Orthogonal Factorizations: LAPACK Computational Routines

This topic describes the LAPACK routines for the QR (RQ) and LQ (QL) factorization of matrices. Routines for the RZ factorization as well as for generalized QR and RQ factorizations are also included.

QR Factorization. Assume that A is an m-by-n matrix to be factored.

If mn, the QR factorization is given by

Equation

where R is an n-by-n upper triangular matrix with real diagonal elements, and Q is an m-by-m orthogonal (or unitary) matrix.

You can use the QR factorization for solving the following least squares problem: minimize ||Ax - b||2 where A is a full-rank m-by-n matrix (mn). After factoring the matrix, compute the solution x by solving Rx = (Q1)Tb.

If m < n, the QR factorization is given by

A = QR = Q(R1R2)

where R is trapezoidal, R1 is upper triangular and R2 is rectangular.

Q is represented as a product of min(m, n) elementary reflectors. Routines are provided to work with Q in this representation.

LQ Factorization LQ factorization of an m-by-n matrix A is as follows. If mn,


Equation

where L is an m-by-m lower triangular matrix with real diagonal elements, and Q is an n-by-n orthogonal (or unitary) matrix.

If m > n, the LQ factorization is


Equation

where L1 is an n-by-n lower triangular matrix, L2 is rectangular, and Q is an n-by-n orthogonal (or unitary) matrix.

You can use the LQ factorization to find the minimum-norm solution of an underdetermined system of linear equations Ax = b where A is an m-by-n matrix of rank m (m < n). After factoring the matrix, compute the solution vector x as follows: solve Ly = b for y, and then compute x = (Q1)Hy.

Table "Computational Routines for Orthogonal Factorization" lists LAPACK routines that perform orthogonal factorization of matrices.

Computational Routines for Orthogonal Factorization

Matrix type, factorization

Factorize without pivoting

Factorize with pivoting

Generate matrix Q

Apply matrix Q

general matrices, QR factorization

geqrf

geqrfp

geqpf

geqp3

orgqr 

ungqr

ormqr 

unmqr

general matrices, blocked QR factorization

geqrt

 

 

gemqrt

general matrices, RQ factorization

gerqf

 

orgrq 

ungrq

ormrq 

unmrq

general matrices, LQ factorization

gelqf

 

orglq 

unglq

ormlq 

unmlq

general matrices, QL factorization

geqlf

 

orgql 

ungql

ormql 

unmql

trapezoidal matrices, RZ factorization

tzrzf

 

 

ormrz 

unmrz

pair of matrices, generalized QR factorization

ggqrf

 

 

 

pair of matrices, generalized RQ factorization

ggrqf

 

 

 

triangular-pentagonal matrices, blocked QR factorization

tpqrt

 

 

tpmqrt

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