1 SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
2 $ JPIV )
3 *
4 * -- LAPACK auxiliary routine (version 3.2.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * June 2010
8 *
9 * .. Scalar Arguments ..
10 INTEGER IJOB, LDZ, N
11 DOUBLE PRECISION RDSCAL, RDSUM
12 * ..
13 * .. Array Arguments ..
14 INTEGER IPIV( * ), JPIV( * )
15 DOUBLE PRECISION RHS( * ), Z( LDZ, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DLATDF uses the LU factorization of the n-by-n matrix Z computed by
22 * DGETC2 and computes a contribution to the reciprocal Dif-estimate
23 * by solving Z * x = b for x, and choosing the r.h.s. b such that
24 * the norm of x is as large as possible. On entry RHS = b holds the
25 * contribution from earlier solved sub-systems, and on return RHS = x.
26 *
27 * The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
28 * where P and Q are permutation matrices. L is lower triangular with
29 * 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 DGECON, e is normalized and solve for
37 * Zx = +-e - f with the sign giving the greater value
38 * of 2-norm(x). About 5 times as expensive as Default.
39 * IJOB .ne. 2: Local look ahead strategy where all entries of
40 * the r.h.s. b is choosen as either +1 or -1 (Default).
41 *
42 * N (input) INTEGER
43 * The number of columns of the matrix Z.
44 *
45 * Z (input) DOUBLE PRECISION array, dimension (LDZ, N)
46 * On entry, the LU part of the factorization of the n-by-n
47 * matrix Z computed by DGETC2: Z = P * L * U * Q
48 *
49 * LDZ (input) INTEGER
50 * The leading dimension of the array Z. LDA >= max(1, N).
51 *
52 * RHS (input/output) DOUBLE PRECISION array, dimension (N)
53 * On entry, RHS contains contributions from other subsystems.
54 * On exit, RHS contains the solution of the subsystem with
55 * entries acoording to the value of IJOB (see above).
56 *
57 * RDSUM (input/output) DOUBLE PRECISION
58 * On entry, the sum of squares of computed contributions to
59 * the Dif-estimate under computation by DTGSYL, where the
60 * scaling factor RDSCAL (see below) has been factored out.
61 * On exit, the corresponding sum of squares updated with the
62 * contributions from the current sub-system.
63 * If TRANS = 'T' RDSUM is not touched.
64 * NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
65 *
66 * RDSCAL (input/output) DOUBLE PRECISION
67 * On entry, scaling factor used to prevent overflow in RDSUM.
68 * On exit, RDSCAL is updated w.r.t. the current contributions
69 * in RDSUM.
70 * If TRANS = 'T', RDSCAL is not touched.
71 * NOTE: RDSCAL only makes sense when DTGSY2 is called by
72 * DTGSYL.
73 *
74 * IPIV (input) INTEGER array, dimension (N).
75 * The pivot indices; for 1 <= i <= N, row i of the
76 * matrix has been interchanged with row IPIV(i).
77 *
78 * JPIV (input) INTEGER array, dimension (N).
79 * The pivot indices; for 1 <= j <= N, column j of the
80 * matrix has been interchanged with column JPIV(j).
81 *
82 * Further Details
83 * ===============
84 *
85 * Based on contributions by
86 * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
87 * Umea University, S-901 87 Umea, Sweden.
88 *
89 * This routine is a further developed implementation of algorithm
90 * BSOLVE in [1] using complete pivoting in the LU factorization.
91 *
92 * [1] Bo Kagstrom and Lars Westin,
93 * Generalized Schur Methods with Condition Estimators for
94 * Solving the Generalized Sylvester Equation, IEEE Transactions
95 * on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
96 *
97 * [2] Peter Poromaa,
98 * On Efficient and Robust Estimators for the Separation
99 * between two Regular Matrix Pairs with Applications in
100 * Condition Estimation. Report IMINF-95.05, Departement of
101 * Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
102 *
103 * =====================================================================
104 *
105 * .. Parameters ..
