Orthogonal Factorizations: LAPACK Computational Routines

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 6/24/2024

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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 m≥n, the QR factorization is given by

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 (m≥n). 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 m≤n,

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

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 geqr geqrfp	geqpf geqp3	orgqr ungqr	ormqr unmqr gemqr
general matrices, blocked QR factorization	geqrt			gemqrt
general matrices, RQ factorization	gerqf		orgrq ungrq	ormrq unmrq
general matrices, LQ factorization	gelqf gelq		orglq unglq	ormlq unmlq gemlq
general matrices, blocked LQ factorization	gelqt			gemlqt
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
triangular-pentagonal matrices, blocked LQ factorization	tplqt			tpmlqt

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Orthogonal Factorizations: LAPACK Computational Routines