LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
sgbsvx.f
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1 *> \brief <b> SGBSVX computes the solution to system of linear equations A * X = B for GB matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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13 *> [ZIP]</a>
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
22 * LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
23 * RCOND, FERR, BERR, WORK, IWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER EQUED, FACT, TRANS
27 * INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
28 * REAL RCOND
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IPIV( * ), IWORK( * )
32 * REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
33 * $ BERR( * ), C( * ), FERR( * ), R( * ),
34 * $ WORK( * ), X( LDX, * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> SGBSVX uses the LU factorization to compute the solution to a real
44 *> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
45 *> where A is a band matrix of order N with KL subdiagonals and KU
46 *> superdiagonals, and X and B are N-by-NRHS matrices.
47 *>
48 *> Error bounds on the solution and a condition estimate are also
49 *> provided.
50 *> \endverbatim
51 *
52 *> \par Description:
53 * =================
54 *>
55 *> \verbatim
56 *>
57 *> The following steps are performed by this subroutine:
58 *>
59 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
60 *> the system:
61 *> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
62 *> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
63 *> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
64 *> Whether or not the system will be equilibrated depends on the
65 *> scaling of the matrix A, but if equilibration is used, A is
66 *> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
67 *> or diag(C)*B (if TRANS = 'T' or 'C').
68 *>
69 *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
70 *> matrix A (after equilibration if FACT = 'E') as
71 *> A = L * U,
72 *> where L is a product of permutation and unit lower triangular
73 *> matrices with KL subdiagonals, and U is upper triangular with
74 *> KL+KU superdiagonals.
75 *>
76 *> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
77 *> returns with INFO = i. Otherwise, the factored form of A is used
78 *> to estimate the condition number of the matrix A. If the
79 *> reciprocal of the condition number is less than machine precision,
80 *> INFO = N+1 is returned as a warning, but the routine still goes on
81 *> to solve for X and compute error bounds as described below.
82 *>
83 *> 4. The system of equations is solved for X using the factored form
84 *> of A.
85 *>
86 *> 5. Iterative refinement is applied to improve the computed solution
87 *> matrix and calculate error bounds and backward error estimates
88 *> for it.
89 *>
90 *> 6. If equilibration was used, the matrix X is premultiplied by
91 *> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
92 *> that it solves the original system before equilibration.
93 *> \endverbatim
94 *
95 * Arguments:
96 * ==========
97 *
98 *> \param[in] FACT
99 *> \verbatim
100 *> FACT is CHARACTER*1
101 *> Specifies whether or not the factored form of the matrix A is
102 *> supplied on entry, and if not, whether the matrix A should be
103 *> equilibrated before it is factored.
104 *> = 'F': On entry, AFB and IPIV contain the factored form of
105 *> A. If EQUED is not 'N', the matrix A has been
106 *> equilibrated with scaling factors given by R and C.
107 *> AB, AFB, and IPIV are not modified.
108 *> = 'N': The matrix A will be copied to AFB and factored.
109 *> = 'E': The matrix A will be equilibrated if necessary, then
110 *> copied to AFB and factored.
111 *> \endverbatim
112 *>
113 *> \param[in] TRANS
114 *> \verbatim
115 *> TRANS is CHARACTER*1
116 *> Specifies the form of the system of equations.
117 *> = 'N': A * X = B (No transpose)
118 *> = 'T': A**T * X = B (Transpose)
119 *> = 'C': A**H * X = B (Transpose)
120 *> \endverbatim
121 *>
122 *> \param[in] N
123 *> \verbatim
124 *> N is INTEGER
125 *> The number of linear equations, i.e., the order of the
126 *> matrix A. N >= 0.
127 *> \endverbatim
128 *>
129 *> \param[in] KL
130 *> \verbatim
131 *> KL is INTEGER
132 *> The number of subdiagonals within the band of A. KL >= 0.
133 *> \endverbatim
134 *>
135 *> \param[in] KU
136 *> \verbatim
137 *> KU is INTEGER
138 *> The number of superdiagonals within the band of A. KU >= 0.
139 *> \endverbatim
140 *>
141 *> \param[in] NRHS
142 *> \verbatim
143 *> NRHS is INTEGER
144 *> The number of right hand sides, i.e., the number of columns
145 *> of the matrices B and X. NRHS >= 0.
