?ggqrf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 6/24/2024

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?ggqrf

Computes the generalized QR factorization of two matrices.

Syntax

call sggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)

call dggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)

call cggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)

call zggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)

call ggqrf(a, b [,taua] [,taub] [,info])

Include Files

mkl.fi, lapack.f90

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

n

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

m

INTEGER. The number of columns in A (m≥ 0).

p

INTEGER. The number of columns in B (p≥ 0).

a, b, work

REAL for sggqrf

DOUBLE PRECISION for dggqrf

COMPLEX for cggqrf

DOUBLE COMPLEX for zggqrf.

Arrays: a(lda,*) contains the matrix A.

The second dimension of a must be at least max(1, m).

b(ldb,*) contains the matrix B.

The second dimension of b must be at least max(1, p).

work is a workspace array, its dimension max(1, lwork).

lda

INTEGER. The leading dimension of a; at least max(1, n).

ldb

INTEGER. The leading dimension of b; at least max(1, n).

lwork

INTEGER. The size of the work array; must be at least max(1, n, m, p).

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

See Application Notes for the suggested value of lwork.

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

REAL for sggqrf

DOUBLE PRECISION for dggqrf

COMPLEX for cggqrf

DOUBLE COMPLEX for zggqrf.

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.

work(1)

If info = 0, on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

INTEGER.

If info = 0, the execution is successful.

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

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine ggqrf interface are the following:

a: Holds the matrix A of size (n,m).
b: Holds the matrix B of size (n,p).
taua: Holds the vector of length min(n,m).
taub: Holds the vector of length min(n,p).

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(i+1:n, i) and τ_a is stored in taua(i)

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:p-k+i-1) and τ_b is stored in taub(i).

For better performance, try using lwork≥ max(n,m, p)*max(nb1,nb2,nb3), where nb1 is the optimal blocksize for the QR factorization of an n-by-m matrix, nb2 is the optimal blocksize for the RQ factorization of an n-by-p matrix, and nb3 is the optimal blocksize for a call of ormqr/unmqr.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.

If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?ggqrf

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes