1 SUBROUTINE ZPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
2 *
3 * -- LAPACK PROTOTYPE routine (version 3.2.2) --
4 * Craig Lucas, University of Manchester / NAG Ltd.
5 * October, 2008
6 *
7 * .. Scalar Arguments ..
8 DOUBLE PRECISION TOL
9 INTEGER INFO, LDA, N, RANK
10 CHARACTER UPLO
11 * ..
12 * .. Array Arguments ..
13 COMPLEX*16 A( LDA, * )
14 DOUBLE PRECISION WORK( 2*N )
15 INTEGER PIV( N )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * ZPSTF2 computes the Cholesky factorization with complete
22 * pivoting of a complex Hermitian positive semidefinite matrix A.
23 *
24 * The factorization has the form
25 * P**T * A * P = U**H * U , if UPLO = 'U',
26 * P**T * A * P = L * L**H, if UPLO = 'L',
27 * where U is an upper triangular matrix and L is lower triangular, and
28 * P is stored as vector PIV.
29 *
30 * This algorithm does not attempt to check that A is positive
31 * semidefinite. This version of the algorithm calls level 2 BLAS.
32 *
33 * Arguments
34 * =========
35 *
36 * UPLO (input) CHARACTER*1
37 * Specifies whether the upper or lower triangular part of the
38 * symmetric matrix A is stored.
39 * = 'U': Upper triangular
40 * = 'L': Lower triangular
41 *
42 * N (input) INTEGER
43 * The order of the matrix A. N >= 0.
44 *
45 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
46 * On entry, the symmetric matrix A. If UPLO = 'U', the leading
47 * n by n upper triangular part of A contains the upper
48 * triangular part of the matrix A, and the strictly lower
49 * triangular part of A is not referenced. If UPLO = 'L', the
50 * leading n by n lower triangular part of A contains the lower
51 * triangular part of the matrix A, and the strictly upper
52 * triangular part of A is not referenced.
53 *
54 * On exit, if INFO = 0, the factor U or L from the Cholesky
55 * factorization as above.
56 *
57 * PIV (output) INTEGER array, dimension (N)
58 * PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
59 *
60 * RANK (output) INTEGER
61 * The rank of A given by the number of steps the algorithm
62 * completed.
63 *
64 * TOL (input) DOUBLE PRECISION
65 * User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
66 * will be used. The algorithm terminates at the (K-1)st step
67 * if the pivot <= TOL.
68 *
69 * LDA (input) INTEGER
70 * The leading dimension of the array A. LDA >= max(1,N).
71 *
72 * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
73 * Work space.
74 *
75 * INFO (output) INTEGER
76 * < 0: If INFO = -K, the K-th argument had an illegal value,
77 * = 0: algorithm completed successfully, and
78 * > 0: the matrix A is either rank deficient with computed rank
79 * as returned in RANK, or is indefinite. See Section 7 of
80 * LAPACK Working Note #161 for further information.
81 *
82 * =====================================================================
83 *
84 * .. Parameters ..
85 DOUBLE PRECISION ONE, ZERO
86 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
87 COMPLEX*16 CONE
88 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
89 * ..
90 * .. Local Scalars ..
91 COMPLEX*16 ZTEMP
92 DOUBLE PRECISION AJJ, DSTOP, DTEMP
93 INTEGER I, ITEMP, J, PVT
94 LOGICAL UPPER
95 * ..
96 * .. External Functions ..
97 DOUBLE PRECISION DLAMCH
98 LOGICAL LSAME, DISNAN
99 EXTERNAL DLAMCH, LSAME, DISNAN
100 * ..
101 * .. External Subroutines ..
102 EXTERNAL ZDSCAL, ZGEMV, ZLACGV, ZSWAP, XERBLA
103 * ..
104 * .. Intrinsic Functions ..
105 INTRINSIC DBLE, DCONJG, MAX, SQRT
106 * ..
107 * .. Executable Statements ..
