenvectors returned, none = 0 | all = 1 | selected = 2, default ↲
= 0
right eigenvector   : Right eigenvectors returned, none = 0 | all = 1 | selected = 2, defaul ↲
t = 0
select_func         : Select_func is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue is selected if PerlInt select_func(PDL::Complex(w)) is t ↲
rue;
Note that a selected complex eigenvalue may no longer
satisfy select_func(PDL::Complex(w)) = 1 after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned).
All  eigenvalues/vectors are selected if select_func is undefined.
sense               : Determines which reciprocal condition numbers will be computed.
0: None are computed
1: Computed for average of selected eigenvalues only
2: Computed for selected right invariant subspace only
3: Computed for both
If select_func is undefined, sense is not used.
backtransform       : Whether or not backtransforms eigenvectors to those of A.
Only supported if schur vector are computed, default = 1
norm                : Whether or not computed eigenvectors are normalized to have Euclidean  ↲
norm equal to
1 and largest component real, default = 1

Returned values     :
Schur form T (SCALAR CONTEXT),
eigenvalues,
Schur vectors if requested,
HASH{VL}: left eigenvectors if requested
HASH{VR}: right eigenvectors if requested
HASH{info}: info output from gees/cgees.
if select_func is defined:
HASH{n}: number of eigenvalues selected,
HASH{rconde}: reciprocal condition numbers for the average of
the selected eigenvalues if requested,
HASH{rcondv}: reciprocal condition numbers for the selected
right invariant subspace if requested.

my \$a = random(10,10);
my \$schur  = mschurx(\$a);
sub select{
my \$m = shift;
# select "discrete time" eigenspace
return \$m->Cabs < 1 ? 1 : 0;
}
my (\$schur,\$eigen, \$vectors,%ret)  = mschurx(\$a,1,0,0,\&select);

mgschur
Computes generalized Schur decomposition of the pair (A,B).

A = Q x S x Z'
B = Q x T x Z'

Uses gges or cgges from Lapack.  Works on transposed array.

( PDL(schur S), PDL(schur T), PDL(alpha), PDL(beta), HASH{result}) = mgschur(PDL(A), PDL(B), ↲
SCALAR(left schur vector),SCALAR(right schur vector),SCALAR(left eigenvector), SCALAR(right eigenvector), SCALAR(select_func), SCALAR(backtransform), SCALAR(scale))
left schur vector   : Left Schur vectors returned, none = 0 | all = 1 | selected = 2, defaul ↲
t = 0
right schur vector  : Right Schur vectors returned, none = 0 | all = 1 | selected = 2, defau ↲
lt = 0
left eigenvector    : Left eigenvectors returned, none = 0 | all = 1 | selected = 2, default ↲
= 0
right eigenvector   : Right eigenvectors returned, none = 0 | all = 1 | selected = 2, defaul ↲
t = 0
select_func         : Select_func is used to select eigenvalues to sort.
to the top left of the Schur form.
An eigenvalue w = wr(j)+sqrt(-1)*wi(j) is selected if
PerlInt select_func(PDL::Complex(alpha),PDL | PDL::Complex (beta)) is  ↲
true;
Note that a selected complex eigenvalue may no longer
satisfy select_func = 1 after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned).
All eigenvalues/vectors are selected if select_func is undefined.
backtransform       : Whether or not backtransforms eigenvectors to those of (A,B).
Only supported if right and/or left schur vector are computed,
scale               : Whether or not computed eigenvectors are scaled so the largest compone ↲
nt
will have abs(real part) + abs(imag. part) = 1, default = 1

Returned values     :
Schur form S,
Schur form T,
alpha,
beta (eigenvalues = alpha/beta),
HASH{info}: info output from gges/cgges.
HASH{SL}: left Schur vectors if requested
HASH{SR}: right Schur vectors if requested
HASH{VL}: left eigenvectors if requested
HASH{VR}: right eigenvectors if requested
HASH{n} : Number of eigenvalues selected if select_func is defined.

my \$a = random(10,10);
my \$b = random(10,10);
my (\$S,\$T) = mgschur(\$a,\$b);
sub select{
my (\$alpha,\$beta) = @_;
return \$alpha->Cabs < abs(\$beta) ? 1 : 0;
}
my (\$S, \$T, \$alpha, \$beta, %res)  = mgschur( \$a, \$b, 1, 1, 1, 1,\&select);

mgschurx
Computes generalized Schur decomposition of the pair (A,B).

