1 SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
2 $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
3 $ WORK, IWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, TRANS
12 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * ), IWORK( * )
17 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ BERR( * ), C( * ), FERR( * ), R( * ),
19 $ WORK( * ), X( LDX, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * DGESVX uses the LU factorization to compute the solution to a real
26 * system of linear equations
27 * A * X = B,
28 * where A is an N-by-N matrix and X and B 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 * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
41 * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
42 * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
43 * Whether or not the system will be equilibrated depends on the
44 * scaling of the matrix A, but if equilibration is used, A is
45 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
46 * or diag(C)*B (if TRANS = 'T' or 'C').
47 *
48 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
49 * matrix A (after equilibration if FACT = 'E') as
50 * A = P * L * U,
51 * where P is a permutation matrix, L is a unit lower triangular
52 * matrix, and U is upper triangular.
53 *
54 * 3. If some U(i,i)=0, so that U is exactly singular, then the routine
55 * returns with INFO = i. Otherwise, the factored form of A is used
56 * to estimate the condition number of the matrix A. If the
57 * reciprocal of the condition number is less than machine precision,
58 * INFO = N+1 is returned as a warning, but the routine still goes on
59 * to solve for X and compute error bounds as described below.
60 *
61 * 4. The system of equations is solved for X using the factored form
62 * of A.
63 *
64 * 5. Iterative refinement is applied to improve the computed solution
65 * matrix and calculate error bounds and backward error estimates
66 * for it.
67 *
68 * 6. If equilibration was used, the matrix X is premultiplied by
69 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
70 * that it solves the original system before equilibration.
71 *
72 * Arguments
73 * =========
74 *
75 * FACT (input) CHARACTER*1
76 * Specifies whether or not the factored form of the matrix A is
77 * supplied on entry, and if not, whether the matrix A should be
78 * equilibrated before it is factored.
79 * = 'F': On entry, AF and IPIV contain the factored form of A.
80 * If EQUED is not 'N', the matrix A has been
81 * equilibrated with scaling factors given by R and C.
82 * A, AF, and IPIV are not modified.
83 * = 'N': The matrix A will be copied to AF and factored.
84 * = 'E': The matrix A will be equilibrated if necessary, then
85 * copied to AF and factored.
86 *
87 * TRANS (input) CHARACTER*1
88 * Specifies the form of the system of equations:
89 * = 'N': A * X = B (No transpose)
90 * = 'T': A**T * X = B (Transpose)
91 * = 'C': A**H * X = B (Transpose)
92 *
93 * N (input) INTEGER
94 * The number of linear equations, i.e., the order of the
95 * matrix A. N >= 0.
96 *
97 * NRHS (input) INTEGER
98 * The number of right hand sides, i.e., the number of columns
99 * of the matrices B and X. NRHS >= 0.
100 *
101 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
102 * On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
103 * not 'N', then A must have been equilibrated by the scaling
104 * factors in R and/or C. A is not modified if FACT = 'F' or
105 * 'N', or if FACT = 'E' and EQUED = 'N' on exit.
106 *
107 * On exit, if EQUED .ne. 'N', A is scaled as follows:
108 * EQUED = 'R': A := diag(R) * A
109 * EQUED = 'C': A := A * diag(C)
110 * EQUED = 'B': A := diag(R) * A * diag(C).
111 *
112 * LDA (input) INTEGER
113 * The leading dimension of the array A. LDA >= max(1,N).
114 *
115 * AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
116 * If FACT = 'F', then AF is an input argument and on entry
117 * contains the factors L and U from the factorization
118 * A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
119 * AF is the factored form of the equilibrated matrix A.
120 *
121 * If FACT = 'N', then AF is an output argument and on exit
122 * returns the factors L and U from the factorization A = P*L*U
123 * of the original matrix A.
124 *
125 * If FACT = 'E', then AF is an output argument and on exit
126 * returns the factors L and U from the factorization A = P*L*U
127 * of the equilibrated matrix A (see the description of A for
128 * the form of the equilibrated matrix).
129 *
130 * LDAF (input) INTEGER
131 * The leading dimension of the array AF. LDAF >= max(1,N).