146 *> \endverbatim
147 *>
148 *> \param[in,out] AB
149 *> \verbatim
150 *> AB is REAL array, dimension (LDAB,N)
151 *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
152 *> The j-th column of A is stored in the j-th column of the
153 *> array AB as follows:
154 *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
155 *>
156 *> If FACT = 'F' and EQUED is not 'N', then A must have been
157 *> equilibrated by the scaling factors in R and/or C. AB is not
158 *> modified if FACT = 'F' or 'N', or if FACT = 'E' and
159 *> EQUED = 'N' on exit.
160 *>
161 *> On exit, if EQUED .ne. 'N', A is scaled as follows:
162 *> EQUED = 'R': A := diag(R) * A
163 *> EQUED = 'C': A := A * diag(C)
164 *> EQUED = 'B': A := diag(R) * A * diag(C).
165 *> \endverbatim
166 *>
167 *> \param[in] LDAB
168 *> \verbatim
169 *> LDAB is INTEGER
170 *> The leading dimension of the array AB. LDAB >= KL+KU+1.
171 *> \endverbatim
172 *>
173 *> \param[in,out] AFB
174 *> \verbatim
175 *> AFB is REAL array, dimension (LDAFB,N)
176 *> If FACT = 'F', then AFB is an input argument and on entry
177 *> contains details of the LU factorization of the band matrix
178 *> A, as computed by SGBTRF. U is stored as an upper triangular
179 *> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
180 *> and the multipliers used during the factorization are stored
181 *> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
182 *> the factored form of the equilibrated matrix A.
183 *>
184 *> If FACT = 'N', then AFB is an output argument and on exit
185 *> returns details of the LU factorization of A.
186 *>
187 *> If FACT = 'E', then AFB is an output argument and on exit
188 *> returns details of the LU factorization of the equilibrated
189 *> matrix A (see the description of AB for the form of the
190 *> equilibrated matrix).
191 *> \endverbatim
192 *>
193 *> \param[in] LDAFB
194 *> \verbatim
195 *> LDAFB is INTEGER
196 *> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
197 *> \endverbatim
198 *>
199 *> \param[in,out] IPIV
200 *> \verbatim
201 *> IPIV is INTEGER array, dimension (N)
202 *> If FACT = 'F', then IPIV is an input argument and on entry
203 *> contains the pivot indices from the factorization A = L*U
204 *> as computed by SGBTRF; row i of the matrix was interchanged
205 *> with row IPIV(i).
206 *>
207 *> If FACT = 'N', then IPIV is an output argument and on exit
208 *> contains the pivot indices from the factorization A = L*U
209 *> of the original matrix A.
210 *>
211 *> If FACT = 'E', then IPIV is an output argument and on exit
212 *> contains the pivot indices from the factorization A = L*U
213 *> of the equilibrated matrix A.
214 *> \endverbatim
215 *>
216 *> \param[in,out] EQUED
217 *> \verbatim
218 *> EQUED is CHARACTER*1
219 *> Specifies the form of equilibration that was done.
220 *> = 'N': No equilibration (always true if FACT = 'N').
221 *> = 'R': Row equilibration, i.e., A has been premultiplied by
222 *> diag(R).
223 *> = 'C': Column equilibration, i.e., A has been postmultiplied
224 *> by diag(C).
225 *> = 'B': Both row and column equilibration, i.e., A has been
226 *> replaced by diag(R) * A * diag(C).
227 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
228 *> output argument.
229 *> \endverbatim
230 *>
231 *> \param[in,out] R
232 *> \verbatim
233 *> R is REAL array, dimension (N)
234 *> The row scale factors for A. If EQUED = 'R' or 'B', A is
235 *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
236 *> is not accessed. R is an input argument if FACT = 'F';
237 *> otherwise, R is an output argument. If FACT = 'F' and
238 *> EQUED = 'R' or 'B', each element of R must be positive.
239 *> \endverbatim
240 *>
241 *> \param[in,out] C
242 *> \verbatim
243 *> C is REAL array, dimension (N)
244 *> The column scale factors for A. If EQUED = 'C' or 'B', A is
245 *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
246 *> is not accessed. C is an input argument if FACT = 'F';
247 *> otherwise, C is an output argument. If FACT = 'F' and
248 *> EQUED = 'C' or 'B', each element of C must be positive.
249 *> \endverbatim
250 *>
251 *> \param[in,out] B
252 *> \verbatim
253 *> B is REAL array, dimension (LDB,NRHS)
254 *> On entry, the right hand side matrix B.
255 *> On exit,
256 *> if EQUED = 'N', B is not modified;
257 *> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
258 *> diag(R)*B;
259 *> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
260 *> overwritten by diag(C)*B.