A = Q x S x Z'
B = Q x T x Z'

Uses ggesx or cggesx from Lapack. Works on transposed array.

( PDL(schur S), PDL(schur T), PDL(alpha), PDL(beta), HASH{result}) = mgschurx(PDL(A), PDL(B) ↲
, SCALAR(left schur vector),SCALAR(right schur vector),SCALAR(left eigenvector), SCALAR(right eigenvector), SCALAR(select_func), SCALAR(sense), SCALAR(backtransform), SCALAR(scale))
left schur vector   : Left Schur vectors returned, none = 0 | all = 1 | selected = 2, defaul ↲
t = 0
right schur vector  : Right Schur vectors returned, none = 0 | all = 1 | selected = 2, defau ↲
lt = 0
left eigenvector    : Left eigenvectors returned, none = 0 | all = 1 | selected = 2, default ↲
= 0
right eigenvector   : Right eigenvectors returned, none = 0 | all = 1 | selected = 2, defaul ↲
t = 0
select_func         : Select_func is used to select eigenvalues to sort.
to the top left of the Schur form.
An eigenvalue w = wr(j)+sqrt(-1)*wi(j) is selected if
PerlInt select_func(PDL::Complex(alpha),PDL | PDL::Complex (beta)) is  ↲
true;
Note that a selected complex eigenvalue may no longer
satisfy select_func = 1 after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned).
All eigenvalues/vectors are selected if select_func is undefined.
sense               : Determines which reciprocal condition numbers will be computed.
0: None are computed
1: Computed for average of selected eigenvalues only
2: Computed for selected deflating subspaces only
3: Computed for both
If select_func is undefined, sense is not used.

backtransform       : Whether or not backtransforms eigenvectors to those of (A,B).
Only supported if right and/or left schur vector are computed, default ↲
= 1
scale               : Whether or not computed eigenvectors are scaled so the largest compone ↲
nt
will have abs(real part) + abs(imag. part) = 1, default = 1

Returned values     :
Schur form S,
Schur form T,
alpha,
beta (eigenvalues = alpha/beta),
HASH{info}: info output from gges/cgges.
HASH{SL}: left Schur vectors if requested
HASH{SR}: right Schur vectors if requested
HASH{VL}: left eigenvectors if requested
HASH{VR}: right eigenvectors if requested
HASH{rconde}: reciprocal condition numbers for average of selected eig ↲
envalues if requested
HASH{rcondv}: reciprocal condition numbers for selected deflating subs ↲
paces if requested
HASH{n} : Number of eigenvalues selected if select_func is defined.

my \$a = random(10,10);
my \$b = random(10,10);
my (\$S,\$T) = mgschurx(\$a,\$b);
sub select{
my (\$alpha,\$beta) = @_;
return \$alpha->Cabs < abs(\$beta) ? 1 : 0;
}
my (\$S, \$T, \$alpha, \$beta, %res)  = mgschurx( \$a, \$b, 1, 1, 1, 1,\&select,3);

mqr
Computes QR decomposition.  For complex number needs object of type PDL::Complex.  Uses
geqrf and orgqr or cgeqrf and cungqr from Lapack and returns "Q" in scalar context. Works
on transposed array.

(PDL(Q), PDL(R), PDL(info)) = mqr(PDL, SCALAR)
SCALAR : ECONOMIC = 0 | FULL = 1, default = 0

my \$a = random(10,10);
my ( \$q, \$r )  = mqr(\$a);
# Can compute full decomposition if nrow > ncol
\$a = random(5,7);
( \$q, \$r )  = \$a->mqr(1);

mrq
Computes RQ decomposition.  For complex number needs object of type PDL::Complex.  Uses
gerqf and orgrq or cgerqf and cungrq from Lapack and returns "Q" in scalar context. Works
on transposed array.

(PDL(R), PDL(Q), PDL(info)) = mrq(PDL, SCALAR)
SCALAR : ECONOMIC = 0 | FULL = 1, default = 0

my \$a = random(10,10);
my ( \$r, \$q )  = mrq(\$a);
# Can compute full decomposition if nrow < ncol
\$a = random(5,7);
( \$r, \$q )  = \$a->mrq(1);

mql
Computes QL decomposition.  For complex number needs object of type PDL::Complex.  Uses
geqlf and orgql or cgeqlf and cungql from Lapack and returns "Q" in scalar context. Works
on transposed array.

