LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
spbsvx.f
Go to the documentation of this file.
1 *> \brief <b> SPBSVX 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 SPBSVX + 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/spbsvx.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spbsvx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPBSVX( 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 * REAL RCOND
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IWORK( * )
32 * REAL 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 *> SPBSVX 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
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 REAL 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 REAL 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 REAL 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 realOTHERsolve
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 spbsvx( 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  REAL RCOND
352 * ..
353 * .. Array Arguments ..
354  INTEGER IWORK( * )
355  REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
356  $ berr( * ), ferr( * ), s( * ), work( * ),
357  $ x( ldx, * )
358 * ..
359 *
360 * =====================================================================
361 *
362 * .. Parameters ..
363  REAL ZERO, ONE
364  PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
365 * ..
366 * .. Local Scalars ..
367  LOGICAL EQUIL, NOFACT, RCEQU, UPPER
368  INTEGER I, INFEQU, J, J1, J2
369  REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
370 * ..
371 * .. External Functions ..
372  LOGICAL LSAME
373  REAL SLAMCH, SLANSB
374  EXTERNAL lsame, slamch, slansb
375 * ..
376 * .. External Subroutines ..
377  EXTERNAL scopy, slacpy, slaqsb, spbcon, spbequ, spbrfs,
378  $ spbtrf, spbtrs, 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 = slamch( '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( 'SPBSVX', -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 spbequ( uplo, n, kd, ab, ldab, s, scond, amax, infequ )
453  IF( infequ.EQ.0 ) THEN
454 *
455 * Equilibrate the matrix.
456 *
457  CALL slaqsb( 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 scopy( 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 scopy( j2-j+1, ab( 1, j ), 1, afb( 1, j ), 1 )
486  50 CONTINUE
487  END IF
488 *
489  CALL spbtrf( 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 = slansb( '1', uplo, n, kd, ab, ldab, work )
502 *
503 * Compute the reciprocal of the condition number of A.
504 *
505  CALL spbcon( uplo, n, kd, afb, ldafb, anorm, rcond, work, iwork,
506  $ info )
507 *
508 * Compute the solution matrix X.
509 *
510  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
511  CALL spbtrs( 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 spbrfs( 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.slamch( 'Epsilon' ) )
536  $ info = n + 1
537 *
538  RETURN
539 *
540 * End of SPBSVX
541 *
542  END
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slaqsb(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
SLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.
Definition: slaqsb.f:140
subroutine spbrfs(UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SPBRFS
Definition: spbrfs.f:189
subroutine spbtrs(UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
SPBTRS
Definition: spbtrs.f:121
subroutine spbequ(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
SPBEQU
Definition: spbequ.f:129
subroutine spbcon(UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, IWORK, INFO)
SPBCON
Definition: spbcon.f:132
subroutine spbtrf(UPLO, N, KD, AB, LDAB, INFO)
SPBTRF
Definition: spbtrf.f:142
subroutine spbsvx(FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices
Definition: spbsvx.f:343
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82