1 SUBROUTINE ZPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
2 $ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
3 $ RWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, UPLO
12 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
17 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
25 * compute the solution to a complex system of linear equations
26 * A * X = B,
27 * where A is an N-by-N Hermitian positive definite matrix and X and B
28 * are N-by-NRHS matrices.
29 *
30 * Error bounds on the solution and a condition estimate are also
31 * provided.
32 *
33 * Description
34 * ===========
35 *
36 * The following steps are performed:
37 *
38 * 1. If FACT = 'E', real scaling factors are computed to equilibrate
39 * the system:
40 * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
41 * Whether or not the system will be equilibrated depends on the
42 * scaling of the matrix A, but if equilibration is used, A is
43 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
44 *
45 * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
46 * factor the matrix A (after equilibration if FACT = 'E') as
47 * A = U**H* U, if UPLO = 'U', or
48 * A = L * L**H, if UPLO = 'L',
49 * where U is an upper triangular matrix and L is a lower triangular
50 * matrix.
51 *
52 * 3. If the leading i-by-i principal minor is not positive definite,
53 * then the routine returns with INFO = i. Otherwise, the factored
54 * form of A is used to estimate the condition number of the matrix
55 * A. If the reciprocal of the condition number is less than machine
56 * precision, INFO = N+1 is returned as a warning, but the routine
57 * still goes on to solve for X and compute error bounds as
58 * described below.
59 *
60 * 4. The system of equations is solved for X using the factored form
61 * of A.
62 *
63 * 5. Iterative refinement is applied to improve the computed solution
64 * matrix and calculate error bounds and backward error estimates
65 * for it.
66 *
67 * 6. If equilibration was used, the matrix X is premultiplied by
68 * diag(S) so that it solves the original system before
69 * equilibration.
70 *
71 * Arguments
72 * =========
73 *
74 * FACT (input) CHARACTER*1
75 * Specifies whether or not the factored form of the matrix A is
76 * supplied on entry, and if not, whether the matrix A should be
77 * equilibrated before it is factored.
78 * = 'F': On entry, AF contains the factored form of A.
79 * If EQUED = 'Y', the matrix A has been equilibrated
80 * with scaling factors given by S. A and AF will not
81 * be modified.
82 * = 'N': The matrix A will be copied to AF and factored.
83 * = 'E': The matrix A will be equilibrated if necessary, then
84 * copied to AF and factored.
85 *
86 * UPLO (input) CHARACTER*1
87 * = 'U': Upper triangle of A is stored;
88 * = 'L': Lower triangle of A is stored.
89 *
90 * N (input) INTEGER
91 * The number of linear equations, i.e., the order of the
92 * matrix A. N >= 0.
93 *
94 * NRHS (input) INTEGER
95 * The number of right hand sides, i.e., the number of columns
96 * of the matrices B and X. NRHS >= 0.
97 *
98 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
99 * On entry, the Hermitian matrix A, except if FACT = 'F' and
100 * EQUED = 'Y', then A must contain the equilibrated matrix
101 * diag(S)*A*diag(S). If UPLO = 'U', the leading
102 * N-by-N upper triangular part of A contains the upper
103 * triangular part of the matrix A, and the strictly lower
104 * triangular part of A is not referenced. If UPLO = 'L', the
105 * leading N-by-N lower triangular part of A contains the lower
106 * triangular part of the matrix A, and the strictly upper
107 * triangular part of A is not referenced. A is not modified if
108 * FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
109 *
110 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
111 * diag(S)*A*diag(S).
112 *
113 * LDA (input) INTEGER
114 * The leading dimension of the array A. LDA >= max(1,N).
115 *
116 * AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
117 * If FACT = 'F', then AF is an input argument and on entry
118 * contains the triangular factor U or L from the Cholesky
119 * factorization A = U**H *U or A = L*L**H, in the same storage
120 * format as A. If EQUED .ne. 'N', then AF is the factored form
121 * of the equilibrated matrix diag(S)*A*diag(S).
122 *
123 * If FACT = 'N', then AF is an output argument and on exit
124 * returns the triangular factor U or L from the Cholesky
125 * factorization A = U**H *U or A = L*L**H of the original
126 * matrix A.