(PDL(Q), PDL(L), PDL(info)) = mql(PDL, SCALAR)
SCALAR : ECONOMIC = 0 | FULL = 1, default = 0

my \$a = random(10,10);
my ( \$q, \$l )  = mql(\$a);
# Can compute full decomposition if nrow > ncol
\$a = random(5,7);
( \$q, \$l )  = \$a->mql(1);

mlq
Computes LQ decomposition.  For complex number needs object of type PDL::Complex.  Uses
gelqf and orglq or cgelqf and cunglq from Lapack and returns "Q" in scalar context. Works
on transposed array.

( PDL(L), PDL(Q), PDL(info) ) = mlq(PDL, SCALAR)
SCALAR : ECONOMIC = 0 | FULL = 1, default = 0

my \$a = random(10,10);
my ( \$l, \$q )  = mlq(\$a);
# Can compute full decomposition if nrow < ncol
\$a = random(5,7);
( \$l, \$q )  = \$a->mlq(1);

msolve
Solves linear system of equations using LU decomposition.

A * X = B

Returns X in scalar context else X, LU, pivot vector and info.  B is overwritten by X if
its inplace flag is set.  Supports threading.  Uses gesv or cgesv from Lapack.  Works on
transposed arrays.

(PDL(X), (PDL(LU), PDL(pivot), PDL(info))) = msolve(PDL(A), PDL(B) )

my \$a = random(5,5);
my \$b = random(10,5);
my \$X = msolve(\$a, \$b);

msolvex
Solves linear system of equations using LU decomposition.

A * X = B

Can optionnally equilibrate the matrix.  Uses gesvx or cgesvx from Lapack.  Works on
transposed arrays.

(PDL, (HASH(result))) = msolvex(PDL(A), PDL(B), HASH(options))
where options are:
transpose:     solves A' * X = B
0: false
1: true
equilibrate:   equilibrates A if necessary.
form equilibration is returned in HASH{'equilibration'}:
0: no equilibration
1: row equilibration
2: column equilibration
row scale factors are returned in HASH{'row'}
column scale factors are returned in HASH{'column'}
0: false
1: true
LU:            returns lu decomposition in HASH{LU}
0: false
1: true
A:             returns scaled A if equilibration was done in HASH{A}
0: false
1: true
B:             returns scaled B if equilibration was done in HASH{B}
0: false
1: true
Returned values:
X (SCALAR CONTEXT),
HASH{'pivot'}:
Pivot indice from LU factorization
HASH{'rcondition'}:
Reciprocal condition of the matrix
HASH{'ferror'}:
Forward error bound
HASH{'berror'}:
Componentwise relative backward error
HASH{'rpvgrw'}:
Reciprocal pivot growth factor
HASH{'info'}:
Info: output from gesvx

my \$a = random(10,10);
my \$b = random(5,10);
my %options = (
LU=>1,
equilibrate => 1,
);
my( \$X, %result) = msolvex(\$a,\$b,%options);

mtrisolve
Solves linear system of equations with triangular matrix A.

A * X = B  or A' * X = B

B is overwritten by X if its inplace flag is set.  Supports threading.  Uses trtrs or
ctrtrs from Lapack.  Work on transposed array(s).

(PDL(X), (PDL(info)) = mtrisolve(PDL(A), SCALAR(uplo), PDL(B), SCALAR(trans), SCALAR(diag))
uplo   : UPPER  = 0 | LOWER = 1
trans  : NOTRANSPOSE  = 0 | TRANSPOSE = 1, default = 0
uplo   : UNITARY DIAGONAL = 1, default = 0

# Assume \$a is upper triagonal
my \$a = random(5,5);
my \$b = random(5,10);
my \$X = mtrisolve(\$a, 0, \$b);

msymsolve
Solves linear system of equations using diagonal pivoting method with symmetric matrix A.

A * X = B

Returns X in scalar context else X, block diagonal matrix D (and the multipliers), pivot
vector an info. B is overwritten by X if its inplace flag is set.  Supports threading.
Uses sysv or csysv from Lapack.  Works on transposed array(s).

(PDL(X), ( PDL(D), PDL(pivot), PDL(info) ) ) = msymsolve(PDL(A), SCALAR(uplo), PDL(B) )
uplo : UPPER  = 0 | LOWER = 1, default = 0

# Assume \$a is symmetric
my \$a = random(5,5);
my \$b = random(5,10);
my \$X = msymsolve(\$a, 0, \$b);

msymsolvex
Solves linear system of equations using diagonal pivoting method with symmetric matrix A.

