1: *> \brief <b> DPBSVX computes the solution to system of linear equations A * X = B for OTHER 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 DPBSVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbsvx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbsvx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbsvx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
22: * EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
23: * WORK, IWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER EQUED, FACT, UPLO
27: * INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
28: * DOUBLE PRECISION RCOND
29: * ..
30: * .. Array Arguments ..
31: * INTEGER IWORK( * )
32: * DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
33: * $ BERR( * ), FERR( * ), S( * ), WORK( * ),
34: * $ X( LDX, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
44: *> compute the solution to a real system of linear equations
45: *> A * X = B,
46: *> where A is an N-by-N symmetric positive definite band matrix and X
47: *> 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:
59: *>
60: *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
61: *> the system:
62: *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
63: *> Whether or not the system will be equilibrated depends on the
64: *> scaling of the matrix A, but if equilibration is used, A is
65: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
66: *>
67: *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
68: *> factor the matrix A (after equilibration if FACT = 'E') as
69: *> A = U**T * U, if UPLO = 'U', or
70: *> A = L * L**T, if UPLO = 'L',
71: *> where U is an upper triangular band matrix, and L is a lower
72: *> triangular band matrix.
73: *>
74: *> 3. If the leading i-by-i principal minor is not positive definite,
75: *> then the routine returns with INFO = i. Otherwise, the factored
76: *> form of A is used to estimate the condition number of the matrix
77: *> A. If the reciprocal of the condition number is less than machine
78: *> precision, INFO = N+1 is returned as a warning, but the routine
79: *> still goes on to solve for X and compute error bounds as
80: *> described below.
81: *>
82: *> 4. The system of equations is solved for X using the factored form
83: *> of A.
84: *>
85: *> 5. Iterative refinement is applied to improve the computed solution
86: *> matrix and calculate error bounds and backward error estimates
87: *> for it.
88: *>
89: *> 6. If equilibration was used, the matrix X is premultiplied by
90: *> diag(S) so that it solves the original system before
91: *> equilibration.
92: *> \endverbatim
93: *
94: * Arguments:
95: * ==========
96: *
97: *> \param[in] FACT
98: *> \verbatim
99: *> FACT is CHARACTER*1
100: *> Specifies whether or not the factored form of the matrix A is
101: *> supplied on entry, and if not, whether the matrix A should be
102: *> equilibrated before it is factored.
103: *> = 'F': On entry, AFB contains the factored form of A.
104: *> If EQUED = 'Y', the matrix A has been equilibrated
105: *> with scaling factors given by S. AB and AFB will not
106: *> be modified.
107: *> = 'N': The matrix A will be copied to AFB and factored.
108: *> = 'E': The matrix A will be equilibrated if necessary, then
109: *> copied to AFB and factored.
110: *> \endverbatim
111: *>
112: *> \param[in] UPLO
113: *> \verbatim
114: *> UPLO is CHARACTER*1
115: *> = 'U': Upper triangle of A is stored;
116: *> = 'L': Lower triangle of A is stored.
117: *> \endverbatim
118: *>
119: *> \param[in] N
120: *> \verbatim
121: *> N is INTEGER
122: *> The number of linear equations, i.e., the order of the
123: *> matrix A. N >= 0.
124: *> \endverbatim
125: *>
126: *> \param[in] KD
127: *> \verbatim
128: *> KD is INTEGER
129: *> The number of superdiagonals of the matrix A if UPLO = 'U',
130: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
131: *> \endverbatim
132: *>
133: *> \param[in] NRHS
134: *> \verbatim
135: *> NRHS is INTEGER
136: *> The number of right-hand sides, i.e., the number of columns
137: *> of the matrices B and X. NRHS >= 0.
138: *> \endverbatim
139: *>
140: *> \param[in,out] AB
141: *> \verbatim
142: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
143: *> On entry, the upper or lower triangle of the symmetric band
144: *> matrix A, stored in the first KD+1 rows of the array, except
145: *> if FACT = 'F' and EQUED = 'Y', then A must contain the
146: *> equilibrated matrix diag(S)*A*diag(S). The j-th column of A
147: *> is stored in the j-th column of the array AB as follows:
148: *> if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
149: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
150: *> See below for further details.
151: *>
152: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
153: *> diag(S)*A*diag(S).
