LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zgbsvx.f
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1*> \brief <b> ZGBSVX 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 ZGBSVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbsvx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbsvx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbsvx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
22* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
23* RCOND, FERR, BERR, WORK, RWORK, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER EQUED, FACT, TRANS
27* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
28* DOUBLE PRECISION RCOND
29* ..
30* .. Array Arguments ..
31* INTEGER IPIV( * )
32* DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
33* $ RWORK( * )
34* COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
35* $ WORK( * ), X( LDX, * )
36* ..
37*
38*
39*> \par Purpose:
40* =============
41*>
42*> \verbatim
43*>
44*> ZGBSVX uses the LU factorization to compute the solution to a complex
45*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
46*> where A is a band matrix of order N with KL subdiagonals and KU
47*> superdiagonals, and X and B are N-by-NRHS matrices.
48*>
49*> Error bounds on the solution and a condition estimate are also
50*> provided.
51*> \endverbatim
52*
53*> \par Description:
54* =================
55*>
56*> \verbatim
57*>
58*> The following steps are performed by this subroutine:
59*>
60*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
61*> the system:
62*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
63*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
64*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
65*> Whether or not the system will be equilibrated depends on the
66*> scaling of the matrix A, but if equilibration is used, A is
67*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
68*> or diag(C)*B (if TRANS = 'T' or 'C').
69*>
70*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
71*> matrix A (after equilibration if FACT = 'E') as
72*> A = L * U,
73*> where L is a product of permutation and unit lower triangular
74*> matrices with KL subdiagonals, and U is upper triangular with
75*> KL+KU superdiagonals.
76*>
77*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
78*> returns with INFO = i. Otherwise, the factored form of A is used
79*> to estimate the condition number of the matrix A. If the
80*> reciprocal of the condition number is less than machine precision,
81*> INFO = N+1 is returned as a warning, but the routine still goes on
82*> to solve for X and compute error bounds as described below.
83*>
84*> 4. The system of equations is solved for X using the factored form
85*> of A.
86*>
87*> 5. Iterative refinement is applied to improve the computed solution
88*> matrix and calculate error bounds and backward error estimates
89*> for it.
90*>
91*> 6. If equilibration was used, the matrix X is premultiplied by
92*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
93*> that it solves the original system before equilibration.
94*> \endverbatim
95*
96* Arguments:
97* ==========
98*
99*> \param[in] FACT
100*> \verbatim
101*> FACT is CHARACTER*1
102*> Specifies whether or not the factored form of the matrix A is
103*> supplied on entry, and if not, whether the matrix A should be
104*> equilibrated before it is factored.
105*> = 'F': On entry, AFB and IPIV contain the factored form of
106*> A. If EQUED is not 'N', the matrix A has been
107*> equilibrated with scaling factors given by R and C.
108*> AB, AFB, and IPIV are not modified.
109*> = 'N': The matrix A will be copied to AFB and factored.
110*> = 'E': The matrix A will be equilibrated if necessary, then
111*> copied to AFB and factored.
112*> \endverbatim
113*>
114*> \param[in] TRANS
115*> \verbatim
116*> TRANS is CHARACTER*1
117*> Specifies the form of the system of equations.
118*> = 'N': A * X = B (No transpose)
119*> = 'T': A**T * X = B (Transpose)
120*> = 'C': A**H * X = B (Conjugate transpose)
121*> \endverbatim
122*>
123*> \param[in] N
124*> \verbatim
125*> N is INTEGER
126*> The number of linear equations, i.e., the order of the
127*> matrix A. N >= 0.
128*> \endverbatim
129*>
130*> \param[in] KL
131*> \verbatim
132*> KL is INTEGER
133*> The number of subdiagonals within the band of A. KL >= 0.
134*> \endverbatim
135*>
136*> \param[in] KU
137*> \verbatim
138*> KU is INTEGER
139*> The number of superdiagonals within the band of A. KU >= 0.
140*> \endverbatim
141*>
142*> \param[in] NRHS
143*> \verbatim
144*> NRHS is INTEGER
145*> The number of right hand sides, i.e., the number of columns
146*> of the matrices B and X. NRHS >= 0.