A * X = B

Uses sysvx or csysvx from Lapack. Works on transposed array.

(PDL, (HASH(result))) = msymsolvex(PDL(A), SCALAR (uplo), PDL(B), SCALAR(d))
uplo : UPPER  = 0 | LOWER = 1, default = 0
d    : whether return diagonal matrix d and pivot vector
FALSE  = 0 | TRUE = 1, default = 0
Returned values:
X (SCALAR CONTEXT),
HASH{'D'}:
Block diagonal matrix D (and the multipliers) (if requested)
HASH{'pivot'}:
Pivot indice from LU factorization (if requested)
HASH{'rcondition'}:
Reciprocal condition of the matrix
HASH{'ferror'}:
Forward error bound
HASH{'berror'}:
Componentwise relative backward error
HASH{'info'}:
Info: output from sysvx

# Assume \$a is symmetric
my \$a = random(10,10);
my \$b = random(5,10);
my (\$X, %result) = msolvex(\$a, 0, \$b);

mpossolve
Solves linear system of equations using Cholesky decomposition with symmetric positive
definite matrix A.

A * X = B

Returns X in scalar context else X, U or L and info.  B is overwritten by X if its inplace
flag is set.  Supports threading.  Uses posv or cposv from Lapack.  Works on transposed
array(s).

(PDL, (PDL, PDL, PDL)) = mpossolve(PDL(A), SCALAR(uplo), PDL(B) )
uplo : UPPER  = 0 | LOWER = 1, default = 0

# asume \$a is symmetric positive definite
my \$a = random(5,5);
my \$b = random(5,10);
my \$X = mpossolve(\$a, 0, \$b);

mpossolvex
Solves linear system of equations using Cholesky decomposition with symmetric positive
definite matrix A

A * X = B

Can optionnally equilibrate the matrix.  Uses posvx or cposvx from Lapack.  Works on
transposed array(s).

(PDL, (HASH(result))) = mpossolvex(PDL(A), SCARA(uplo), PDL(B), HASH(options))
uplo : UPPER  = 0 | LOWER = 1, default = 0
where options are:
equilibrate:   equilibrates A if necessary.
form equilibration is returned in HASH{'equilibration'}:
0: no equilibration
1: equilibration
scale factors are returned in HASH{'scale'}
0: false
1: true
U|L:           returns Cholesky factorization in HASH{U} or HASH{L}
0: false
1: true
A:             returns scaled A if equilibration was done in HASH{A}
0: false
1: true
B:             returns scaled B if equilibration was done in HASH{B}
0: false
1: true
Returned values:
X (SCALAR CONTEXT),
HASH{'rcondition'}:
Reciprocal condition of the matrix
HASH{'ferror'}:
Forward error bound
HASH{'berror'}:
Componentwise relative backward error
HASH{'info'}:
Info: output from gesvx

# Assume \$a is symmetric positive definite
my \$a = random(10,10);
my \$b = random(5,10);
my %options = (U=>1,
equilibrate => 1,
);
my (\$X, %result) = msolvex(\$a, 0, \$b,%opt);

mlls
Solves overdetermined or underdetermined real linear systems using QR or LQ factorization.

If M > N in the M-by-N matrix A, returns the residual sum of squares too.  Uses gels or
cgels from Lapack.  Works on transposed arrays.

PDL(X) = mlls(PDL(A), PDL(B), SCALAR(trans))
trans : NOTRANSPOSE  = 0 | TRANSPOSE/CONJUGATE = 1, default = 0

\$a = random(4,5);
\$b = random(3,5);
(\$x, \$res) = mlls(\$a, \$b);

mllsy
Computes the minimum-norm solution to a real linear least squares problem using a complete
orthogonal factorization.

Uses gelsy or cgelsy from Lapack. Works on tranposed arrays.

( PDL(X), ( HASH(result) ) ) = mllsy(PDL(A), PDL(B))
Returned values:
X (SCALAR CONTEXT),
HASH{'A'}:
complete orthogonal factorization of A
HASH{'jpvt'}:
details of columns interchanges
HASH{'rank'}:
effective rank of A

my \$a = random(10,10);
my \$b = random(10,10);
\$X = mllsy(\$a, \$b);

mllss
Computes the minimum-norm solution to a real linear least squares problem using a singular
value decomposition.