154: *> \endverbatim
155: *>
156: *> \param[in] LDAB
157: *> \verbatim
158: *> LDAB is INTEGER
159: *> The leading dimension of the array A. LDAB >= KD+1.
160: *> \endverbatim
161: *>
162: *> \param[in,out] AFB
163: *> \verbatim
164: *> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
165: *> If FACT = 'F', then AFB is an input argument and on entry
166: *> contains the triangular factor U or L from the Cholesky
167: *> factorization A = U**T*U or A = L*L**T of the band matrix
168: *> A, in the same storage format as A (see AB). If EQUED = 'Y',
169: *> then AFB is the factored form of the equilibrated matrix A.
170: *>
171: *> If FACT = 'N', then AFB is an output argument and on exit
172: *> returns the triangular factor U or L from the Cholesky
173: *> factorization A = U**T*U or A = L*L**T.
174: *>
175: *> If FACT = 'E', then AFB is an output argument and on exit
176: *> returns the triangular factor U or L from the Cholesky
177: *> factorization A = U**T*U or A = L*L**T of the equilibrated
178: *> matrix A (see the description of A for the form of the
179: *> equilibrated matrix).
180: *> \endverbatim
181: *>
182: *> \param[in] LDAFB
183: *> \verbatim
184: *> LDAFB is INTEGER
185: *> The leading dimension of the array AFB. LDAFB >= KD+1.
186: *> \endverbatim
187: *>
188: *> \param[in,out] EQUED
189: *> \verbatim
190: *> EQUED is CHARACTER*1
191: *> Specifies the form of equilibration that was done.
192: *> = 'N': No equilibration (always true if FACT = 'N').
193: *> = 'Y': Equilibration was done, i.e., A has been replaced by
194: *> diag(S) * A * diag(S).
195: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
196: *> output argument.
197: *> \endverbatim
198: *>
199: *> \param[in,out] S
200: *> \verbatim
201: *> S is DOUBLE PRECISION array, dimension (N)
202: *> The scale factors for A; not accessed if EQUED = 'N'. S is
203: *> an input argument if FACT = 'F'; otherwise, S is an output
204: *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
205: *> must be positive.
206: *> \endverbatim
207: *>
208: *> \param[in,out] B
209: *> \verbatim
210: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
211: *> On entry, the N-by-NRHS right hand side matrix B.
212: *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
213: *> B is overwritten by diag(S) * B.
214: *> \endverbatim
215: *>
216: *> \param[in] LDB
217: *> \verbatim
218: *> LDB is INTEGER
219: *> The leading dimension of the array B. LDB >= max(1,N).
220: *> \endverbatim
221: *>
222: *> \param[out] X
223: *> \verbatim
224: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
225: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
226: *> the original system of equations. Note that if EQUED = 'Y',
227: *> A and B are modified on exit, and the solution to the
228: *> equilibrated system is inv(diag(S))*X.
229: *> \endverbatim
230: *>
231: *> \param[in] LDX
232: *> \verbatim
233: *> LDX is INTEGER
234: *> The leading dimension of the array X. LDX >= max(1,N).
235: *> \endverbatim
236: *>
237: *> \param[out] RCOND
238: *> \verbatim
239: *> RCOND is DOUBLE PRECISION
240: *> The estimate of the reciprocal condition number of the matrix
241: *> A after equilibration (if done). If RCOND is less than the
242: *> machine precision (in particular, if RCOND = 0), the matrix
243: *> is singular to working precision. This condition is
244: *> indicated by a return code of INFO > 0.
245: *> \endverbatim
246: *>
247: *> \param[out] FERR
248: *> \verbatim
249: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
250: *> The estimated forward error bound for each solution vector
251: *> X(j) (the j-th column of the solution matrix X).
252: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
253: *> is an estimated upper bound for the magnitude of the largest
254: *> element in (X(j) - XTRUE) divided by the magnitude of the
255: *> largest element in X(j). The estimate is as reliable as
256: *> the estimate for RCOND, and is almost always a slight
257: *> overestimate of the true error.
258: *> \endverbatim
259: *>
260: *> \param[out] BERR
261: *> \verbatim
262: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
263: *> The componentwise relative backward error of each solution
264: *> vector X(j) (i.e., the smallest relative change in
265: *> any element of A or B that makes X(j) an exact solution).
