1: SUBROUTINE ZLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
2: $ VN2, AUXV, F, LDF )
3: *
4: * -- LAPACK auxiliary routine (version 3.3.1) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * -- April 2011 --
8: *
9: * .. Scalar Arguments ..
10: INTEGER KB, LDA, LDF, M, N, NB, OFFSET
11: * ..
12: * .. Array Arguments ..
13: INTEGER JPVT( * )
14: DOUBLE PRECISION VN1( * ), VN2( * )
15: COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * )
16: * ..
17: *
18: * Purpose
19: * =======
20: *
21: * ZLAQPS computes a step of QR factorization with column pivoting
22: * of a complex M-by-N matrix A by using Blas-3. It tries to factorize
23: * NB columns from A starting from the row OFFSET+1, and updates all
24: * of the matrix with Blas-3 xGEMM.
25: *
26: * In some cases, due to catastrophic cancellations, it cannot
27: * factorize NB columns. Hence, the actual number of factorized
28: * columns is returned in KB.
29: *
30: * Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
31: *
32: * Arguments
33: * =========
34: *
35: * M (input) INTEGER
36: * The number of rows of the matrix A. M >= 0.
37: *
38: * N (input) INTEGER
39: * The number of columns of the matrix A. N >= 0
40: *
41: * OFFSET (input) INTEGER
42: * The number of rows of A that have been factorized in
43: * previous steps.
44: *
45: * NB (input) INTEGER
46: * The number of columns to factorize.
47: *
48: * KB (output) INTEGER
49: * The number of columns actually factorized.
50: *
51: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
52: * On entry, the M-by-N matrix A.
53: * On exit, block A(OFFSET+1:M,1:KB) is the triangular
54: * factor obtained and block A(1:OFFSET,1:N) has been
55: * accordingly pivoted, but no factorized.
56: * The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
57: * been updated.
58: *
59: * LDA (input) INTEGER
60: * The leading dimension of the array A. LDA >= max(1,M).
61: *
62: * JPVT (input/output) INTEGER array, dimension (N)
63: * JPVT(I) = K <==> Column K of the full matrix A has been
64: * permuted into position I in AP.
65: *
66: * TAU (output) COMPLEX*16 array, dimension (KB)
67: * The scalar factors of the elementary reflectors.
68: *
69: * VN1 (input/output) DOUBLE PRECISION array, dimension (N)
70: * The vector with the partial column norms.
71: *
72: * VN2 (input/output) DOUBLE PRECISION array, dimension (N)
73: * The vector with the exact column norms.
74: *
75: * AUXV (input/output) COMPLEX*16 array, dimension (NB)
76: * Auxiliar vector.
77: *
78: * F (input/output) COMPLEX*16 array, dimension (LDF,NB)
79: * Matrix F**H = L * Y**H * A.
80: *
81: * LDF (input) INTEGER
82: * The leading dimension of the array F. LDF >= max(1,N).
83: *
84: * Further Details
85: * ===============
86: *
87: * Based on contributions by
88: * G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
89: * X. Sun, Computer Science Dept., Duke University, USA
90: *
91: * Partial column norm updating strategy modified by
92: * Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
93: * University of Zagreb, Croatia.
94: * -- April 2011 --
95: * For more details see LAPACK Working Note 176.
96: * =====================================================================
97: *
98: * .. Parameters ..
99: DOUBLE PRECISION ZERO, ONE
100: COMPLEX*16 CZERO, CONE
101: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0,
102: $ CZERO = ( 0.0D+0, 0.0D+0 ),
103: $ CONE = ( 1.0D+0, 0.0D+0 ) )
104: * ..
105: * .. Local Scalars ..
106: INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
107: DOUBLE PRECISION TEMP, TEMP2, TOL3Z
108: COMPLEX*16 AKK
109: * ..
110: * .. External Subroutines ..
111: EXTERNAL ZGEMM, ZGEMV, ZLARFG, ZSWAP
112: * ..
113: * .. Intrinsic Functions ..
114: INTRINSIC ABS, DBLE, DCONJG, MAX, MIN, NINT, SQRT
115: * ..
116: * .. External Functions ..
117: INTEGER IDAMAX
118: DOUBLE PRECISION DLAMCH, DZNRM2
119: EXTERNAL IDAMAX, DLAMCH, DZNRM2
120: * ..
121: * .. Executable Statements ..
122: *
123: LASTRK = MIN( M, N+OFFSET )
124: LSTICC = 0
125: K = 0
126: TOL3Z = SQRT(DLAMCH('Epsilon'))
127: *
128: * Beginning of while loop.