132 *
133 * IPIV (input or output) INTEGER array, dimension (N)
134 * If FACT = 'F', then IPIV is an input argument and on entry
135 * contains the pivot indices from the factorization A = P*L*U
136 * as computed by DGETRF; row i of the matrix was interchanged
137 * with row IPIV(i).
138 *
139 * If FACT = 'N', then IPIV is an output argument and on exit
140 * contains the pivot indices from the factorization A = P*L*U
141 * of the original matrix A.
142 *
143 * If FACT = 'E', then IPIV is an output argument and on exit
144 * contains the pivot indices from the factorization A = P*L*U
145 * of the equilibrated matrix A.
146 *
147 * EQUED (input or output) CHARACTER*1
148 * Specifies the form of equilibration that was done.
149 * = 'N': No equilibration (always true if FACT = 'N').
150 * = 'R': Row equilibration, i.e., A has been premultiplied by
151 * diag(R).
152 * = 'C': Column equilibration, i.e., A has been postmultiplied
153 * by diag(C).
154 * = 'B': Both row and column equilibration, i.e., A has been
155 * replaced by diag(R) * A * diag(C).
156 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
157 * output argument.
158 *
159 * R (input or output) DOUBLE PRECISION array, dimension (N)
160 * The row scale factors for A. If EQUED = 'R' or 'B', A is
161 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
162 * is not accessed. R is an input argument if FACT = 'F';
163 * otherwise, R is an output argument. If FACT = 'F' and
164 * EQUED = 'R' or 'B', each element of R must be positive.
165 *
166 * C (input or output) DOUBLE PRECISION array, dimension (N)
167 * The column scale factors for A. If EQUED = 'C' or 'B', A is
168 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
169 * is not accessed. C is an input argument if FACT = 'F';
170 * otherwise, C is an output argument. If FACT = 'F' and
171 * EQUED = 'C' or 'B', each element of C must be positive.
172 *
173 * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
174 * On entry, the N-by-NRHS right hand side matrix B.
175 * On exit,
176 * if EQUED = 'N', B is not modified;
177 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
178 * diag(R)*B;
179 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
180 * overwritten by diag(C)*B.
181 *
182 * LDB (input) INTEGER
183 * The leading dimension of the array B. LDB >= max(1,N).
184 *
185 * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
186 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
187 * to the original system of equations. Note that A and B are
188 * modified on exit if EQUED .ne. 'N', and the solution to the
189 * equilibrated system is inv(diag(C))*X if TRANS = 'N' and
190 * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
191 * and EQUED = 'R' or 'B'.
192 *
193 * LDX (input) INTEGER
194 * The leading dimension of the array X. LDX >= max(1,N).
195 *
196 * RCOND (output) DOUBLE PRECISION
197 * The estimate of the reciprocal condition number of the matrix
198 * A after equilibration (if done). If RCOND is less than the
199 * machine precision (in particular, if RCOND = 0), the matrix
200 * is singular to working precision. This condition is
201 * indicated by a return code of INFO > 0.
202 *
203 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
204 * The estimated forward error bound for each solution vector
205 * X(j) (the j-th column of the solution matrix X).
206 * If XTRUE is the true solution corresponding to X(j), FERR(j)
207 * is an estimated upper bound for the magnitude of the largest
208 * element in (X(j) - XTRUE) divided by the magnitude of the
209 * largest element in X(j). The estimate is as reliable as
210 * the estimate for RCOND, and is almost always a slight
211 * overestimate of the true error.
212 *
213 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
214 * The componentwise relative backward error of each solution
215 * vector X(j) (i.e., the smallest relative change in
216 * any element of A or B that makes X(j) an exact solution).
217 *
218 * WORK (workspace/output) DOUBLE PRECISION array, dimension (4*N)
219 * On exit, WORK(1) contains the reciprocal pivot growth
220 * factor norm(A)/norm(U). The "max absolute element" norm is
221 * used. If WORK(1) is much less than 1, then the stability
222 * of the LU factorization of the (equilibrated) matrix A
223 * could be poor. This also means that the solution X, condition
224 * estimator RCOND, and forward error bound FERR could be
225 * unreliable. If factorization fails with 0<INFO<=N, then
226 * WORK(1) contains the reciprocal pivot growth factor for the
227 * leading INFO columns of A.