108 *
109 * Test the input parameters
110 *
111 INFO = 0
112 UPPER = LSAME( UPLO, 'U' )
113 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
114 INFO = -1
115 ELSE IF( N.LT.0 ) THEN
116 INFO = -2
117 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
118 INFO = -4
119 END IF
120 IF( INFO.NE.0 ) THEN
121 CALL XERBLA( 'ZPSTF2', -INFO )
122 RETURN
123 END IF
124 *
125 * Quick return if possible
126 *
127 IF( N.EQ.0 )
128 $ RETURN
129 *
130 * Initialize PIV
131 *
132 DO 100 I = 1, N
133 PIV( I ) = I
134 100 CONTINUE
135 *
136 * Compute stopping value
137 *
138 DO 110 I = 1, N
139 WORK( I ) = DBLE( A( I, I ) )
140 110 CONTINUE
141 PVT = MAXLOC( WORK( 1:N ), 1 )
142 AJJ = DBLE( A( PVT, PVT ) )
143 IF( AJJ.EQ.ZERO.OR.DISNAN( AJJ ) ) THEN
144 RANK = 0
145 INFO = 1
146 GO TO 200
147 END IF
148 *
149 * Compute stopping value if not supplied
150 *
151 IF( TOL.LT.ZERO ) THEN
152 DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
153 ELSE
154 DSTOP = TOL
155 END IF
156 *
157 * Set first half of WORK to zero, holds dot products
158 *
159 DO 120 I = 1, N
160 WORK( I ) = 0
161 120 CONTINUE
162 *
163 IF( UPPER ) THEN
164 *
165 * Compute the Cholesky factorization P**T * A * P = U**H* U
166 *
167 DO 150 J = 1, N
168 *
169 * Find pivot, test for exit, else swap rows and columns
170 * Update dot products, compute possible pivots which are
171 * stored in the second half of WORK
172 *
173 DO 130 I = J, N
174 *
175 IF( J.GT.1 ) THEN
176 WORK( I ) = WORK( I ) +
177 $ DBLE( DCONJG( A( J-1, I ) )*
178 $ A( J-1, I ) )
179 END IF
180 WORK( N+I ) = DBLE( A( I, I ) ) - WORK( I )
181 *
182 130 CONTINUE
183 *
184 IF( J.GT.1 ) THEN
185 ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
186 PVT = ITEMP + J - 1
187 AJJ = WORK( N+PVT )
188 IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
189 A( J, J ) = AJJ
190 GO TO 190
191 END IF
192 END IF
193 *
194 IF( J.NE.PVT ) THEN
195 *
196 * Pivot OK, so can now swap pivot rows and columns
197 *
198 A( PVT, PVT ) = A( J, J )
199 CALL ZSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
200 IF( PVT.LT.N )
201 $ CALL ZSWAP( N-PVT, A( J, PVT+1 ), LDA,
202 $ A( PVT, PVT+1 ), LDA )
203 DO 140 I = J + 1, PVT - 1
204 ZTEMP = DCONJG( A( J, I ) )
205 A( J, I ) = DCONJG( A( I, PVT ) )
206 A( I, PVT ) = ZTEMP
207 140 CONTINUE
208 A( J, PVT ) = DCONJG( A( J, PVT ) )
209 *
210 * Swap dot products and PIV
211 *
212 DTEMP = WORK( J )
213 WORK( J ) = WORK( PVT )
214 WORK( PVT ) = DTEMP
215 ITEMP = PIV( PVT )
216 PIV( PVT ) = PIV( J )
217 PIV( J ) = ITEMP
218 END IF
219 *
220 AJJ = SQRT( AJJ )
221 A( J, J ) = AJJ
222 *
223 * Compute elements J+1:N of row J
224 *
225 IF( J.LT.N ) THEN
226 CALL ZLACGV( J-1, A( 1, J ), 1 )
227 CALL ZGEMV( 'Trans', J-1, N-J, -CONE, A( 1, J+1 ), LDA,
228 $ A( 1, J ), 1, CONE, A( J, J+1 ), LDA )
229 CALL ZLACGV( J-1, A( 1, J ), 1 )
230 CALL ZDSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
231 END IF
232 *
233 150 CONTINUE
234 *
235 ELSE
236 *
237 * Compute the Cholesky factorization P**T * A * P = L * L**H
238 *
239 DO 180 J = 1, N
240 *
241 * Find pivot, test for exit, else swap rows and columns
242 * Update dot products, compute possible pivots which are
243 * stored in the second half of WORK
244 *
245 DO 160 I = J, N
246 *
247 IF( J.GT.1 ) THEN
248 WORK( I ) = WORK( I ) +
249 $ DBLE( DCONJG( A( I, J-1 ) )*
250 $ A( I, J-1 ) )
251 END IF
252 WORK( N+I ) = DBLE( A( I, I ) ) - WORK( I )
253 *
254 160 CONTINUE
255 *
256 IF( J.GT.1 ) THEN
257 ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
258 PVT = ITEMP + J - 1
259 AJJ = WORK( N+PVT )
