1: *> \brief \b ZGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGEQRT3 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeqrt3.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgeqrt3.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgeqrt3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * RECURSIVE SUBROUTINE ZGEQRT3( M, N, A, LDA, T, LDT, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, M, N, LDT
25: * ..
26: * .. Array Arguments ..
27: * COMPLEX*16 A( LDA, * ), T( LDT, * )
28: * ..
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> ZGEQRT3 recursively computes a QR factorization of a complex M-by-N
37: *> matrix A, using the compact WY representation of Q.
38: *>
39: *> Based on the algorithm of Elmroth and Gustavson,
40: *> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
41: *> \endverbatim
42: *
43: * Arguments:
44: * ==========
45: *
46: *> \param[in] M
47: *> \verbatim
48: *> M is INTEGER
49: *> The number of rows of the matrix A. M >= N.
50: *> \endverbatim
51: *>
52: *> \param[in] N
53: *> \verbatim
54: *> N is INTEGER
55: *> The number of columns of the matrix A. N >= 0.
56: *> \endverbatim
57: *>
58: *> \param[in,out] A
59: *> \verbatim
60: *> A is COMPLEX*16 array, dimension (LDA,N)
61: *> On entry, the complex M-by-N matrix A. On exit, the elements on
62: *> and above the diagonal contain the N-by-N upper triangular matrix R;
63: *> the elements below the diagonal are the columns of V. See below for
64: *> further details.
65: *> \endverbatim
66: *>
67: *> \param[in] LDA
68: *> \verbatim
69: *> LDA is INTEGER
70: *> The leading dimension of the array A. LDA >= max(1,M).
71: *> \endverbatim
72: *>
73: *> \param[out] T
74: *> \verbatim
75: *> T is COMPLEX*16 array, dimension (LDT,N)
76: *> The N-by-N upper triangular factor of the block reflector.
77: *> The elements on and above the diagonal contain the block
78: *> reflector T; the elements below the diagonal are not used.
79: *> See below for further details.
80: *> \endverbatim
81: *>
82: *> \param[in] LDT
83: *> \verbatim
84: *> LDT is INTEGER
85: *> The leading dimension of the array T. LDT >= max(1,N).
86: *> \endverbatim
87: *>
88: *> \param[out] INFO
89: *> \verbatim
90: *> INFO is INTEGER
91: *> = 0: successful exit
92: *> < 0: if INFO = -i, the i-th argument had an illegal value
93: *> \endverbatim
94: *
95: * Authors:
96: * ========
97: *
98: *> \author Univ. of Tennessee
99: *> \author Univ. of California Berkeley
100: *> \author Univ. of Colorado Denver
101: *> \author NAG Ltd.
102: *
103: *> \date June 2016
104: *
105: *> \ingroup complex16GEcomputational
106: *
107: *> \par Further Details:
108: * =====================
109: *>
110: *> \verbatim
111: *>
112: *> The matrix V stores the elementary reflectors H(i) in the i-th column
113: *> below the diagonal. For example, if M=5 and N=3, the matrix V is
114: *>
115: *> V = ( 1 )
116: *> ( v1 1 )
117: *> ( v1 v2 1 )
118: *> ( v1 v2 v3 )
119: *> ( v1 v2 v3 )
120: *>
121: *> where the vi's represent the vectors which define H(i), which are returned
122: *> in the matrix A. The 1's along the diagonal of V are not stored in A. The
123: *> block reflector H is then given by
124: *>
125: *> H = I - V * T * V**H
126: *>
127: *> where V**H is the conjugate transpose of V.
128: *>
129: *> For details of the algorithm, see Elmroth and Gustavson (cited above).
130: *> \endverbatim
131: *>
132: * =====================================================================
133: RECURSIVE SUBROUTINE ZGEQRT3( M, N, A, LDA, T, LDT, INFO )
134: *
135: * -- LAPACK computational routine (version 3.6.1) --
136: * -- LAPACK is a software package provided by Univ. of Tennessee, --
137: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
138: * June 2016
139: *
140: * .. Scalar Arguments ..
141: INTEGER INFO, LDA, M, N, LDT
142: * ..
143: * .. Array Arguments ..
144: COMPLEX*16 A( LDA, * ), T( LDT, * )
145: * ..
146: *
147: * =====================================================================
148: *
149: * .. Parameters ..
150: COMPLEX*16 ONE
151: PARAMETER ( ONE = (1.0D+00,0.0D+00) )
152: * ..