228 *
229 * IWORK (workspace) INTEGER array, dimension (N)
230 *
231 * INFO (output) INTEGER
232 * = 0: successful exit
233 * < 0: if INFO = -i, the i-th argument had an illegal value
234 * > 0: if INFO = i, and i is
235 * <= N: U(i,i) is exactly zero. The factorization has
236 * been completed, but the factor U is exactly
237 * singular, so the solution and error bounds
238 * could not be computed. RCOND = 0 is returned.
239 * = N+1: U is nonsingular, but RCOND is less than machine
240 * precision, meaning that the matrix is singular
241 * to working precision. Nevertheless, the
242 * solution and error bounds are computed because
243 * there are a number of situations where the
244 * computed solution can be more accurate than the
245 * value of RCOND would suggest.
246 *
247 * =====================================================================
248 *
249 * .. Parameters ..
250 DOUBLE PRECISION ZERO, ONE
251 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
252 * ..
253 * .. Local Scalars ..
254 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
255 CHARACTER NORM
256 INTEGER I, INFEQU, J
257 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
258 $ ROWCND, RPVGRW, SMLNUM
259 * ..
260 * .. External Functions ..
261 LOGICAL LSAME
262 DOUBLE PRECISION DLAMCH, DLANGE, DLANTR
263 EXTERNAL LSAME, DLAMCH, DLANGE, DLANTR
264 * ..
265 * .. External Subroutines ..
266 EXTERNAL DGECON, DGEEQU, DGERFS, DGETRF, DGETRS, DLACPY,
267 $ DLAQGE, XERBLA
268 * ..
269 * .. Intrinsic Functions ..
270 INTRINSIC MAX, MIN
271 * ..
272 * .. Executable Statements ..
273 *
274 INFO = 0
275 NOFACT = LSAME( FACT, 'N' )
276 EQUIL = LSAME( FACT, 'E' )
277 NOTRAN = LSAME( TRANS, 'N' )
278 IF( NOFACT .OR. EQUIL ) THEN
279 EQUED = 'N'
280 ROWEQU = .FALSE.
281 COLEQU = .FALSE.
282 ELSE
283 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
284 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
285 SMLNUM = DLAMCH( 'Safe minimum' )
286 BIGNUM = ONE / SMLNUM
287 END IF
288 *
289 * Test the input parameters.
290 *
291 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
292 $ THEN
293 INFO = -1
294 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
295 $ LSAME( TRANS, 'C' ) ) THEN
296 INFO = -2
297 ELSE IF( N.LT.0 ) THEN
298 INFO = -3
299 ELSE IF( NRHS.LT.0 ) THEN
300 INFO = -4
301 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
302 INFO = -6
303 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
304 INFO = -8
305 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
306 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
307 INFO = -10
308 ELSE
309 IF( ROWEQU ) THEN
310 RCMIN = BIGNUM
311 RCMAX = ZERO
312 DO 10 J = 1, N
313 RCMIN = MIN( RCMIN, R( J ) )
314 RCMAX = MAX( RCMAX, R( J ) )
315 10 CONTINUE
316 IF( RCMIN.LE.ZERO ) THEN
317 INFO = -11
318 ELSE IF( N.GT.0 ) THEN
319 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
320 ELSE
321 ROWCND = ONE
322 END IF
323 END IF
324 IF( COLEQU .AND. INFO.EQ.0 ) THEN
325 RCMIN = BIGNUM
326 RCMAX = ZERO
327 DO 20 J = 1, N
328 RCMIN = MIN( RCMIN, C( J ) )
329 RCMAX = MAX( RCMAX, C( J ) )
330 20 CONTINUE
331 IF( RCMIN.LE.ZERO ) THEN
332 INFO = -12
333 ELSE IF( N.GT.0 ) THEN
334 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
335 ELSE
336 COLCND = ONE
337 END IF
338 END IF
339 IF( INFO.EQ.0 ) THEN
340 IF( LDB.LT.MAX( 1, N ) ) THEN
341 INFO = -14
342 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
343 INFO = -16
344 END IF
345 END IF
346 END IF
347 *
348 IF( INFO.NE.0 ) THEN
349 CALL XERBLA( 'DGESVX', -INFO )
350 RETURN
351 END IF
352 *
353 IF( EQUIL ) THEN
354 *
355 * Compute row and column scalings to equilibrate the matrix A.
