Annotation of rpl/lapack/lapack/dgbsvx.f, revision 1.13
1.8 bertrand 1: *> \brief <b> DGBSVX 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 DGBSVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsvx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsvx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsvx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGBSVX( 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: * DOUBLE PRECISION RCOND
29: * ..
30: * .. Array Arguments ..
31: * INTEGER IPIV( * ), IWORK( * )
32: * DOUBLE PRECISION 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: *> DGBSVX 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 DOUBLE PRECISION 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
1.10 bertrand 175: *> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
1.8 bertrand 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 DGBTRF. 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
1.10 bertrand 201: *> IPIV is INTEGER array, dimension (N)
1.8 bertrand 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 DGBTRF; 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
1.10 bertrand 218: *> EQUED is CHARACTER*1
1.8 bertrand 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
1.10 bertrand 233: *> R is DOUBLE PRECISION array, dimension (N)
1.8 bertrand 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
1.10 bertrand 243: *> C is DOUBLE PRECISION array, dimension (N)
1.8 bertrand 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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: *> value of RCOND would suggest.
353: *> \endverbatim
354: *
355: * Authors:
356: * ========
357: *
358: *> \author Univ. of Tennessee
359: *> \author Univ. of California Berkeley
360: *> \author Univ. of Colorado Denver
361: *> \author NAG Ltd.
362: *
1.10 bertrand 363: *> \date April 2012
1.8 bertrand 364: *
365: *> \ingroup doubleGBsolve
366: *
367: * =====================================================================
1.1 bertrand 368: SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
369: $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
370: $ RCOND, FERR, BERR, WORK, IWORK, INFO )
371: *
1.10 bertrand 372: * -- LAPACK driver routine (version 3.4.1) --
1.1 bertrand 373: * -- LAPACK is a software package provided by Univ. of Tennessee, --
374: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.10 bertrand 375: * April 2012
1.1 bertrand 376: *
377: * .. Scalar Arguments ..
378: CHARACTER EQUED, FACT, TRANS
379: INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
380: DOUBLE PRECISION RCOND
381: * ..
382: * .. Array Arguments ..
383: INTEGER IPIV( * ), IWORK( * )
384: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
385: $ BERR( * ), C( * ), FERR( * ), R( * ),
386: $ WORK( * ), X( LDX, * )
387: * ..
388: *
389: * =====================================================================
390: *
391: * .. Parameters ..
392: DOUBLE PRECISION ZERO, ONE
393: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
394: * ..
395: * .. Local Scalars ..
396: LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
397: CHARACTER NORM
398: INTEGER I, INFEQU, J, J1, J2
399: DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
400: $ ROWCND, RPVGRW, SMLNUM
401: * ..
402: * .. External Functions ..
403: LOGICAL LSAME
404: DOUBLE PRECISION DLAMCH, DLANGB, DLANTB
405: EXTERNAL LSAME, DLAMCH, DLANGB, DLANTB
406: * ..
407: * .. External Subroutines ..
408: EXTERNAL DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF, DGBTRS,
409: $ DLACPY, DLAQGB, XERBLA
410: * ..
411: * .. Intrinsic Functions ..
412: INTRINSIC ABS, MAX, MIN
413: * ..
414: * .. Executable Statements ..
415: *
416: INFO = 0
417: NOFACT = LSAME( FACT, 'N' )
418: EQUIL = LSAME( FACT, 'E' )
419: NOTRAN = LSAME( TRANS, 'N' )
420: IF( NOFACT .OR. EQUIL ) THEN
421: EQUED = 'N'
422: ROWEQU = .FALSE.
423: COLEQU = .FALSE.
424: ELSE
425: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
426: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
427: SMLNUM = DLAMCH( 'Safe minimum' )
428: BIGNUM = ONE / SMLNUM
429: END IF
430: *
431: * Test the input parameters.
432: *
433: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
434: $ THEN
435: INFO = -1
436: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
437: $ LSAME( TRANS, 'C' ) ) THEN
438: INFO = -2
439: ELSE IF( N.LT.0 ) THEN
440: INFO = -3
441: ELSE IF( KL.LT.0 ) THEN
442: INFO = -4
443: ELSE IF( KU.LT.0 ) THEN
444: INFO = -5
445: ELSE IF( NRHS.LT.0 ) THEN
446: INFO = -6
447: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
448: INFO = -8
449: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
450: INFO = -10
451: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
452: $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
453: INFO = -12
454: ELSE
455: IF( ROWEQU ) THEN
456: RCMIN = BIGNUM
457: RCMAX = ZERO
458: DO 10 J = 1, N
459: RCMIN = MIN( RCMIN, R( J ) )
460: RCMAX = MAX( RCMAX, R( J ) )
461: 10 CONTINUE
462: IF( RCMIN.LE.ZERO ) THEN
463: INFO = -13
464: ELSE IF( N.GT.0 ) THEN
465: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
466: ELSE
467: ROWCND = ONE
468: END IF
469: END IF
470: IF( COLEQU .AND. INFO.EQ.0 ) THEN
471: RCMIN = BIGNUM
472: RCMAX = ZERO
473: DO 20 J = 1, N
474: RCMIN = MIN( RCMIN, C( J ) )
475: RCMAX = MAX( RCMAX, C( J ) )
476: 20 CONTINUE
477: IF( RCMIN.LE.ZERO ) THEN
478: INFO = -14
479: ELSE IF( N.GT.0 ) THEN
480: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
481: ELSE
482: COLCND = ONE
483: END IF
484: END IF
485: IF( INFO.EQ.0 ) THEN
486: IF( LDB.LT.MAX( 1, N ) ) THEN
487: INFO = -16
488: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
489: INFO = -18
490: END IF
491: END IF
492: END IF
493: *
494: IF( INFO.NE.0 ) THEN
495: CALL XERBLA( 'DGBSVX', -INFO )
496: RETURN
497: END IF
498: *
499: IF( EQUIL ) THEN
500: *
501: * Compute row and column scalings to equilibrate the matrix A.