106 INTEGER MAXDIM
107 PARAMETER ( MAXDIM = 8 )
108 DOUBLE PRECISION ZERO, ONE
109 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
110 * ..
111 * .. Local Scalars ..
112 INTEGER I, INFO, J, K
113 DOUBLE PRECISION BM, BP, PMONE, SMINU, SPLUS, TEMP
114 * ..
115 * .. Local Arrays ..
116 INTEGER IWORK( MAXDIM )
117 DOUBLE PRECISION WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
118 * ..
119 * .. External Subroutines ..
120 EXTERNAL DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP,
121 $ DSCAL
122 * ..
123 * .. External Functions ..
124 DOUBLE PRECISION DASUM, DDOT
125 EXTERNAL DASUM, DDOT
126 * ..
127 * .. Intrinsic Functions ..
128 INTRINSIC ABS, SQRT
129 * ..
130 * .. Executable Statements ..
131 *
132 IF( IJOB.NE.2 ) THEN
133 *
134 * Apply permutations IPIV to RHS
135 *
136 CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
137 *
138 * Solve for L-part choosing RHS either to +1 or -1.
139 *
140 PMONE = -ONE
141 *
142 DO 10 J = 1, N - 1
143 BP = RHS( J ) + ONE
144 BM = RHS( J ) - ONE
145 SPLUS = ONE
146 *
147 * Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
148 * SMIN computed more efficiently than in BSOLVE [1].
149 *
150 SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 )
151 SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
152 SPLUS = SPLUS*RHS( J )
153 IF( SPLUS.GT.SMINU ) THEN
154 RHS( J ) = BP
155 ELSE IF( SMINU.GT.SPLUS ) THEN
156 RHS( J ) = BM
157 ELSE
158 *
159 * In this case the updating sums are equal and we can
160 * choose RHS(J) +1 or -1. The first time this happens
161 * we choose -1, thereafter +1. This is a simple way to
162 * get good estimates of matrices like Byers well-known
163 * example (see [1]). (Not done in BSOLVE.)
164 *
165 RHS( J ) = RHS( J ) + PMONE
166 PMONE = ONE
167 END IF
168 *
169 * Compute the remaining r.h.s.
170 *
171 TEMP = -RHS( J )
172 CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
173 *
174 10 CONTINUE
175 *
176 * Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
177 * in BSOLVE and will hopefully give us a better estimate because
178 * any ill-conditioning of the original matrix is transfered to U
179 * and not to L. U(N, N) is an approximation to sigma_min(LU).
180 *
181 CALL DCOPY( N-1, RHS, 1, XP, 1 )
182 XP( N ) = RHS( N ) + ONE
183 RHS( N ) = RHS( N ) - ONE
184 SPLUS = ZERO
185 SMINU = ZERO
186 DO 30 I = N, 1, -1
187 TEMP = ONE / Z( I, I )
188 XP( I ) = XP( I )*TEMP
189 RHS( I ) = RHS( I )*TEMP
190 DO 20 K = I + 1, N
191 XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
192 RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
193 20 CONTINUE
194 SPLUS = SPLUS + ABS( XP( I ) )
195 SMINU = SMINU + ABS( RHS( I ) )
196 30 CONTINUE
197 IF( SPLUS.GT.SMINU )
198 $ CALL DCOPY( N, XP, 1, RHS, 1 )
199 *
200 * Apply the permutations JPIV to the computed solution (RHS)
201 *
202 CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
203 *
204 * Compute the sum of squares
205 *
206 CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
207 *
208 ELSE
209 *
210 * IJOB = 2, Compute approximate nullvector XM of Z
211 *
212 CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
213 CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
214 *
215 * Compute RHS
216 *
217 CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
218 TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
219 CALL DSCAL( N, TEMP, XM, 1 )
220 CALL DCOPY( N, XM, 1, XP, 1 )
221 CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
222 CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
223 CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
224 CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
225 IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
226 $ CALL DCOPY( N, XP, 1, RHS, 1 )
227 *
228 * Compute the sum of squares
229 *
230 CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
231 *
232 END IF
233 *
234 RETURN
235 *
236 * End of DLATDF
237 *
238 END
2 $ JPIV )
3 *
4 * -- LAPACK auxiliary routine (version 3.2.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * June 2010
8 *
9 * .. Scalar Arguments ..
