LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
cpbsvx.f
Go to the documentation of this file.
1 *> \brief <b> CPBSVX 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 CPBSVX + dependencies
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11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cpbsvx.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cpbsvx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
22 * EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
23 * WORK, RWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER EQUED, FACT, UPLO
27 * INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
28 * REAL RCOND
29 * ..
30 * .. Array Arguments ..
31 * REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
32 * COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
33 * $ WORK( * ), X( LDX, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> CPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
43 *> compute the solution to a complex system of linear equations
44 *> A * X = B,
45 *> where A is an N-by-N Hermitian positive definite band matrix and X
46 *> 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:
58 *>
59 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
60 *> the system:
61 *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
62 *> Whether or not the system will be equilibrated depends on the
63 *> scaling of the matrix A, but if equilibration is used, A is
64 *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
65 *>
66 *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
67 *> factor the matrix A (after equilibration if FACT = 'E') as
68 *> A = U**H * U, if UPLO = 'U', or
69 *> A = L * L**H, if UPLO = 'L',
70 *> where U is an upper triangular band matrix, and L is a lower
71 *> triangular band matrix.
72 *>
73 *> 3. If the leading i-by-i principal minor is not positive definite,
74 *> then the routine returns with INFO = i. Otherwise, the factored
75 *> form of A is used to estimate the condition number of the matrix
76 *> A. If the reciprocal of the condition number is less than machine
77 *> precision, INFO = N+1 is returned as a warning, but the routine
78 *> still goes on to solve for X and compute error bounds as
79 *> described below.
80 *>
81 *> 4. The system of equations is solved for X using the factored form
82 *> of A.
83 *>
84 *> 5. Iterative refinement is applied to improve the computed solution
85 *> matrix and calculate error bounds and backward error estimates
86 *> for it.
87 *>
88 *> 6. If equilibration was used, the matrix X is premultiplied by
89 *> diag(S) so that it solves the original system before
90 *> equilibration.
91 *> \endverbatim
92 *
93 * Arguments:
94 * ==========
95 *
96 *> \param[in] FACT
97 *> \verbatim
98 *> FACT is CHARACTER*1
99 *> Specifies whether or not the factored form of the matrix A is
100 *> supplied on entry, and if not, whether the matrix A should be
101 *> equilibrated before it is factored.
102 *> = 'F': On entry, AFB contains the factored form of A.
103 *> If EQUED = 'Y', the matrix A has been equilibrated
104 *> with scaling factors given by S. AB and AFB will not
105 *> be modified.
106 *> = 'N': The matrix A will be copied to AFB and factored.
107 *> = 'E': The matrix A will be equilibrated if necessary, then
108 *> copied to AFB and factored.
109 *> \endverbatim
110 *>
111 *> \param[in] UPLO
112 *> \verbatim
113 *> UPLO is CHARACTER*1
114 *> = 'U': Upper triangle of A is stored;
115 *> = 'L': Lower triangle of A is stored.
116 *> \endverbatim
117 *>
118 *> \param[in] N
119 *> \verbatim
120 *> N is INTEGER
121 *> The number of linear equations, i.e., the order of the
122 *> matrix A. N >= 0.
123 *> \endverbatim
124 *>
125 *> \param[in] KD
126 *> \verbatim
127 *> KD is INTEGER
128 *> The number of superdiagonals of the matrix A if UPLO = 'U',
129 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
130 *> \endverbatim
131 *>
132 *> \param[in] NRHS
133 *> \verbatim
134 *> NRHS is INTEGER
135 *> The number of right-hand sides, i.e., the number of columns
136 *> of the matrices B and X. NRHS >= 0.
137 *> \endverbatim
138 *>
139 *> \param[in,out] AB
140 *> \verbatim
141 *> AB is COMPLEX array, dimension (LDAB,N)
142 *> On entry, the upper or lower triangle of the Hermitian band
143 *> matrix A, stored in the first KD+1 rows of the array, except
144 *> if FACT = 'F' and EQUED = 'Y', then A must contain the
145 *> equilibrated matrix diag(S)*A*diag(S). The j-th column of A
146 *> is stored in the j-th column of the array AB as follows:
147 *> if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
148 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
149 *> See below for further details.
150 *>
151 *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
152 *> diag(S)*A*diag(S).
153 *> \endverbatim
154 *>
155 *> \param[in] LDAB
156 *> \verbatim
157 *> LDAB is INTEGER
158 *> The leading dimension of the array A. LDAB >= KD+1.