261 *> \endverbatim
262 *>
263 *> \param[in] LDB
264 *> \verbatim
265 *> LDB is INTEGER
266 *> The leading dimension of the array B. LDB >= max(1,N).
267 *> \endverbatim
268 *>
269 *> \param[out] X
270 *> \verbatim
271 *> X is REAL array, dimension (LDX,NRHS)
272 *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
273 *> to the original system of equations. Note that A and B are
274 *> modified on exit if EQUED .ne. 'N', and the solution to the
275 *> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
276 *> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
277 *> and EQUED = 'R' or 'B'.
278 *> \endverbatim
279 *>
280 *> \param[in] LDX
281 *> \verbatim
282 *> LDX is INTEGER
283 *> The leading dimension of the array X. LDX >= max(1,N).
284 *> \endverbatim
285 *>
286 *> \param[out] RCOND
287 *> \verbatim
288 *> RCOND is REAL
289 *> The estimate of the reciprocal condition number of the matrix
290 *> A after equilibration (if done). If RCOND is less than the
291 *> machine precision (in particular, if RCOND = 0), the matrix
292 *> is singular to working precision. This condition is
293 *> indicated by a return code of INFO > 0.
294 *> \endverbatim
295 *>
296 *> \param[out] FERR
297 *> \verbatim
298 *> FERR is REAL array, dimension (NRHS)
299 *> The estimated forward error bound for each solution vector
300 *> X(j) (the j-th column of the solution matrix X).
301 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
302 *> is an estimated upper bound for the magnitude of the largest
303 *> element in (X(j) - XTRUE) divided by the magnitude of the
304 *> largest element in X(j). The estimate is as reliable as
305 *> the estimate for RCOND, and is almost always a slight
306 *> overestimate of the true error.
307 *> \endverbatim
308 *>
309 *> \param[out] BERR
310 *> \verbatim
311 *> BERR is REAL array, dimension (NRHS)
312 *> The componentwise relative backward error of each solution
313 *> vector X(j) (i.e., the smallest relative change in
314 *> any element of A or B that makes X(j) an exact solution).
315 *> \endverbatim
316 *>
317 *> \param[out] WORK
318 *> \verbatim
319 *> WORK is REAL array, dimension (3*N)
320 *> On exit, WORK(1) contains the reciprocal pivot growth
321 *> factor norm(A)/norm(U). The "max absolute element" norm is
322 *> used. If WORK(1) is much less than 1, then the stability
323 *> of the LU factorization of the (equilibrated) matrix A
324 *> could be poor. This also means that the solution X, condition
325 *> estimator RCOND, and forward error bound FERR could be
326 *> unreliable. If factorization fails with 0<INFO<=N, then
327 *> WORK(1) contains the reciprocal pivot growth factor for the
328 *> leading INFO columns of A.
329 *> \endverbatim
330 *>
331 *> \param[out] IWORK
332 *> \verbatim
333 *> IWORK is INTEGER array, dimension (N)
334 *> \endverbatim
335 *>
336 *> \param[out] INFO
337 *> \verbatim
338 *> INFO is INTEGER
339 *> = 0: successful exit
340 *> < 0: if INFO = -i, the i-th argument had an illegal value
341 *> > 0: if INFO = i, and i is
342 *> <= N: U(i,i) is exactly zero. The factorization
343 *> has been completed, but the factor U is exactly
344 *> singular, so the solution and error bounds
345 *> could not be computed. RCOND = 0 is returned.
346 *> = N+1: U is nonsingular, but RCOND is less than machine
347 *> precision, meaning that the matrix is singular
348 *> to working precision. Nevertheless, the
349 *> solution and error bounds are computed because
350 *> there are a number of situations where the
351 *> computed solution can be more accurate than the
352 *> \endverbatim
353 *
354 * Authors:
355 * ========
356 *
357 *> \author Univ. of Tennessee
358 *> \author Univ. of California Berkeley
359 *> \author Univ. of Colorado Denver
360 *> \author NAG Ltd.
361 *
362 *> \date April 2012
363 *
364 *> \ingroup realGBsolve
365 *
366 * =====================================================================
367  SUBROUTINE sgbsvx( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
368  $ ldafb, ipiv, equed, r, c, b, ldb, x, ldx,
369  $ rcond, ferr, berr, work, iwork, info )
370 *
371 * -- LAPACK driver routine (version 3.4.1) --
372 * -- LAPACK is a software package provided by Univ. of Tennessee, --
373 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
374 * April 2012
375 *
376 * .. Scalar Arguments ..
