QRMatrix (i.e. QR decomposition, QR factorisation or orthogonal-triangular decomposition) decomposes a scalar/complex matrix A into the following matrix product: More...

Public Types
enum	outputTypes : uint8_t { FULL_R = 1, FULL_QR = 2, REDUCED_R = 3 }

enum	storeMethods : uint8_t { IN_PLACE = 1, OUT_OF_PLACE = 2 }

enum	colPivoting : bool { FALSE = false, TRUE = true }

typedef MatrixType::cmptType	cmptType

typedef SquareMatrix< cmptType >	SMatrix

typedef RectangularMatrix< cmptType >	RMatrix

Public Member Functions
	QRMatrix ()

	QRMatrix (const outputTypes outputType, const storeMethods storeMethod=storeMethods::OUT_OF_PLACE, const colPivoting colPivot=colPivoting::FALSE)

	QRMatrix (MatrixType &A, const outputTypes outputType, const storeMethods storeMethod=storeMethods::OUT_OF_PLACE, const colPivoting colPivot=colPivoting::FALSE)

	QRMatrix (const MatrixType &A, const outputTypes outputType, const colPivoting colPivot=colPivoting::FALSE)

const MatrixType &	Q () const

const MatrixType &	R () const

const labelList &	orderP () const

SMatrix	P () const

void	decompose (MatrixType &A)

void	decompose (const MatrixType &A)

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

SMatrix	inv () const

RMatrix	backSubstitution (const SMatrix &A, const RMatrix &b)

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

Protected Member Functions
RMatrix	householderReflector (RMatrix u)

void	applyLeftReflector (MatrixType &A, const RMatrix &reflector, const label k=0, const label k1=0)

void	applyRightReflector (MatrixType &A, const RMatrix &reflector, const label k=0)

Detailed Description

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

QRMatrix (i.e. QR decomposition, QR factorisation or orthogonal-triangular decomposition) decomposes a scalar/complex matrix A into the following matrix product:

    A = Q*R,

where Q is a unitary similarity matrix, R is an upper triangular matrix.

Reference:

    QR decomposition:
        mathworld.wolfram.com/QRDecomposition.html (Retrieved:17-06-19)
        w.wiki/52X (Retrieved: 17-06-19)

    QR decomposition with Householder reflector (tag:M):
        Monahan, J. F. (2011).
        Numerical methods of statistics.
        Cambridge: Cambridge University Press.
        DOI:10.1017/CBO9780511977176

    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

    Operands of QR decomposition:
        mathworld.wolfram.com/UnitaryMatrix.html (Retrieved:13-06-19)
        mathworld.wolfram.com/UpperTriangularMatrix.html (Retrieved:16-06-19)
        mathworld.wolfram.com/PermutationMatrix.html (Retrieved:20-06-19)

    Back substitution:
        mathworld.wolfram.com/GaussianElimination.html (Retrieved:15-06-19)

Usage

Input types:

A can be a SquareMatrix<Type> or RectangularMatrix<Type>

Output types:

Q is always of the type of the matrix A
R is always of the type of the matrix A

Options for the output forms of QRMatrix (for an (m-by-n) input matrix A with k = min(m, n)):

outputTypes::FULL_R: computes only R (m-by-n)
outputTypes::FULL_QR: computes both R and Q (m-by-m)
outputTypes::REDUCED_R: computes only reduced R (k-by-n)

Options where to store R:

storeMethods::IN_PLACE: replaces input matrix content with R
storeMethods::OUT_OF_PLACE: creates new object of R

Options for the computation of column pivoting:

colPivoting::FALSE: switches off column pivoting
colPivoting::TRUE: switches on column pivoting

Direct solution of linear systems A x = b is possible by solve() alongside the following limitations:

A = a scalar square matrix
output type = outputTypes::FULL_QR
store method = storeMethods::IN_PLACE

Notes

QR decomposition is not unique if R is not positive diagonal R.
The option combination:
- outputTypes::REDUCED_R
- storeMethods::IN_PLACE will not modify the rows of input matrix A after its nth row.
Both FULL_R and REDUCED_R QR decompositions execute the same number of operations. Yet REDUCED_R QR decomposition returns only the first n rows of R if m > n for an input m-by-n matrix A.
For m <= n, FULL_R and REDUCED_R will produce the same matrices.

See also: Test-QRMatrix.C

Source files

Definition at line 148 of file QRMatrix.H.

Member Typedef Documentation

◆ cmptType

typedef MatrixType::cmptType cmptType

Definition at line 153 of file QRMatrix.H.

◆ SMatrix

typedef SquareMatrix<cmptType> SMatrix

Definition at line 154 of file QRMatrix.H.

◆ RMatrix

typedef RectangularMatrix<cmptType> RMatrix

Definition at line 155 of file QRMatrix.H.

Member Enumeration Documentation

◆ outputTypes

enum outputTypes : uint8_t

Enumerator
FULL_R	computes only `R`
FULL_QR	computes both `R` and `Q`
REDUCED_R	computes only reduced `R`

Definition at line 158 of file QRMatrix.H.

◆ storeMethods

enum storeMethods : uint8_t

Enumerator
IN_PLACE	replaces input matrix content with `R`
OUT_OF_PLACE	creates new object of `R`

Definition at line 166 of file QRMatrix.H.

◆ colPivoting

enum colPivoting : bool

Enumerator
FALSE	switches off column pivoting
TRUE	switches on column pivoting

Definition at line 173 of file QRMatrix.H.

Constructor & Destructor Documentation

◆ QRMatrix() [1/4]

QRMatrix

Definition at line 179 of file QRMatrix.C.

◆ QRMatrix() [2/4]

QRMatrix	(	const outputTypes	outputType,
		const storeMethods	storeMethod = `storeMethods::OUT_OF_PLACE`,
		const colPivoting	colPivot = `colPivoting::FALSE`
	)

Definition at line 189 of file QRMatrix.C.

◆ QRMatrix() [3/4]

QRMatrix	(	MatrixType &	A,
		const outputTypes	outputType,
		const storeMethods	storeMethod = `storeMethods::OUT_OF_PLACE`,
		const colPivoting	colPivot = `colPivoting::FALSE`
	)