153: * .. Local Scalars ..
154: INTEGER I, I1, J, J1, N1, N2, IINFO
155: * ..
156: * .. External Subroutines ..
157: EXTERNAL ZLARFG, ZTRMM, ZGEMM, XERBLA
158: * ..
159: * .. Executable Statements ..
160: *
161: INFO = 0
162: IF( N .LT. 0 ) THEN
163: INFO = -2
164: ELSE IF( M .LT. N ) THEN
165: INFO = -1
166: ELSE IF( LDA .LT. MAX( 1, M ) ) THEN
167: INFO = -4
168: ELSE IF( LDT .LT. MAX( 1, N ) ) THEN
169: INFO = -6
170: END IF
171: IF( INFO.NE.0 ) THEN
172: CALL XERBLA( 'ZGEQRT3', -INFO )
173: RETURN
174: END IF
175: *
176: IF( N.EQ.1 ) THEN
177: *
178: * Compute Householder transform when N=1
179: *
180: CALL ZLARFG( M, A(1,1), A( MIN( 2, M ), 1 ), 1, T(1,1) )
181: *
182: ELSE
183: *
184: * Otherwise, split A into blocks...
185: *
186: N1 = N/2
187: N2 = N-N1
188: J1 = MIN( N1+1, N )
189: I1 = MIN( N+1, M )
190: *
191: * Compute A(1:M,1:N1) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
192: *
193: CALL ZGEQRT3( M, N1, A, LDA, T, LDT, IINFO )
194: *
195: * Compute A(1:M,J1:N) = Q1^H A(1:M,J1:N) [workspace: T(1:N1,J1:N)]
196: *
197: DO J=1,N2
198: DO I=1,N1
199: T( I, J+N1 ) = A( I, J+N1 )
200: END DO
201: END DO
202: CALL ZTRMM( 'L', 'L', 'C', 'U', N1, N2, ONE,
203: & A, LDA, T( 1, J1 ), LDT )
204: *
205: CALL ZGEMM( 'C', 'N', N1, N2, M-N1, ONE, A( J1, 1 ), LDA,
206: & A( J1, J1 ), LDA, ONE, T( 1, J1 ), LDT)
207: *
208: CALL ZTRMM( 'L', 'U', 'C', 'N', N1, N2, ONE,
209: & T, LDT, T( 1, J1 ), LDT )
210: *
211: CALL ZGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( J1, 1 ), LDA,
212: & T( 1, J1 ), LDT, ONE, A( J1, J1 ), LDA )
213: *
214: CALL ZTRMM( 'L', 'L', 'N', 'U', N1, N2, ONE,
215: & A, LDA, T( 1, J1 ), LDT )
216: *
217: DO J=1,N2
218: DO I=1,N1
219: A( I, J+N1 ) = A( I, J+N1 ) - T( I, J+N1 )
220: END DO
221: END DO
222: *
223: * Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
224: *
225: CALL ZGEQRT3( M-N1, N2, A( J1, J1 ), LDA,
226: & T( J1, J1 ), LDT, IINFO )
227: *
228: * Compute T3 = T(1:N1,J1:N) = -T1 Y1^H Y2 T2
229: *
230: DO I=1,N1
231: DO J=1,N2
232: T( I, J+N1 ) = CONJG(A( J+N1, I ))
233: END DO
234: END DO
235: *
236: CALL ZTRMM( 'R', 'L', 'N', 'U', N1, N2, ONE,
237: & A( J1, J1 ), LDA, T( 1, J1 ), LDT )
238: *
239: CALL ZGEMM( 'C', 'N', N1, N2, M-N, ONE, A( I1, 1 ), LDA,
240: & A( I1, J1 ), LDA, ONE, T( 1, J1 ), LDT )
241: *
242: CALL ZTRMM( 'L', 'U', 'N', 'N', N1, N2, -ONE, T, LDT,
243: & T( 1, J1 ), LDT )
244: *
245: CALL ZTRMM( 'R', 'U', 'N', 'N', N1, N2, ONE,
246: & T( J1, J1 ), LDT, T( 1, J1 ), LDT )
247: *
248: * Y = (Y1,Y2); R = [ R1 A(1:N1,J1:N) ]; T = [T1 T3]
249: * [ 0 R2 ] [ 0 T2]
250: *
251: END IF
252: *
253: RETURN
254: *
255: * End of ZGEQRT3
256: *
257: END
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