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
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