Uses gelss or gelsd from Lapack.  Works on transposed arrays.

( PDL(X), ( HASH(result) ) )= mllss(PDL(A), PDL(B), SCALAR(method))
method: specifie which method to use (see Lapack for further details)
'(c)gelss' or '(c)gelsd', default = '(c)gelsd'
Returned values:
X (SCALAR CONTEXT),
HASH{'V'}:
if method = (c)gelss, the right singular vectors, stored columnwise
HASH{'s'}:
singular values from SVD
HASH{'res'}:
if A has full rank the residual sum-of-squares for the solution
HASH{'rank'}:
effective rank of A
HASH{'info'}:
info output from method

my \$a = random(10,10);
my \$b = random(10,10);
\$X = mllss(\$a, \$b);

mglm
Solves a general Gauss-Markov Linear Model (GLM) problem.  Supports threading.  Uses ggglm
or cggglm from Lapack. Works on transposed arrays.

(PDL(x), PDL(y)) = mglm(PDL(a), PDL(b), PDL(d))
where d is the left hand side of the GLM equation

my \$a = random(8,10);
my \$b = random(7,10);
my \$d = random(10);
my (\$x, \$y) = mglm(\$a, \$b, \$d);

mlse
Solves a linear equality-constrained least squares (LSE) problem.  Uses gglse or cgglse
from Lapack. Works on transposed arrays.

(PDL(x), PDL(res2)) = mlse(PDL(a), PDL(b), PDL(c), PDL(d))
where
c      : The right hand side vector for the
least squares part of the LSE problem.
d      : The right hand side vector for the
constrained equation.
x      : The solution of the LSE problem.
res2   : The residual sum of squares for the solution
(returned only in array context)

my \$a = random(5,4);
my \$b = random(5,3);
my \$c = random(4);
my \$d = random(3);
my (\$x, \$res2) = mlse(\$a, \$b, \$c, \$d);

meigen
Computes eigenvalues and, optionally, the left and/or right eigenvectors of a general
square matrix (spectral decomposition).  Eigenvectors are normalized (Euclidean norm = 1)
and largest component real.  The eigenvalues and eigenvectors returned are object of type
PDL::Complex.  If only eigenvalues are requested, info is returned in array context.
Supports threading.  Uses geev or cgeev from Lapack.  Works on transposed arrays.

(PDL(values), (PDL(LV),  (PDL(RV)), (PDL(info))) = meigen(PDL, SCALAR(left vector), SCALAR(r ↲
ight vector))
left vector  : FALSE = 0 | TRUE = 1, default = 0
right vector : FALSE = 0 | TRUE = 1, default = 0

my \$a = random(10,10);
my ( \$eigenvalues, \$left_eigenvectors, \$right_eigenvectors )  = meigen(\$a,1,1);

meigenx
Computes eigenvalues, one-norm and, optionally, the left and/or right eigenvectors of a
general square matrix (spectral decomposition).  Eigenvectors are normalized (Euclidean
norm = 1) and largest component real.  The eigenvalues and eigenvectors returned are
object of type PDL::Complex.  Uses geevx or cgeevx from Lapack.  Works on transposed
arrays.

(PDL(value), (PDL(lv),  (PDL(rv)), HASH(result)), HASH(result)) = meigenx(PDL, HASH(options) ↲
)
where options are:
vector:     eigenvectors to compute
'left':  computes left eigenvectors
'right': computes right eigenvectors
'all':   computes left and right eigenvectors
0:     doesn't compute (default)
rcondition: reciprocal condition numbers to compute (returned in HASH{'rconde'} for eigenval ↲
ues and HASH{'rcondv'} for eigenvectors)
'value':  computes reciprocal condition numbers for eigenvalues
'vector': computes reciprocal condition numbers for eigenvectors
'all':    computes reciprocal condition numbers for eigenvalues and eigenvect ↲
ors
0:      doesn't compute (default)
error:      specifie whether or not it computes the error bounds (returned in HASH{'eerror'} ↲
and HASH{'verror'})
error bound = EPS * One-norm / rcond(e|v)
(reciprocal condition numbers for eigenvalues or eigenvectors must be computed).
1: returns error bounds
0: not computed
scale:      specifie whether or not it diagonaly scales the entry matrix
(scale details returned in HASH : 'scale')
1: scales
0: Doesn't scale (default)
permute:    specifie whether or not it permutes row and columns
(permute details returned in HASH{'balance'})
1: permutes
0: Doesn't permute (default)
schur:      specifie whether or not it returns the Schur form (returned in HASH{'schur'})
1: returns Schur form
0: not returned
Returned values:
eigenvalues (SCALAR CONTEXT),
left eigenvectors if requested,
right eigenvectors if requested,
HASH{'norm'}:
One-norm of the matrix
HASH{'info'}:
Info: if > 0, the QR algorithm failed to compute all the eigenvalues
(see syevx for further details)

my \$a = random(10,10);
my %options = ( rcondition => 'all',
vector => 'all',
error => 1,
scale => 1,
permute=>1,
shur => 1
);
my ( \$eigenvalues, \$left_eigenvectors, \$right_eigenvectors, %result)  = meigenx(\$a,%options) ↲
;
print "Error bounds for eigenvalues:\n \$eigenvalues\n are:\n". transpose(\$result{'eerror'})  ↲
unless \$info;

mgeigen
Computes generalized eigenvalues and, optionally, the left and/or right generalized
eigenvectors for a pair of N-by-N real nonsymmetric matrices (A,B) .  The alpha from ratio
alpha/beta is object of type PDL::Complex.  Supports threading. Uses ggev or cggev from
Lapack.  Works on transposed arrays.

( PDL(alpha), PDL(beta), ( PDL(LV),  (PDL(RV) ), PDL(info)) = mgeigen(PDL(A),PDL(B) SCALAR(l ↲
eft vector), SCALAR(right vector))
left vector  : FALSE = 0 | TRUE = 1, default = 0
right vector : FALSE = 0 | TRUE = 1, default = 0

my \$a = random(10,10);
my \$b = random(10,10);
my ( \$alpha, \$beta, \$left_eigenvectors, \$right_eigenvectors )  = mgeigen(\$a, \$b,1, 1);

mgeigenx
Computes generalized eigenvalues, one-norms and, optionally, the left and/or right
generalized eigenvectors for a pair of N-by-N real nonsymmetric matrices (A,B).  The alpha
from ratio alpha/beta is object of type PDL::Complex.  Uses ggevx or cggevx from Lapack.
Works on transposed arrays.

(PDL(alpha), PDL(beta), PDL(lv),  PDL(rv), HASH(result) ) = mgeigenx(PDL(a), PDL(b), HASH(op ↲
tions))
where options are:
vector:     eigenvectors to compute
'left':  computes left eigenvectors
'right': computes right eigenvectors
'all':   computes left and right eigenvectors
0:     doesn't compute (default)
rcondition: reciprocal condition numbers to compute (returned in HASH{'rconde'} for eigenval ↲
ues and HASH{'rcondv'} for eigenvectors)
'value':  computes reciprocal condition numbers for eigenvalues
'vector': computes reciprocal condition numbers for eigenvectors
'all':    computes reciprocal condition numbers for eigenvalues and eigenvect ↲
ors
0:      doesn't compute (default)
error:      specifie whether or not it computes the error bounds (returned in HASH{'eerror'} ↲
and HASH{'verror'})
error bound = EPS * sqrt(one-norm(a)**2 + one-norm(b)**2) / rcond(e|v)
(reciprocal condition numbers for eigenvalues or eigenvectors must be computed).
1: returns error bounds
0: not computed
scale:      specifie whether or not it diagonaly scales the entry matrix
(scale details returned in HASH : 'lscale' and 'rscale')
1: scales
0: doesn't scale (default)
permute:    specifie whether or not it permutes row and columns
(permute details returned in HASH{'balance'})
1: permutes
0: Doesn't permute (default)
schur:      specifie whether or not it returns the Schur forms (returned in HASH{'aschur'} a ↲
nd HASH{'bschur'})
(right or left eigenvectors must be computed).
1: returns Schur forms
0: not returned
Returned values:
alpha,
beta,
left eigenvectors if requested,
right eigenvectors if requested,
HASH{'anorm'}, HASH{'bnorm'}:
One-norm of the matrix A and B
HASH{'info'}:
Info: if > 0, the QR algorithm failed to compute all the eigenvalues
(see syevx for further details)

\$a = random(10,10);
\$b = random(10,10);
%options = (rcondition => 'all',
vector => 'all',
error => 1,
scale => 1,
permute=>1,
shur => 1
);
(\$alpha, \$beta, \$left_eigenvectors, \$right_eigenvectors, %result)  = mgeigenx(\$a, \$b,%option ↲
s);
print "Error bounds for eigenvalues:\n \$eigenvalues\n are:\n". transpose(\$result{'eerror'})  ↲
unless \$info;

msymeigen
Computes eigenvalues and, optionally eigenvectors of a real symmetric square or complex
Hermitian matrix (spectral decomposition).  The eigenvalues are computed from lower or
upper triangular matrix.  If only eigenvalues are requested, info is returned in array
context.  Supports threading and works inplace if eigenvectors are requested.  From
Lapack, uses syev or syevd for real and cheev or cheevd for complex.  Works on transposed
array(s).

