?ggqrf

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 6/24/2024

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?ggqrf

Computes the generalized QR factorization of two matrices.

Syntax

lapack_int LAPACKE_sggqrf (int matrix_layout, lapack_int n, lapack_int m, lapack_int p, float* a, lapack_int lda, float* taua, float* b, lapack_int ldb, float* taub);

lapack_int LAPACKE_dggqrf (int matrix_layout, lapack_int n, lapack_int m, lapack_int p, double* a, lapack_int lda, double* taua, double* b, lapack_int ldb, double* taub);

lapack_int LAPACKE_cggqrf (int matrix_layout, lapack_int n, lapack_int m, lapack_int p, lapack_complex_float* a, lapack_int lda, lapack_complex_float* taua, lapack_complex_float* b, lapack_int ldb, lapack_complex_float* taub);

lapack_int LAPACKE_zggqrf (int matrix_layout, lapack_int n, lapack_int m, lapack_int p, lapack_complex_double* a, lapack_int lda, lapack_complex_double* taua, lapack_complex_double* b, lapack_int ldb, lapack_complex_double* taub);

Include Files

mkl.h

Description

The routine forms the generalized QR factorization of an n-by-m matrix A and an n-by-p matrix B as A = Q*R, B = Q*T*Z, where Q is an n-by-n orthogonal/unitary matrix, Z is a p-by-p orthogonal/unitary matrix, and R and T assume one of the forms:

where R₁₁ is upper triangular, and

where T₁₂ or T₂₁ is a p-by-p upper triangular matrix.

In particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of B^-1A as:

B^-1*A = Z^T*(T^-1*R) (for real flavors) or B^-1*A = Z^H*(T^-1*R) (for complex flavors).

Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

n

The number of rows of the matrices A and B (n≥ 0).

m

The number of columns in A (m≥ 0).

p

The number of columns in B (p≥ 0).

a, b

Array a of size max(1, lda*m) for column major layout and max(1, lda*n) for row major layout contains the matrix A.

Array b of size max(1, ldb*p) for column major layout and max(1, ldb*n) for row major layout contains the matrix B.

lda

The leading dimension of a; at least max(1, n) for column major layout and at least max(1, m) for row major layout.

ldb

The leading dimension of b; at least max(1, n) for column major layout and at least max(1, p) for row major layout.

Output Parameters

a, b

Overwritten by the factorization data as follows:

on exit, the elements on and above the diagonal of the array a contain the min(n,m)-by-m upper trapezoidal matrix R (R is upper triangular if n≥m);the elements below the diagonal, with the array taua, represent the orthogonal/unitary matrix Q as a product of min(n,m) elementary reflectors ;

if n≤p, the upper triangle of the subarray b(1:n, p-n+1:p ) contains the n-by-n upper triangular matrix T;

if n > p, the elements on and above the (n-p)th subdiagonal contain the n-by-p upper trapezoidal matrix T; the remaining elements, with the array taub, represent the orthogonal/unitary matrix Z as a product of elementary reflectors.

taua, taub

Arrays, size at least max (1, min(n, m)) for taua and at least max (1, min(n, p)) for taub. The array taua contains the scalar factors of the elementary reflectors which represent the orthogonal/unitary matrix Q.

The array taub contains the scalar factors of the elementary reflectors which represent the orthogonal/unitary matrix Z.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

Application Notes

The matrix Q is represented as a product of elementary reflectors

Q = H(1)H(2)...H(k), where k = min(n,m).

Each H(i) has the form

H(i) = I - τ_a*v*v^T for real flavors, or

H(i) = I - τ_a*v*v^H for complex flavors,

where τ_a is a real/complex scalar, and v is a real/complex vector with v_j = 0 for 1 ≤j≤i - 1, v_i = 1.

On exit, fori + 1 ≤j≤n, v_j is stored in a[(j - 1) + (i - 1)*lda] for column major layout and in a[(j - 1)*lda + (i - 1)] for row major layout and τ_a is stored in taua[i - 1]

The matrix Z is represented as a product of elementary reflectors

Z = H(1)H(2)...H(k), where k = min(n,p).

Each H(i) has the form

H(i) = I - τ_b*v*v^T for real flavors, or

H(i) = I - τ_b*v*v^H for complex flavors,

where τ_b is a real/complex scalar, and v is a real/complex vector with v_{p - k + 1} = 1, v_j = 0 for p - k + 1 ≤j≤p - 1, .

On exit, for 1 ≤j≤p - k + i - 1, v_j is stored in b[(n - k + i - 1) + (j - 1)*ldb] for column major layout and in b[(n - k + i - 1)*ldb + (j - 1)] for row major layout and τ_b is stored in taub[i - 1].

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?ggqrf

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

Application Notes