LAPACK  3.8.0
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 *> \date April 2012
305 *
306 *> \ingroup complexOTHERsolve
307 *
308 *> \par Further Details:
309 * =====================
310 *>
311 *> \verbatim
312 *>
313 *> The band storage scheme is illustrated by the following example, when
314 *> N = 6, KD = 2, and UPLO = 'U':
315 *>
316 *> Two-dimensional storage of the Hermitian matrix A:
317 *>
318 *> a11 a12 a13
319 *> a22 a23 a24
320 *> a33 a34 a35
321 *> a44 a45 a46
322 *> a55 a56
323 *> (aij=conjg(aji)) a66
324 *>
325 *> Band storage of the upper triangle of A:
326 *>
327 *> * * a13 a24 a35 a46
328 *> * a12 a23 a34 a45 a56
329 *> a11 a22 a33 a44 a55 a66
330 *>
331 *> Similarly, if UPLO = 'L' the format of A is as follows:
332 *>
333 *> a11 a22 a33 a44 a55 a66
334 *> a21 a32 a43 a54 a65 *
335 *> a31 a42 a53 a64 * *
336 *>
337 *> Array elements marked * are not used by the routine.
338 *> \endverbatim
339 *>
340 * =====================================================================
341  SUBROUTINE cpbsvx( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
342  $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
343  $ WORK, RWORK, INFO )
344 *
345 * -- LAPACK driver routine (version 3.7.0) --
346 * -- LAPACK is a software package provided by Univ. of Tennessee, --
347 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
348 * April 2012
349 *
350 * .. Scalar Arguments ..
351  CHARACTER EQUED, FACT, UPLO
352  INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
353  REAL RCOND
354 * ..
355 * .. Array Arguments ..
356  REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
357  COMPLEX AB( ldab, * ), AFB( ldafb, * ), B( ldb, * ),
358  $ work( * ), x( ldx, * )
359 * ..
360 *
361 * =====================================================================
362 *
363 * .. Parameters ..
364  REAL ZERO, ONE
365  parameter( zero = 0.0e+0, one = 1.0e+0 )
366 * ..
367 * .. Local Scalars ..
368  LOGICAL EQUIL, NOFACT, RCEQU, UPPER
369  INTEGER I, INFEQU, J, J1, J2
370  REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
371 * ..
372 * .. External Functions ..
373  LOGICAL LSAME
374  REAL CLANHB, SLAMCH
375  EXTERNAL lsame, clanhb, slamch
376 * ..
377 * .. External Subroutines ..
378  EXTERNAL ccopy, clacpy, claqhb, cpbcon, cpbequ, cpbrfs,
379  $ cpbtrf, cpbtrs, xerbla
380 * ..
381 * .. Intrinsic Functions ..
382  INTRINSIC max, min
383 * ..
384 * .. Executable Statements ..
385 *
386  info = 0
387  nofact = lsame( fact, 'N' )
388  equil = lsame( fact, 'E' )
389  upper = lsame( uplo, 'U' )
390  IF( nofact .OR. equil ) THEN
391  equed = 'N'
392  rcequ = .false.
393  ELSE
394  rcequ = lsame( equed, 'Y' )
395  smlnum = slamch( 'Safe minimum' )
396  bignum = one / smlnum
397  END IF
398 *
399 * Test the input parameters.
400 *
401  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
402  $ THEN
403  info = -1
404  ELSE IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
405  info = -2
406  ELSE IF( n.LT.0 ) THEN
407  info = -3
408  ELSE IF( kd.LT.0 ) THEN
409  info = -4
410  ELSE IF( nrhs.LT.0 ) THEN
411  info = -5
412  ELSE IF( ldab.LT.kd+1 ) THEN
413  info = -7
414  ELSE IF( ldafb.LT.kd+1 ) THEN
415  info = -9
416  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
417  $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
418  info = -10
419  ELSE
420  IF( rcequ ) THEN
421  smin = bignum
422  smax = zero
423  DO 10 j = 1, n
424  smin = min( smin, s( j ) )
425  smax = max( smax, s( j ) )
426  10 CONTINUE
427  IF( smin.LE.zero ) THEN
428  info = -11
429  ELSE IF( n.GT.0 ) THEN
430  scond = max( smin, smlnum ) / min( smax, bignum )
431  ELSE
432  scond = one
433  END IF
434  END IF
435  IF( info.EQ.0 ) THEN
436  IF( ldb.LT.max( 1, n ) ) THEN
437  info = -13
438  ELSE IF( ldx.LT.max( 1, n ) ) THEN
439  info = -15
440  END IF
441  END IF
442  END IF
443 *
444  IF( info.NE.0 ) THEN
445  CALL xerbla( 'CPBSVX', -info )
446  RETURN
447  END IF
448 *
449  IF( equil ) THEN
450 *
451 * Compute row and column scalings to equilibrate the matrix A.
