# p?ggrqf

Computes the generalized RQ factorization.

## Syntax

call psggrqf(m, p, n, a, ia, ja, desca, taua, b, ib, jb, descb, taub, work, lwork, info)

call pdggrqf(m, p, n, a, ia, ja, desca, taua, b, ib, jb, descb, taub, work, lwork, info)

call pcggrqf(m, p, n, a, ia, ja, desca, taua, b, ib, jb, descb, taub, work, lwork, info)

call pzggrqf(m, p, n, a, ia, ja, desca, taua, b, ib, jb, descb, taub, work, lwork, info)

## Description

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

or

where R11 or R21 is upper triangular, and

or

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 (conjygate transpose) of matrix Z.

## Input Parameters

m

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

p

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

n

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

a

  (local)

  REAL for psggrqf

  DOUBLE PRECISION for pdggrqf

  COMPLEX for pcggrqf

  DOUBLE COMPLEX for pzggrqf.

  Pointer into the local memory to an array of dimension `(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) INTEGER. The row and column indices in the global array a indicating the first row and the first column of the submatrix A, respectively.

desca

  (global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix A.

b

  (local)

  REAL for psggrqf

  DOUBLE PRECISION for pdggrqf

  COMPLEX for pcggrqf

  DOUBLE COMPLEX for pzggrqf.

  Pointer into the local memory to an array of dimension `(lld_b, LOCc(jb+n-1))`.

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

ib, jb

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

descb

  (global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix B.

work

  (local)

  REAL for psggrqf

  DOUBLE PRECISION for pdggrqf

  COMPLEX for pcggrqf

  DOUBLE COMPLEX for pzggrqf.

  Workspace array of dimension of lwork.

lwork

  (local or global) INTEGER.

  Dimension 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)`

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

  If `lwork = -1`, then lwork is global input and a workspace query is assumed; the routine 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)

  REAL for psggqrf

  DOUBLE PRECISION for pdggqrf

  COMPLEX for pcggqrf

  DOUBLE COMPLEX for pzggqrf.

  Arrays, DIMENSION `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(1)`

  On exit `work(1)` contains the minimum value of lwork required for optimum performance.

info

  (global) INTEGER.

  `= 0`: the execution is successful.

  `< 0`: if the i-th argument is an array and the j-entry 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-1)`. To form Q explicitly, use ScaLAPACK subroutine p?orgrq/p?ungrq. To use Q to update another matrix, use ScaLAPACK subroutine 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-1)`. To form Z explicitly, use ScaLAPACK subroutine p?orgqr/p?ungqr. To use Z to update another matrix, use ScaLAPACK subroutine p?ormqr/p?unmqr.

Parent topic: Orthogonal Factorizations