(PDL(values), (PDL(VECTORS)), PDL(info)) = msymeigen(PDL, SCALAR(uplo), SCALAR(vector), SCAL ↲
AR(method))
uplo : UPPER  = 0 | LOWER = 1, default = 0
vector : FALSE = 0 | TRUE = 1, default = 0
method : 'syev' | 'syevd' | 'cheev' | 'cheevd', default = 'syevd'|'cheevd'

# Assume \$a is symmetric
my \$a = random(10,10);
my ( \$eigenvalues, \$eigenvectors )  = msymeigen(\$a,0,1, 'syev');

msymeigenx
Computes eigenvalues and, optionally eigenvectors of a symmetric square matrix (spectral
decomposition).  The eigenvalues are computed from lower or upper triangular matrix and
can be selected by specifying a range. From Lapack, uses syevx or syevr for real and
cheevx or cheevr for complex. Works on transposed arrays.

(PDL(value), (PDL(vector)), PDL(n), PDL(info), (PDL(support)) ) = msymeigenx(PDL, SCALAR(upl ↲
o), SCALAR(vector), HASH(options))
uplo : UPPER  = 0 | LOWER = 1, default = 0
vector : FALSE = 0 | TRUE = 1, default = 0
where options are:
range_type:    method for selecting eigenvalues
indice:  range of indices
interval: range of values
0: find all eigenvalues and optionally all vectors
range:         PDL(2), lower and upper bounds interval or smallest and largest indices
1<=range<=N for indice
abstol:        specifie error tolerance for eigenvalues
method:        specifie which method to use (see Lapack for further details)
'syevx' (default)
'syevr'
'cheevx' (default)
'cheevr'
Returned values:
eigenvalues (SCALAR CONTEXT),
eigenvectors if requested,
total number of eigenvalues found (n),
info
issupz or ifail (support) according to method used and returned info,
for (sy|che)evx returns support only if info != 0

# Assume \$a is symmetric
my \$a = random(10,10);
my \$overflow = lamch(9);
my \$range = cat pdl(0),\$overflow;
my \$abstol = pdl(1.e-5);
my %options = (range_type=>'interval',
range => \$range,
abstol => \$abstol,
method=>'syevd');
my ( \$eigenvalues, \$eigenvectors, \$n, \$isuppz )  = msymeigenx(\$a,0,1, %options);

msymgeigen
Computes eigenvalues and, optionally eigenvectors of a real generalized symmetric-definite
or Hermitian-definite eigenproblem.  The eigenvalues are computed from lower or upper
triangular matrix If only eigenvalues are requested, info is returned in array context.
Supports threading. From Lapack, uses sygv or sygvd for real or chegv or chegvd for
complex.  Works on transposed array(s).

(PDL(values), (PDL(vectors)), PDL(info)) = msymgeigen(PDL(a), PDL(b),SCALAR(uplo), SCALAR(ve ↲
ctor), SCALAR(type), SCALAR(method))
uplo : UPPER  = 0 | LOWER = 1, default = 0
vector : FALSE = 0 | TRUE = 1, default = 0
type :
1: A * x = (lambda) * B * x
2: A * B * x = (lambda) * x
3: B * A * x = (lambda) * x
default = 1
method : 'sygv' | 'sygvd' for real or  ,'chegv' | 'chegvd' for complex,  default = 'sygvd' | ↲
'chegvd'

# Assume \$a is symmetric
my \$a = random(10,10);
my \$b = random(10,10);
\$b = \$b->crossprod(\$b);
my ( \$eigenvalues, \$eigenvectors )  = msymgeigen(\$a, \$b, 0, 1, 1, 'sygv');

msymgeigenx
Computes eigenvalues and, optionally eigenvectors of a real generalized symmetric-definite
or Hermitian eigenproblem.  The eigenvalues are computed from lower or upper triangular
matrix and can be selected by specifying a range. Uses sygvx or cheevx from Lapack. Works
on transposed arrays.