377  CHARACTER EQUED, FACT, TRANS
378  INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
379  REAL RCOND
380 * ..
381 * .. Array Arguments ..
382  INTEGER IPIV( * ), IWORK( * )
383  REAL AB( ldab, * ), AFB( ldafb, * ), B( ldb, * ),
384  $ berr( * ), c( * ), ferr( * ), r( * ),
385  $ work( * ), x( ldx, * )
386 * ..
387 *
388 * =====================================================================
389 * Moved setting of INFO = N+1 so INFO does not subsequently get
390 * overwritten. Sven, 17 Mar 05.
391 * =====================================================================
392 *
393 * .. Parameters ..
394  REAL ZERO, ONE
395  parameter( zero = 0.0e+0, one = 1.0e+0 )
396 * ..
397 * .. Local Scalars ..
398  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
399  CHARACTER NORM
400  INTEGER I, INFEQU, J, J1, J2
401  REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
402  $ rowcnd, rpvgrw, smlnum
403 * ..
404 * .. External Functions ..
405  LOGICAL LSAME
406  REAL SLAMCH, SLANGB, SLANTB
407  EXTERNAL lsame, slamch, slangb, slantb
408 * ..
409 * .. External Subroutines ..
410  EXTERNAL scopy, sgbcon, sgbequ, sgbrfs, sgbtrf, sgbtrs,
411  $ slacpy, slaqgb, xerbla
412 * ..
413 * .. Intrinsic Functions ..
414  INTRINSIC abs, max, min
415 * ..
416 * .. Executable Statements ..
417 *
418  info = 0
419  nofact = lsame( fact, 'N' )
420  equil = lsame( fact, 'E' )
421  notran = lsame( trans, 'N' )
422  IF( nofact .OR. equil ) THEN
423  equed = 'N'
424  rowequ = .false.
425  colequ = .false.
426  ELSE
427  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
428  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
429  smlnum = slamch( 'Safe minimum' )
430  bignum = one / smlnum
431  END IF
432 *
433 * Test the input parameters.
434 *
435  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
436  $ THEN
437  info = -1
438  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
439  $ lsame( trans, 'C' ) ) THEN
440  info = -2
441  ELSE IF( n.LT.0 ) THEN
442  info = -3
443  ELSE IF( kl.LT.0 ) THEN
444  info = -4
445  ELSE IF( ku.LT.0 ) THEN
446  info = -5
447  ELSE IF( nrhs.LT.0 ) THEN
448  info = -6
449  ELSE IF( ldab.LT.kl+ku+1 ) THEN
450  info = -8
451  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
452  info = -10
453  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
454  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
455  info = -12
456  ELSE
457  IF( rowequ ) THEN
458  rcmin = bignum
459  rcmax = zero
460  DO 10 j = 1, n
461  rcmin = min( rcmin, r( j ) )
462  rcmax = max( rcmax, r( j ) )
463  10 CONTINUE
464  IF( rcmin.LE.zero ) THEN
465  info = -13
466  ELSE IF( n.GT.0 ) THEN
467  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
468  ELSE
469  rowcnd = one
470  END IF
471  END IF
472  IF( colequ .AND. info.EQ.0 ) THEN
473  rcmin = bignum
474  rcmax = zero
475  DO 20 j = 1, n
476  rcmin = min( rcmin, c( j ) )
477  rcmax = max( rcmax, c( j ) )
478  20 CONTINUE
479  IF( rcmin.LE.zero ) THEN
480  info = -14
481  ELSE IF( n.GT.0 ) THEN
482  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
483  ELSE
484  colcnd = one
485  END IF
486  END IF
487  IF( info.EQ.0 ) THEN
488  IF( ldb.LT.max( 1, n ) ) THEN
489  info = -16
490  ELSE IF( ldx.LT.max( 1, n ) ) THEN
491  info = -18
492  END IF
493  END IF
494  END IF
495 *
496  IF( info.NE.0 ) THEN
497  CALL xerbla( 'SGBSVX', -info )
498  RETURN
499  END IF
500 *
501  IF( equil ) THEN
502 *
503 * Compute row and column scalings to equilibrate the matrix A.
504 *
505  CALL sgbequ( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
506  $ amax, infequ )
507  IF( infequ.EQ.0 ) THEN
508 *
509 * Equilibrate the matrix.
510 *
511  CALL slaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
512  $ amax, equed )
513  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
514  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
515  END IF
516  END IF
517 *
518 * Scale the right hand side.