159 *> \endverbatim
160 *>
161 *> \param[in,out] AFB
162 *> \verbatim
163 *> AFB is COMPLEX array, dimension (LDAFB,N)
164 *> If FACT = 'F', then AFB is an input argument and on entry
165 *> contains the triangular factor U or L from the Cholesky
166 *> factorization A = U**H*U or A = L*L**H of the band matrix
167 *> A, in the same storage format as A (see AB). If EQUED = 'Y',
168 *> then AFB is the factored form of the equilibrated matrix A.
169 *>
170 *> If FACT = 'N', then AFB is an output argument and on exit
171 *> returns the triangular factor U or L from the Cholesky
172 *> factorization A = U**H*U or A = L*L**H.
173 *>
174 *> If FACT = 'E', then AFB is an output argument and on exit
175 *> returns the triangular factor U or L from the Cholesky
176 *> factorization A = U**H*U or A = L*L**H of the equilibrated
177 *> matrix A (see the description of A for the form of the
178 *> equilibrated matrix).
179 *> \endverbatim
180 *>
181 *> \param[in] LDAFB
182 *> \verbatim
183 *> LDAFB is INTEGER
184 *> The leading dimension of the array AFB. LDAFB >= KD+1.
185 *> \endverbatim
186 *>
187 *> \param[in,out] EQUED
188 *> \verbatim
189 *> EQUED is CHARACTER*1
190 *> Specifies the form of equilibration that was done.
191 *> = 'N': No equilibration (always true if FACT = 'N').
192 *> = 'Y': Equilibration was done, i.e., A has been replaced by
193 *> diag(S) * A * diag(S).
194 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
195 *> output argument.
196 *> \endverbatim
197 *>
198 *> \param[in,out] S
199 *> \verbatim
200 *> S is REAL array, dimension (N)
201 *> The scale factors for A; not accessed if EQUED = 'N'. S is
202 *> an input argument if FACT = 'F'; otherwise, S is an output
203 *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
204 *> must be positive.
205 *> \endverbatim
206 *>
207 *> \param[in,out] B
208 *> \verbatim
209 *> B is COMPLEX array, dimension (LDB,NRHS)
210 *> On entry, the N-by-NRHS right hand side matrix B.
211 *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
212 *> B is overwritten by diag(S) * B.
213 *> \endverbatim
214 *>
215 *> \param[in] LDB
216 *> \verbatim
217 *> LDB is INTEGER
218 *> The leading dimension of the array B. LDB >= max(1,N).
219 *> \endverbatim
220 *>
221 *> \param[out] X
222 *> \verbatim
223 *> X is COMPLEX array, dimension (LDX,NRHS)
224 *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
225 *> the original system of equations. Note that if EQUED = 'Y',
226 *> A and B are modified on exit, and the solution to the
227 *> equilibrated system is inv(diag(S))*X.
228 *> \endverbatim
229 *>
230 *> \param[in] LDX
231 *> \verbatim
232 *> LDX is INTEGER
233 *> The leading dimension of the array X. LDX >= max(1,N).
234 *> \endverbatim
235 *>
236 *> \param[out] RCOND
237 *> \verbatim
238 *> RCOND is REAL
239 *> The estimate of the reciprocal condition number of the matrix
240 *> A after equilibration (if done). If RCOND is less than the
241 *> machine precision (in particular, if RCOND = 0), the matrix
242 *> is singular to working precision. This condition is
243 *> indicated by a return code of INFO > 0.
244 *> \endverbatim
245 *>
246 *> \param[out] FERR
247 *> \verbatim
248 *> FERR is REAL array, dimension (NRHS)
249 *> The estimated forward error bound for each solution vector
250 *> X(j) (the j-th column of the solution matrix X).
251 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
252 *> is an estimated upper bound for the magnitude of the largest
253 *> element in (X(j) - XTRUE) divided by the magnitude of the
254 *> largest element in X(j). The estimate is as reliable as
255 *> the estimate for RCOND, and is almost always a slight
256 *> overestimate of the true error.
257 *> \endverbatim
258 *>
259 *> \param[out] BERR
260 *> \verbatim
261 *> BERR is REAL array, dimension (NRHS)
262 *> The componentwise relative backward error of each solution
263 *> vector X(j) (i.e., the smallest relative change in
264 *> any element of A or B that makes X(j) an exact solution).
265 *> \endverbatim
266 *>
267 *> \param[out] WORK
268 *> \verbatim
269 *> WORK is COMPLEX array, dimension (2*N)
270 *> \endverbatim
271 *>
272 *> \param[out] RWORK
273 *> \verbatim
274 *> RWORK is REAL array, dimension (N)
275 *> \endverbatim
276 *>
277 *> \param[out] INFO
278 *> \verbatim
279 *> INFO is INTEGER
280 *> = 0: successful exit
281 *> < 0: if INFO = -i, the i-th argument had an illegal value
282 *> > 0: if INFO = i, and i is
283 *> <= N: the leading minor of order i of A is
284 *> not positive definite, so the factorization
285 *> could not be completed, and the solution has not
286 *> been computed. RCOND = 0 is returned.