127 *
128 * If FACT = 'E', then AF is an output argument and on exit
129 * returns the triangular factor U or L from the Cholesky
130 * factorization A = U**H *U or A = L*L**H of the equilibrated
131 * matrix A (see the description of A for the form of the
132 * equilibrated matrix).
133 *
134 * LDAF (input) INTEGER
135 * The leading dimension of the array AF. LDAF >= max(1,N).
136 *
137 * EQUED (input or output) CHARACTER*1
138 * Specifies the form of equilibration that was done.
139 * = 'N': No equilibration (always true if FACT = 'N').
140 * = 'Y': Equilibration was done, i.e., A has been replaced by
141 * diag(S) * A * diag(S).
142 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
143 * output argument.
144 *
145 * S (input or output) DOUBLE PRECISION array, dimension (N)
146 * The scale factors for A; not accessed if EQUED = 'N'. S is
147 * an input argument if FACT = 'F'; otherwise, S is an output
148 * argument. If FACT = 'F' and EQUED = 'Y', each element of S
149 * must be positive.
150 *
151 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
152 * On entry, the N-by-NRHS righthand side matrix B.
153 * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
154 * B is overwritten by diag(S) * B.
155 *
156 * LDB (input) INTEGER
157 * The leading dimension of the array B. LDB >= max(1,N).
158 *
159 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
160 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
161 * the original system of equations. Note that if EQUED = 'Y',
162 * A and B are modified on exit, and the solution to the
163 * equilibrated system is inv(diag(S))*X.
164 *
165 * LDX (input) INTEGER
166 * The leading dimension of the array X. LDX >= max(1,N).
167 *
168 * RCOND (output) DOUBLE PRECISION
169 * The estimate of the reciprocal condition number of the matrix
170 * A after equilibration (if done). If RCOND is less than the
171 * machine precision (in particular, if RCOND = 0), the matrix
172 * is singular to working precision. This condition is
173 * indicated by a return code of INFO > 0.
174 *
175 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
176 * The estimated forward error bound for each solution vector
177 * X(j) (the j-th column of the solution matrix X).
178 * If XTRUE is the true solution corresponding to X(j), FERR(j)
179 * is an estimated upper bound for the magnitude of the largest
180 * element in (X(j) - XTRUE) divided by the magnitude of the
181 * largest element in X(j). The estimate is as reliable as
182 * the estimate for RCOND, and is almost always a slight
183 * overestimate of the true error.
184 *
185 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
186 * The componentwise relative backward error of each solution
187 * vector X(j) (i.e., the smallest relative change in
188 * any element of A or B that makes X(j) an exact solution).
189 *
190 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
191 *
192 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
193 *
194 * INFO (output) INTEGER
195 * = 0: successful exit
196 * < 0: if INFO = -i, the i-th argument had an illegal value
197 * > 0: if INFO = i, and i is
198 * <= N: the leading minor of order i of A is
199 * not positive definite, so the factorization
200 * could not be completed, and the solution has not
201 * been computed. RCOND = 0 is returned.
202 * = N+1: U is nonsingular, but RCOND is less than machine
203 * precision, meaning that the matrix is singular
204 * to working precision. Nevertheless, the
205 * solution and error bounds are computed because
206 * there are a number of situations where the
207 * computed solution can be more accurate than the
208 * value of RCOND would suggest.
209 *
210 * =====================================================================
211 *
212 * .. Parameters ..
213 DOUBLE PRECISION ZERO, ONE
214 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
215 * ..
216 * .. Local Scalars ..
217 LOGICAL EQUIL, NOFACT, RCEQU
218 INTEGER I, INFEQU, J
219 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
220 * ..
221 * .. External Functions ..
222 LOGICAL LSAME
223 DOUBLE PRECISION DLAMCH, ZLANHE
224 EXTERNAL LSAME, DLAMCH, ZLANHE
225 * ..
226 * .. External Subroutines ..
227 EXTERNAL XERBLA, ZLACPY, ZLAQHE, ZPOCON, ZPOEQU, ZPORFS,
228 $ ZPOTRF, ZPOTRS
229 * ..
230 * .. Intrinsic Functions ..