356 *
357 CALL DGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
358 IF( INFEQU.EQ.0 ) THEN
359 *
360 * Equilibrate the matrix.
361 *
362 CALL DLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
363 $ EQUED )
364 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
365 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
366 END IF
367 END IF
368 *
369 * Scale the right hand side.
370 *
371 IF( NOTRAN ) THEN
372 IF( ROWEQU ) THEN
373 DO 40 J = 1, NRHS
374 DO 30 I = 1, N
375 B( I, J ) = R( I )*B( I, J )
376 30 CONTINUE
377 40 CONTINUE
378 END IF
379 ELSE IF( COLEQU ) THEN
380 DO 60 J = 1, NRHS
381 DO 50 I = 1, N
382 B( I, J ) = C( I )*B( I, J )
383 50 CONTINUE
384 60 CONTINUE
385 END IF
386 *
387 IF( NOFACT .OR. EQUIL ) THEN
388 *
389 * Compute the LU factorization of A.
390 *
391 CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF )
392 CALL DGETRF( N, N, AF, LDAF, IPIV, INFO )
393 *
394 * Return if INFO is non-zero.
395 *
396 IF( INFO.GT.0 ) THEN
397 *
398 * Compute the reciprocal pivot growth factor of the
399 * leading rank-deficient INFO columns of A.
400 *
401 RPVGRW = DLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
402 $ WORK )
403 IF( RPVGRW.EQ.ZERO ) THEN
404 RPVGRW = ONE
405 ELSE
406 RPVGRW = DLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW
407 END IF
408 WORK( 1 ) = RPVGRW
409 RCOND = ZERO
410 RETURN
411 END IF
412 END IF
413 *
414 * Compute the norm of the matrix A and the
415 * reciprocal pivot growth factor RPVGRW.
416 *
417 IF( NOTRAN ) THEN
418 NORM = '1'
419 ELSE
420 NORM = 'I'
421 END IF
422 ANORM = DLANGE( NORM, N, N, A, LDA, WORK )
423 RPVGRW = DLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK )
424 IF( RPVGRW.EQ.ZERO ) THEN
425 RPVGRW = ONE
426 ELSE
427 RPVGRW = DLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW
428 END IF
429 *
430 * Compute the reciprocal of the condition number of A.
431 *
432 CALL DGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
433 *
434 * Compute the solution matrix X.
435 *
436 CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
437 CALL DGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
438 *
439 * Use iterative refinement to improve the computed solution and
440 * compute error bounds and backward error estimates for it.
441 *
442 CALL DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
443 $ LDX, FERR, BERR, WORK, IWORK, INFO )
444 *
445 * Transform the solution matrix X to a solution of the original
446 * system.
447 *
448 IF( NOTRAN ) THEN
449 IF( COLEQU ) THEN
450 DO 80 J = 1, NRHS
451 DO 70 I = 1, N
452 X( I, J ) = C( I )*X( I, J )
453 70 CONTINUE
454 80 CONTINUE
455 DO 90 J = 1, NRHS
456 FERR( J ) = FERR( J ) / COLCND
457 90 CONTINUE
458 END IF
459 ELSE IF( ROWEQU ) THEN
460 DO 110 J = 1, NRHS
461 DO 100 I = 1, N
462 X( I, J ) = R( I )*X( I, J )
463 100 CONTINUE
464 110 CONTINUE
465 DO 120 J = 1, NRHS
466 FERR( J ) = FERR( J ) / ROWCND
467 120 CONTINUE
468 END IF
469 *
470 WORK( 1 ) = RPVGRW
471 *
472 * Set INFO = N+1 if the matrix is singular to working precision.
473 *
474 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
475 $ INFO = N + 1
476 RETURN
477 *
478 * End of DGESVX
479 *
480 END
2 $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
3 $ WORK, IWORK, INFO )
4 *
5 * -- LAPACK driver routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * .. Scalar Arguments ..