260 IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
261 A( J, J ) = AJJ
262 GO TO 190
263 END IF
264 END IF
265 *
266 IF( J.NE.PVT ) THEN
267 *
268 * Pivot OK, so can now swap pivot rows and columns
269 *
270 A( PVT, PVT ) = A( J, J )
271 CALL ZSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
272 IF( PVT.LT.N )
273 $ CALL ZSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1, PVT ),
274 $ 1 )
275 DO 170 I = J + 1, PVT - 1
276 ZTEMP = DCONJG( A( I, J ) )
277 A( I, J ) = DCONJG( A( PVT, I ) )
278 A( PVT, I ) = ZTEMP
279 170 CONTINUE
280 A( PVT, J ) = DCONJG( A( PVT, J ) )
281 *
282 * Swap dot products and PIV
283 *
284 DTEMP = WORK( J )
285 WORK( J ) = WORK( PVT )
286 WORK( PVT ) = DTEMP
287 ITEMP = PIV( PVT )
288 PIV( PVT ) = PIV( J )
289 PIV( J ) = ITEMP
290 END IF
291 *
292 AJJ = SQRT( AJJ )
293 A( J, J ) = AJJ
294 *
295 * Compute elements J+1:N of column J
296 *
297 IF( J.LT.N ) THEN
298 CALL ZLACGV( J-1, A( J, 1 ), LDA )
299 CALL ZGEMV( 'No Trans', N-J, J-1, -CONE, A( J+1, 1 ),
300 $ LDA, A( J, 1 ), LDA, CONE, A( J+1, J ), 1 )
301 CALL ZLACGV( J-1, A( J, 1 ), LDA )
302 CALL ZDSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
303 END IF
304 *
305 180 CONTINUE
306 *
307 END IF
308 *
309 * Ran to completion, A has full rank
310 *
311 RANK = N
312 *
313 GO TO 200
314 190 CONTINUE
315 *
316 * Rank is number of steps completed. Set INFO = 1 to signal
317 * that the factorization cannot be used to solve a system.
318 *
319 RANK = J - 1
320 INFO = 1
321 *
322 200 CONTINUE
323 RETURN
324 *
325 * End of ZPSTF2
326 *
327 END
2 *
3 * -- LAPACK PROTOTYPE routine (version 3.2.2) --
4 * Craig Lucas, University of Manchester / NAG Ltd.
5 * October, 2008
6 *
7 * .. Scalar Arguments ..
8 DOUBLE PRECISION TOL
9 INTEGER INFO, LDA, N, RANK
10 CHARACTER UPLO
11 * ..
12 * .. Array Arguments ..
13 COMPLEX*16 A( LDA, * )
14 DOUBLE PRECISION WORK( 2*N )
15 INTEGER PIV( N )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * ZPSTF2 computes the Cholesky factorization with complete
22 * pivoting of a complex Hermitian positive semidefinite matrix A.
23 *
24 * The factorization has the form
25 * P**T * A * P = U**H * U , if UPLO = 'U',
26 * P**T * A * P = L * L**H, if UPLO = 'L',
27 * where U is an upper triangular matrix and L is lower triangular, and
28 * P is stored as vector PIV.
29 *
30 * This algorithm does not attempt to check that A is positive
31 * semidefinite. This version of the algorithm calls level 2 BLAS.
32 *
33 * Arguments
34 * =========
35 *
36 * UPLO (input) CHARACTER*1
37 * Specifies whether the upper or lower triangular part of the
38 * symmetric matrix A is stored.
39 * = 'U': Upper triangular
40 * = 'L': Lower triangular
41 *
42 * N (input) INTEGER
43 * The order of the matrix A. N >= 0.
44 *
45 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
46 * On entry, the symmetric matrix A. If UPLO = 'U', the leading
47 * n by n upper triangular part of A contains the upper
48 * triangular part of the matrix A, and the strictly lower
49 * triangular part of A is not referenced. If UPLO = 'L', the
50 * leading n by n lower triangular part of A contains the lower
51 * triangular part of the matrix A, and the strictly upper
52 * triangular part of A is not referenced.
53 *
54 * On exit, if INFO = 0, the factor U or L from the Cholesky
55 * factorization as above.
56 *
57 * PIV (output) INTEGER array, dimension (N)
58 * PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
59 *
60 * RANK (output) INTEGER
61 * The rank of A given by the number of steps the algorithm
62 * completed.