231 INTRINSIC MAX, MIN
232 * ..
233 * .. Executable Statements ..
234 *
235 INFO = 0
236 NOFACT = LSAME( FACT, 'N' )
237 EQUIL = LSAME( FACT, 'E' )
238 IF( NOFACT .OR. EQUIL ) THEN
239 EQUED = 'N'
240 RCEQU = .FALSE.
241 ELSE
242 RCEQU = LSAME( EQUED, 'Y' )
243 SMLNUM = DLAMCH( 'Safe minimum' )
244 BIGNUM = ONE / SMLNUM
245 END IF
246 *
247 * Test the input parameters.
248 *
249 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
250 $ THEN
251 INFO = -1
252 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
253 $ THEN
254 INFO = -2
255 ELSE IF( N.LT.0 ) THEN
256 INFO = -3
257 ELSE IF( NRHS.LT.0 ) THEN
258 INFO = -4
259 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
260 INFO = -6
261 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
262 INFO = -8
263 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
264 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
265 INFO = -9
266 ELSE
267 IF( RCEQU ) THEN
268 SMIN = BIGNUM
269 SMAX = ZERO
270 DO 10 J = 1, N
271 SMIN = MIN( SMIN, S( J ) )
272 SMAX = MAX( SMAX, S( J ) )
273 10 CONTINUE
274 IF( SMIN.LE.ZERO ) THEN
275 INFO = -10
276 ELSE IF( N.GT.0 ) THEN
277 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
278 ELSE
279 SCOND = ONE
280 END IF
281 END IF
282 IF( INFO.EQ.0 ) THEN
283 IF( LDB.LT.MAX( 1, N ) ) THEN
284 INFO = -12
285 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
286 INFO = -14
287 END IF
288 END IF
289 END IF
290 *
291 IF( INFO.NE.0 ) THEN
292 CALL XERBLA( 'ZPOSVX', -INFO )
293 RETURN
294 END IF
295 *
296 IF( EQUIL ) THEN
297 *
298 * Compute row and column scalings to equilibrate the matrix A.
299 *
300 CALL ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU )
301 IF( INFEQU.EQ.0 ) THEN
302 *
303 * Equilibrate the matrix.
304 *
305 CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
306 RCEQU = LSAME( EQUED, 'Y' )
307 END IF
308 END IF
309 *
310 * Scale the right hand side.
311 *
312 IF( RCEQU ) THEN
313 DO 30 J = 1, NRHS
314 DO 20 I = 1, N
315 B( I, J ) = S( I )*B( I, J )
316 20 CONTINUE
317 30 CONTINUE
318 END IF
319 *
320 IF( NOFACT .OR. EQUIL ) THEN
321 *
322 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
323 *
324 CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
325 CALL ZPOTRF( UPLO, N, AF, LDAF, INFO )
326 *
327 * Return if INFO is non-zero.
328 *
329 IF( INFO.GT.0 )THEN
330 RCOND = ZERO
331 RETURN
332 END IF
333 END IF
334 *
335 * Compute the norm of the matrix A.
336 *
337 ANORM = ZLANHE( '1', UPLO, N, A, LDA, RWORK )
338 *
339 * Compute the reciprocal of the condition number of A.
340 *
341 CALL ZPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
342 *
343 * Compute the solution matrix X.
344 *
345 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
346 CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
347 *
348 * Use iterative refinement to improve the computed solution and
349 * compute error bounds and backward error estimates for it.
350 *
351 CALL ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
352 $ FERR, BERR, WORK, RWORK, INFO )
353 *
354 * Transform the solution matrix X to a solution of the original
355 * system.
356 *
357 IF( RCEQU ) THEN
358 DO 50 J = 1, NRHS
359 DO 40 I = 1, N
360 X( I, J ) = S( I )*X( I, J )
361 40 CONTINUE
362 50 CONTINUE
363 DO 60 J = 1, NRHS
364 FERR( J ) = FERR( J ) / SCOND
365 60 CONTINUE
366 END IF
367 *
368 * Set INFO = N+1 if the matrix is singular to working precision.
