1: *> \brief \b ZGGQRF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGGQRF( N, M, P, 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: * COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
29: * $ WORK( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZGGQRF computes a generalized QR factorization of an N-by-M matrix A
39: *> and an N-by-P matrix B:
40: *>
41: *> A = Q*R, B = Q*T*Z,
42: *>
43: *> where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix,
44: *> and R and T assume one of the forms:
45: *>
46: *> if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
47: *> ( 0 ) N-M N M-N
48: *> M
49: *>
50: *> where R11 is upper triangular, and
51: *>
52: *> if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
53: *> P-N N ( T21 ) P
54: *> P
55: *>
56: *> where T12 or T21 is upper triangular.
57: *>
58: *> In particular, if B is square and nonsingular, the GQR factorization
59: *> of A and B implicitly gives the QR factorization of inv(B)*A:
60: *>
61: *> inv(B)*A = Z**H * (inv(T)*R)
62: *>
63: *> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
64: *> conjugate transpose of matrix Z.
65: *> \endverbatim
66: *
67: * Arguments:
68: * ==========
69: *
70: *> \param[in] N
71: *> \verbatim
72: *> N is INTEGER
73: *> The number of rows of the matrices A and B. N >= 0.
74: *> \endverbatim
75: *>
76: *> \param[in] M
77: *> \verbatim
78: *> M is INTEGER
79: *> The number of columns of the matrix A. M >= 0.
80: *> \endverbatim
81: *>
82: *> \param[in] P
83: *> \verbatim
84: *> P is INTEGER
85: *> The number of columns of the matrix B. P >= 0.
86: *> \endverbatim
87: *>
88: *> \param[in,out] A
89: *> \verbatim
90: *> A is COMPLEX*16 array, dimension (LDA,M)
91: *> On entry, the N-by-M matrix A.
92: *> On exit, the elements on and above the diagonal of the array
93: *> contain the min(N,M)-by-M upper trapezoidal matrix R (R is
94: *> upper triangular if N >= M); the elements below the diagonal,
95: *> with the array TAUA, represent the unitary matrix Q as a
96: *> product of min(N,M) elementary reflectors (see Further
97: *> Details).
98: *> \endverbatim
99: *>
100: *> \param[in] LDA
101: *> \verbatim
102: *> LDA is INTEGER
103: *> The leading dimension of the array A. LDA >= max(1,N).
104: *> \endverbatim
105: *>
106: *> \param[out] TAUA
107: *> \verbatim
108: *> TAUA is COMPLEX*16 array, dimension (min(N,M))
109: *> The scalar factors of the elementary reflectors which
110: *> represent the unitary matrix Q (see Further Details).
111: *> \endverbatim
112: *>
113: *> \param[in,out] B
114: *> \verbatim
115: *> B is COMPLEX*16 array, dimension (LDB,P)
116: *> On entry, the N-by-P matrix B.
117: *> On exit, if N <= P, the upper triangle of the subarray
118: *> B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
119: *> if N > P, the elements on and above the (N-P)-th subdiagonal
120: *> contain the N-by-P upper trapezoidal matrix T; the remaining
121: *> elements, with the array TAUB, represent the unitary
122: *> matrix Z as a product of elementary reflectors (see Further
123: *> Details).
124: *> \endverbatim
125: *>
126: *> \param[in] LDB
127: *> \verbatim
128: *> LDB is INTEGER
129: *> The leading dimension of the array B. LDB >= max(1,N).
130: *> \endverbatim
131: *>
132: *> \param[out] TAUB
133: *> \verbatim
134: *> TAUB is COMPLEX*16 array, dimension (min(N,P))
135: *> The scalar factors of the elementary reflectors which
136: *> represent the unitary matrix Z (see Further Details).
137: *> \endverbatim
138: *>
139: *> \param[out] WORK
140: *> \verbatim
141: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
142: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
143: *> \endverbatim
144: *>
145: *> \param[in] LWORK
146: *> \verbatim
147: *> LWORK is INTEGER
148: *> The dimension of the array WORK. LWORK >= max(1,N,M,P).
149: *> For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
150: *> where NB1 is the optimal blocksize for the QR factorization
151: *> of an N-by-M matrix, NB2 is the optimal blocksize for the
152: *> RQ factorization of an N-by-P matrix, and NB3 is the optimal
153: *> blocksize for a call of ZUNMQR.
154: *>
155: *> If LWORK = -1, then a workspace query is assumed; the routine
156: *> only calculates the optimal size of the WORK array, returns
157: *> this value as the first entry of the WORK array, and no error
158: *> message related to LWORK is issued by XERBLA.
159: *> \endverbatim
160: *>
161: *> \param[out] INFO
162: *> \verbatim
163: *> INFO is INTEGER
164: *> = 0: successful exit
165: *> < 0: if INFO = -i, the i-th argument had an illegal value.
166: *> \endverbatim
167: *
168: * Authors:
169: * ========
170: *
171: *> \author Univ. of Tennessee
172: *> \author Univ. of California Berkeley
173: *> \author Univ. of Colorado Denver
174: *> \author NAG Ltd.
175: *
176: *> \ingroup complex16OTHERcomputational
177: *
178: *> \par Further Details:
179: * =====================
180: *>
181: *> \verbatim
182: *>
183: *> The matrix Q is represented as a product of elementary reflectors
184: *>
185: *> Q = H(1) H(2) . . . H(k), where k = min(n,m).
186: *>
187: *> Each H(i) has the form
188: *>
189: *> H(i) = I - taua * v * v**H
190: *>
191: *> where taua is a complex scalar, and v is a complex vector with
192: *> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
193: *> and taua in TAUA(i).
194: *> To form Q explicitly, use LAPACK subroutine ZUNGQR.
195: *> To use Q to update another matrix, use LAPACK subroutine ZUNMQR.
196: *>
197: *> The matrix Z is represented as a product of elementary reflectors
198: *>
199: *> Z = H(1) H(2) . . . H(k), where k = min(n,p).
200: *>
201: *> Each H(i) has the form
202: *>
203: *> H(i) = I - taub * v * v**H
204: *>
205: *> where taub is a complex scalar, and v is a complex vector with
206: *> v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
207: *> B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
208: *> To form Z explicitly, use LAPACK subroutine ZUNGRQ.
209: *> To use Z to update another matrix, use LAPACK subroutine ZUNMRQ.
210: *> \endverbatim
211: *>
212: * =====================================================================
213: SUBROUTINE ZGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
214: $ LWORK, INFO )
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: COMPLEX*16 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 XERBLA, ZGEQRF, ZGERQF, ZUNMQR
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, 'ZGEQRF', ' ', N, M, -1, -1 )
250: NB2 = ILAENV( 1, 'ZGERQF', ' ', N, P, -1, -1 )
251: NB3 = ILAENV( 1, 'ZUNMQR', ' ', 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( 'ZGGQRF', -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 ZGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO )
279: LOPT = INT( WORK( 1 ) )
280: *
281: * Update B := Q**H*B.
282: *
283: CALL ZUNMQR( 'Left', 'Conjugate Transpose', N, P, MIN( N, M ), A,
284: $ LDA, TAUA, 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 ZGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO )
290: WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
291: *
292: RETURN
293: *
294: * End of ZGGQRF
295: *
296: END
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