1 SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
2 $ JPIV )
3 *
4 * -- LAPACK auxiliary routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 INTEGER IJOB, LDZ, N
11 DOUBLE PRECISION RDSCAL, RDSUM
12 * ..
13 * .. Array Arguments ..
14 INTEGER IPIV( * ), JPIV( * )
15 COMPLEX*16 RHS( * ), Z( LDZ, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * ZLATDF computes the contribution to the reciprocal Dif-estimate
22 * by solving for x in Z * x = b, where b is chosen such that the norm
23 * of x is as large as possible. It is assumed that LU decomposition
24 * of Z has been computed by ZGETC2. On entry RHS = f holds the
25 * contribution from earlier solved sub-systems, and on return RHS = x.
26 *
27 * The factorization of Z returned by ZGETC2 has the form
28 * Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
29 * triangular with unit diagonal elements and U is upper triangular.
30 *
31 * Arguments
32 * =========
33 *
34 * IJOB (input) INTEGER
35 * IJOB = 2: First compute an approximative null-vector e
36 * of Z using ZGECON, e is normalized and solve for
37 * Zx = +-e - f with the sign giving the greater value of
38 * 2-norm(x). About 5 times as expensive as Default.
39 * IJOB .ne. 2: Local look ahead strategy where
40 * all entries of the r.h.s. b is choosen as either +1 or
41 * -1. Default.
42 *
43 * N (input) INTEGER
44 * The number of columns of the matrix Z.
45 *
46 * Z (input) DOUBLE PRECISION array, dimension (LDZ, N)
47 * On entry, the LU part of the factorization of the n-by-n
48 * matrix Z computed by ZGETC2: Z = P * L * U * Q
49 *
50 * LDZ (input) INTEGER
51 * The leading dimension of the array Z. LDA >= max(1, N).
52 *
53 * RHS (input/output) DOUBLE PRECISION array, dimension (N).
54 * On entry, RHS contains contributions from other subsystems.
55 * On exit, RHS contains the solution of the subsystem with
56 * entries according to the value of IJOB (see above).
57 *
58 * RDSUM (input/output) DOUBLE PRECISION
59 * On entry, the sum of squares of computed contributions to
60 * the Dif-estimate under computation by ZTGSYL, where the
61 * scaling factor RDSCAL (see below) has been factored out.
62 * On exit, the corresponding sum of squares updated with the
63 * contributions from the current sub-system.
64 * If TRANS = 'T' RDSUM is not touched.
65 * NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.
66 *
67 * RDSCAL (input/output) DOUBLE PRECISION
68 * On entry, scaling factor used to prevent overflow in RDSUM.
69 * On exit, RDSCAL is updated w.r.t. the current contributions
70 * in RDSUM.
71 * If TRANS = 'T', RDSCAL is not touched.
72 * NOTE: RDSCAL only makes sense when ZTGSY2 is called by
73 * ZTGSYL.
74 *
75 * IPIV (input) INTEGER array, dimension (N).
76 * The pivot indices; for 1 <= i <= N, row i of the
77 * matrix has been interchanged with row IPIV(i).
78 *
79 * JPIV (input) INTEGER array, dimension (N).
80 * The pivot indices; for 1 <= j <= N, column j of the
81 * matrix has been interchanged with column JPIV(j).
82 *
83 * Further Details
84 * ===============
85 *
86 * Based on contributions by
87 * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
88 * Umea University, S-901 87 Umea, Sweden.
89 *
90 * This routine is a further developed implementation of algorithm
91 * BSOLVE in [1] using complete pivoting in the LU factorization.
92 *
93 * [1] Bo Kagstrom and Lars Westin,
94 * Generalized Schur Methods with Condition Estimators for
95 * Solving the Generalized Sylvester Equation, IEEE Transactions
96 * on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
97 *
98 * [2] Peter Poromaa,
99 * On Efficient and Robust Estimators for the Separation
100 * between two Regular Matrix Pairs with Applications in
101 * Condition Estimation. Report UMINF-95.05, Department of
102 * Computing Science, Umea University, S-901 87 Umea, Sweden,
103 * 1995.
104 *
105 * =====================================================================
106 *
107 * .. Parameters ..
108 INTEGER MAXDIM
109 PARAMETER ( MAXDIM = 2 )
110 DOUBLE PRECISION ZERO, ONE
111 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
112 COMPLEX*16 CONE
113 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
114 * ..
115 * .. Local Scalars ..
116 INTEGER I, INFO, J, K
117 DOUBLE PRECISION RTEMP, SCALE, SMINU, SPLUS
118 COMPLEX*16 BM, BP, PMONE, TEMP
119 * ..
120 * .. Local Arrays ..