147*> \endverbatim
148*>
149*> \param[in,out] AB
150*> \verbatim
151*> AB is COMPLEX*16 array, dimension (LDAB,N)
152*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
153*> The j-th column of A is stored in the j-th column of the
154*> array AB as follows:
155*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
156*>
157*> If FACT = 'F' and EQUED is not 'N', then A must have been
158*> equilibrated by the scaling factors in R and/or C. AB is not
159*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
160*> EQUED = 'N' on exit.
161*>
162*> On exit, if EQUED .ne. 'N', A is scaled as follows:
163*> EQUED = 'R': A := diag(R) * A
164*> EQUED = 'C': A := A * diag(C)
165*> EQUED = 'B': A := diag(R) * A * diag(C).
166*> \endverbatim
167*>
168*> \param[in] LDAB
169*> \verbatim
170*> LDAB is INTEGER
171*> The leading dimension of the array AB. LDAB >= KL+KU+1.
172*> \endverbatim
173*>
174*> \param[in,out] AFB
175*> \verbatim
176*> AFB is COMPLEX*16 array, dimension (LDAFB,N)
177*> If FACT = 'F', then AFB is an input argument and on entry
178*> contains details of the LU factorization of the band matrix
179*> A, as computed by ZGBTRF. U is stored as an upper triangular
180*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
181*> and the multipliers used during the factorization are stored
182*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
183*> the factored form of the equilibrated matrix A.
184*>
185*> If FACT = 'N', then AFB is an output argument and on exit
186*> returns details of the LU factorization of A.
187*>
188*> If FACT = 'E', then AFB is an output argument and on exit
189*> returns details of the LU factorization of the equilibrated
190*> matrix A (see the description of AB for the form of the
191*> equilibrated matrix).
192*> \endverbatim
193*>
194*> \param[in] LDAFB
195*> \verbatim
196*> LDAFB is INTEGER
197*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
198*> \endverbatim
199*>
200*> \param[in,out] IPIV
201*> \verbatim
202*> IPIV is INTEGER array, dimension (N)
203*> If FACT = 'F', then IPIV is an input argument and on entry
204*> contains the pivot indices from the factorization A = L*U
205*> as computed by ZGBTRF; row i of the matrix was interchanged
206*> with row IPIV(i).
207*>
208*> If FACT = 'N', then IPIV is an output argument and on exit
209*> contains the pivot indices from the factorization A = L*U
210*> of the original matrix A.
211*>
212*> If FACT = 'E', then IPIV is an output argument and on exit
213*> contains the pivot indices from the factorization A = L*U
214*> of the equilibrated matrix A.
215*> \endverbatim
216*>
217*> \param[in,out] EQUED
218*> \verbatim
219*> EQUED is CHARACTER*1
220*> Specifies the form of equilibration that was done.
221*> = 'N': No equilibration (always true if FACT = 'N').
222*> = 'R': Row equilibration, i.e., A has been premultiplied by
223*> diag(R).
224*> = 'C': Column equilibration, i.e., A has been postmultiplied
225*> by diag(C).
226*> = 'B': Both row and column equilibration, i.e., A has been
227*> replaced by diag(R) * A * diag(C).
228*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
229*> output argument.
230*> \endverbatim
231*>
232*> \param[in,out] R
233*> \verbatim
234*> R is DOUBLE PRECISION array, dimension (N)
235*> The row scale factors for A. If EQUED = 'R' or 'B', A is
236*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
237*> is not accessed. R is an input argument if FACT = 'F';
238*> otherwise, R is an output argument. If FACT = 'F' and
239*> EQUED = 'R' or 'B', each element of R must be positive.
240*> \endverbatim
241*>
242*> \param[in,out] C
243*> \verbatim
244*> C is DOUBLE PRECISION array, dimension (N)
245*> The column scale factors for A. If EQUED = 'C' or 'B', A is
246*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
247*> is not accessed. C is an input argument if FACT = 'F';
248*> otherwise, C is an output argument. If FACT = 'F' and
249*> EQUED = 'C' or 'B', each element of C must be positive.