11 CHARACTER EQUED, FACT, TRANS
12 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
13 DOUBLE PRECISION RCOND
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * ), IWORK( * )
17 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ BERR( * ), C( * ), FERR( * ), R( * ),
19 $ WORK( * ), X( LDX, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * DGESVX uses the LU factorization to compute the solution to a real
26 * system of linear equations
27 * A * X = B,
28 * where A is an N-by-N matrix and X and B 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 * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
41 * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
42 * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
43 * Whether or not the system will be equilibrated depends on the
44 * scaling of the matrix A, but if equilibration is used, A is
45 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
46 * or diag(C)*B (if TRANS = 'T' or 'C').
47 *
48 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
49 * matrix A (after equilibration if FACT = 'E') as
50 * A = P * L * U,
51 * where P is a permutation matrix, L is a unit lower triangular
52 * matrix, and U is upper triangular.
53 *
54 * 3. If some U(i,i)=0, so that U is exactly singular, then the routine
55 * returns with INFO = i. Otherwise, the factored form of A is used
56 * to estimate the condition number of the matrix A. If the
57 * reciprocal of the condition number is less than machine precision,
58 * INFO = N+1 is returned as a warning, but the routine still goes on
59 * to solve for X and compute error bounds as described below.
60 *
61 * 4. The system of equations is solved for X using the factored form
62 * of A.
63 *
64 * 5. Iterative refinement is applied to improve the computed solution
65 * matrix and calculate error bounds and backward error estimates
66 * for it.
67 *
68 * 6. If equilibration was used, the matrix X is premultiplied by
69 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
70 * that it solves the original system before equilibration.
71 *
72 * Arguments
73 * =========
74 *
75 * FACT (input) CHARACTER*1
76 * Specifies whether or not the factored form of the matrix A is
77 * supplied on entry, and if not, whether the matrix A should be
78 * equilibrated before it is factored.
79 * = 'F': On entry, AF and IPIV contain the factored form of A.
80 * If EQUED is not 'N', the matrix A has been
81 * equilibrated with scaling factors given by R and C.
82 * A, AF, and IPIV are not modified.
83 * = 'N': The matrix A will be copied to AF and factored.
84 * = 'E': The matrix A will be equilibrated if necessary, then
85 * copied to AF and factored.
86 *
87 * TRANS (input) CHARACTER*1
88 * Specifies the form of the system of equations:
89 * = 'N': A * X = B (No transpose)
90 * = 'T': A**T * X = B (Transpose)
91 * = 'C': A**H * X = B (Transpose)
92 *
93 * N (input) INTEGER
94 * The number of linear equations, i.e., the order of the
95 * matrix A. N >= 0.
96 *
97 * NRHS (input) INTEGER
98 * The number of right hand sides, i.e., the number of columns
99 * of the matrices B and X. NRHS >= 0.
100 *
101 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
102 * On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
103 * not 'N', then A must have been equilibrated by the scaling
104 * factors in R and/or C. A is not modified if FACT = 'F' or
105 * 'N', or if FACT = 'E' and EQUED = 'N' on exit.
106 *
107 * On exit, if EQUED .ne. 'N', A is scaled as follows:
108 * EQUED = 'R': A := diag(R) * A
109 * EQUED = 'C': A := A * diag(C)
110 * EQUED = 'B': A := diag(R) * A * diag(C).
111 *
112 * LDA (input) INTEGER
113 * The leading dimension of the array A. LDA >= max(1,N).
114 *
115 * AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
116 * If FACT = 'F', then AF is an input argument and on entry
117 * contains the factors L and U from the factorization
118 * A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then
119 * AF is the factored form of the equilibrated matrix A.
120 *
121 * If FACT = 'N', then AF is an output argument and on exit
122 * returns the factors L and U from the factorization A = P*L*U
123 * of the original matrix A.
124 *
125 * If FACT = 'E', then AF is an output argument and on exit
126 * returns the factors L and U from the factorization A = P*L*U
127 * of the equilibrated matrix A (see the description of A for
128 * the form of the equilibrated matrix).
129 *
130 * LDAF (input) INTEGER
131 * The leading dimension of the array AF. LDAF >= max(1,N).