287 *> = N+1: U is nonsingular, but RCOND is less than machine
288 *> precision, meaning that the matrix is singular
289 *> to working precision. Nevertheless, the
290 *> solution and error bounds are computed because
291 *> there are a number of situations where the
292 *> computed solution can be more accurate than the
293 *> value of RCOND would suggest.
294 *> \endverbatim
295 *
296 * Authors:
297 * ========
298 *
299 *> \author Univ. of Tennessee
300 *> \author Univ. of California Berkeley
301 *> \author Univ. of Colorado Denver
302 *> \author NAG Ltd.
303 *
304 *> \ingroup complexOTHERsolve
305 *
306 *> \par Further Details:
307 * =====================
308 *>
309 *> \verbatim
310 *>
311 *> The band storage scheme is illustrated by the following example, when
312 *> N = 6, KD = 2, and UPLO = 'U':
313 *>
314 *> Two-dimensional storage of the Hermitian matrix A:
315 *>
316 *> a11 a12 a13
317 *> a22 a23 a24
318 *> a33 a34 a35
319 *> a44 a45 a46
320 *> a55 a56
321 *> (aij=conjg(aji)) a66
322 *>
323 *> Band storage of the upper triangle of A:
324 *>
325 *> * * a13 a24 a35 a46
326 *> * a12 a23 a34 a45 a56
327 *> a11 a22 a33 a44 a55 a66
328 *>
329 *> Similarly, if UPLO = 'L' the format of A is as follows:
330 *>
331 *> a11 a22 a33 a44 a55 a66
332 *> a21 a32 a43 a54 a65 *
333 *> a31 a42 a53 a64 * *
334 *>
335 *> Array elements marked * are not used by the routine.
336 *> \endverbatim
337 *>
338 * =====================================================================
339  SUBROUTINE cpbsvx( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
340  $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
341  $ WORK, RWORK, INFO )
342 *
343 * -- LAPACK driver routine --
344 * -- LAPACK is a software package provided by Univ. of Tennessee, --
345 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
346 *
347 * .. Scalar Arguments ..
348  CHARACTER EQUED, FACT, UPLO
349  INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
350  REAL RCOND
351 * ..
352 * .. Array Arguments ..
353  REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
354  COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
355  $ work( * ), x( ldx, * )
356 * ..
357 *
358 * =====================================================================
359 *
360 * .. Parameters ..
361  REAL ZERO, ONE
362  PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
363 * ..
364 * .. Local Scalars ..
365  LOGICAL EQUIL, NOFACT, RCEQU, UPPER
366  INTEGER I, INFEQU, J, J1, J2
367  REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
368 * ..
369 * .. External Functions ..
370  LOGICAL LSAME
371  REAL CLANHB, SLAMCH
372  EXTERNAL lsame, clanhb, slamch
373 * ..
374 * .. External Subroutines ..
375  EXTERNAL ccopy, clacpy, claqhb, cpbcon, cpbequ, cpbrfs,
376  $ cpbtrf, cpbtrs, xerbla
377 * ..
378 * .. Intrinsic Functions ..
379  INTRINSIC max, min
380 * ..
381 * .. Executable Statements ..
382 *
383  info = 0
384  nofact = lsame( fact, 'N' )
385  equil = lsame( fact, 'E' )
386  upper = lsame( uplo, 'U' )
387  IF( nofact .OR. equil ) THEN
388  equed = 'N'
389  rcequ = .false.
390  ELSE
391  rcequ = lsame( equed, 'Y' )
392  smlnum = slamch( 'Safe minimum' )
393  bignum = one / smlnum
394  END IF
395 *
396 * Test the input parameters.