519 *
520  IF( notran ) THEN
521  IF( rowequ ) THEN
522  DO 40 j = 1, nrhs
523  DO 30 i = 1, n
524  b( i, j ) = r( i )*b( i, j )
525  30 CONTINUE
526  40 CONTINUE
527  END IF
528  ELSE IF( colequ ) THEN
529  DO 60 j = 1, nrhs
530  DO 50 i = 1, n
531  b( i, j ) = c( i )*b( i, j )
532  50 CONTINUE
533  60 CONTINUE
534  END IF
535 *
536  IF( nofact .OR. equil ) THEN
537 *
538 * Compute the LU factorization of the band matrix A.
539 *
540  DO 70 j = 1, n
541  j1 = max( j-ku, 1 )
542  j2 = min( j+kl, n )
543  CALL scopy( j2-j1+1, ab( ku+1-j+j1, j ), 1,
544  $ afb( kl+ku+1-j+j1, j ), 1 )
545  70 CONTINUE
546 *
547  CALL sgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
548 *
549 * Return if INFO is non-zero.
550 *
551  IF( info.GT.0 ) THEN
552 *
553 * Compute the reciprocal pivot growth factor of the
554 * leading rank-deficient INFO columns of A.
555 *
556  anorm = zero
557  DO 90 j = 1, info
558  DO 80 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
559  anorm = max( anorm, abs( ab( i, j ) ) )
560  80 CONTINUE
561  90 CONTINUE
562  rpvgrw = slantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
563  $ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
564  $ work )
565  IF( rpvgrw.EQ.zero ) THEN
566  rpvgrw = one
567  ELSE
568  rpvgrw = anorm / rpvgrw
569  END IF
570  work( 1 ) = rpvgrw
571  rcond = zero
572  RETURN
573  END IF
574  END IF
575 *
576 * Compute the norm of the matrix A and the
577 * reciprocal pivot growth factor RPVGRW.
578 *
579  IF( notran ) THEN
580  norm = '1'
581  ELSE
582  norm = 'I'
583  END IF
584  anorm = slangb( norm, n, kl, ku, ab, ldab, work )
585  rpvgrw = slantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, work )
586  IF( rpvgrw.EQ.zero ) THEN
587  rpvgrw = one
588  ELSE
589  rpvgrw = slangb( 'M', n, kl, ku, ab, ldab, work ) / rpvgrw
590  END IF
591 *
592 * Compute the reciprocal of the condition number of A.
593 *
594  CALL sgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
595  $ work, iwork, info )
596 *
597 * Compute the solution matrix X.
598 *
599  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
600  CALL sgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
601  $ info )
602 *
603 * Use iterative refinement to improve the computed solution and
604 * compute error bounds and backward error estimates for it.
605 *
606  CALL sgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
607  $ b, ldb, x, ldx, ferr, berr, work, iwork, info )
608 *
609 * Transform the solution matrix X to a solution of the original
610 * system.
611 *
612  IF( notran ) THEN
613  IF( colequ ) THEN
614  DO 110 j = 1, nrhs
615  DO 100 i = 1, n
616  x( i, j ) = c( i )*x( i, j )
617  100 CONTINUE
618  110 CONTINUE
619  DO 120 j = 1, nrhs
620  ferr( j ) = ferr( j ) / colcnd
621  120 CONTINUE
622  END IF
623  ELSE IF( rowequ ) THEN
624  DO 140 j = 1, nrhs
625  DO 130 i = 1, n
626  x( i, j ) = r( i )*x( i, j )
627  130 CONTINUE
628  140 CONTINUE
629  DO 150 j = 1, nrhs
630  ferr( j ) = ferr( j ) / rowcnd
631  150 CONTINUE
632  END IF
633 *
634 * Set INFO = N+1 if the matrix is singular to working precision.
635 *
636  IF( rcond.LT.slamch( 'Epsilon' ) )
637  $ info = n + 1
638 *
639  work( 1 ) = rpvgrw
640  RETURN
641 *
642 * End of SGBSVX
643 *
644  END
subroutine sgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
SGBTRF
Definition: sgbtrf.f:146
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
SLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ...
Definition: slaqgb.f:161
subroutine sgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
SGBEQU
Definition: sgbequ.f:155
subroutine sgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
SGBTRS
Definition: sgbtrs.f:140
subroutine sgbsvx(FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SGBSVX computes the solution to system of linear equations A * X = B for GB matrices ...
Definition: sgbsvx.f:370
subroutine sgbrfs(TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGBRFS
Definition: sgbrfs.f:207
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine sgbcon(NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SGBCON
Definition: sgbcon.f:148