132 *
133 * IPIV (input or output) INTEGER array, dimension (N)
134 * If FACT = 'F', then IPIV is an input argument and on entry
135 * contains the pivot indices from the factorization A = P*L*U
136 * as computed by DGETRF; row i of the matrix was interchanged
137 * with row IPIV(i).
138 *
139 * If FACT = 'N', then IPIV is an output argument and on exit
140 * contains the pivot indices from the factorization A = P*L*U
141 * of the original matrix A.
142 *
143 * If FACT = 'E', then IPIV is an output argument and on exit
144 * contains the pivot indices from the factorization A = P*L*U
145 * of the equilibrated matrix A.
146 *
147 * EQUED (input or output) CHARACTER*1
148 * Specifies the form of equilibration that was done.
149 * = 'N': No equilibration (always true if FACT = 'N').
150 * = 'R': Row equilibration, i.e., A has been premultiplied by
151 * diag(R).
152 * = 'C': Column equilibration, i.e., A has been postmultiplied
153 * by diag(C).
154 * = 'B': Both row and column equilibration, i.e., A has been
155 * replaced by diag(R) * A * diag(C).
156 * EQUED is an input argument if FACT = 'F'; otherwise, it is an
157 * output argument.
158 *
159 * R (input or output) DOUBLE PRECISION array, dimension (N)
160 * The row scale factors for A. If EQUED = 'R' or 'B', A is
161 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
162 * is not accessed. R is an input argument if FACT = 'F';
163 * otherwise, R is an output argument. If FACT = 'F' and
164 * EQUED = 'R' or 'B', each element of R must be positive.
165 *
166 * C (input or output) DOUBLE PRECISION array, dimension (N)
167 * The column scale factors for A. If EQUED = 'C' or 'B', A is
168 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
169 * is not accessed. C is an input argument if FACT = 'F';
170 * otherwise, C is an output argument. If FACT = 'F' and
171 * EQUED = 'C' or 'B', each element of C must be positive.
172 *
173 * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
174 * On entry, the N-by-NRHS right hand side matrix B.
175 * On exit,
176 * if EQUED = 'N', B is not modified;
177 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
178 * diag(R)*B;
179 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
180 * overwritten by diag(C)*B.
181 *
182 * LDB (input) INTEGER
183 * The leading dimension of the array B. LDB >= max(1,N).
184 *
185 * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
186 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
187 * to the original system of equations. Note that A and B are
188 * modified on exit if EQUED .ne. 'N', and the solution to the
189 * equilibrated system is inv(diag(C))*X if TRANS = 'N' and
190 * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
191 * and EQUED = 'R' or 'B'.
192 *
193 * LDX (input) INTEGER
194 * The leading dimension of the array X. LDX >= max(1,N).
195 *
196 * RCOND (output) DOUBLE PRECISION
197 * The estimate of the reciprocal condition number of the matrix
198 * A after equilibration (if done). If RCOND is less than the
199 * machine precision (in particular, if RCOND = 0), the matrix
200 * is singular to working precision. This condition is
201 * indicated by a return code of INFO > 0.
202 *
203 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
204 * The estimated forward error bound for each solution vector
205 * X(j) (the j-th column of the solution matrix X).
206 * If XTRUE is the true solution corresponding to X(j), FERR(j)
207 * is an estimated upper bound for the magnitude of the largest
208 * element in (X(j) - XTRUE) divided by the magnitude of the
209 * largest element in X(j). The estimate is as reliable as
210 * the estimate for RCOND, and is almost always a slight
211 * overestimate of the true error.
212 *
213 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
214 * The componentwise relative backward error of each solution
215 * vector X(j) (i.e., the smallest relative change in
216 * any element of A or B that makes X(j) an exact solution).
217 *
218 * WORK (workspace/output) DOUBLE PRECISION array, dimension (4*N)
219 * On exit, WORK(1) contains the reciprocal pivot growth
220 * factor norm(A)/norm(U). The "max absolute element" norm is
221 * used. If WORK(1) is much less than 1, then the stability
222 * of the LU factorization of the (equilibrated) matrix A
223 * could be poor. This also means that the solution X, condition
224 * estimator RCOND, and forward error bound FERR could be
225 * unreliable. If factorization fails with 0<INFO<=N, then
226 * WORK(1) contains the reciprocal pivot growth factor for the
227 * leading INFO columns of A.
