 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ dggqrf()

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

DGGQRF

Download DGGQRF + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` DGGQRF 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DORMQR. 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.```
Date
December 2016
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 DORGQR.
To use Q to update another matrix, use LAPACK subroutine DORMQR.

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 DORGRQ.
To use Z to update another matrix, use LAPACK subroutine DORMRQ.```

Definition at line 217 of file dggqrf.f.

217 *
218 * -- LAPACK computational routine (version 3.7.0) --
219 * -- LAPACK is a software package provided by Univ. of Tennessee, --
220 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
221 * December 2016
222 *
223 * .. Scalar Arguments ..
224  INTEGER info, lda, ldb, lwork, m, n, p
225 * ..
226 * .. Array Arguments ..
227  DOUBLE PRECISION a( lda, * ), b( ldb, * ), taua( * ), taub( * ),
228  \$ work( * )
229 * ..
230 *
231 * =====================================================================
232 *
233 * .. Local Scalars ..
234  LOGICAL lquery
235  INTEGER lopt, lwkopt, nb, nb1, nb2, nb3
236 * ..
237 * .. External Subroutines ..
238  EXTERNAL dgeqrf, dgerqf, dormqr, xerbla
239 * ..
240 * .. External Functions ..
241  INTEGER ilaenv
242  EXTERNAL ilaenv
243 * ..
244 * .. Intrinsic Functions ..
245  INTRINSIC int, max, min
246 * ..
247 * .. Executable Statements ..
248 *
249 * Test the input parameters
250 *
251  info = 0
252  nb1 = ilaenv( 1, 'DGEQRF', ' ', n, m, -1, -1 )
253  nb2 = ilaenv( 1, 'DGERQF', ' ', n, p, -1, -1 )
254  nb3 = ilaenv( 1, 'DORMQR', ' ', n, m, p, -1 )
255  nb = max( nb1, nb2, nb3 )
256  lwkopt = max( n, m, p )*nb
257  work( 1 ) = lwkopt
258  lquery = ( lwork.EQ.-1 )
259  IF( n.LT.0 ) THEN
260  info = -1
261  ELSE IF( m.LT.0 ) THEN
262  info = -2
263  ELSE IF( p.LT.0 ) THEN
264  info = -3
265  ELSE IF( lda.LT.max( 1, n ) ) THEN
266  info = -5
267  ELSE IF( ldb.LT.max( 1, n ) ) THEN
268  info = -8
269  ELSE IF( lwork.LT.max( 1, n, m, p ) .AND. .NOT.lquery ) THEN
270  info = -11
271  END IF
272  IF( info.NE.0 ) THEN
273  CALL xerbla( 'DGGQRF', -info )
274  RETURN
275  ELSE IF( lquery ) THEN
276  RETURN
277  END IF
278 *
279 * QR factorization of N-by-M matrix A: A = Q*R
280 *
281  CALL dgeqrf( n, m, a, lda, taua, work, lwork, info )
282  lopt = work( 1 )
283 *
284 * Update B := Q**T*B.
285 *
286  CALL dormqr( 'Left', 'Transpose', n, p, min( n, m ), a, lda, taua,
287  \$ b, ldb, work, lwork, info )
288  lopt = max( lopt, int( work( 1 ) ) )
289 *
290 * RQ factorization of N-by-P matrix B: B = T*Z.
291 *
292  CALL dgerqf( n, p, b, ldb, taub, work, lwork, info )
293  work( 1 ) = max( lopt, int( work( 1 ) ) )
294 *
295  RETURN
296 *
297 * End of DGGQRF
298 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: tstiee.f:83
subroutine dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:169
subroutine dgerqf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGERQF
Definition: dgerqf.f:140
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:138
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