(PDL(value), (PDL(vector)), PDL(info), PDL(n), (PDL(support)) ) = msymeigenx(PDL(a), PDL(b), ↲
SCALAR(uplo), SCALAR(vector), HASH(options))
uplo : UPPER  = 0 | LOWER = 1, default = 0
vector : FALSE = 0 | TRUE = 1, default = 0
where options are:
type :         Specifies the problem type to be solved
1: A * x = (lambda) * B * x
2: A * B * x = (lambda) * x
3: B * A * x = (lambda) * x
default = 1
range_type:    method for selecting eigenvalues
indice:  range of indices
interval: range of values
0: find all eigenvalues and optionally all vectors
range:         PDL(2), lower and upper bounds interval or smallest and largest indices
1<=range<=N for indice
abstol:        specifie error tolerance for eigenvalues
Returned values:
eigenvalues (SCALAR CONTEXT),
eigenvectors if requested,
total number of eigenvalues found (n),
info
ifail according to returned info (support).

# Assume \$a is symmetric
my \$a = random(10,10);
my \$b = random(10,10);
\$b = \$b->crossprod(\$b);
my \$overflow = lamch(9);
my \$range = cat pdl(0),\$overflow;
my \$abstol = pdl(1.e-5);
my %options = (range_type=>'interval',
range => \$range,
abstol => \$abstol,
type => 1);
my ( \$eigenvalues, \$eigenvectors, \$n, \$isuppz )  = msymgeigenx(\$a, \$b, 0,1, %options);

mdsvd
Computes SVD using Coppen's divide and conquer algorithm.  Return singular values in
scalar context else left (U), singular values, right (V' (hermitian for complex)) singular
vectors and info.  Supports threading.  If only singulars values are requested, info is
only returned in array context.  Uses gesdd or cgesdd from Lapack.

(PDL(U), (PDL(s), PDL(V)), PDL(info)) = mdsvd(PDL, SCALAR(job))
job :  0 = computes only singular values
1 = computes full SVD (square U and V)
2 = computes SVD (singular values, right and left singular vectors)
default = 1

my \$a = random(5,10);
my (\$u, \$s, \$v) = mdsvd(\$a);

msvd
Computes SVD.  Can compute singular values, either U or V or neither.  Return singulars
values in scalar context else left (U), singular values, right (V' (hermitian for complex)
singulars vector and info.  Supports threading.  If only singulars values are requested,
info is returned in array context.  Uses gesvd or cgesvd from Lapack.

( (PDL(U)), PDL(s), (PDL(V), PDL(info)) = msvd(PDL, SCALAR(jobu), SCALAR(jobv))
jobu : 0 = Doesn't compute U
1 = computes full SVD (square U)
2 = computes right singular vectors
default = 1
jobv : 0 = Doesn't compute V
1 = computes full SVD (square V)
2 = computes left singular vectors
default = 1

my \$a = random(10,10);
my (\$u, \$s, \$v) = msvd(\$a);

mgsvd
Computes generalized (or quotient) singular value decomposition.  If the effective rank of
(A',B')' is 0 return only unitary V, U, Q.  For complex number, needs object of type
PDL::Complex.  Uses ggsvd or cggsvd from Lapack. Works on transposed arrays.

(PDL(sa), PDL(sb), %ret) = mgsvd(PDL(a), PDL(b), %HASH(options))
where options are:
V:    whether or not computes V (boolean, returned in HASH{'V'})
U:    whether or not computes U (boolean, returned in HASH{'U'})
Q:    whether or not computes Q (boolean, returned in HASH{'Q'})
D1:   whether or not computes D1 (boolean, returned in HASH{'D1'})
D2:   whether or not computes D2 (boolean, returned in HASH{'D2'})
0R:   whether or not computes 0R (boolean, returned in HASH{'0R'})
R:    whether or not computes R (boolean, returned in HASH{'R'})
X:    whether or not computes X (boolean, returned in HASH{'X'})
all:  whether or not computes all the above.
Returned value:
sa,sb          : singular value pairs of A and B (generalized singular values = sa/s ↲
b)
\$ret{'rank'}   : effective numerical rank of (A',B')'
\$ret{'info'}   : info from (c)ggsvd

my \$a = random(5,5);
my \$b = random(5,7);
my (\$c, \$s, %ret) = mgsvd(\$a, \$b, X => 1);

AUTHOR