228 *
229 * IWORK (workspace) INTEGER array, dimension (N)
230 *
231 * INFO (output) INTEGER
232 * = 0: successful exit
233 * < 0: if INFO = -i, the i-th argument had an illegal value
234 * > 0: if INFO = i, and i is
235 * <= N: U(i,i) is exactly zero. The factorization has
236 * been completed, but the factor U is exactly
237 * singular, so the solution and error bounds
238 * could not be computed. RCOND = 0 is returned.
239 * = N+1: U is nonsingular, but RCOND is less than machine
240 * precision, meaning that the matrix is singular
241 * to working precision. Nevertheless, the
242 * solution and error bounds are computed because
243 * there are a number of situations where the
244 * computed solution can be more accurate than the
245 * value of RCOND would suggest.
246 *
247 * =====================================================================
248 *
249 * .. Parameters ..
250 DOUBLE PRECISION ZERO, ONE
251 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
252 * ..
253 * .. Local Scalars ..
254 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
255 CHARACTER NORM
256 INTEGER I, INFEQU, J
257 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
258 $ ROWCND, RPVGRW, SMLNUM
259 * ..
260 * .. External Functions ..
261 LOGICAL LSAME
262 DOUBLE PRECISION DLAMCH, DLANGE, DLANTR
263 EXTERNAL LSAME, DLAMCH, DLANGE, DLANTR
264 * ..
265 * .. External Subroutines ..
266 EXTERNAL DGECON, DGEEQU, DGERFS, DGETRF, DGETRS, DLACPY,
267 $ DLAQGE, XERBLA
268 * ..
269 * .. Intrinsic Functions ..
270 INTRINSIC MAX, MIN
271 * ..
272 * .. Executable Statements ..
273 *
274 INFO = 0
275 NOFACT = LSAME( FACT, 'N' )
276 EQUIL = LSAME( FACT, 'E' )
277 NOTRAN = LSAME( TRANS, 'N' )
278 IF( NOFACT .OR. EQUIL ) THEN
279 EQUED = 'N'
280 ROWEQU = .FALSE.
281 COLEQU = .FALSE.
282 ELSE
283 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
284 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
285 SMLNUM = DLAMCH( 'Safe minimum' )
286 BIGNUM = ONE / SMLNUM
287 END IF
288 *
289 * Test the input parameters.
290 *
291 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
292 $ THEN
293 INFO = -1
294 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
295 $ LSAME( TRANS, 'C' ) ) THEN
296 INFO = -2
297 ELSE IF( N.LT.0 ) THEN
298 INFO = -3
299 ELSE IF( NRHS.LT.0 ) THEN
300 INFO = -4
301 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
302 INFO = -6
303 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
304 INFO = -8
305 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
306 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
307 INFO = -10
308 ELSE
309 IF( ROWEQU ) THEN
310 RCMIN = BIGNUM
311 RCMAX = ZERO
312 DO 10 J = 1, N
313 RCMIN = MIN( RCMIN, R( J ) )
314 RCMAX = MAX( RCMAX, R( J ) )
315 10 CONTINUE
316 IF( RCMIN.LE.ZERO ) THEN
317 INFO = -11
318 ELSE IF( N.GT.0 ) THEN
319 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
320 ELSE
321 ROWCND = ONE
322 END IF
323 END IF
324 IF( COLEQU .AND. INFO.EQ.0 ) THEN
325 RCMIN = BIGNUM
326 RCMAX = ZERO
327 DO 20 J = 1, N
328 RCMIN = MIN( RCMIN, C( J ) )
329 RCMAX = MAX( RCMAX, C( J ) )
330 20 CONTINUE
331 IF( RCMIN.LE.ZERO ) THEN
332 INFO = -12
333 ELSE IF( N.GT.0 ) THEN
334 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
335 ELSE
336 COLCND = ONE
337 END IF
338 END IF
339 IF( INFO.EQ.0 ) THEN
340 IF( LDB.LT.MAX( 1, N ) ) THEN
341 INFO = -14
342 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
343 INFO = -16
344 END IF
345 END IF
346 END IF
347 *
348 IF( INFO.NE.0 ) THEN
349 CALL XERBLA( 'DGESVX', -INFO )
350 RETURN
351 END IF
352 *
353 IF( EQUIL ) THEN
354 *
355 * Compute row and column scalings to equilibrate the matrix A.
