# p?ggqrf

Computes the generalized QR factorization.

## Syntax

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

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

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

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

## Description

The p?ggqrf routine forms the generalized QR factorization of an n-by-m matrix

sub(A) = A(ia:ia+n-1, ja:ja+m-1)

and an n-by-p matrix

sub(B) = B(ib:ib+n-1, jb:jb+p-1):

as

sub(A) = Q*R, sub(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:

If nm

or if n < m

where R11 is upper triangular, and

where T12 or T21 is an upper triangular matrix.

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

inv(sub(B))*sub(A) = ZH*(inv(T)*R)

## Input Parameters

n

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

m

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

p

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

a

(local)

REAL for psggqrf

DOUBLE PRECISION for pdggqrf

COMPLEX for pcggqrf

DOUBLE COMPLEX for pzggqrf.

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

ia, ja

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

b

(local)

REAL for psggqrf

DOUBLE PRECISION for pdggqrf

COMPLEX for pcggqrf

DOUBLE COMPLEX for pzggqrf.

Pointer into the local memory to an array of size `(lld_b,LOCc(jb+p-1))`. Contains the local pieces of the n-by-p matrix sub(B) to be factored.

ib, jb
(global) INTEGER. 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) INTEGER array of size dlen_. The array descriptor for the distributed matrix B.

work

(local)

REAL for psggqrf

DOUBLE PRECISION for pdggqrf

COMPLEX for pcggqrf

DOUBLE COMPLEX for pzggqrf.

Workspace array of size of lwork.

lwork

(local or global) INTEGER. Sze of work, must be at least

`lwork ≥ max(nb_a*(npa0+mqa0+nb_a), max((nb_a*(nb_a-1))/2, (pqb0+npb0)*nb_a)+nb_a*nb_a, mb_b*(npb0+pqb0+mb_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)`,

`npa0 = numroc (n+iroffa, mb_a, MYROW, iarow, NPROW)`,

`mqa0 = 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)`,

`npb0 = numroc (n+iroffa, mb_b, MYROW, Ibrow, NPROW)`,

`pqb0 = numroc(m+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 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, the elements on and above the diagonal of sub (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. (See Application Notes below).

taua, taub

(local)

REAL for psggqrf

DOUBLE PRECISION for pdggqrf

COMPLEX for pcggqrf

DOUBLE COMPLEX for pzggqrf.

Arrays of size `LOCc(ja+min(n,m)-1)` for taua and `LOCr(ib+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-th 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(ja)*H(ja+1)*...*H(ja+k-1),

where k `= min(n,m)`.

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(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(ia+i:ia+n-1, ja+i-1) , and taua in taua(ja+i-1).To form Q explicitly, use ScaLAPACK subroutine p?orgqr/p?ungqr. To use Q to update another matrix, use ScaLAPACK subroutine p?ormqr/p?unmqr.

The matrix Z is represented as a product of elementary reflectors

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

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(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in taub(ib+n-k+i-1). To form Z explicitly, use ScaLAPACK subroutine p?orgrq/p?ungrq. To use Z to update another matrix, use ScaLAPACK subroutine p?ormrq/p?unmrq.