369 *
370 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
371 $ INFO = N + 1
372 *
373 RETURN
374 *
375 * End of ZPOSVX
376 *
377 END
2 $ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
3 $ RWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, UPLO
12 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
17 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
25 * compute the solution to a complex system of linear equations
26 * A * X = B,
27 * where A is an N-by-N Hermitian positive definite matrix and X and B
28 * are N-by-NRHS matrices.
29 *
30 * Error bounds on the solution and a condition estimate are also
31 * provided.
32 *
33 * Description
34 * ===========
35 *
36 * The following steps are performed:
37 *
38 * 1. If FACT = 'E', real scaling factors are computed to equilibrate
39 * the system:
40 * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
41 * Whether or not the system will be equilibrated depends on the
42 * scaling of the matrix A, but if equilibration is used, A is
43 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
44 *
45 * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
46 * factor the matrix A (after equilibration if FACT = 'E') as
47 * A = U**H* U, if UPLO = 'U', or
48 * A = L * L**H, if UPLO = 'L',
49 * where U is an upper triangular matrix and L is a lower triangular
50 * matrix.
51 *
52 * 3. If the leading i-by-i principal minor is not positive definite,
53 * then the routine returns with INFO = i. Otherwise, the factored
54 * form of A is used to estimate the condition number of the matrix
55 * A. If the reciprocal of the condition number is less than machine
56 * precision, INFO = N+1 is returned as a warning, but the routine
57 * still goes on to solve for X and compute error bounds as
58 * described below.
59 *
60 * 4. The system of equations is solved for X using the factored form
61 * of A.
62 *
63 * 5. Iterative refinement is applied to improve the computed solution
64 * matrix and calculate error bounds and backward error estimates
65 * for it.
66 *
67 * 6. If equilibration was used, the matrix X is premultiplied by
68 * diag(S) so that it solves the original system before
69 * equilibration.
70 *
71 * Arguments
72 * =========
73 *
74 * FACT (input) CHARACTER*1
75 * Specifies whether or not the factored form of the matrix A is
76 * supplied on entry, and if not, whether the matrix A should be
77 * equilibrated before it is factored.
78 * = 'F': On entry, AF contains the factored form of A.
79 * If EQUED = 'Y', the matrix A has been equilibrated
80 * with scaling factors given by S. A and AF will not
81 * be modified.
82 * = 'N': The matrix A will be copied to AF and factored.
83 * = 'E': The matrix A will be equilibrated if necessary, then
84 * copied to AF and factored.
85 *
86 * UPLO (input) CHARACTER*1
87 * = 'U': Upper triangle of A is stored;
88 * = 'L': Lower triangle of A is stored.
89 *
90 * N (input) INTEGER
91 * The number of linear equations, i.e., the order of the
92 * matrix A. N >= 0.
93 *
94 * NRHS (input) INTEGER
95 * The number of right hand sides, i.e., the number of columns
96 * of the matrices B and X. NRHS >= 0.
97 *
98 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
99 * On entry, the Hermitian matrix A, except if FACT = 'F' and
100 * EQUED = 'Y', then A must contain the equilibrated matrix
101 * diag(S)*A*diag(S). If UPLO = 'U', the leading
102 * N-by-N upper triangular part of A contains the upper
103 * triangular part of the matrix A, and the strictly lower
104 * triangular part of A is not referenced. If UPLO = 'L', the
105 * leading N-by-N lower triangular part of A contains the lower
106 * triangular part of the matrix A, and the strictly upper
107 * triangular part of A is not referenced. A is not modified if
108 * FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
109 *
110 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
111 * diag(S)*A*diag(S).
112 *
113 * LDA (input) INTEGER
114 * The leading dimension of the array A. LDA >= max(1,N).
115 *
116 * AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
117 * If FACT = 'F', then AF is an input argument and on entry
118 * contains the triangular factor U or L from the Cholesky
119 * factorization A = U**H *U or A = L*L**H, in the same storage
120 * format as A. If EQUED .ne. 'N', then AF is the factored form
121 * of the equilibrated matrix diag(S)*A*diag(S).
122 *
123 * If FACT = 'N', then AF is an output argument and on exit
124 * returns the triangular factor U or L from the Cholesky
125 * factorization A = U**H *U or A = L*L**H of the original
126 * matrix A.