452 *
453  CALL cpbequ( uplo, n, kd, ab, ldab, s, scond, amax, infequ )
454  IF( infequ.EQ.0 ) THEN
455 *
456 * Equilibrate the matrix.
457 *
458  CALL claqhb( uplo, n, kd, ab, ldab, s, scond, amax, equed )
459  rcequ = lsame( equed, 'Y' )
460  END IF
461  END IF
462 *
463 * Scale the right-hand side.
464 *
465  IF( rcequ ) THEN
466  DO 30 j = 1, nrhs
467  DO 20 i = 1, n
468  b( i, j ) = s( i )*b( i, j )
469  20 CONTINUE
470  30 CONTINUE
471  END IF
472 *
473  IF( nofact .OR. equil ) THEN
474 *
475 * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
476 *
477  IF( upper ) THEN
478  DO 40 j = 1, n
479  j1 = max( j-kd, 1 )
480  CALL ccopy( j-j1+1, ab( kd+1-j+j1, j ), 1,
481  $ afb( kd+1-j+j1, j ), 1 )
482  40 CONTINUE
483  ELSE
484  DO 50 j = 1, n
485  j2 = min( j+kd, n )
486  CALL ccopy( j2-j+1, ab( 1, j ), 1, afb( 1, j ), 1 )
487  50 CONTINUE
488  END IF
489 *
490  CALL cpbtrf( uplo, n, kd, afb, ldafb, info )
491 *
492 * Return if INFO is non-zero.
493 *
494  IF( info.GT.0 )THEN
495  rcond = zero
496  RETURN
497  END IF
498  END IF
499 *
500 * Compute the norm of the matrix A.
501 *
502  anorm = clanhb( '1', uplo, n, kd, ab, ldab, rwork )
503 *
504 * Compute the reciprocal of the condition number of A.
505 *
506  CALL cpbcon( uplo, n, kd, afb, ldafb, anorm, rcond, work, rwork,
507  $ info )
508 *
509 * Compute the solution matrix X.
510 *
511  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
512  CALL cpbtrs( uplo, n, kd, nrhs, afb, ldafb, x, ldx, info )
513 *
514 * Use iterative refinement to improve the computed solution and
515 * compute error bounds and backward error estimates for it.
516 *
517  CALL cpbrfs( uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x,
518  $ ldx, ferr, berr, work, rwork, info )
519 *
520 * Transform the solution matrix X to a solution of the original
521 * system.
522 *
523  IF( rcequ ) THEN
524  DO 70 j = 1, nrhs
525  DO 60 i = 1, n
526  x( i, j ) = s( i )*x( i, j )
527  60 CONTINUE
528  70 CONTINUE
529  DO 80 j = 1, nrhs
530  ferr( j ) = ferr( j ) / scond
531  80 CONTINUE
532  END IF
533 *
534 * Set INFO = N+1 if the matrix is singular to working precision.
535 *
536  IF( rcond.LT.slamch( 'Epsilon' ) )
537  $ info = n + 1
538 *
539  RETURN
540 *
541 * End of CPBSVX
542 *
543  END
subroutine cpbtrs(UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
CPBTRS
Definition: cpbtrs.f:123
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:143
subroutine cpbrfs(UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPBRFS
Definition: cpbrfs.f:191
subroutine cpbequ(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
CPBEQU
Definition: cpbequ.f:132
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:344
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:83
subroutine cpbtrf(UPLO, N, KD, AB, LDAB, INFO)
CPBTRF
Definition: cpbtrf.f:144
subroutine cpbcon(UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, RWORK, INFO)
CPBCON
Definition: cpbcon.f:135