10 INTEGER IJOB, LDZ, N
11 DOUBLE PRECISION RDSCAL, RDSUM
12 * ..
13 * .. Array Arguments ..
14 INTEGER IPIV( * ), JPIV( * )
15 DOUBLE PRECISION RHS( * ), Z( LDZ, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DLATDF uses the LU factorization of the n-by-n matrix Z computed by
22 * DGETC2 and computes a contribution to the reciprocal Dif-estimate
23 * by solving Z * x = b for x, and choosing the r.h.s. b such that
24 * the norm of x is as large as possible. On entry RHS = b holds the
25 * contribution from earlier solved sub-systems, and on return RHS = x.
26 *
27 * The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
28 * where P and Q are permutation matrices. L is lower triangular with
29 * 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 DGECON, e is normalized and solve for
37 * Zx = +-e - f with the sign giving the greater value
38 * of 2-norm(x). About 5 times as expensive as Default.
39 * IJOB .ne. 2: Local look ahead strategy where all entries of
40 * the r.h.s. b is choosen as either +1 or -1 (Default).
41 *
42 * N (input) INTEGER
43 * The number of columns of the matrix Z.
44 *
45 * Z (input) DOUBLE PRECISION array, dimension (LDZ, N)
46 * On entry, the LU part of the factorization of the n-by-n
47 * matrix Z computed by DGETC2: Z = P * L * U * Q
48 *
49 * LDZ (input) INTEGER
50 * The leading dimension of the array Z. LDA >= max(1, N).
51 *
52 * RHS (input/output) DOUBLE PRECISION array, dimension (N)
53 * On entry, RHS contains contributions from other subsystems.
54 * On exit, RHS contains the solution of the subsystem with
55 * entries acoording to the value of IJOB (see above).
56 *
57 * RDSUM (input/output) DOUBLE PRECISION
58 * On entry, the sum of squares of computed contributions to
59 * the Dif-estimate under computation by DTGSYL, where the
60 * scaling factor RDSCAL (see below) has been factored out.
61 * On exit, the corresponding sum of squares updated with the
62 * contributions from the current sub-system.
63 * If TRANS = 'T' RDSUM is not touched.
64 * NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
65 *
66 * RDSCAL (input/output) DOUBLE PRECISION
67 * On entry, scaling factor used to prevent overflow in RDSUM.
68 * On exit, RDSCAL is updated w.r.t. the current contributions
69 * in RDSUM.
70 * If TRANS = 'T', RDSCAL is not touched.
71 * NOTE: RDSCAL only makes sense when DTGSY2 is called by
72 * DTGSYL.
73 *
74 * IPIV (input) INTEGER array, dimension (N).
75 * The pivot indices; for 1 <= i <= N, row i of the
76 * matrix has been interchanged with row IPIV(i).
77 *
78 * JPIV (input) INTEGER array, dimension (N).
79 * The pivot indices; for 1 <= j <= N, column j of the
80 * matrix has been interchanged with column JPIV(j).
81 *
82 * Further Details
83 * ===============
84 *
85 * Based on contributions by
86 * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
87 * Umea University, S-901 87 Umea, Sweden.
88 *
89 * This routine is a further developed implementation of algorithm
90 * BSOLVE in [1] using complete pivoting in the LU factorization.
91 *
92 * [1] Bo Kagstrom and Lars Westin,
93 * Generalized Schur Methods with Condition Estimators for
94 * Solving the Generalized Sylvester Equation, IEEE Transactions
95 * on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
96 *
97 * [2] Peter Poromaa,
98 * On Efficient and Robust Estimators for the Separation
99 * between two Regular Matrix Pairs with Applications in
100 * Condition Estimation. Report IMINF-95.05, Departement of
101 * Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
102 *
103 * =====================================================================
104 *
105 * .. Parameters ..
106 INTEGER MAXDIM
107 PARAMETER ( MAXDIM = 8 )
108 DOUBLE PRECISION ZERO, ONE
109 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
110 * ..
