LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
sgbsvx.f
Go to the documentation of this file.
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 *
8 *> \htmlonly
9 *> Download SGBSVX + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgbsvx.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgbsvx.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgbsvx.f">
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 *> \ingroup realGBsolve
363 *
364 * =====================================================================
365  SUBROUTINE sgbsvx( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
366  $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
367  $ RCOND, FERR, BERR, WORK, IWORK, INFO )
368 *
369 * -- LAPACK driver routine --
370 * -- LAPACK is a software package provided by Univ. of Tennessee, --
371 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
372 *
373 * .. Scalar Arguments ..
374  CHARACTER EQUED, FACT, TRANS
375  INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
376  REAL RCOND
377 * ..
378 * .. Array Arguments ..
379  INTEGER IPIV( * ), IWORK( * )
380  REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
381  $ berr( * ), c( * ), ferr( * ), r( * ),
382  $ work( * ), x( ldx, * )
383 * ..
384 *
385 * =====================================================================
386 * Moved setting of INFO = N+1 so INFO does not subsequently get
387 * overwritten. Sven, 17 Mar 05.
388 * =====================================================================
389 *
390 * .. Parameters ..
391  REAL ZERO, ONE
392  PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
393 * ..
394 * .. Local Scalars ..
395  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
396  CHARACTER NORM
397  INTEGER I, INFEQU, J, J1, J2
398  REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
399  $ rowcnd, rpvgrw, smlnum
400 * ..
401 * .. External Functions ..
402  LOGICAL LSAME
403  REAL SLAMCH, SLANGB, SLANTB
404  EXTERNAL lsame, slamch, slangb, slantb
405 * ..
406 * .. External Subroutines ..
407  EXTERNAL scopy, sgbcon, sgbequ, sgbrfs, sgbtrf, sgbtrs,
408  $ slacpy, slaqgb, xerbla
409 * ..
410 * .. Intrinsic Functions ..
411  INTRINSIC abs, max, min
412 * ..
413 * .. Executable Statements ..
414 *
415  info = 0
416  nofact = lsame( fact, 'N' )
417  equil = lsame( fact, 'E' )
418  notran = lsame( trans, 'N' )
419  IF( nofact .OR. equil ) THEN
420  equed = 'N'
421  rowequ = .false.
422  colequ = .false.
423  ELSE
424  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
425  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
426  smlnum = slamch( 'Safe minimum' )
427  bignum = one / smlnum
428  END IF
429 *
430 * Test the input parameters.
431 *
432  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
433  $ THEN
434  info = -1
435  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
436  $ lsame( trans, 'C' ) ) THEN
437  info = -2
438  ELSE IF( n.LT.0 ) THEN
439  info = -3
440  ELSE IF( kl.LT.0 ) THEN
441  info = -4
442  ELSE IF( ku.LT.0 ) THEN
443  info = -5
444  ELSE IF( nrhs.LT.0 ) THEN
445  info = -6
446  ELSE IF( ldab.LT.kl+ku+1 ) THEN
447  info = -8
448  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
449  info = -10
450  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
451  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
452  info = -12
453  ELSE
454  IF( rowequ ) THEN
455  rcmin = bignum
456  rcmax = zero
457  DO 10 j = 1, n
458  rcmin = min( rcmin, r( j ) )
459  rcmax = max( rcmax, r( j ) )
460  10 CONTINUE
461  IF( rcmin.LE.zero ) THEN
462  info = -13
463  ELSE IF( n.GT.0 ) THEN
464  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
465  ELSE
466  rowcnd = one
467  END IF
468  END IF
469  IF( colequ .AND. info.EQ.0 ) THEN
470  rcmin = bignum
471  rcmax = zero
472  DO 20 j = 1, n
473  rcmin = min( rcmin, c( j ) )
474  rcmax = max( rcmax, c( j ) )
475  20 CONTINUE
476  IF( rcmin.LE.zero ) THEN
477  info = -14
478  ELSE IF( n.GT.0 ) THEN
479  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
480  ELSE
481  colcnd = one
482  END IF
483  END IF
484  IF( info.EQ.0 ) THEN
485  IF( ldb.LT.max( 1, n ) ) THEN
486  info = -16
487  ELSE IF( ldx.LT.max( 1, n ) ) THEN
488  info = -18
489  END IF
490  END IF
491  END IF
492 *
493  IF( info.NE.0 ) THEN
494  CALL xerbla( 'SGBSVX', -info )
495  RETURN
496  END IF
497 *
498  IF( equil ) THEN
499 *
500 * Compute row and column scalings to equilibrate the matrix A.
501 *
502  CALL sgbequ( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
503  $ amax, infequ )
504  IF( infequ.EQ.0 ) THEN
505 *
506 * Equilibrate the matrix.
