p?ggrqf

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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p?ggrqf

Computes the generalized RQ factorization.

Syntax

void psggrqf (MKL_INT *m , MKL_INT *p , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *taua , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , float *taub , float *work , MKL_INT *lwork , MKL_INT *info );

void pdggrqf (MKL_INT *m , MKL_INT *p , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *taua , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , double *taub , double *work , MKL_INT *lwork , MKL_INT *info );

void pcggrqf (MKL_INT *m , MKL_INT *p , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *taua , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex8 *taub , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );

void pzggrqf (MKL_INT *m , MKL_INT *p , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *taua , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex16 *taub , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );

Include Files

mkl_scalapack.h

Description

The p?ggrqffunction forms the generalized RQ factorization of an m-by-n matrix sub(A) = A(ia:ia+m-1, ja:ja+n-1) and a p-by-n matrix sub(B) = B(ib:ib+p-1, jb:jb+n-1):

sub(A) = R*Q, sub(B) = Z*T*Q,

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 or R₂₁ is upper triangular, and

where T₁₁ is upper triangular.

In particular, if sub(B) is square and nonsingular, the GRQ factorization of sub(A) and sub(B) implicitly gives the RQ factorization of sub (A)*inv(sub(B)):

sub(A)*inv(sub(B))= (R*inv(T))*Z'

where inv(sub(B)) denotes the inverse of the matrix sub(B), and Z' denotes the transpose (conjugate transpose) of matrix Z.

Input Parameters

m

(global) The number of rows in the distributed matrices sub (A) (m≥0).

p

The number of rows in the distributed matrix sub(B) (p≥0).

n

(global) The number of columns in the distributed matrices sub(A) and sub(B) (n≥0).

a

(local)

Pointer into the local memory to an array of size lld_a*LOCc(ja+n-1). Contains the local pieces of the m-by-n distributed matrix sub(A) to be factored.

ia, ja

(global) The row and column indices in the global matrix A indicating the first row and the first column of the submatrix A, respectively.

desca

(global and local) array of size dlen_. The array descriptor for the distributed matrix A.

b

(local)

Pointer into the local memory to an array of size lld_b*LOCc(jb+n-1).

Contains the local pieces of the p-by-n matrix sub(B) to be factored.

ib, jb

(global) The row and column indices in the global matrix B indicating the first row and the first column of the submatrix B, respectively.

descb

(global and local) array of size dlen_. The array descriptor for the distributed matrix B.

work

(local)

Workspace array of size of lwork.

lwork

(local or global)

Size of work, must be at least lwork≥max(mb_a*(mpa0+nqa0+mb_a), max((mb_a*(mb_a-1))/2, (ppb0+nqb0)*mb_a) + mb_a*mb_a, nb_b*(ppb0+nqb0+nb_b)), where

iroffa = mod(ia-1, mb_A),

icoffa = mod(ja-1, nb_a),

iarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),

iacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),

mpa0 = numroc (m+iroffa, mb_a, MYROW, iarow, NPROW),

nqa0 = numroc (m+icoffa, nb_a, MYCOL, iacol, NPCOL)

iroffb = mod(ib-1, mb_b),

icoffb = mod(jb-1, nb_b),

ibrow = indxg2p(ib, mb_b, MYROW, rsrc_b, NPROW ),

ibcol = indxg2p(jb, nb_b, MYCOL, csrc_b, NPCOL ),

ppb0 = numroc (p+iroffb, mb_b, MYROW, ibrow,NPROW),

nqb0 = numroc (n+icoffb, nb_b, MYCOL, ibcol,NPCOL)

NOTE:

mod(x,y) is the integer remainder of x/y.

and numroc, indxg2p are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the function blacs_gridinfo.

If lwork = -1, then lwork is global input and a workspace query is assumed; the function only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by pxerbla.

Output Parameters

a

On exit, if m≤n, the upper triangle of A(ia:ia+m-1, ja+n-m:ja+n-1) contains the m-by-m upper triangular matrix R; if m≥n, the elements on and above the (m-n)-th subdiagonal contain the m-by-n upper trapezoidal matrix R; the remaining elements, with the array taua, represent the orthogonal/unitary matrix Q as a product of min(n,m) elementary reflectors (see Application Notes below).

taua, taub

(local)

Arrays of size LOCr(ia+m-1)for taua and LOCc(jb+min(p,n)-1) for taub.

The array taua contains the scalar factors of the elementary reflectors which represent the orthogonal/unitary matrix Q. taua is tied to the distributed matrix A.(See Application Notes below).

The array taub contains the scalar factors of the elementary reflectors which represent the orthogonal/unitary matrix Z. taub is tied to the distributed matrix B. (See Application Notes below).

work[0]

On exit work[0] contains the minimum value of lwork required for optimum performance.

info

(global)

= 0: the execution is successful.

< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had an illegal value, then info = -(i*100+j); if the i-th argument is a scalar and had an illegal value, then info = -i.

Application Notes

The matrix Q is represented as a product of elementary reflectors

Q = H(ia)*H(ia+1)*...*H(ia+k-1),

where k= min(m,n).

Each H(i) has the form

H(i) = i - taua*v*v'

where taua is a real/complex scalar, and v is a real/complex vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in A(ia+m-k+i-1, ja:ja+n-k+i-2), and taua in taua[ia+m-k+i-2]. To form Q explicitly, use ScaLAPACK function p?orgrq/p?ungrq. To use Q to update another matrix, use ScaLAPACK function p?ormrq/p?unmrq.

The matrix Z is represented as a product of elementary reflectors

Z = H(jb)*H(jb+1)*...*H(jb+k-1), where k= min(p,n).

Each H(i) has the form

H(i) = i - taub*v*v'

where taub is a real/complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i)= 1; v(i+1:p) is stored on exit in B(ib+i:ib+p-1,jb+i-1), and taub in taub[jb+i-2]. To form Z explicitly, use ScaLAPACK function p?orgqr/p?ungqr. To use Z to update another matrix, use ScaLAPACK function p?ormqr/p?unmqr.

Parent topic: Orthogonal Factorizations: ScaLAPACK Computational Routines

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