QRMatrix computes QR decomposition of a given scalar/complex matrix A into the following: More...

Public Types
enum	modes : uint8_t { FULL = 1 , ECONOMY = 2 }
	Options for the decomposition mode. More...

enum	outputs : uint8_t { ONLY_Q = 1 , ONLY_R = 2 , BOTH_QR = 3 }
	Options for the output types. More...

enum	pivoting : bool { FALSE = false , TRUE = true }
	Options for the column pivoting. More...

typedef MatrixType::cmptType	cmptType

typedef SquareMatrix< cmptType >	SMatrix

typedef RectangularMatrix< cmptType >	RMatrix

Public Member Functions
	QRMatrix ()=delete
	Construct null. More...

	QRMatrix (const modes mode, const outputs output, const bool pivoting, MatrixType &A)
	Construct with a matrix and perform QR decomposition. More...

	QRMatrix (const MatrixType &A, const modes mode, const outputs output=outputs::BOTH_QR, const bool pivoting=false)
	Construct with a const matrix and perform QR decomposition. More...

const MatrixType &	Q () const noexcept
	Return const reference to Q. More...

MatrixType &	Q () noexcept
	Return reference to Q. More...

const MatrixType &	R () const noexcept
	Return const reference to R. More...

MatrixType &	R () noexcept
	Return reference to R. More...

const labelList &	p () const noexcept
	Return const reference to p. More...

SMatrix	P () const
	Create and return the permutation matrix. More...

void	solve (List< cmptType > &x, const UList< cmptType > &source) const

template<class Addr >
void	solve (List< cmptType > &x, const IndirectListBase< cmptType, Addr > &source) const

tmp< Field< cmptType > >	solve (const UList< cmptType > &source) const

template<class Addr >
tmp< Field< cmptType > >	solve (const IndirectListBase< cmptType, Addr > &source) const

RMatrix	solve (const RMatrix &b)

SMatrix	inv () const
	Return the inverse of (QR), solving x = (QR).inv()*source. More...

template<class Addr >
Foam::tmp< Foam::Field< typename MatrixType::cmptType > >	solve (const IndirectListBase< cmptType, Addr > &source) const

Detailed Description

template<class MatrixType>
class Foam::QRMatrix< MatrixType >

QRMatrix computes QR decomposition of a given scalar/complex matrix A into the following:

    A = Q R

or in case of QR decomposition with column pivoting:

    A P = Q R

where

\( Q \)	=	Unitary/orthogonal matrix
\( R \)	=	Upper triangular matrix
\( P \)	=	Permutation matrix

References:

        TNT implementation:
            Pozo, R. (1997).
            Template Numerical Toolkit for linear algebra:
            High performance programming with C++
            and the Standard Template Library.
            The International Journal of Supercomputer Applications
            and High Performance Computing, 11(3), 251-263.
            DOI:10.1177/109434209701100307

        QR decomposition with column pivoting (tag:QSB):
            Quintana-Ortí, G., Sun, X., & Bischof, C. H. (1998).
            A BLAS-3 version of the QR factorization with column pivoting.
            SIAM Journal on Scientific Computing, 19(5), 1486-1494.
            DOI:10.1137/S1064827595296732

        Moore-Penrose inverse algorithm (tags:KP; KPP):
            Katsikis, V. N., & Pappas, D. (2008).
            Fast computing of the Moore-Penrose inverse matrix.
            Electronic Journal of Linear Algebra, 17(1), 637-650.
            DOI:10.13001/1081-3810.1287

            Katsikis, V. N., Pappas, D., & Petralias, A. (2011).
            An improved method for the computation of
            the Moore–Penrose inverse matrix.
            Applied Mathematics and Computation, 217(23), 9828-9834.
            DOI:10.1016/j.amc.2011.04.080

        Tolerance for the Moore-Penrose inverse algorithm (tag:TA):
            Toutounian, F., & Ataei, A. (2009).
            A new method for computing Moore–Penrose inverse matrices.
            Journal of Computational and applied Mathematics, 228(1), 412-417.
            DOI:10.1016/j.cam.2008.10.008

Usage

Input:

\( A \)

=

RectangularMatrix<Type> or SquareMatrix<Type>

Options for the decomposition mode:

\( modes::FULL \)	=	compute full-size decomposition
\( modes::ECONOMY \)	=	compute economy-size decomposition

Options for the output types:

\( outputs::ONLY\_Q \)	=	compute only Q
\( outputs::ONLY\_R \)	=	compute only R
\( outputs::BOTH\_QR \)	=	compute both Q and R

Options for the column pivoting:

\( pivoting::FALSE \)	=	switch off column pivoting
\( pivoting::TRUE \)	=	switch on column pivoting

Output:

\( Q \)	=	m-by-m (FULL) or m-by-k (ECONOMY) with k = min(m,n)
\( R \)	=	m-by-n (FULL) or k-by-n (ECONOMY) with k = min(m,n)
\( p \)	=	n-element label list
\( P \)	=	n-by-n permutation matrix

Notes

QRMatrix involves modified implementations of the public-domain library TNT, which is available at https://math.nist.gov/tnt/index.html.
Q and R are always of the same type of the matrix A.
Type can be scalar or complex.

See also: Test-QRMatrix.C

Source files

Definition at line 211 of file QRMatrix.H.

Member Typedef Documentation

◆ cmptType

typedef MatrixType::cmptType cmptType

Definition at line 215 of file QRMatrix.H.

◆ SMatrix

typedef SquareMatrix<cmptType> SMatrix

Definition at line 216 of file QRMatrix.H.

◆ RMatrix

typedef RectangularMatrix<cmptType> RMatrix

Definition at line 217 of file QRMatrix.H.

Member Enumeration Documentation

◆ modes

enum modes : uint8_t

Options for the decomposition mode.

Enumerator
FULL	compute full-size decomposition
ECONOMY	compute economy-size decomposition

Definition at line 220 of file QRMatrix.H.

◆ outputs

enum outputs : uint8_t

Options for the output types.

Enumerator
ONLY_Q	compute only Q
ONLY_R	compute only R
BOTH_QR	compute both Q and R

Definition at line 227 of file QRMatrix.H.

◆ pivoting

enum pivoting : bool

Options for the column pivoting.

Enumerator
FALSE	switch off column pivoting
TRUE	switch on column pivoting

Definition at line 235 of file QRMatrix.H.

Constructor & Destructor Documentation

◆ QRMatrix() [1/3]

QRMatrix ( )

delete

Construct null.

◆ QRMatrix() [2/3]

QRMatrix	(	const modes	mode,
		const outputs	output,
		const bool	pivoting,
		MatrixType &	A
	)

explicit

Construct with a matrix and perform QR decomposition.

Definition at line 382 of file QRMatrix.C.

References A, and Foam::mode().

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◆ QRMatrix() [3/3]

QRMatrix	(	const MatrixType &	A,
		const modes	mode,
		const outputs	output = `outputs::BOTH_QR`,
		const bool	pivoting = `false`
	)

explicit

Construct with a const matrix and perform QR decomposition.

Definition at line 415 of file QRMatrix.C.

References A.

Member Function Documentation

◆ Q() [1/2]

const MatrixType & Q ( ) const

inlinenoexcept

Return const reference to Q.

Definition at line 341 of file QRMatrix.H.

Referenced by Foam::MatrixTools::pinv().

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◆ Q() [2/2]

MatrixType & Q ( )

inlinenoexcept

Return reference to Q.

Definition at line 347 of file QRMatrix.H.

◆ R() [1/2]

const MatrixType & R ( ) const

inlinenoexcept

Return const reference to R.

Definition at line 353 of file QRMatrix.H.

Referenced by Foam::MatrixTools::pinv().

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◆ R() [2/2]

MatrixType & R ( )

inlinenoexcept

Return reference to R.

Definition at line 359 of file QRMatrix.H.

◆ p()

const labelList & p ( ) const

inlinenoexcept

Return const reference to p.

Definition at line 365 of file QRMatrix.H.

◆ P()

Foam::SquareMatrix< typename MatrixType::cmptType > P

inline

Create and return the permutation matrix.

Definition at line 33 of file QRMatrixI.H.

References forAll.

Referenced by Foam::MatrixTools::pinv().

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◆ solve() [1/6]

void solve	(	List< cmptType > &	x,
		const UList< cmptType > &	source
	)		const

Solve the linear system with the given source and return the solution in the argument x

Definition at line 448 of file QRMatrix.C.

References x.

Referenced by Foam::MatrixTools::pinv().

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◆ solve() [2/6]

void solve	(	List< cmptType > &	x,
		const IndirectListBase< cmptType, Addr > &	source
	)		const

Solve the linear system with the given source and return the solution in the argument x

Definition at line 460 of file QRMatrix.C.

References x.

◆ solve() [3/6]

Foam::tmp< Foam::Field< typename MatrixType::cmptType > > solve ( const UList< cmptType > & source ) const

Solve the linear system with the given source and return the solution

Definition at line 472 of file QRMatrix.C.

◆ solve() [4/6]

tmp< Field< cmptType > > solve ( const IndirectListBase< cmptType, Addr > & source ) const

Solve the linear system with the given source and return the solution

◆ solve() [5/6]

Foam::RectangularMatrix< typename MatrixType::cmptType > solve ( const RMatrix & b )

Solve a row-echelon-form linear system (Ax = b) starting from the bottom by back substitution

A = R: Non-singular upper-triangular square matrix (m-by-m) b: Source (m-by-p) x: Solution (m-by-p)

Definition at line 503 of file QRMatrix.C.

References Foam::abort(), alpha, b, Foam::diag(), Foam::endl(), Foam::FatalError, FatalErrorInFunction, k, Matrix< Form, Type >::m(), Foam::mag(), Matrix< Form, Type >::n(), Foam::tab, WarningInFunction, and Foam::Zero.

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◆ inv()

Foam::QRMatrix< MatrixType >::SMatrix inv

Return the inverse of (Q*R), solving x = (Q*R).inv()*source.

Definition at line 559 of file QRMatrix.C.

References Foam::inv(), Matrix< Form, Type >::m(), and x.

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◆ solve() [6/6]

Foam::tmp< Foam::Field< typename MatrixType::cmptType > > solve ( const IndirectListBase< cmptType, Addr > & source ) const

Definition at line 488 of file QRMatrix.C.

The documentation for this class was generated from the following files:

src/OpenFOAM/matrices/QRMatrix/QRMatrix.H
src/OpenFOAM/matrices/QRMatrix/QRMatrix.C
src/OpenFOAM/matrices/QRMatrix/QRMatrixI.H

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ cmptType

◆ SMatrix

◆ RMatrix

Member Enumeration Documentation

◆ modes

◆ outputs

◆ pivoting

Constructor & Destructor Documentation

◆ QRMatrix() [1/3]

◆ QRMatrix() [2/3]

◆ QRMatrix() [3/3]

Member Function Documentation

◆ Q() [1/2]

◆ Q() [2/2]

◆ R() [1/2]

◆ R() [2/2]

◆ p()

◆ P()

◆ solve() [1/6]

◆ solve() [2/6]

◆ solve() [3/6]

◆ solve() [4/6]

◆ solve() [5/6]

◆ inv()

◆ solve() [6/6]