Developer Reference for Intel® oneAPI Math Kernel Library for C
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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.
Matrix type, factorization |
Factorize without pivoting |
Factorize with pivoting |
Generate matrix Q |
Apply matrix Q |
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general matrices, QR factorization |
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general matrices, blocked QR factorization |
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general matrices, RQ factorization |
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general matrices, LQ factorization |
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general matrices, QL factorization |
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trapezoidal matrices, RZ factorization |
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pair of matrices, generalized QR factorization |
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pair of matrices, generalized RQ factorization |
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triangular-pentagonal matrices, blocked QR factorization |
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