129: *
130: 10 CONTINUE
131: IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN
132: K = K + 1
133: RK = OFFSET + K
134: *
135: * Determine ith pivot column and swap if necessary
136: *
137: PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 )
138: IF( PVT.NE.K ) THEN
139: CALL ZSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 )
140: CALL ZSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF )
141: ITEMP = JPVT( PVT )
142: JPVT( PVT ) = JPVT( K )
143: JPVT( K ) = ITEMP
144: VN1( PVT ) = VN1( K )
145: VN2( PVT ) = VN2( K )
146: END IF
147: *
148: * Apply previous Householder reflectors to column K:
149: * A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**H.
150: *
151: IF( K.GT.1 ) THEN
152: DO 20 J = 1, K - 1
153: F( K, J ) = DCONJG( F( K, J ) )
154: 20 CONTINUE
155: CALL ZGEMV( 'No transpose', M-RK+1, K-1, -CONE, A( RK, 1 ),
156: $ LDA, F( K, 1 ), LDF, CONE, A( RK, K ), 1 )
157: DO 30 J = 1, K - 1
158: F( K, J ) = DCONJG( F( K, J ) )
159: 30 CONTINUE
160: END IF
161: *
162: * Generate elementary reflector H(k).
163: *
164: IF( RK.LT.M ) THEN
165: CALL ZLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) )
166: ELSE
167: CALL ZLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) )
168: END IF
169: *
170: AKK = A( RK, K )
171: A( RK, K ) = CONE
172: *
173: * Compute Kth column of F:
174: *
175: * Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**H*A(RK:M,K).
176: *
177: IF( K.LT.N ) THEN
178: CALL ZGEMV( 'Conjugate transpose', M-RK+1, N-K, TAU( K ),
179: $ A( RK, K+1 ), LDA, A( RK, K ), 1, CZERO,
180: $ F( K+1, K ), 1 )
181: END IF
182: *
183: * Padding F(1:K,K) with zeros.
184: *
185: DO 40 J = 1, K
186: F( J, K ) = CZERO
187: 40 CONTINUE
188: *
189: * Incremental updating of F:
190: * F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**H
191: * *A(RK:M,K).
192: *
193: IF( K.GT.1 ) THEN
194: CALL ZGEMV( 'Conjugate transpose', M-RK+1, K-1, -TAU( K ),
195: $ A( RK, 1 ), LDA, A( RK, K ), 1, CZERO,
196: $ AUXV( 1 ), 1 )
197: *
198: CALL ZGEMV( 'No transpose', N, K-1, CONE, F( 1, 1 ), LDF,
199: $ AUXV( 1 ), 1, CONE, F( 1, K ), 1 )
200: END IF
201: *
202: * Update the current row of A:
203: * A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**H.
204: *
205: IF( K.LT.N ) THEN
206: CALL ZGEMM( 'No transpose', 'Conjugate transpose', 1, N-K,
207: $ K, -CONE, A( RK, 1 ), LDA, F( K+1, 1 ), LDF,
208: $ CONE, A( RK, K+1 ), LDA )
209: END IF
210: *
211: * Update partial column norms.
212: *
213: IF( RK.LT.LASTRK ) THEN
214: DO 50 J = K + 1, N
215: IF( VN1( J ).NE.ZERO ) THEN
216: *
217: * NOTE: The following 4 lines follow from the analysis in
218: * Lapack Working Note 176.
219: *
220: TEMP = ABS( A( RK, J ) ) / VN1( J )
221: TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
222: TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
223: IF( TEMP2 .LE. TOL3Z ) THEN
224: VN2( J ) = DBLE( LSTICC )
225: LSTICC = J
226: ELSE
227: VN1( J ) = VN1( J )*SQRT( TEMP )
228: END IF
229: END IF
230: 50 CONTINUE
231: END IF
232: *
233: A( RK, K ) = AKK
234: *
235: * End of while loop.
236: *
237: GO TO 10
238: END IF
239: KB = K
240: RK = OFFSET + KB
241: *
242: * Apply the block reflector to the rest of the matrix:
243: * A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
244: * A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**H.
245: *
246: IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
247: CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-RK, N-KB,
248: $ KB, -CONE, A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF,
249: $ CONE, A( RK+1, KB+1 ), LDA )
250: END IF
251: *
252: * Recomputation of difficult columns.
253: *
254: 60 CONTINUE
255: IF( LSTICC.GT.0 ) THEN
256: ITEMP = NINT( VN2( LSTICC ) )
257: VN1( LSTICC ) = DZNRM2( M-RK, A( RK+1, LSTICC ), 1 )
258: *
259: * NOTE: The computation of VN1( LSTICC ) relies on the fact that
260: * SNRM2 does not fail on vectors with norm below the value of
261: * SQRT(DLAMCH('S'))
262: *
263: VN2( LSTICC ) = VN1( LSTICC )
264: LSTICC = ITEMP
265: GO TO 60
266: END IF
267: *
268: RETURN
269: *
270: * End of ZLAQPS
271: *
272: END
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