Annotation of rpl/lapack/lapack/dggrqf.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DGGRQF
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DGGRQF + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggrqf.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggrqf.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggrqf.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
! 22: * LWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * INTEGER INFO, LDA, LDB, LWORK, M, N, P
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
! 29: * $ WORK( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> DGGRQF computes a generalized RQ factorization of an M-by-N matrix A
! 39: *> and a P-by-N matrix B:
! 40: *>
! 41: *> A = R*Q, B = Z*T*Q,
! 42: *>
! 43: *> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
! 44: *> matrix, and R and T assume one of the forms:
! 45: *>
! 46: *> if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
! 47: *> N-M M ( R21 ) N
! 48: *> N
! 49: *>
! 50: *> where R12 or R21 is upper triangular, and
! 51: *>
! 52: *> if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
! 53: *> ( 0 ) P-N P N-P
! 54: *> N
! 55: *>
! 56: *> where T11 is upper triangular.
! 57: *>
! 58: *> In particular, if B is square and nonsingular, the GRQ factorization
! 59: *> of A and B implicitly gives the RQ factorization of A*inv(B):
! 60: *>
! 61: *> A*inv(B) = (R*inv(T))*Z**T
! 62: *>
! 63: *> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
! 64: *> transpose of the matrix Z.
! 65: *> \endverbatim
! 66: *
! 67: * Arguments:
! 68: * ==========
! 69: *
! 70: *> \param[in] M
! 71: *> \verbatim
! 72: *> M is INTEGER
! 73: *> The number of rows of the matrix A. M >= 0.
! 74: *> \endverbatim
! 75: *>
! 76: *> \param[in] P
! 77: *> \verbatim
! 78: *> P is INTEGER
! 79: *> The number of rows of the matrix B. P >= 0.
! 80: *> \endverbatim
! 81: *>
! 82: *> \param[in] N
! 83: *> \verbatim
! 84: *> N is INTEGER
! 85: *> The number of columns of the matrices A and B. N >= 0.
! 86: *> \endverbatim
! 87: *>
! 88: *> \param[in,out] A
! 89: *> \verbatim
! 90: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 91: *> On entry, the M-by-N matrix A.
! 92: *> On exit, if M <= N, the upper triangle of the subarray
! 93: *> A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
! 94: *> if M > N, the elements on and above the (M-N)-th subdiagonal
! 95: *> contain the M-by-N upper trapezoidal matrix R; the remaining
! 96: *> elements, with the array TAUA, represent the orthogonal
! 97: *> matrix Q as a product of elementary reflectors (see Further
! 98: *> Details).
! 99: *> \endverbatim
! 100: *>
! 101: *> \param[in] LDA
! 102: *> \verbatim
! 103: *> LDA is INTEGER
! 104: *> The leading dimension of the array A. LDA >= max(1,M).
! 105: *> \endverbatim
! 106: *>
! 107: *> \param[out] TAUA
! 108: *> \verbatim
! 109: *> TAUA is DOUBLE PRECISION array, dimension (min(M,N))
! 110: *> The scalar factors of the elementary reflectors which
! 111: *> represent the orthogonal matrix Q (see Further Details).
! 112: *> \endverbatim
! 113: *>
! 114: *> \param[in,out] B
! 115: *> \verbatim
! 116: *> B is DOUBLE PRECISION array, dimension (LDB,N)
! 117: *> On entry, the P-by-N matrix B.
! 118: *> On exit, the elements on and above the diagonal of the array
! 119: *> contain the min(P,N)-by-N upper trapezoidal matrix T (T is
! 120: *> upper triangular if P >= N); the elements below the diagonal,
! 121: *> with the array TAUB, represent the orthogonal matrix Z as a
! 122: *> product of elementary reflectors (see Further Details).
! 123: *> \endverbatim
! 124: *>
! 125: *> \param[in] LDB
! 126: *> \verbatim
! 127: *> LDB is INTEGER
! 128: *> The leading dimension of the array B. LDB >= max(1,P).
! 129: *> \endverbatim
! 130: *>
! 131: *> \param[out] TAUB
! 132: *> \verbatim
! 133: *> TAUB is DOUBLE PRECISION array, dimension (min(P,N))
! 134: *> The scalar factors of the elementary reflectors which
! 135: *> represent the orthogonal matrix Z (see Further Details).
! 136: *> \endverbatim
! 137: *>
! 138: *> \param[out] WORK
! 139: *> \verbatim
! 140: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 141: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 142: *> \endverbatim
! 143: *>
! 144: *> \param[in] LWORK
! 145: *> \verbatim
! 146: *> LWORK is INTEGER
! 147: *> The dimension of the array WORK. LWORK >= max(1,N,M,P).
! 148: *> For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
! 149: *> where NB1 is the optimal blocksize for the RQ factorization
! 150: *> of an M-by-N matrix, NB2 is the optimal blocksize for the
! 151: *> QR factorization of a P-by-N matrix, and NB3 is the optimal
! 152: *> blocksize for a call of DORMRQ.