356 *
357 CALL DGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU )
358 IF( INFEQU.EQ.0 ) THEN
359 *
360 * Equilibrate the matrix.
361 *
362 CALL DLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
363 $ EQUED )
364 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
365 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
366 END IF
367 END IF
368 *
369 * Scale the right hand side.
370 *
371 IF( NOTRAN ) THEN
372 IF( ROWEQU ) THEN
373 DO 40 J = 1, NRHS
374 DO 30 I = 1, N
375 B( I, J ) = R( I )*B( I, J )
376 30 CONTINUE
377 40 CONTINUE
378 END IF
379 ELSE IF( COLEQU ) THEN
380 DO 60 J = 1, NRHS
381 DO 50 I = 1, N
382 B( I, J ) = C( I )*B( I, J )
383 50 CONTINUE
384 60 CONTINUE
385 END IF
386 *
387 IF( NOFACT .OR. EQUIL ) THEN
388 *
389 * Compute the LU factorization of A.
390 *
391 CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF )
392 CALL DGETRF( N, N, AF, LDAF, IPIV, INFO )
393 *
394 * Return if INFO is non-zero.
395 *
396 IF( INFO.GT.0 ) THEN
397 *
398 * Compute the reciprocal pivot growth factor of the
399 * leading rank-deficient INFO columns of A.
400 *
401 RPVGRW = DLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF,
402 $ WORK )
403 IF( RPVGRW.EQ.ZERO ) THEN
404 RPVGRW = ONE
405 ELSE
406 RPVGRW = DLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW
407 END IF
408 WORK( 1 ) = RPVGRW
409 RCOND = ZERO
410 RETURN
411 END IF
412 END IF
413 *
414 * Compute the norm of the matrix A and the
415 * reciprocal pivot growth factor RPVGRW.
416 *
417 IF( NOTRAN ) THEN
418 NORM = '1'
419 ELSE
420 NORM = 'I'
421 END IF
422 ANORM = DLANGE( NORM, N, N, A, LDA, WORK )
423 RPVGRW = DLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK )
424 IF( RPVGRW.EQ.ZERO ) THEN
425 RPVGRW = ONE
426 ELSE
427 RPVGRW = DLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW
428 END IF
429 *
430 * Compute the reciprocal of the condition number of A.
431 *
432 CALL DGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
433 *
434 * Compute the solution matrix X.
435 *
436 CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
437 CALL DGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
438 *
439 * Use iterative refinement to improve the computed solution and
440 * compute error bounds and backward error estimates for it.
441 *
442 CALL DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
443 $ LDX, FERR, BERR, WORK, IWORK, INFO )
444 *
445 * Transform the solution matrix X to a solution of the original
446 * system.
447 *
448 IF( NOTRAN ) THEN
449 IF( COLEQU ) THEN
450 DO 80 J = 1, NRHS
451 DO 70 I = 1, N
452 X( I, J ) = C( I )*X( I, J )
453 70 CONTINUE
454 80 CONTINUE
455 DO 90 J = 1, NRHS
456 FERR( J ) = FERR( J ) / COLCND
457 90 CONTINUE
458 END IF
459 ELSE IF( ROWEQU ) THEN
460 DO 110 J = 1, NRHS
461 DO 100 I = 1, N
462 X( I, J ) = R( I )*X( I, J )
463 100 CONTINUE
464 110 CONTINUE
465 DO 120 J = 1, NRHS
466 FERR( J ) = FERR( J ) / ROWCND
467 120 CONTINUE
468 END IF
469 *
470 WORK( 1 ) = RPVGRW
471 *
472 * Set INFO = N+1 if the matrix is singular to working precision.
473 *
474 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
475 $ INFO = N + 1
476 RETURN
477 *
478 * End of DGESVX
479 *
480 END