266: *> \endverbatim
267: *>
268: *> \param[out] WORK
269: *> \verbatim
270: *> WORK is DOUBLE PRECISION array, dimension (3*N)
271: *> \endverbatim
272: *>
273: *> \param[out] IWORK
274: *> \verbatim
275: *> IWORK is INTEGER array, dimension (N)
276: *> \endverbatim
277: *>
278: *> \param[out] INFO
279: *> \verbatim
280: *> INFO is INTEGER
281: *> = 0: successful exit
282: *> < 0: if INFO = -i, the i-th argument had an illegal value
283: *> > 0: if INFO = i, and i is
284: *> <= N: the leading minor of order i of A is
285: *> not positive definite, so the factorization
286: *> could not be completed, and the solution has not
287: *> been computed. RCOND = 0 is returned.
288: *> = N+1: U is nonsingular, but RCOND is less than machine
289: *> precision, meaning that the matrix is singular
290: *> to working precision. Nevertheless, the
291: *> solution and error bounds are computed because
292: *> there are a number of situations where the
293: *> computed solution can be more accurate than the
294: *> value of RCOND would suggest.
295: *> \endverbatim
296: *
297: * Authors:
298: * ========
299: *
300: *> \author Univ. of Tennessee
301: *> \author Univ. of California Berkeley
302: *> \author Univ. of Colorado Denver
303: *> \author NAG Ltd.
304: *
305: *> \ingroup doubleOTHERsolve
306: *
307: *> \par Further Details:
308: * =====================
309: *>
310: *> \verbatim
311: *>
312: *> The band storage scheme is illustrated by the following example, when
313: *> N = 6, KD = 2, and UPLO = 'U':
314: *>
315: *> Two-dimensional storage of the symmetric matrix A:
316: *>
317: *> a11 a12 a13
318: *> a22 a23 a24
319: *> a33 a34 a35
320: *> a44 a45 a46
321: *> a55 a56
322: *> (aij=conjg(aji)) a66
323: *>
324: *> Band storage of the upper triangle of A:
325: *>
326: *> * * a13 a24 a35 a46
327: *> * a12 a23 a34 a45 a56
328: *> a11 a22 a33 a44 a55 a66
329: *>
330: *> Similarly, if UPLO = 'L' the format of A is as follows:
331: *>
332: *> a11 a22 a33 a44 a55 a66
333: *> a21 a32 a43 a54 a65 *
334: *> a31 a42 a53 a64 * *
335: *>
336: *> Array elements marked * are not used by the routine.
337: *> \endverbatim
338: *>
339: * =====================================================================
340: SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
341: $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
342: $ WORK, IWORK, INFO )
343: *
344: * -- LAPACK driver routine --
345: * -- LAPACK is a software package provided by Univ. of Tennessee, --
346: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
347: *
348: * .. Scalar Arguments ..
349: CHARACTER EQUED, FACT, UPLO
350: INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
351: DOUBLE PRECISION RCOND
352: * ..
353: * .. Array Arguments ..
354: INTEGER IWORK( * )
355: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
356: $ BERR( * ), FERR( * ), S( * ), WORK( * ),
357: $ X( LDX, * )
358: * ..
359: *
360: * =====================================================================
361: *
362: * .. Parameters ..
363: DOUBLE PRECISION ZERO, ONE
364: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
365: * ..
366: * .. Local Scalars ..
367: LOGICAL EQUIL, NOFACT, RCEQU, UPPER
368: INTEGER I, INFEQU, J, J1, J2
369: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
370: * ..
371: * .. External Functions ..
372: LOGICAL LSAME
373: DOUBLE PRECISION DLAMCH, DLANSB
374: EXTERNAL LSAME, DLAMCH, DLANSB
375: * ..
376: * .. External Subroutines ..
377: EXTERNAL DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU, DPBRFS,
378: $ DPBTRF, DPBTRS, XERBLA
379: * ..
380: * .. Intrinsic Functions ..
381: INTRINSIC MAX, MIN
382: * ..
383: * .. Executable Statements ..
384: *
385: INFO = 0
386: NOFACT = LSAME( FACT, 'N' )
387: EQUIL = LSAME( FACT, 'E' )
388: UPPER = LSAME( UPLO, 'U' )
389: IF( NOFACT .OR. EQUIL ) THEN
390: EQUED = 'N'
391: RCEQU = .FALSE.