397 *
398  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
399  $ THEN
400  info = -1
401  ELSE IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
402  info = -2
403  ELSE IF( n.LT.0 ) THEN
404  info = -3
405  ELSE IF( kd.LT.0 ) THEN
406  info = -4
407  ELSE IF( nrhs.LT.0 ) THEN
408  info = -5
409  ELSE IF( ldab.LT.kd+1 ) THEN
410  info = -7
411  ELSE IF( ldafb.LT.kd+1 ) THEN
412  info = -9
413  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
414  $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
415  info = -10
416  ELSE
417  IF( rcequ ) THEN
418  smin = bignum
419  smax = zero
420  DO 10 j = 1, n
421  smin = min( smin, s( j ) )
422  smax = max( smax, s( j ) )
423  10 CONTINUE
424  IF( smin.LE.zero ) THEN
425  info = -11
426  ELSE IF( n.GT.0 ) THEN
427  scond = max( smin, smlnum ) / min( smax, bignum )
428  ELSE
429  scond = one
430  END IF
431  END IF
432  IF( info.EQ.0 ) THEN
433  IF( ldb.LT.max( 1, n ) ) THEN
434  info = -13
435  ELSE IF( ldx.LT.max( 1, n ) ) THEN
436  info = -15
437  END IF
438  END IF
439  END IF
440 *
441  IF( info.NE.0 ) THEN
442  CALL xerbla( 'CPBSVX', -info )
443  RETURN
444  END IF
445 *
446  IF( equil ) THEN
447 *
448 * Compute row and column scalings to equilibrate the matrix A.
449 *
450  CALL cpbequ( uplo, n, kd, ab, ldab, s, scond, amax, infequ )
451  IF( infequ.EQ.0 ) THEN
452 *
453 * Equilibrate the matrix.
454 *
455  CALL claqhb( uplo, n, kd, ab, ldab, s, scond, amax, equed )
456  rcequ = lsame( equed, 'Y' )
457  END IF
458  END IF
459 *
460 * Scale the right-hand side.
461 *
462  IF( rcequ ) THEN
463  DO 30 j = 1, nrhs
464  DO 20 i = 1, n
465  b( i, j ) = s( i )*b( i, j )
466  20 CONTINUE
467  30 CONTINUE
468  END IF
469 *
470  IF( nofact .OR. equil ) THEN
471 *
472 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
473 *
474  IF( upper ) THEN
475  DO 40 j = 1, n
476  j1 = max( j-kd, 1 )
477  CALL ccopy( j-j1+1, ab( kd+1-j+j1, j ), 1,
478  $ afb( kd+1-j+j1, j ), 1 )
479  40 CONTINUE
480  ELSE
481  DO 50 j = 1, n
482  j2 = min( j+kd, n )
483  CALL ccopy( j2-j+1, ab( 1, j ), 1, afb( 1, j ), 1 )
484  50 CONTINUE
485  END IF
486 *
487  CALL cpbtrf( uplo, n, kd, afb, ldafb, info )
488 *
489 * Return if INFO is non-zero.
490 *
491  IF( info.GT.0 )THEN
492  rcond = zero
493  RETURN
494  END IF
495  END IF
496 *
497 * Compute the norm of the matrix A.
498 *
499  anorm = clanhb( '1', uplo, n, kd, ab, ldab, rwork )
500 *
501 * Compute the reciprocal of the condition number of A.
502 *
503  CALL cpbcon( uplo, n, kd, afb, ldafb, anorm, rcond, work, rwork,
504  $ info )
505 *
506 * Compute the solution matrix X.
507 *
508  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
509  CALL cpbtrs( uplo, n, kd, nrhs, afb, ldafb, x, ldx, info )
510 *
511 * Use iterative refinement to improve the computed solution and
512 * compute error bounds and backward error estimates for it.
513 *
514  CALL cpbrfs( uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x,
515  $ ldx, ferr, berr, work, rwork, info )
516 *
517 * Transform the solution matrix X to a solution of the original
518 * system.
519 *
520  IF( rcequ ) THEN
521  DO 70 j = 1, nrhs
522  DO 60 i = 1, n
523  x( i, j ) = s( i )*x( i, j )
524  60 CONTINUE
525  70 CONTINUE
526  DO 80 j = 1, nrhs
527  ferr( j ) = ferr( j ) / scond
528  80 CONTINUE
529  END IF
530 *
531 * Set INFO = N+1 if the matrix is singular to working precision.
532 *
533  IF( rcond.LT.slamch( 'Epsilon' ) )
534  $ info = n + 1
535 *
536  RETURN
537 *
538 * End of CPBSVX
539 *
540  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine claqhb(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
CLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.
Definition: claqhb.f:141
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine cpbequ(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
CPBEQU
Definition: cpbequ.f:130
subroutine cpbrfs(UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPBRFS
Definition: cpbrfs.f:189
subroutine cpbcon(UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, RWORK, INFO)
CPBCON
Definition: cpbcon.f:133
subroutine cpbtrs(UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
CPBTRS
Definition: cpbtrs.f:121
subroutine cpbtrf(UPLO, N, KD, AB, LDAB, INFO)
CPBTRF
Definition: cpbtrf.f:142
subroutine cpbsvx(FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
CPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices
Definition: cpbsvx.f:342