63 *
64 * TOL (input) DOUBLE PRECISION
65 * User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
66 * will be used. The algorithm terminates at the (K-1)st step
67 * if the pivot <= TOL.
68 *
69 * LDA (input) INTEGER
70 * The leading dimension of the array A. LDA >= max(1,N).
71 *
72 * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
73 * Work space.
74 *
75 * INFO (output) INTEGER
76 * < 0: If INFO = -K, the K-th argument had an illegal value,
77 * = 0: algorithm completed successfully, and
78 * > 0: the matrix A is either rank deficient with computed rank
79 * as returned in RANK, or is indefinite. See Section 7 of
80 * LAPACK Working Note #161 for further information.
81 *
82 * =====================================================================
83 *
84 * .. Parameters ..
85 DOUBLE PRECISION ONE, ZERO
86 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
87 COMPLEX*16 CONE
88 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
89 * ..
90 * .. Local Scalars ..
91 COMPLEX*16 ZTEMP
92 DOUBLE PRECISION AJJ, DSTOP, DTEMP
93 INTEGER I, ITEMP, J, PVT
94 LOGICAL UPPER
95 * ..
96 * .. External Functions ..
97 DOUBLE PRECISION DLAMCH
98 LOGICAL LSAME, DISNAN
99 EXTERNAL DLAMCH, LSAME, DISNAN
100 * ..
101 * .. External Subroutines ..
102 EXTERNAL ZDSCAL, ZGEMV, ZLACGV, ZSWAP, XERBLA
103 * ..
104 * .. Intrinsic Functions ..
105 INTRINSIC DBLE, DCONJG, MAX, SQRT
106 * ..
107 * .. Executable Statements ..
108 *
109 * Test the input parameters
110 *
111 INFO = 0
112 UPPER = LSAME( UPLO, 'U' )
113 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
114 INFO = -1
115 ELSE IF( N.LT.0 ) THEN
116 INFO = -2
117 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
118 INFO = -4
119 END IF
120 IF( INFO.NE.0 ) THEN
121 CALL XERBLA( 'ZPSTF2', -INFO )
122 RETURN
123 END IF
124 *
125 * Quick return if possible
126 *
127 IF( N.EQ.0 )
128 $ RETURN
129 *
130 * Initialize PIV
131 *
132 DO 100 I = 1, N
133 PIV( I ) = I
134 100 CONTINUE
135 *
136 * Compute stopping value
137 *
138 DO 110 I = 1, N
139 WORK( I ) = DBLE( A( I, I ) )
140 110 CONTINUE
141 PVT = MAXLOC( WORK( 1:N ), 1 )
142 AJJ = DBLE( A( PVT, PVT ) )
143 IF( AJJ.EQ.ZERO.OR.DISNAN( AJJ ) ) THEN
144 RANK = 0
145 INFO = 1
146 GO TO 200
147 END IF
148 *
149 * Compute stopping value if not supplied
150 *
151 IF( TOL.LT.ZERO ) THEN
152 DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
153 ELSE
154 DSTOP = TOL
155 END IF
156 *
157 * Set first half of WORK to zero, holds dot products
158 *
159 DO 120 I = 1, N
160 WORK( I ) = 0
161 120 CONTINUE
162 *
163 IF( UPPER ) THEN
164 *
165 * Compute the Cholesky factorization P**T * A * P = U**H* U
166 *
167 DO 150 J = 1, N
168 *
169 * Find pivot, test for exit, else swap rows and columns
170 * Update dot products, compute possible pivots which are
171 * stored in the second half of WORK
172 *
173 DO 130 I = J, N
174 *
175 IF( J.GT.1 ) THEN
176 WORK( I ) = WORK( I ) +
177 $ DBLE( DCONJG( A( J-1, I ) )*
178 $ A( J-1, I ) )
179 END IF
180 WORK( N+I ) = DBLE( A( I, I ) ) - WORK( I )
181 *
182 130 CONTINUE
183 *
184 IF( J.GT.1 ) THEN
185 ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
186 PVT = ITEMP + J - 1
187 AJJ = WORK( N+PVT )
188 IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
189 A( J, J ) = AJJ
190 GO TO 190
