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: *
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9: *> Download ZGBSVX + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
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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,
411: $ ZGBTRS, ZLACPY, ZLAQGB
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
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