127 *
128 * If FACT = 'E', then AF is an output argument and on exit
129 * returns the triangular factor U or L from the Cholesky
130 * factorization A = U**H *U or A = L*L**H of the equilibrated
131 * matrix A (see the description of A for the form of the
132 * equilibrated matrix).
133 *
134 * LDAF (input) INTEGER
135 * The leading dimension of the array AF. LDAF >= max(1,N).
136 *
137 * EQUED (input or output) CHARACTER*1
138 * Specifies the form of equilibration that was done.
139 * = 'N': No equilibration (always true if FACT = 'N').
140 * = 'Y': Equilibration was done, i.e., A has been replaced by
141 * diag(S) * A * diag(S).
142 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
143 * output argument.
144 *
145 * S (input or output) DOUBLE PRECISION array, dimension (N)
146 * The scale factors for A; not accessed if EQUED = 'N'. S is
147 * an input argument if FACT = 'F'; otherwise, S is an output
148 * argument. If FACT = 'F' and EQUED = 'Y', each element of S
149 * must be positive.
150 *
151 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
152 * On entry, the N-by-NRHS righthand side matrix B.
153 * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
154 * B is overwritten by diag(S) * B.
155 *
156 * LDB (input) INTEGER
157 * The leading dimension of the array B. LDB >= max(1,N).
158 *
159 * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
160 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
161 * the original system of equations. Note that if EQUED = 'Y',
162 * A and B are modified on exit, and the solution to the
163 * equilibrated system is inv(diag(S))*X.
164 *
165 * LDX (input) INTEGER
166 * The leading dimension of the array X. LDX >= max(1,N).
167 *
168 * RCOND (output) DOUBLE PRECISION
169 * The estimate of the reciprocal condition number of the matrix
170 * A after equilibration (if done). If RCOND is less than the
171 * machine precision (in particular, if RCOND = 0), the matrix
172 * is singular to working precision. This condition is
173 * indicated by a return code of INFO > 0.
174 *
175 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
176 * The estimated forward error bound for each solution vector
177 * X(j) (the j-th column of the solution matrix X).
178 * If XTRUE is the true solution corresponding to X(j), FERR(j)
179 * is an estimated upper bound for the magnitude of the largest
180 * element in (X(j) - XTRUE) divided by the magnitude of the
181 * largest element in X(j). The estimate is as reliable as
182 * the estimate for RCOND, and is almost always a slight
183 * overestimate of the true error.
184 *
185 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
186 * The componentwise relative backward error of each solution
187 * vector X(j) (i.e., the smallest relative change in
188 * any element of A or B that makes X(j) an exact solution).
189 *
190 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
191 *
192 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
193 *
194 * INFO (output) INTEGER
195 * = 0: successful exit
196 * < 0: if INFO = -i, the i-th argument had an illegal value
197 * > 0: if INFO = i, and i is
198 * <= N: the leading minor of order i of A is
199 * not positive definite, so the factorization
200 * could not be completed, and the solution has not
201 * been computed. RCOND = 0 is returned.
202 * = N+1: U is nonsingular, but RCOND is less than machine
203 * precision, meaning that the matrix is singular
204 * to working precision. Nevertheless, the
205 * solution and error bounds are computed because
206 * there are a number of situations where the
207 * computed solution can be more accurate than the
208 * value of RCOND would suggest.
209 *
210 * =====================================================================
211 *
212 * .. Parameters ..
213 DOUBLE PRECISION ZERO, ONE
214 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
215 * ..
216 * .. Local Scalars ..
217 LOGICAL EQUIL, NOFACT, RCEQU
218 INTEGER I, INFEQU, J
219 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
220 * ..
221 * .. External Functions ..
222 LOGICAL LSAME
223 DOUBLE PRECISION DLAMCH, ZLANHE
224 EXTERNAL LSAME, DLAMCH, ZLANHE
225 * ..
226 * .. External Subroutines ..
227 EXTERNAL XERBLA, ZLACPY, ZLAQHE, ZPOCON, ZPOEQU, ZPORFS,
228 $ ZPOTRF, ZPOTRS
229 * ..
230 * .. Intrinsic Functions ..
231 INTRINSIC MAX, MIN
232 * ..