111 * .. Local Scalars ..
112 INTEGER I, INFO, J, K
113 DOUBLE PRECISION BM, BP, PMONE, SMINU, SPLUS, TEMP
114 * ..
115 * .. Local Arrays ..
116 INTEGER IWORK( MAXDIM )
117 DOUBLE PRECISION WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
118 * ..
119 * .. External Subroutines ..
120 EXTERNAL DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP,
121 $ DSCAL
122 * ..
123 * .. External Functions ..
124 DOUBLE PRECISION DASUM, DDOT
125 EXTERNAL DASUM, DDOT
126 * ..
127 * .. Intrinsic Functions ..
128 INTRINSIC ABS, SQRT
129 * ..
130 * .. Executable Statements ..
131 *
132 IF( IJOB.NE.2 ) THEN
133 *
134 * Apply permutations IPIV to RHS
135 *
136 CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
137 *
138 * Solve for L-part choosing RHS either to +1 or -1.
139 *
140 PMONE = -ONE
141 *
142 DO 10 J = 1, N - 1
143 BP = RHS( J ) + ONE
144 BM = RHS( J ) - ONE
145 SPLUS = ONE
146 *
147 * Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
148 * SMIN computed more efficiently than in BSOLVE [1].
149 *
150 SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 )
151 SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
152 SPLUS = SPLUS*RHS( J )
153 IF( SPLUS.GT.SMINU ) THEN
154 RHS( J ) = BP
155 ELSE IF( SMINU.GT.SPLUS ) THEN
156 RHS( J ) = BM
157 ELSE
158 *
159 * In this case the updating sums are equal and we can
160 * choose RHS(J) +1 or -1. The first time this happens
161 * we choose -1, thereafter +1. This is a simple way to
162 * get good estimates of matrices like Byers well-known
163 * example (see [1]). (Not done in BSOLVE.)
164 *
165 RHS( J ) = RHS( J ) + PMONE
166 PMONE = ONE
167 END IF
168 *
169 * Compute the remaining r.h.s.
170 *
171 TEMP = -RHS( J )
172 CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
173 *
174 10 CONTINUE
175 *
176 * Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
177 * in BSOLVE and will hopefully give us a better estimate because
178 * any ill-conditioning of the original matrix is transfered to U
179 * and not to L. U(N, N) is an approximation to sigma_min(LU).
180 *
181 CALL DCOPY( N-1, RHS, 1, XP, 1 )
182 XP( N ) = RHS( N ) + ONE
183 RHS( N ) = RHS( N ) - ONE
184 SPLUS = ZERO
185 SMINU = ZERO
186 DO 30 I = N, 1, -1
187 TEMP = ONE / Z( I, I )
188 XP( I ) = XP( I )*TEMP
189 RHS( I ) = RHS( I )*TEMP
190 DO 20 K = I + 1, N
191 XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
192 RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
193 20 CONTINUE
194 SPLUS = SPLUS + ABS( XP( I ) )
195 SMINU = SMINU + ABS( RHS( I ) )
196 30 CONTINUE
197 IF( SPLUS.GT.SMINU )
198 $ CALL DCOPY( N, XP, 1, RHS, 1 )
199 *
200 * Apply the permutations JPIV to the computed solution (RHS)
201 *
202 CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
203 *
204 * Compute the sum of squares
205 *
206 CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
207 *
208 ELSE
209 *
210 * IJOB = 2, Compute approximate nullvector XM of Z
211 *
212 CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
213 CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
214 *
215 * Compute RHS
216 *
217 CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
218 TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
219 CALL DSCAL( N, TEMP, XM, 1 )
220 CALL DCOPY( N, XM, 1, XP, 1 )
221 CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
222 CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
223 CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
224 CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
225 IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
226 $ CALL DCOPY( N, XP, 1, RHS, 1 )
227 *
228 * Compute the sum of squares
229 *
230 CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
231 *
232 END IF
233 *
234 RETURN
235 *
236 * End of DLATDF
237 *
238 END