1: *> \brief \b ZLATSQR
2: *
3: * Definition:
4: * ===========
5: *
6: * SUBROUTINE ZLATSQR( M, N, MB, NB, A, LDA, T, LDT, WORK,
7: * LWORK, INFO)
8: *
9: * .. Scalar Arguments ..
10: * INTEGER INFO, LDA, M, N, MB, NB, LDT, LWORK
11: * ..
12: * .. Array Arguments ..
13: * COMPLEX*16 A( LDA, * ), T( LDT, * ), WORK( * )
14: * ..
15: *
16: *
17: *> \par Purpose:
18: * =============
19: *>
20: *> \verbatim
21: *>
22: *> ZLATSQR computes a blocked Tall-Skinny QR factorization of
23: *> a complex M-by-N matrix A for M >= N:
24: *>
25: *> A = Q * ( R ),
26: *> ( 0 )
27: *>
28: *> where:
29: *>
30: *> Q is a M-by-M orthogonal matrix, stored on exit in an implicit
31: *> form in the elements below the diagonal of the array A and in
32: *> the elements of the array T;
33: *>
34: *> R is an upper-triangular N-by-N matrix, stored on exit in
35: *> the elements on and above the diagonal of the array A.
36: *>
37: *> 0 is a (M-N)-by-N zero matrix, and is not stored.
38: *>
39: *> \endverbatim
40: *
41: * Arguments:
42: * ==========
43: *
44: *> \param[in] M
45: *> \verbatim
46: *> M is INTEGER
47: *> The number of rows of the matrix A. M >= 0.
48: *> \endverbatim
49: *>
50: *> \param[in] N
51: *> \verbatim
52: *> N is INTEGER
53: *> The number of columns of the matrix A. M >= N >= 0.
54: *> \endverbatim
55: *>
56: *> \param[in] MB
57: *> \verbatim
58: *> MB is INTEGER
59: *> The row block size to be used in the blocked QR.
60: *> MB > N.
61: *> \endverbatim
62: *>
63: *> \param[in] NB
64: *> \verbatim
65: *> NB is INTEGER
66: *> The column block size to be used in the blocked QR.
67: *> N >= NB >= 1.
68: *> \endverbatim
69: *>
70: *> \param[in,out] A
71: *> \verbatim
72: *> A is COMPLEX*16 array, dimension (LDA,N)
73: *> On entry, the M-by-N matrix A.
74: *> On exit, the elements on and above the diagonal
75: *> of the array contain the N-by-N upper triangular matrix R;
76: *> the elements below the diagonal represent Q by the columns
77: *> of blocked V (see Further Details).
78: *> \endverbatim
79: *>
80: *> \param[in] LDA
81: *> \verbatim
82: *> LDA is INTEGER
83: *> The leading dimension of the array A. LDA >= max(1,M).
84: *> \endverbatim
85: *>
86: *> \param[out] T
87: *> \verbatim
88: *> T is COMPLEX*16 array,
89: *> dimension (LDT, N * Number_of_row_blocks)
90: *> where Number_of_row_blocks = CEIL((M-N)/(MB-N))
91: *> The blocked upper triangular block reflectors stored in compact form
92: *> as a sequence of upper triangular blocks.
93: *> See Further Details below.
94: *> \endverbatim
95: *>
96: *> \param[in] LDT
97: *> \verbatim
98: *> LDT is INTEGER
99: *> The leading dimension of the array T. LDT >= NB.
100: *> \endverbatim
101: *>
102: *> \param[out] WORK
103: *> \verbatim
104: *> (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))
105: *> \endverbatim
106: *>
107: *> \param[in] LWORK
108: *> \verbatim
109: *> The dimension of the array WORK. LWORK >= NB*N.
110: *> If LWORK = -1, then a workspace query is assumed; the routine
111: *> only calculates the optimal size of the WORK array, returns
112: *> this value as the first entry of the WORK array, and no error
113: *> message related to LWORK is issued by XERBLA.
114: *> \endverbatim
115: *>
116: *> \param[out] INFO
117: *> \verbatim
118: *> INFO is INTEGER
119: *> = 0: successful exit
120: *> < 0: if INFO = -i, the i-th argument had an illegal value
121: *> \endverbatim
122: *
123: * Authors:
124: * ========
125: *
126: *> \author Univ. of Tennessee
127: *> \author Univ. of California Berkeley
128: *> \author Univ. of Colorado Denver
129: *> \author NAG Ltd.
130: *
131: *> \par Further Details:
132: * =====================
133: *>
134: *> \verbatim
135: *> Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
136: *> representing Q as a product of other orthogonal matrices
137: *> Q = Q(1) * Q(2) * . . . * Q(k)
138: *> where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
139: *> Q(1) zeros out the subdiagonal entries of rows 1:MB of A
140: *> Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
141: *> Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
142: *> . . .
