 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ sggqrf()

 subroutine sggqrf ( integer N, integer M, integer P, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAUA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) TAUB, real, dimension( * ) WORK, integer LWORK, integer INFO )

SGGQRF

Purpose:
``` SGGQRF computes a generalized QR factorization of an N-by-M matrix A
and an N-by-P matrix B:

A = Q*R,        B = Q*T*Z,

where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:

if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
(  0  ) N-M                         N   M-N
M

where R11 is upper triangular, and

if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
P-N  N                           ( T21 ) P
P

where T12 or T21 is upper triangular.

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

inv(B)*A = Z**T*(inv(T)*R)

where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
transpose of the matrix Z.```
Parameters
 [in] N ``` N is INTEGER The number of rows of the matrices A and B. N >= 0.``` [in] M ``` M is INTEGER The number of columns of the matrix A. M >= 0.``` [in] P ``` P is INTEGER The number of columns of the matrix B. P >= 0.``` [in,out] A ``` A is REAL array, dimension (LDA,M) On entry, the N-by-M matrix A. On exit, the elements on and above the diagonal of the array 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 matrix Q as a product of min(N,M) elementary reflectors (see Further Details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] TAUA ``` TAUA is REAL array, dimension (min(N,M)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q (see Further Details).``` [in,out] B ``` B is REAL array, dimension (LDB,P) On entry, the N-by-P matrix B. On exit, 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 matrix Z as a product of elementary reflectors (see Further Details).``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] TAUB ``` TAUB is REAL array, dimension (min(N,P)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Z (see Further Details).``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N,M,P). For optimum performance 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 SORMQR. 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.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value.```
Further Details:
```  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 - taua * v * v**T

where taua is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine SORGQR.
To use Q to update another matrix, use LAPACK subroutine SORMQR.

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 - taub * v * v**T

where taub is a real scalar, and v is a real 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(n-k+i,1:p-k+i-1), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine SORGRQ.
To use Z to update another matrix, use LAPACK subroutine SORMRQ.```

Definition at line 213 of file sggqrf.f.

215 *
216 * -- LAPACK computational routine --
217 * -- LAPACK is a software package provided by Univ. of Tennessee, --
218 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
219 *
220 * .. Scalar Arguments ..
221  INTEGER INFO, LDA, LDB, LWORK, M, N, P
222 * ..
223 * .. Array Arguments ..
224  REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
225  \$ WORK( * )
226 * ..
227 *
228 * =====================================================================
229 *
230 * .. Local Scalars ..
231  LOGICAL LQUERY
232  INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
233 * ..
234 * .. External Subroutines ..
235  EXTERNAL sgeqrf, sgerqf, sormqr, xerbla
236 * ..
237 * .. External Functions ..
238  INTEGER ILAENV
239  EXTERNAL ilaenv
240 * ..
241 * .. Intrinsic Functions ..
242  INTRINSIC int, max, min
243 * ..
244 * .. Executable Statements ..
245 *
246 * Test the input parameters
247 *
248  info = 0
249  nb1 = ilaenv( 1, 'SGEQRF', ' ', n, m, -1, -1 )
250  nb2 = ilaenv( 1, 'SGERQF', ' ', n, p, -1, -1 )
251  nb3 = ilaenv( 1, 'SORMQR', ' ', n, m, p, -1 )
252  nb = max( nb1, nb2, nb3 )
253  lwkopt = max( n, m, p )*nb
254  work( 1 ) = lwkopt
255  lquery = ( lwork.EQ.-1 )
256  IF( n.LT.0 ) THEN
257  info = -1
258  ELSE IF( m.LT.0 ) THEN
259  info = -2
260  ELSE IF( p.LT.0 ) THEN
261  info = -3
262  ELSE IF( lda.LT.max( 1, n ) ) THEN
263  info = -5
264  ELSE IF( ldb.LT.max( 1, n ) ) THEN
265  info = -8
266  ELSE IF( lwork.LT.max( 1, n, m, p ) .AND. .NOT.lquery ) THEN
267  info = -11
268  END IF
269  IF( info.NE.0 ) THEN
270  CALL xerbla( 'SGGQRF', -info )
271  RETURN
272  ELSE IF( lquery ) THEN
273  RETURN
274  END IF
275 *
276 * QR factorization of N-by-M matrix A: A = Q*R
277 *
278  CALL sgeqrf( n, m, a, lda, taua, work, lwork, info )
279  lopt = work( 1 )
280 *
281 * Update B := Q**T*B.
282 *
283  CALL sormqr( 'Left', 'Transpose', n, p, min( n, m ), a, lda, taua,
284  \$ b, ldb, work, lwork, info )
285  lopt = max( lopt, int( work( 1 ) ) )
286 *
287 * RQ factorization of N-by-P matrix B: B = T*Z.
288 *
289  CALL sgerqf( n, p, b, ldb, taub, work, lwork, info )
290  work( 1 ) = max( lopt, int( work( 1 ) ) )
291 *
292  RETURN
293 *
294 * End of SGGQRF
295 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgerqf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGERQF
Definition: sgerqf.f:138
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:145
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:168
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