121 DOUBLE PRECISION RWORK( MAXDIM )
122 COMPLEX*16 WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
123 * ..
124 * .. External Subroutines ..
125 EXTERNAL ZAXPY, ZCOPY, ZGECON, ZGESC2, ZLASSQ, ZLASWP,
126 $ ZSCAL
127 * ..
128 * .. External Functions ..
129 DOUBLE PRECISION DZASUM
130 COMPLEX*16 ZDOTC
131 EXTERNAL DZASUM, ZDOTC
132 * ..
133 * .. Intrinsic Functions ..
134 INTRINSIC ABS, DBLE, SQRT
135 * ..
136 * .. Executable Statements ..
137 *
138 IF( IJOB.NE.2 ) THEN
139 *
140 * Apply permutations IPIV to RHS
141 *
142 CALL ZLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
143 *
144 * Solve for L-part choosing RHS either to +1 or -1.
145 *
146 PMONE = -CONE
147 DO 10 J = 1, N - 1
148 BP = RHS( J ) + CONE
149 BM = RHS( J ) - CONE
150 SPLUS = ONE
151 *
152 * Lockahead for L- part RHS(1:N-1) = +-1
153 * SPLUS and SMIN computed more efficiently than in BSOLVE[1].
154 *
155 SPLUS = SPLUS + DBLE( ZDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
156 $ J ), 1 ) )
157 SMINU = DBLE( ZDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
158 SPLUS = SPLUS*DBLE( RHS( J ) )
159 IF( SPLUS.GT.SMINU ) THEN
160 RHS( J ) = BP
161 ELSE IF( SMINU.GT.SPLUS ) THEN
162 RHS( J ) = BM
163 ELSE
164 *
165 * In this case the updating sums are equal and we can
166 * choose RHS(J) +1 or -1. The first time this happens we
167 * choose -1, thereafter +1. This is a simple way to get
168 * good estimates of matrices like Byers well-known example
169 * (see [1]). (Not done in BSOLVE.)
170 *
171 RHS( J ) = RHS( J ) + PMONE
172 PMONE = CONE
173 END IF
174 *
175 * Compute the remaining r.h.s.
176 *
177 TEMP = -RHS( J )
178 CALL ZAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
179 10 CONTINUE
180 *
181 * Solve for U- part, lockahead for RHS(N) = +-1. This is not done
182 * In BSOLVE and will hopefully give us a better estimate because
183 * any ill-conditioning of the original matrix is transfered to U
184 * and not to L. U(N, N) is an approximation to sigma_min(LU).
185 *
186 CALL ZCOPY( N-1, RHS, 1, WORK, 1 )
187 WORK( N ) = RHS( N ) + CONE
188 RHS( N ) = RHS( N ) - CONE
189 SPLUS = ZERO
190 SMINU = ZERO
191 DO 30 I = N, 1, -1
192 TEMP = CONE / Z( I, I )
193 WORK( I ) = WORK( I )*TEMP
194 RHS( I ) = RHS( I )*TEMP
195 DO 20 K = I + 1, N
196 WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
197 RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
198 20 CONTINUE
199 SPLUS = SPLUS + ABS( WORK( I ) )
200 SMINU = SMINU + ABS( RHS( I ) )
201 30 CONTINUE
202 IF( SPLUS.GT.SMINU )
203 $ CALL ZCOPY( N, WORK, 1, RHS, 1 )
204 *
205 * Apply the permutations JPIV to the computed solution (RHS)
206 *
207 CALL ZLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
208 *
209 * Compute the sum of squares
210 *
211 CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
212 RETURN
213 END IF
214 *
215 * ENTRY IJOB = 2
216 *
217 * Compute approximate nullvector XM of Z
218 *
219 CALL ZGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
220 CALL ZCOPY( N, WORK( N+1 ), 1, XM, 1 )
221 *
222 * Compute RHS
223 *
224 CALL ZLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
225 TEMP = CONE / SQRT( ZDOTC( N, XM, 1, XM, 1 ) )
226 CALL ZSCAL( N, TEMP, XM, 1 )
227 CALL ZCOPY( N, XM, 1, XP, 1 )
228 CALL ZAXPY( N, CONE, RHS, 1, XP, 1 )
229 CALL ZAXPY( N, -CONE, XM, 1, RHS, 1 )
230 CALL ZGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
231 CALL ZGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
232 IF( DZASUM( N, XP, 1 ).GT.DZASUM( N, RHS, 1 ) )
233 $ CALL ZCOPY( N, XP, 1, RHS, 1 )
234 *
235 * Compute the sum of squares
236 *
237 CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
238 RETURN
239 *
240 * End of ZLATDF
241 *
242 END
2 $ JPIV )
3 *
4 * -- LAPACK auxiliary routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 INTEGER IJOB, LDZ, N
11 DOUBLE PRECISION RDSCAL, RDSUM
12 * ..
