LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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zpbsvx.f
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1*> \brief <b> ZPBSVX 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 ZPBSVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpbsvx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpbsvx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpbsvx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZPBSVX( 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* DOUBLE PRECISION RCOND
29* ..
30* .. Array Arguments ..
31* DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
32* COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
33* $ WORK( * ), X( LDX, * )
34* ..
35*
36*
37*> \par Purpose:
38* =============
39*>
40*> \verbatim
41*>
42*> ZPBSVX 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 principal minor of order i is not positive,
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*16 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*16 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 DOUBLE PRECISION 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*16 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*16 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 DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 array, dimension (2*N)
270*> \endverbatim
271*>
272*> \param[out] RWORK
273*> \verbatim
274*> RWORK is DOUBLE PRECISION 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 principal minor of order i of A
284*> is not positive, so the factorization could not
285*> be completed, and the solution has not been
286*> 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 pbsvx
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 zpbsvx( 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 DOUBLE PRECISION RCOND
351* ..
352* .. Array Arguments ..
353 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
354 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
355 $ work( * ), x( ldx, * )
356* ..
357*
358* =====================================================================
359*
360* .. Parameters ..
361 DOUBLE PRECISION ZERO, ONE
362 PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0 )
363* ..
364* .. Local Scalars ..
365 LOGICAL EQUIL, NOFACT, RCEQU, UPPER
366 INTEGER I, INFEQU, J, J1, J2
367 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
368* ..
369* .. External Functions ..
370 LOGICAL LSAME
371 DOUBLE PRECISION DLAMCH, ZLANHB
372 EXTERNAL lsame, dlamch, zlanhb
373* ..
374* .. External Subroutines ..
375 EXTERNAL xerbla, zcopy, zlacpy, zlaqhb, zpbcon, zpbequ,
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 = dlamch( '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( 'ZPBSVX', -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 zpbequ( uplo, n, kd, ab, ldab, s, scond, amax, infequ )
451 IF( infequ.EQ.0 ) THEN
452*
453* Equilibrate the matrix.
454*
455 CALL zlaqhb( 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 zcopy( 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 zcopy( j2-j+1, ab( 1, j ), 1, afb( 1, j ), 1 )
484 50 CONTINUE
485 END IF
486*
487 CALL zpbtrf( 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 = zlanhb( '1', uplo, n, kd, ab, ldab, rwork )
500*
501* Compute the reciprocal of the condition number of A.
502*
503 CALL zpbcon( uplo, n, kd, afb, ldafb, anorm, rcond, work, rwork,
504 $ info )
505*
506* Compute the solution matrix X.
507*
508 CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
509 CALL zpbtrs( 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 zpbrfs( 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.dlamch( 'Epsilon' ) )
534 $ info = n + 1
535*
536 RETURN
537*
538* End of ZPBSVX
539*
540 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
subroutine zlacpy(uplo, m, n, a, lda, b, ldb)
ZLACPY copies all or part of one two-dimensional array to another.
Definition zlacpy.f:103
subroutine zlaqhb(uplo, n, kd, ab, ldab, s, scond, amax, equed)
ZLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.
Definition zlaqhb.f:141
subroutine zpbcon(uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info)
ZPBCON
Definition zpbcon.f:133
subroutine zpbequ(uplo, n, kd, ab, ldab, s, scond, amax, info)
ZPBEQU
Definition zpbequ.f:130
subroutine zpbrfs(uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZPBRFS
Definition zpbrfs.f:189
subroutine zpbsvx(fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices
Definition zpbsvx.f:342
subroutine zpbtrf(uplo, n, kd, ab, ldab, info)
ZPBTRF
Definition zpbtrf.f:142
subroutine zpbtrs(uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
ZPBTRS
Definition zpbtrs.f:121