! 153: *>
! 154: *> If LWORK = -1, then a workspace query is assumed; the routine
! 155: *> only calculates the optimal size of the WORK array, returns
! 156: *> this value as the first entry of the WORK array, and no error
! 157: *> message related to LWORK is issued by XERBLA.
! 158: *> \endverbatim
! 159: *>
! 160: *> \param[out] INFO
! 161: *> \verbatim
! 162: *> INFO is INTEGER
! 163: *> = 0: successful exit
! 164: *> < 0: if INF0= -i, the i-th argument had an illegal value.
! 165: *> \endverbatim
! 166: *
! 167: * Authors:
! 168: * ========
! 169: *
! 170: *> \author Univ. of Tennessee
! 171: *> \author Univ. of California Berkeley
! 172: *> \author Univ. of Colorado Denver
! 173: *> \author NAG Ltd.
! 174: *
! 175: *> \date November 2011
! 176: *
! 177: *> \ingroup doubleOTHERcomputational
! 178: *
! 179: *> \par Further Details:
! 180: * =====================
! 181: *>
! 182: *> \verbatim
! 183: *>
! 184: *> The matrix Q is represented as a product of elementary reflectors
! 185: *>
! 186: *> Q = H(1) H(2) . . . H(k), where k = min(m,n).
! 187: *>
! 188: *> Each H(i) has the form
! 189: *>
! 190: *> H(i) = I - taua * v * v**T
! 191: *>
! 192: *> where taua is a real scalar, and v is a real vector with
! 193: *> v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
! 194: *> A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
! 195: *> To form Q explicitly, use LAPACK subroutine DORGRQ.
! 196: *> To use Q to update another matrix, use LAPACK subroutine DORMRQ.
! 197: *>
! 198: *> The matrix Z is represented as a product of elementary reflectors
! 199: *>
! 200: *> Z = H(1) H(2) . . . H(k), where k = min(p,n).
! 201: *>
! 202: *> Each H(i) has the form
! 203: *>
! 204: *> H(i) = I - taub * v * v**T
! 205: *>
! 206: *> where taub is a real scalar, and v is a real vector with
! 207: *> v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
! 208: *> and taub in TAUB(i).
! 209: *> To form Z explicitly, use LAPACK subroutine DORGQR.
! 210: *> To use Z to update another matrix, use LAPACK subroutine DORMQR.
! 211: *> \endverbatim
! 212: *>
! 213: * =====================================================================
1.1 bertrand 214: SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
215: $ LWORK, INFO )
216: *
1.9 ! bertrand 217: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 218: * -- LAPACK is a software package provided by Univ. of Tennessee, --
219: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 220: * November 2011
1.1 bertrand 221: *
222: * .. Scalar Arguments ..
223: INTEGER INFO, LDA, LDB, LWORK, M, N, P
224: * ..
225: * .. Array Arguments ..
226: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
227: $ WORK( * )
228: * ..
229: *
230: * =====================================================================
231: *
232: * .. Local Scalars ..
233: LOGICAL LQUERY
234: INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
235: * ..
236: * .. External Subroutines ..
237: EXTERNAL DGEQRF, DGERQF, DORMRQ, XERBLA
238: * ..
239: * .. External Functions ..
240: INTEGER ILAENV
241: EXTERNAL ILAENV
242: * ..
243: * .. Intrinsic Functions ..
244: INTRINSIC INT, MAX, MIN
245: * ..
246: * .. Executable Statements ..
247: *
248: * Test the input parameters
249: *
250: INFO = 0
251: NB1 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
252: NB2 = ILAENV( 1, 'DGEQRF', ' ', P, N, -1, -1 )
253: NB3 = ILAENV( 1, 'DORMRQ', ' ', M, N, P, -1 )
254: NB = MAX( NB1, NB2, NB3 )
255: LWKOPT = MAX( N, M, P )*NB
256: WORK( 1 ) = LWKOPT
257: LQUERY = ( LWORK.EQ.-1 )
258: IF( M.LT.0 ) THEN
259: INFO = -1
260: ELSE IF( P.LT.0 ) THEN
261: INFO = -2
262: ELSE IF( N.LT.0 ) THEN
263: INFO = -3
264: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
265: INFO = -5
266: ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
267: INFO = -8
268: ELSE IF( LWORK.LT.MAX( 1, M, P, N ) .AND. .NOT.LQUERY ) THEN
269: INFO = -11
270: END IF
271: IF( INFO.NE.0 ) THEN
272: CALL XERBLA( 'DGGRQF', -INFO )
273: RETURN
274: ELSE IF( LQUERY ) THEN
275: RETURN
276: END IF
277: *
278: * RQ factorization of M-by-N matrix A: A = R*Q
279: *
280: CALL DGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO )
281: LOPT = WORK( 1 )
282: *
1.8 bertrand 283: * Update B := B*Q**T
1.1 bertrand 284: *
285: CALL DORMRQ( 'Right', 'Transpose', P, N, MIN( M, N ),
286: $ A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK,
287: $ LWORK, INFO )
288: LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
289: *
290: * QR factorization of P-by-N matrix B: B = Z*T
291: *
292: CALL DGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO )
293: WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
294: *
295: RETURN
296: *
297: * End of DGGRQF
298: *
299: END
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