143: *>
144: *> Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
145: *> stored under the diagonal of rows 1:MB of A, and by upper triangular
146: *> block reflectors, stored in array T(1:LDT,1:N).
147: *> For more information see Further Details in GEQRT.
148: *>
149: *> Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
150: *> stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
151: *> block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
152: *> The last Q(k) may use fewer rows.
153: *> For more information see Further Details in TPQRT.
154: *>
155: *> For more details of the overall algorithm, see the description of
156: *> Sequential TSQR in Section 2.2 of [1].
157: *>
158: *> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
159: *> J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
160: *> SIAM J. Sci. Comput, vol. 34, no. 1, 2012
161: *> \endverbatim
162: *>
163: * =====================================================================
164: SUBROUTINE ZLATSQR( M, N, MB, NB, A, LDA, T, LDT, WORK,
165: $ LWORK, INFO)
166: *
167: * -- LAPACK computational routine --
168: * -- LAPACK is a software package provided by Univ. of Tennessee, --
169: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
170: *
171: * .. Scalar Arguments ..
172: INTEGER INFO, LDA, M, N, MB, NB, LDT, LWORK
173: * ..
174: * .. Array Arguments ..
175: COMPLEX*16 A( LDA, * ), WORK( * ), T(LDT, *)
176: * ..
177: *
178: * =====================================================================
179: *
180: * ..
181: * .. Local Scalars ..
182: LOGICAL LQUERY
183: INTEGER I, II, KK, CTR
184: * ..
185: * .. EXTERNAL FUNCTIONS ..
186: LOGICAL LSAME
187: EXTERNAL LSAME
188: * .. EXTERNAL SUBROUTINES ..
189: EXTERNAL ZGEQRT, ZTPQRT, XERBLA
190: * .. INTRINSIC FUNCTIONS ..
191: INTRINSIC MAX, MIN, MOD
192: * ..
193: * .. EXECUTABLE STATEMENTS ..
194: *
195: * TEST THE INPUT ARGUMENTS
196: *
197: INFO = 0
198: *
199: LQUERY = ( LWORK.EQ.-1 )
200: *
201: IF( M.LT.0 ) THEN
202: INFO = -1
203: ELSE IF( N.LT.0 .OR. M.LT.N ) THEN
204: INFO = -2
205: ELSE IF( MB.LT.1 ) THEN
206: INFO = -3
207: ELSE IF( NB.LT.1 .OR. ( NB.GT.N .AND. N.GT.0 )) THEN
208: INFO = -4
209: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
210: INFO = -6
211: ELSE IF( LDT.LT.NB ) THEN
212: INFO = -8
213: ELSE IF( LWORK.LT.(N*NB) .AND. (.NOT.LQUERY) ) THEN
214: INFO = -10
215: END IF
216: IF( INFO.EQ.0) THEN
217: WORK(1) = NB*N
218: END IF
219: IF( INFO.NE.0 ) THEN
220: CALL XERBLA( 'ZLATSQR', -INFO )
221: RETURN
222: ELSE IF (LQUERY) THEN
223: RETURN
224: END IF
225: *
226: * Quick return if possible
227: *
228: IF( MIN(M,N).EQ.0 ) THEN
229: RETURN
230: END IF
231: *
232: * The QR Decomposition
233: *
234: IF ((MB.LE.N).OR.(MB.GE.M)) THEN
235: CALL ZGEQRT( M, N, NB, A, LDA, T, LDT, WORK, INFO)
236: RETURN
237: END IF
238: KK = MOD((M-N),(MB-N))
239: II=M-KK+1
240: *
241: * Compute the QR factorization of the first block A(1:MB,1:N)
242: *
243: CALL ZGEQRT( MB, N, NB, A(1,1), LDA, T, LDT, WORK, INFO )
244: CTR = 1
245: *
246: DO I = MB+1, II-MB+N , (MB-N)
247: *
248: * Compute the QR factorization of the current block A(I:I+MB-N,1:N)
249: *
250: CALL ZTPQRT( MB-N, N, 0, NB, A(1,1), LDA, A( I, 1 ), LDA,
251: $ T(1, CTR * N + 1),
252: $ LDT, WORK, INFO )
253: CTR = CTR + 1
254: END DO
255: *
256: * Compute the QR factorization of the last block A(II:M,1:N)
257: *
258: IF (II.LE.M) THEN
259: CALL ZTPQRT( KK, N, 0, NB, A(1,1), LDA, A( II, 1 ), LDA,
260: $ T(1,CTR * N + 1), LDT,
261: $ WORK, INFO )
262: END IF
263: *
264: work( 1 ) = N*NB
265: RETURN
266: *
267: * End of ZLATSQR
268: *
269: END
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