13 * .. Array Arguments ..
14 INTEGER IPIV( * ), JPIV( * )
15 COMPLEX*16 RHS( * ), Z( LDZ, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * ZLATDF computes the contribution to the reciprocal Dif-estimate
22 * by solving for x in Z * x = b, where b is chosen such that the norm
23 * of x is as large as possible. It is assumed that LU decomposition
24 * of Z has been computed by ZGETC2. On entry RHS = f holds the
25 * contribution from earlier solved sub-systems, and on return RHS = x.
26 *
27 * The factorization of Z returned by ZGETC2 has the form
28 * Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
29 * triangular with unit diagonal elements and U is upper triangular.
30 *
31 * Arguments
32 * =========
33 *
34 * IJOB (input) INTEGER
35 * IJOB = 2: First compute an approximative null-vector e
36 * of Z using ZGECON, e is normalized and solve for
37 * Zx = +-e - f with the sign giving the greater value of
38 * 2-norm(x). About 5 times as expensive as Default.
39 * IJOB .ne. 2: Local look ahead strategy where
40 * all entries of the r.h.s. b is choosen as either +1 or
41 * -1. Default.
42 *
43 * N (input) INTEGER
44 * The number of columns of the matrix Z.
45 *
46 * Z (input) DOUBLE PRECISION array, dimension (LDZ, N)
47 * On entry, the LU part of the factorization of the n-by-n
48 * matrix Z computed by ZGETC2: Z = P * L * U * Q
49 *
50 * LDZ (input) INTEGER
51 * The leading dimension of the array Z. LDA >= max(1, N).
52 *
53 * RHS (input/output) DOUBLE PRECISION array, dimension (N).
54 * On entry, RHS contains contributions from other subsystems.
55 * On exit, RHS contains the solution of the subsystem with
56 * entries according to the value of IJOB (see above).
57 *
58 * RDSUM (input/output) DOUBLE PRECISION
59 * On entry, the sum of squares of computed contributions to
60 * the Dif-estimate under computation by ZTGSYL, where the
61 * scaling factor RDSCAL (see below) has been factored out.
62 * On exit, the corresponding sum of squares updated with the
63 * contributions from the current sub-system.
64 * If TRANS = 'T' RDSUM is not touched.
65 * NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.
66 *
67 * RDSCAL (input/output) DOUBLE PRECISION
68 * On entry, scaling factor used to prevent overflow in RDSUM.
69 * On exit, RDSCAL is updated w.r.t. the current contributions
70 * in RDSUM.
71 * If TRANS = 'T', RDSCAL is not touched.
72 * NOTE: RDSCAL only makes sense when ZTGSY2 is called by
73 * ZTGSYL.
74 *
75 * IPIV (input) INTEGER array, dimension (N).
76 * The pivot indices; for 1 <= i <= N, row i of the
77 * matrix has been interchanged with row IPIV(i).
78 *
79 * JPIV (input) INTEGER array, dimension (N).
80 * The pivot indices; for 1 <= j <= N, column j of the
81 * matrix has been interchanged with column JPIV(j).
82 *
83 * Further Details
84 * ===============
85 *
86 * Based on contributions by
87 * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
88 * Umea University, S-901 87 Umea, Sweden.
89 *
90 * This routine is a further developed implementation of algorithm
91 * BSOLVE in [1] using complete pivoting in the LU factorization.
92 *
93 * [1] Bo Kagstrom and Lars Westin,
94 * Generalized Schur Methods with Condition Estimators for
95 * Solving the Generalized Sylvester Equation, IEEE Transactions
96 * on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
97 *
98 * [2] Peter Poromaa,
99 * On Efficient and Robust Estimators for the Separation
100 * between two Regular Matrix Pairs with Applications in
101 * Condition Estimation. Report UMINF-95.05, Department of
102 * Computing Science, Umea University, S-901 87 Umea, Sweden,
103 * 1995.
104 *
105 * =====================================================================
106 *
107 * .. Parameters ..
108 INTEGER MAXDIM
109 PARAMETER ( MAXDIM = 2 )
110 DOUBLE PRECISION ZERO, ONE
111 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
112 COMPLEX*16 CONE
113 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
114 * ..
115 * .. Local Scalars ..
116 INTEGER I, INFO, J, K
117 DOUBLE PRECISION RTEMP, SCALE, SMINU, SPLUS
118 COMPLEX*16 BM, BP, PMONE, TEMP
119 * ..
120 * .. Local Arrays ..