250*> \endverbatim
251*>
252*> \param[in,out] B
253*> \verbatim
254*> B is COMPLEX*16 array, dimension (LDB,NRHS)
255*> On entry, the right hand side matrix B.
256*> On exit,
257*> if EQUED = 'N', B is not modified;
258*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
259*> diag(R)*B;
260*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
261*> overwritten by diag(C)*B.
262*> \endverbatim
263*>
264*> \param[in] LDB
265*> \verbatim
266*> LDB is INTEGER
267*> The leading dimension of the array B. LDB >= max(1,N).
268*> \endverbatim
269*>
270*> \param[out] X
271*> \verbatim
272*> X is COMPLEX*16 array, dimension (LDX,NRHS)
273*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
274*> to the original system of equations. Note that A and B are
275*> modified on exit if EQUED .ne. 'N', and the solution to the
276*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
277*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
278*> and EQUED = 'R' or 'B'.
279*> \endverbatim
280*>
281*> \param[in] LDX
282*> \verbatim
283*> LDX is INTEGER
284*> The leading dimension of the array X. LDX >= max(1,N).
285*> \endverbatim
286*>
287*> \param[out] RCOND
288*> \verbatim
289*> RCOND is DOUBLE PRECISION
290*> The estimate of the reciprocal condition number of the matrix
291*> A after equilibration (if done). If RCOND is less than the
292*> machine precision (in particular, if RCOND = 0), the matrix
293*> is singular to working precision. This condition is
294*> indicated by a return code of INFO > 0.
295*> \endverbatim
296*>
297*> \param[out] FERR
298*> \verbatim
299*> FERR is DOUBLE PRECISION array, dimension (NRHS)
300*> The estimated forward error bound for each solution vector
301*> X(j) (the j-th column of the solution matrix X).
302*> If XTRUE is the true solution corresponding to X(j), FERR(j)
303*> is an estimated upper bound for the magnitude of the largest
304*> element in (X(j) - XTRUE) divided by the magnitude of the
305*> largest element in X(j). The estimate is as reliable as
306*> the estimate for RCOND, and is almost always a slight
307*> overestimate of the true error.
308*> \endverbatim
309*>
310*> \param[out] BERR
311*> \verbatim
312*> BERR is DOUBLE PRECISION array, dimension (NRHS)
313*> The componentwise relative backward error of each solution
314*> vector X(j) (i.e., the smallest relative change in
315*> any element of A or B that makes X(j) an exact solution).
316*> \endverbatim
317*>
318*> \param[out] WORK
319*> \verbatim
320*> WORK is COMPLEX*16 array, dimension (2*N)
321*> \endverbatim
322*>
323*> \param[out] RWORK
324*> \verbatim
325*> RWORK is DOUBLE PRECISION array, dimension (N)
326*> On exit, RWORK(1) contains the reciprocal pivot growth
327*> factor norm(A)/norm(U). The "max absolute element" norm is
328*> used. If RWORK(1) is much less than 1, then the stability
329*> of the LU factorization of the (equilibrated) matrix A
330*> could be poor. This also means that the solution X, condition
331*> estimator RCOND, and forward error bound FERR could be
332*> unreliable. If factorization fails with 0<INFO<=N, then
333*> RWORK(1) contains the reciprocal pivot growth factor for the
334*> leading INFO columns of A.
335*> \endverbatim
336*>
337*> \param[out] INFO
338*> \verbatim
339*> INFO is INTEGER
340*> = 0: successful exit
341*> < 0: if INFO = -i, the i-th argument had an illegal value
342*> > 0: if INFO = i, and i is
343*> <= N: U(i,i) is exactly zero. The factorization
344*> has been completed, but the factor U is exactly
345*> singular, so the solution and error bounds
346*> could not be computed. RCOND = 0 is returned.