191 END IF
192 END IF
193 *
194 IF( J.NE.PVT ) THEN
195 *
196 * Pivot OK, so can now swap pivot rows and columns
197 *
198 A( PVT, PVT ) = A( J, J )
199 CALL ZSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
200 IF( PVT.LT.N )
201 $ CALL ZSWAP( N-PVT, A( J, PVT+1 ), LDA,
202 $ A( PVT, PVT+1 ), LDA )
203 DO 140 I = J + 1, PVT - 1
204 ZTEMP = DCONJG( A( J, I ) )
205 A( J, I ) = DCONJG( A( I, PVT ) )
206 A( I, PVT ) = ZTEMP
207 140 CONTINUE
208 A( J, PVT ) = DCONJG( A( J, PVT ) )
209 *
210 * Swap dot products and PIV
211 *
212 DTEMP = WORK( J )
213 WORK( J ) = WORK( PVT )
214 WORK( PVT ) = DTEMP
215 ITEMP = PIV( PVT )
216 PIV( PVT ) = PIV( J )
217 PIV( J ) = ITEMP
218 END IF
219 *
220 AJJ = SQRT( AJJ )
221 A( J, J ) = AJJ
222 *
223 * Compute elements J+1:N of row J
224 *
225 IF( J.LT.N ) THEN
226 CALL ZLACGV( J-1, A( 1, J ), 1 )
227 CALL ZGEMV( 'Trans', J-1, N-J, -CONE, A( 1, J+1 ), LDA,
228 $ A( 1, J ), 1, CONE, A( J, J+1 ), LDA )
229 CALL ZLACGV( J-1, A( 1, J ), 1 )
230 CALL ZDSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
231 END IF
232 *
233 150 CONTINUE
234 *
235 ELSE
236 *
237 * Compute the Cholesky factorization P**T * A * P = L * L**H
238 *
239 DO 180 J = 1, N
240 *
241 * Find pivot, test for exit, else swap rows and columns
242 * Update dot products, compute possible pivots which are
243 * stored in the second half of WORK
244 *
245 DO 160 I = J, N
246 *
247 IF( J.GT.1 ) THEN
248 WORK( I ) = WORK( I ) +
249 $ DBLE( DCONJG( A( I, J-1 ) )*
250 $ A( I, J-1 ) )
251 END IF
252 WORK( N+I ) = DBLE( A( I, I ) ) - WORK( I )
253 *
254 160 CONTINUE
255 *
256 IF( J.GT.1 ) THEN
257 ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
258 PVT = ITEMP + J - 1
259 AJJ = WORK( N+PVT )
260 IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
261 A( J, J ) = AJJ
262 GO TO 190
263 END IF
264 END IF
265 *
266 IF( J.NE.PVT ) THEN
267 *
268 * Pivot OK, so can now swap pivot rows and columns
269 *
270 A( PVT, PVT ) = A( J, J )
271 CALL ZSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
272 IF( PVT.LT.N )
273 $ CALL ZSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1, PVT ),
274 $ 1 )
275 DO 170 I = J + 1, PVT - 1
276 ZTEMP = DCONJG( A( I, J ) )
277 A( I, J ) = DCONJG( A( PVT, I ) )
278 A( PVT, I ) = ZTEMP
279 170 CONTINUE
280 A( PVT, J ) = DCONJG( A( PVT, J ) )
281 *
282 * Swap dot products and PIV
283 *
284 DTEMP = WORK( J )
285 WORK( J ) = WORK( PVT )
286 WORK( PVT ) = DTEMP
287 ITEMP = PIV( PVT )
288 PIV( PVT ) = PIV( J )
289 PIV( J ) = ITEMP
290 END IF
291 *
292 AJJ = SQRT( AJJ )
293 A( J, J ) = AJJ
294 *
295 * Compute elements J+1:N of column J
296 *
297 IF( J.LT.N ) THEN
298 CALL ZLACGV( J-1, A( J, 1 ), LDA )
299 CALL ZGEMV( 'No Trans', N-J, J-1, -CONE, A( J+1, 1 ),
300 $ LDA, A( J, 1 ), LDA, CONE, A( J+1, J ), 1 )
301 CALL ZLACGV( J-1, A( J, 1 ), LDA )
302 CALL ZDSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
303 END IF
304 *
305 180 CONTINUE
306 *
307 END IF
308 *
309 * Ran to completion, A has full rank
310 *
311 RANK = N
312 *
313 GO TO 200
314 190 CONTINUE
315 *
316 * Rank is number of steps completed. Set INFO = 1 to signal
317 * that the factorization cannot be used to solve a system.
318 *
319 RANK = J - 1
320 INFO = 1
321 *
322 200 CONTINUE
323 RETURN
324 *
325 * End of ZPSTF2
326 *
327 END