507 *
508  CALL slaqgb( n, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd,
509  $ amax, equed )
510  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
511  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
512  END IF
513  END IF
514 *
515 * Scale the right hand side.
516 *
517  IF( notran ) THEN
518  IF( rowequ ) THEN
519  DO 40 j = 1, nrhs
520  DO 30 i = 1, n
521  b( i, j ) = r( i )*b( i, j )
522  30 CONTINUE
523  40 CONTINUE
524  END IF
525  ELSE IF( colequ ) THEN
526  DO 60 j = 1, nrhs
527  DO 50 i = 1, n
528  b( i, j ) = c( i )*b( i, j )
529  50 CONTINUE
530  60 CONTINUE
531  END IF
532 *
533  IF( nofact .OR. equil ) THEN
534 *
535 * Compute the LU factorization of the band matrix A.
536 *
537  DO 70 j = 1, n
538  j1 = max( j-ku, 1 )
539  j2 = min( j+kl, n )
540  CALL scopy( j2-j1+1, ab( ku+1-j+j1, j ), 1,
541  $ afb( kl+ku+1-j+j1, j ), 1 )
542  70 CONTINUE
543 *
544  CALL sgbtrf( n, n, kl, ku, afb, ldafb, ipiv, info )
545 *
546 * Return if INFO is non-zero.
547 *
548  IF( info.GT.0 ) THEN
549 *
550 * Compute the reciprocal pivot growth factor of the
551 * leading rank-deficient INFO columns of A.
552 *
553  anorm = zero
554  DO 90 j = 1, info
555  DO 80 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
556  anorm = max( anorm, abs( ab( i, j ) ) )
557  80 CONTINUE
558  90 CONTINUE
559  rpvgrw = slantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
560  $ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
561  $ work )
562  IF( rpvgrw.EQ.zero ) THEN
563  rpvgrw = one
564  ELSE
565  rpvgrw = anorm / rpvgrw
566  END IF
567  work( 1 ) = rpvgrw
568  rcond = zero
569  RETURN
570  END IF
571  END IF
572 *
573 * Compute the norm of the matrix A and the
574 * reciprocal pivot growth factor RPVGRW.
575 *
576  IF( notran ) THEN
577  norm = '1'
578  ELSE
579  norm = 'I'
580  END IF
581  anorm = slangb( norm, n, kl, ku, ab, ldab, work )
582  rpvgrw = slantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, work )
583  IF( rpvgrw.EQ.zero ) THEN
584  rpvgrw = one
585  ELSE
586  rpvgrw = slangb( 'M', n, kl, ku, ab, ldab, work ) / rpvgrw
587  END IF
588 *
589 * Compute the reciprocal of the condition number of A.
590 *
591  CALL sgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
592  $ work, iwork, info )
593 *
594 * Compute the solution matrix X.
595 *
596  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
597  CALL sgbtrs( trans, n, kl, ku, nrhs, afb, ldafb, ipiv, x, ldx,
598  $ info )
599 *
600 * Use iterative refinement to improve the computed solution and
601 * compute error bounds and backward error estimates for it.
602 *
603  CALL sgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
604  $ b, ldb, x, ldx, ferr, berr, work, iwork, info )
605 *
606 * Transform the solution matrix X to a solution of the original
607 * system.
608 *
609  IF( notran ) THEN
610  IF( colequ ) THEN
611  DO 110 j = 1, nrhs
612  DO 100 i = 1, n
613  x( i, j ) = c( i )*x( i, j )
614  100 CONTINUE
615  110 CONTINUE
616  DO 120 j = 1, nrhs
617  ferr( j ) = ferr( j ) / colcnd
618  120 CONTINUE
619  END IF
620  ELSE IF( rowequ ) THEN
621  DO 140 j = 1, nrhs
622  DO 130 i = 1, n
623  x( i, j ) = r( i )*x( i, j )
624  130 CONTINUE
625  140 CONTINUE
626  DO 150 j = 1, nrhs
627  ferr( j ) = ferr( j ) / rowcnd
628  150 CONTINUE
629  END IF
630 *
631 * Set INFO = N+1 if the matrix is singular to working precision.
632 *
633  IF( rcond.LT.slamch( 'Epsilon' ) )
634  $ info = n + 1
635 *
636  work( 1 ) = rpvgrw
637  RETURN
638 *
639 * End of SGBSVX
640 *
641  END
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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:159
subroutine sgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
SGBTRS
Definition: sgbtrs.f:138
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:205
subroutine sgbcon(NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SGBCON
Definition: sgbcon.f:146
subroutine sgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
SGBTRF
Definition: sgbtrf.f:144
subroutine sgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
SGBEQU
Definition: sgbequ.f:153
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:368
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82