Linear Algebra

Collaboration diagram for Linear Algebra:

Detailed Description

Todo:

Have implemented

LU-decompostion
LQ-decompostion
PTLQ-decompostion
QR-decompostion
QRPT-decompostion
SV-decompostion
Cholesky-decompostion

implement the rest these classes, that is

Tridiagnoal decomposition for real-symmetric/Hermitian matrices
Bi-diagonalization
Householder transforms
Householder solver for linear systems
Tri-diagonal systems

Modules

LU decomposition

LQ decomposition

QR decomposition

Single value decomposition

Cholesky decomposition

Functions

template<>

matrix< complex< double > > gslmm::operator/ (const matrix< complex< double > > &lhs, const matrix< complex< double > > &rhs)

template<>

matrix< double > gslmm::operator/ (const matrix< double > &lhs, const matrix< double > &rhs)

Function Documentation

template<>

matrix<double> gslmm::operator/ ( const matrix< double > & lhs,

const matrix< double > & rhs

) [inline]

right / operation between matricies (double) Note that for real matrices, $A / B = AB^{-1} = \left({B^{T}}^{-1}A^T\right)^T$ In any case, the division can be computed by doing an decomposition of $ B^T$ or and perform a back-substitution of the columns of $ A^T$ , and computing the transpose the resulting matrix.
If the matrix $ B$ is square, then an LU decomposition is used. If it has more columns than rows, then a QR decomposition is used. Finally, if $ B$ has more rows than columns, a LQ decomposition is used. Note, that the two matrices $ A, B$ must have the same number of columns. The returned matrix is square with dimension of the input matrices column size.

Parameters:

lhs The left hand operand $ A $

rhs The right hand operand $ B $

Returns:
new matrix, $m_{ij} = A_{ij} B^{-1}_{ij}$

template<>

matrix<complex<double> > gslmm::operator/ ( const matrix< complex< double > > & lhs,

const matrix< complex< double > > & rhs

) [inline]

right / operation between matricies (complex<double>) For complex matrices, $A / B = AB^{-1} = \left({B^{H}}^{-1}A^H\right)^T$ .
In any case, the division can be computed by doing an LU decomposition of $ B^H$ and perform a back-substitution of the columns of $ A^H$ , and computing the transpose hermitian conjugate to the resulting matrix. Note, that the LU decomposition requires that the matrix $ B$ is square.

Parameters:

lhs The left hand operand $ A $

rhs The right hand operand $ B $

Returns:
new matrix, $m_{ij} = A_{ij} B^{-1}_{ij}$

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Christian Holm	Created by DoxyGen 1.4.6