502: *
503: CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
504: $ AMAX, INFEQU )
505: IF( INFEQU.EQ.0 ) THEN
506: *
507: * Equilibrate the matrix.
508: *
509: CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
510: $ AMAX, EQUED )
511: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
512: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
513: END IF
514: END IF
515: *
516: * Scale the right hand side.
517: *
518: IF( NOTRAN ) THEN
519: IF( ROWEQU ) THEN
520: DO 40 J = 1, NRHS
521: DO 30 I = 1, N
522: B( I, J ) = R( I )*B( I, J )
523: 30 CONTINUE
524: 40 CONTINUE
525: END IF
526: ELSE IF( COLEQU ) THEN
527: DO 60 J = 1, NRHS
528: DO 50 I = 1, N
529: B( I, J ) = C( I )*B( I, J )
530: 50 CONTINUE
531: 60 CONTINUE
532: END IF
533: *
534: IF( NOFACT .OR. EQUIL ) THEN
535: *
536: * Compute the LU factorization of the band matrix A.
537: *
538: DO 70 J = 1, N
539: J1 = MAX( J-KU, 1 )
540: J2 = MIN( J+KL, N )
541: CALL DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
542: $ AFB( KL+KU+1-J+J1, J ), 1 )
543: 70 CONTINUE
544: *
545: CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
546: *
547: * Return if INFO is non-zero.
548: *
549: IF( INFO.GT.0 ) THEN
550: *
551: * Compute the reciprocal pivot growth factor of the
552: * leading rank-deficient INFO columns of A.
553: *
554: ANORM = ZERO
555: DO 90 J = 1, INFO
556: DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
557: ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
558: 80 CONTINUE
559: 90 CONTINUE
560: RPVGRW = DLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
561: $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
562: $ WORK )
563: IF( RPVGRW.EQ.ZERO ) THEN
564: RPVGRW = ONE
565: ELSE
566: RPVGRW = ANORM / RPVGRW
567: END IF
568: WORK( 1 ) = RPVGRW
569: RCOND = ZERO
570: RETURN
571: END IF
572: END IF
573: *
574: * Compute the norm of the matrix A and the
575: * reciprocal pivot growth factor RPVGRW.
576: *
577: IF( NOTRAN ) THEN
578: NORM = '1'
579: ELSE
580: NORM = 'I'
581: END IF
582: ANORM = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
583: RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK )
584: IF( RPVGRW.EQ.ZERO ) THEN
585: RPVGRW = ONE
586: ELSE
587: RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW
588: END IF
589: *
590: * Compute the reciprocal of the condition number of A.
591: *
592: CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
593: $ WORK, IWORK, INFO )
594: *
595: * Compute the solution matrix X.
596: *
597: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
598: CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
599: $ INFO )
600: *
601: * Use iterative refinement to improve the computed solution and
602: * compute error bounds and backward error estimates for it.
603: *
604: CALL DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
605: $ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
606: *
607: * Transform the solution matrix X to a solution of the original
608: * system.
609: *
610: IF( NOTRAN ) THEN
611: IF( COLEQU ) THEN
612: DO 110 J = 1, NRHS
613: DO 100 I = 1, N
614: X( I, J ) = C( I )*X( I, J )
615: 100 CONTINUE
616: 110 CONTINUE
617: DO 120 J = 1, NRHS
618: FERR( J ) = FERR( J ) / COLCND
619: 120 CONTINUE
620: END IF
621: ELSE IF( ROWEQU ) THEN
622: DO 140 J = 1, NRHS
623: DO 130 I = 1, N
624: X( I, J ) = R( I )*X( I, J )
625: 130 CONTINUE
626: 140 CONTINUE
627: DO 150 J = 1, NRHS
628: FERR( J ) = FERR( J ) / ROWCND
629: 150 CONTINUE
630: END IF
631: *
632: * Set INFO = N+1 if the matrix is singular to working precision.
633: *
634: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
635: $ INFO = N + 1
636: *
637: WORK( 1 ) = RPVGRW
638: RETURN
639: *
640: * End of DGBSVX
641: *
642: END
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