121 DOUBLE PRECISION RWORK( MAXDIM )
122 COMPLEX*16 WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
123 * ..
124 * .. External Subroutines ..
125 EXTERNAL ZAXPY, ZCOPY, ZGECON, ZGESC2, ZLASSQ, ZLASWP,
126 $ ZSCAL
127 * ..
128 * .. External Functions ..
129 DOUBLE PRECISION DZASUM
130 COMPLEX*16 ZDOTC
131 EXTERNAL DZASUM, ZDOTC
132 * ..
133 * .. Intrinsic Functions ..
134 INTRINSIC ABS, DBLE, SQRT
135 * ..
136 * .. Executable Statements ..
137 *
138 IF( IJOB.NE.2 ) THEN
139 *
140 * Apply permutations IPIV to RHS
141 *
142 CALL ZLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
143 *
144 * Solve for L-part choosing RHS either to +1 or -1.
145 *
146 PMONE = -CONE
147 DO 10 J = 1, N - 1
148 BP = RHS( J ) + CONE
149 BM = RHS( J ) - CONE
150 SPLUS = ONE
151 *
152 * Lockahead for L- part RHS(1:N-1) = +-1
153 * SPLUS and SMIN computed more efficiently than in BSOLVE[1].
154 *
155 SPLUS = SPLUS + DBLE( ZDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
156 $ J ), 1 ) )
157 SMINU = DBLE( ZDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
158 SPLUS = SPLUS*DBLE( RHS( J ) )
159 IF( SPLUS.GT.SMINU ) THEN
160 RHS( J ) = BP
161 ELSE IF( SMINU.GT.SPLUS ) THEN
162 RHS( J ) = BM
163 ELSE
164 *
165 * In this case the updating sums are equal and we can
166 * choose RHS(J) +1 or -1. The first time this happens we
167 * choose -1, thereafter +1. This is a simple way to get
168 * good estimates of matrices like Byers well-known example
169 * (see [1]). (Not done in BSOLVE.)
170 *
171 RHS( J ) = RHS( J ) + PMONE
172 PMONE = CONE
173 END IF
174 *
175 * Compute the remaining r.h.s.
176 *
177 TEMP = -RHS( J )
178 CALL ZAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
179 10 CONTINUE
180 *
181 * Solve for U- part, lockahead for RHS(N) = +-1. This is not done
182 * In BSOLVE and will hopefully give us a better estimate because
183 * any ill-conditioning of the original matrix is transfered to U
184 * and not to L. U(N, N) is an approximation to sigma_min(LU).
185 *
186 CALL ZCOPY( N-1, RHS, 1, WORK, 1 )
187 WORK( N ) = RHS( N ) + CONE
188 RHS( N ) = RHS( N ) - CONE
189 SPLUS = ZERO
190 SMINU = ZERO
191 DO 30 I = N, 1, -1
192 TEMP = CONE / Z( I, I )
193 WORK( I ) = WORK( I )*TEMP
194 RHS( I ) = RHS( I )*TEMP
195 DO 20 K = I + 1, N
196 WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
197 RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
198 20 CONTINUE
199 SPLUS = SPLUS + ABS( WORK( I ) )
200 SMINU = SMINU + ABS( RHS( I ) )
201 30 CONTINUE
202 IF( SPLUS.GT.SMINU )
203 $ CALL ZCOPY( N, WORK, 1, RHS, 1 )
204 *
205 * Apply the permutations JPIV to the computed solution (RHS)
206 *
207 CALL ZLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
208 *
209 * Compute the sum of squares
210 *
211 CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
212 RETURN
213 END IF
214 *
215 * ENTRY IJOB = 2
216 *
217 * Compute approximate nullvector XM of Z
218 *
219 CALL ZGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
220 CALL ZCOPY( N, WORK( N+1 ), 1, XM, 1 )
221 *
222 * Compute RHS
223 *
224 CALL ZLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
225 TEMP = CONE / SQRT( ZDOTC( N, XM, 1, XM, 1 ) )
226 CALL ZSCAL( N, TEMP, XM, 1 )
227 CALL ZCOPY( N, XM, 1, XP, 1 )
228 CALL ZAXPY( N, CONE, RHS, 1, XP, 1 )
229 CALL ZAXPY( N, -CONE, XM, 1, RHS, 1 )
230 CALL ZGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
231 CALL ZGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
232 IF( DZASUM( N, XP, 1 ).GT.DZASUM( N, RHS, 1 ) )
233 $ CALL ZCOPY( N, XP, 1, RHS, 1 )
234 *
235 * Compute the sum of squares
236 *
237 CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
238 RETURN
239 *
240 * End of ZLATDF
241 *
242 END