233 * .. Executable Statements ..
234 *
235 INFO = 0
236 NOFACT = LSAME( FACT, 'N' )
237 EQUIL = LSAME( FACT, 'E' )
238 IF( NOFACT .OR. EQUIL ) THEN
239 EQUED = 'N'
240 RCEQU = .FALSE.
241 ELSE
242 RCEQU = LSAME( EQUED, 'Y' )
243 SMLNUM = DLAMCH( 'Safe minimum' )
244 BIGNUM = ONE / SMLNUM
245 END IF
246 *
247 * Test the input parameters.
248 *
249 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
250 $ THEN
251 INFO = -1
252 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
253 $ THEN
254 INFO = -2
255 ELSE IF( N.LT.0 ) THEN
256 INFO = -3
257 ELSE IF( NRHS.LT.0 ) THEN
258 INFO = -4
259 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
260 INFO = -6
261 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
262 INFO = -8
263 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
264 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
265 INFO = -9
266 ELSE
267 IF( RCEQU ) THEN
268 SMIN = BIGNUM
269 SMAX = ZERO
270 DO 10 J = 1, N
271 SMIN = MIN( SMIN, S( J ) )
272 SMAX = MAX( SMAX, S( J ) )
273 10 CONTINUE
274 IF( SMIN.LE.ZERO ) THEN
275 INFO = -10
276 ELSE IF( N.GT.0 ) THEN
277 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
278 ELSE
279 SCOND = ONE
280 END IF
281 END IF
282 IF( INFO.EQ.0 ) THEN
283 IF( LDB.LT.MAX( 1, N ) ) THEN
284 INFO = -12
285 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
286 INFO = -14
287 END IF
288 END IF
289 END IF
290 *
291 IF( INFO.NE.0 ) THEN
292 CALL XERBLA( 'ZPOSVX', -INFO )
293 RETURN
294 END IF
295 *
296 IF( EQUIL ) THEN
297 *
298 * Compute row and column scalings to equilibrate the matrix A.
299 *
300 CALL ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU )
301 IF( INFEQU.EQ.0 ) THEN
302 *
303 * Equilibrate the matrix.
304 *
305 CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
306 RCEQU = LSAME( EQUED, 'Y' )
307 END IF
308 END IF
309 *
310 * Scale the right hand side.
311 *
312 IF( RCEQU ) THEN
313 DO 30 J = 1, NRHS
314 DO 20 I = 1, N
315 B( I, J ) = S( I )*B( I, J )
316 20 CONTINUE
317 30 CONTINUE
318 END IF
319 *
320 IF( NOFACT .OR. EQUIL ) THEN
321 *
322 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
323 *
324 CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
325 CALL ZPOTRF( UPLO, N, AF, LDAF, INFO )
326 *
327 * Return if INFO is non-zero.
328 *
329 IF( INFO.GT.0 )THEN
330 RCOND = ZERO
331 RETURN
332 END IF
333 END IF
334 *
335 * Compute the norm of the matrix A.
336 *
337 ANORM = ZLANHE( '1', UPLO, N, A, LDA, RWORK )
338 *
339 * Compute the reciprocal of the condition number of A.
340 *
341 CALL ZPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
342 *
343 * Compute the solution matrix X.
344 *
345 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
346 CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
347 *
348 * Use iterative refinement to improve the computed solution and
349 * compute error bounds and backward error estimates for it.
350 *
351 CALL ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
352 $ FERR, BERR, WORK, RWORK, INFO )
353 *
354 * Transform the solution matrix X to a solution of the original
355 * system.
356 *
357 IF( RCEQU ) THEN
358 DO 50 J = 1, NRHS
359 DO 40 I = 1, N
360 X( I, J ) = S( I )*X( I, J )
361 40 CONTINUE
362 50 CONTINUE
363 DO 60 J = 1, NRHS
364 FERR( J ) = FERR( J ) / SCOND
365 60 CONTINUE
366 END IF
367 *
368 * Set INFO = N+1 if the matrix is singular to working precision.
369 *
370 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
371 $ INFO = N + 1
372 *
373 RETURN
374 *
375 * End of ZPOSVX
376 *
377 END