GEL: /Users/jab/Documents/Teaching/02585/GEL2_and_demos/GEL/src/LinAlg/LapackFunc.h File Reference

Interface to some of the LAPACK functionality. More...

#include "Matrix.h"
#include "Vector.h"

Go to the source code of this file.

Namespaces

namespace LinAlg

The Linear Algebra Implementation/Module/Package.

Functions

void LinAlg::LinearSolveSym (const CMatrix &A, const CVector &b, CVector &x)

Singular Value Decomposition SVD

These functions perform the Singular Value Decomposition SVD of the MxN matrix A. The SVD is defined by:

A=U*S*V^T

where:

U is a M by M orthogonal matrix
V is a N by N orthogonal matrix
S is a M by N diaggonal matrix. The values in the diagonal are the singular values

Parameters:

A	the matrix to perform SVD on

Returns:: U will be resized if it is does not have the correct dimensions; V will be resized if it is does not have the correct dimensions; S will be resized if it is does not have the correct dimensions.

Exceptions:

assert(info==0) for Lapack. Add a throw statement later.

Version:: Aug 2001

Author:: Henrik Aans

void LinAlg::SVD (const CMatrix &A, CMatrix &U, CMatrix &S, CMatrix &V)

SVD of A, where the singular values are returned in a 'diagonal' Matrix.

CVector LinAlg::SVD (const CMatrix &A)

SVD of A, returning only the singular values in a Vector.

Linear Equations

These functions solve the system of linear equations

A*x=b

for x, where:

A is a N by N matrix
b is a N vector
x is a N vector

There a speceilaized functions for symetric positive definite (SPD) matrices yeilding better performance. These are denote by SPD in there function name.

Parameters:

A	the NxN square matrix
b	the N vector

Returns:: x will be resized if it is does not have the correct dimensions

Exceptions:

assert(info==0)	for Lapack. Add a throw statement later.
assert(A.Row()==A.Col()).	Add a throw statement later.
assert(A.Row()==b.Length()).	Add a throw statement later.

Version:: Aug 2001

Author:: Henrik Aans

void LinAlg::LinearSolve (const CMatrix &A, const CVector &b, CVector &x)

Solves Ax=b for x.

CVector LinAlg::LinearSolve (const CMatrix &A, const CVector &b)

Solves Ax=b for x and returns x.

void LinAlg::LinearSolveSPD (const CMatrix &A, const CVector &b, CVector &x)

Solves Ax=b for x, where A is SPD.

CVector LinAlg::LinearSolveSPD (const CMatrix &A, const CVector &b)

Solves Ax=b for x and returns x, where A is SPD.

Linear Least Squares

These functions solve the Linear Least Squares problem:

min_x ||Ax-b||^2

for x, where:

|| || denotes the 2-norm
A is a M by N matrix. For a well formed M>=N and rank (A)=N. See below.
b is a M vector.
x is a N vector

If the solution is not well formed the algorithm provided will find a solution, x, which is not unique, but which sets the objective function to 0. The reson being that the underlining algorithm works by SVD.

Parameters:

A	the MxN matrix
b	the M vector

Returns:: x will be resized if it is does not have the correct dimensions

Exceptions:

assert(info==0)	for Lapack. Add a throw statement later.
assert(A.Rows()==b.Length());.	Add a throw statement later.

Version:: Aug 2001

Author:: Henrik Aans

void LinAlg::LinearLSSolve (const CMatrix &A, const CVector &b, CVector &x)

Solves the Linear Least Squares problem min_x ||Ax=b||^2 for x.

CVector LinAlg::LinearLSSolve (const CMatrix &A, const CVector &b)

Solves the Linear Least Squares problem min_x ||Ax=b||^2 for x, and returnes x.

Matrix Inversion

These functions inverts the square matrix A. This matrix A must have full rank.

Parameters:

A	square matrix

Returns:: InvA the invers of A for one instance.

Exceptions:

assert(info==0)	for Lapack. This wil among others happen if A is rank deficient. Add a throw statement later.
assert(A.Rows()==A.Cols()).	Add a throw statement later.

Version:: Aug 2001

Author:: Henrik Aans

void LinAlg::Invert (CMatrix &A)

Invertes the square matrix A. That is here A is altered as opposed to the other Invert functions.

void LinAlg::Inverted (const CMatrix &A, CMatrix &InvA)

Returns the inverse of the square matrix A in InvA.

CMatrix LinAlg::Inverted (const CMatrix &A)

Returns the inverse of the square matrix A.

QR Factorization

This function returns the QR factorization of A, such that Q*R=A where Q is a orthonormal matrix and R is an upper triangular matrix. However, in the case of A.Col()>A.Row(), the last A.Col-A.Row columns of Q are 'carbage' and as such not part of a orthonormal matrix.

Parameters:

A	the input matrix

Returns:: Q an orthonormal matrix. (See above); R an upper triangular matrix.

Exceptions:

assert(info==0)	for Lapack. This wil among others happen if A is rank deficient. Add a throw statement later.
assert(A.Rows()>0	&& A.Cols()>0). Add a throw statement later.

Version:: Aug 2001

Author:: Henrik Aans

void LinAlg::QRfact (const CMatrix &A, CMatrix &Q, CMatrix &R)

RQ Factorization

This function returns the RQ factorization of A, such that R*Q=A where Q is a orthonormal matrix and R is an upper triangular matrix. However, in the case of A not beeing a square matrix, there might be some fuck up of Q.

Parameters:

A	the input matrix

Returns:: Q an orthonormal matrix. (See above); R an upper triangular matrix.

Exceptions:

assert(info==0)	for Lapack. This wil among others happen if A is rank deficient. Add a throw statement later.
assert(A.Rows()>0	&& A.Cols()>0). Add a throw statement later.

Version:: Aug 2001

Author:: Henrik Aans

void LinAlg::RQfact (const CMatrix &A, CMatrix &R, CMatrix &Q)

Find eigensolutions of a symmetric real matrix.

This function accepts a real symmetric matrix Q and a vector b. When the function returns, the eigenvalues of the matrix Q will be stored in b and the eigenvectors form the columns of Q. This function is based on the Lapack function dsyev, and returns its info code. A code of 0 indicates success, and a code < 0 indicates an error. Probably Q is not a real symmetric matrix. If the code is > 0 "the algorithm failed to converge; code off-diagonal elements of an intermediate tridiagonal form did not converge to zero." Presumably this means that code contains the number of eigenvalues which are ok.

Author:: Andreas Brentzen.

int LinAlg::EigenSolutionsSym (CMatrix &Q, CVector &b)

Detailed Description

Interface to some of the LAPACK functionality.

These are functions which more or less directly interface with the Lapack provided algorithms.

For indepth reference to the LAPACK functions see: LAPACK Users' Guide - 3rd Edition, by E. Anderson et al., ISBN 0-89871-447-8, Published by SIAM,

This book is also available at: {http://www.netlib.org/lapack/lug/lapack_lug.html}

The official LAPACK sites where from the source can be downloaded are: {http://www.netlib.org/clapack/} and {http://www.netlib.org/lapack/}

NB: When running this in MS Visual C++ it is usually required to set the multithread "\MD" compiler option. This is to ensure correct linkage to the precompiled library "clapack.lib" and/or "clapackDB.lib".

Author:: Henrik Aans

Version:: Aug 2001