347*> = N+1: U is nonsingular, but RCOND is less than machine
348*> precision, meaning that the matrix is singular
349*> to working precision. Nevertheless, the
350*> solution and error bounds are computed because
351*> there are a number of situations where the
352*> computed solution can be more accurate than the
353*> value of RCOND would suggest.
354*> \endverbatim
355*
356* Authors:
357* ========
358*
359*> \author Univ. of Tennessee
360*> \author Univ. of California Berkeley
361*> \author Univ. of Colorado Denver
362*> \author NAG Ltd.
363*
364*> \ingroup complex16GBsolve
365*
366* =====================================================================
367 SUBROUTINE zgbsvx( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
368 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
369 $ RCOND, FERR, BERR, WORK, RWORK, INFO )
370*
371* -- LAPACK driver routine --
372* -- LAPACK is a software package provided by Univ. of Tennessee, --
373* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
374*
375* .. Scalar Arguments ..
376 CHARACTER EQUED, FACT, TRANS
377 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
378 DOUBLE PRECISION RCOND
379* ..
380* .. Array Arguments ..
381 INTEGER IPIV( * )
382 DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
383 $ rwork( * )
384 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
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 DOUBLE PRECISION ZERO, ONE
395 PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0 )
396* ..
397* .. Local Scalars ..
398 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
399 CHARACTER NORM
400 INTEGER I, INFEQU, J, J1, J2
401 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
402 $ rowcnd, rpvgrw, smlnum
403* ..
404* .. External Functions ..
405 LOGICAL LSAME
406 DOUBLE PRECISION DLAMCH, ZLANGB, ZLANTB
407 EXTERNAL lsame, dlamch, zlangb, zlantb
408* ..
409* .. External Subroutines ..
410 EXTERNAL xerbla, zcopy, zgbcon, zgbequ, zgbrfs, zgbtrf,
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 = dlamch( '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( 'ZGBSVX', -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 zgbequ( 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 zlaqgb( 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 zcopy( 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 zgbtrf( 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 = zlantb( 'M', 'U', 'N', info, min( info-1, kl+ku ),
563 $ afb( max( 1, kl+ku+2-info ), 1 ), ldafb,
564 $ rwork )
565 IF( rpvgrw.EQ.zero ) THEN
566 rpvgrw = one
567 ELSE
568 rpvgrw = anorm / rpvgrw
569 END IF
570 rwork( 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 = zlangb( norm, n, kl, ku, ab, ldab, rwork )
585 rpvgrw = zlantb( 'M', 'U', 'N', n, kl+ku, afb, ldafb, rwork )
586 IF( rpvgrw.EQ.zero ) THEN
587 rpvgrw = one
588 ELSE
589 rpvgrw = zlangb( 'M', n, kl, ku, ab, ldab, rwork ) / rpvgrw
590 END IF
591*
592* Compute the reciprocal of the condition number of A.
593*
594 CALL zgbcon( norm, n, kl, ku, afb, ldafb, ipiv, anorm, rcond,
595 $ work, rwork, info )
596*
597* Compute the solution matrix X.
598*
599 CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
600 CALL zgbtrs( 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 zgbrfs( trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv,
607 $ b, ldb, x, ldx, ferr, berr, work, rwork, 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.dlamch( 'Epsilon' ) )
637 $ info = n + 1
638*
639 rwork( 1 ) = rpvgrw
640 RETURN
641*
642* End of ZGBSVX
643*
644 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zlaqgb(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
ZLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.
Definition: zlaqgb.f:160
subroutine zgbrfs(TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZGBRFS
Definition: zgbrfs.f:206
subroutine zgbcon(NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, RWORK, INFO)
ZGBCON
Definition: zgbcon.f:147
subroutine zgbtrf(M, N, KL, KU, AB, LDAB, IPIV, INFO)
ZGBTRF
Definition: zgbtrf.f:144
subroutine zgbequ(M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
ZGBEQU
Definition: zgbequ.f:154
subroutine zgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
ZGBTRS
Definition: zgbtrs.f:138
subroutine zgbsvx(FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
ZGBSVX computes the solution to system of linear equations A * X = B for GB matrices
Definition: zgbsvx.f:370
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103