392: ELSE
393: RCEQU = LSAME( EQUED, 'Y' )
394: SMLNUM = DLAMCH( 'Safe minimum' )
395: BIGNUM = ONE / SMLNUM
396: END IF
397: *
398: * Test the input parameters.
399: *
400: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
401: $ THEN
402: INFO = -1
403: ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
404: INFO = -2
405: ELSE IF( N.LT.0 ) THEN
406: INFO = -3
407: ELSE IF( KD.LT.0 ) THEN
408: INFO = -4
409: ELSE IF( NRHS.LT.0 ) THEN
410: INFO = -5
411: ELSE IF( LDAB.LT.KD+1 ) THEN
412: INFO = -7
413: ELSE IF( LDAFB.LT.KD+1 ) THEN
414: INFO = -9
415: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
416: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
417: INFO = -10
418: ELSE
419: IF( RCEQU ) THEN
420: SMIN = BIGNUM
421: SMAX = ZERO
422: DO 10 J = 1, N
423: SMIN = MIN( SMIN, S( J ) )
424: SMAX = MAX( SMAX, S( J ) )
425: 10 CONTINUE
426: IF( SMIN.LE.ZERO ) THEN
427: INFO = -11
428: ELSE IF( N.GT.0 ) THEN
429: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
430: ELSE
431: SCOND = ONE
432: END IF
433: END IF
434: IF( INFO.EQ.0 ) THEN
435: IF( LDB.LT.MAX( 1, N ) ) THEN
436: INFO = -13
437: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
438: INFO = -15
439: END IF
440: END IF
441: END IF
442: *
443: IF( INFO.NE.0 ) THEN
444: CALL XERBLA( 'DPBSVX', -INFO )
445: RETURN
446: END IF
447: *
448: IF( EQUIL ) THEN
449: *
450: * Compute row and column scalings to equilibrate the matrix A.
451: *
452: CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
453: IF( INFEQU.EQ.0 ) THEN
454: *
455: * Equilibrate the matrix.
456: *
457: CALL DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
458: RCEQU = LSAME( EQUED, 'Y' )
459: END IF
460: END IF
461: *
462: * Scale the right-hand side.
463: *
464: IF( RCEQU ) THEN
465: DO 30 J = 1, NRHS
466: DO 20 I = 1, N
467: B( I, J ) = S( I )*B( I, J )
468: 20 CONTINUE
469: 30 CONTINUE
470: END IF
471: *
472: IF( NOFACT .OR. EQUIL ) THEN
473: *
474: * Compute the Cholesky factorization A = U**T *U or A = L*L**T.
475: *
476: IF( UPPER ) THEN
477: DO 40 J = 1, N
478: J1 = MAX( J-KD, 1 )
479: CALL DCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
480: $ AFB( KD+1-J+J1, J ), 1 )
481: 40 CONTINUE
482: ELSE
483: DO 50 J = 1, N
484: J2 = MIN( J+KD, N )
485: CALL DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
486: 50 CONTINUE
487: END IF
488: *
489: CALL DPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
490: *
491: * Return if INFO is non-zero.
492: *
493: IF( INFO.GT.0 )THEN
494: RCOND = ZERO
495: RETURN
496: END IF
497: END IF
498: *
499: * Compute the norm of the matrix A.
500: *
501: ANORM = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK )
502: *
503: * Compute the reciprocal of the condition number of A.
504: *
505: CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK,
506: $ INFO )
507: *
508: * Compute the solution matrix X.
509: *
510: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
511: CALL DPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
512: *
513: * Use iterative refinement to improve the computed solution and
514: * compute error bounds and backward error estimates for it.
515: *
516: CALL DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
517: $ LDX, FERR, BERR, WORK, IWORK, INFO )
518: *
519: * Transform the solution matrix X to a solution of the original
520: * system.
521: *
522: IF( RCEQU ) THEN
523: DO 70 J = 1, NRHS
524: DO 60 I = 1, N
525: X( I, J ) = S( I )*X( I, J )
526: 60 CONTINUE
527: 70 CONTINUE
528: DO 80 J = 1, NRHS
529: FERR( J ) = FERR( J ) / SCOND
530: 80 CONTINUE
531: END IF
532: *
533: * Set INFO = N+1 if the matrix is singular to working precision.
534: *
535: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
536: $ INFO = N + 1
537: *
538: RETURN
539: *
540: * End of DPBSVX
541: *
542: END
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