LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ cpbsvx()

 subroutine cpbsvx ( character fact, character uplo, integer n, integer kd, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, character equed, real, dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info )

CPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Download CPBSVX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` CPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex system of linear equations
A * X = B,
where A is an N-by-N Hermitian positive definite band matrix and X
and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also
provided.```
Description:
``` The following steps are performed:

1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**H * U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',
where U is an upper triangular band matrix, and L is a lower
triangular band matrix.

3. If the leading principal minor of order i is not positive,
then the routine returns with INFO = i. Otherwise, the factored
form of A is used to estimate the condition number of the matrix
A.  If the reciprocal of the condition number is less than machine
precision, INFO = N+1 is returned as a warning, but the routine
still goes on to solve for X and compute error bounds as
described below.

4. The system of equations is solved for X using the factored form
of A.

5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

6. If equilibration was used, the matrix X is premultiplied by
diag(S) so that it solves the original system before
equilibration.```
Parameters
 [in] FACT ``` FACT is CHARACTER*1 Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. = 'F': On entry, AFB contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AB and AFB will not be modified. = 'N': The matrix A will be copied to AFB and factored. = 'E': The matrix A will be equilibrated if necessary, then copied to AFB and factored.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right-hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.``` [in,out] AB ``` AB is COMPLEX array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD). See below for further details. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array A. LDAB >= KD+1.``` [in,out] AFB ``` AFB is COMPLEX array, dimension (LDAFB,N) If FACT = 'F', then AFB is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, in the same storage format as A (see AB). If EQUED = 'Y', then AFB is the factored form of the equilibrated matrix A. If FACT = 'N', then AFB is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H. If FACT = 'E', then AFB is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).``` [in] LDAFB ``` LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= KD+1.``` [in,out] EQUED ``` EQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.``` [in,out] S ``` S is REAL array, dimension (N) The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive.``` [in,out] B ``` B is COMPLEX array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] X ``` X is COMPLEX array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] RCOND ``` RCOND is REAL The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.``` [out] FERR ``` FERR is REAL array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.``` [out] BERR ``` BERR is REAL array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).``` [out] WORK ` WORK is COMPLEX array, dimension (2*N)` [out] RWORK ` RWORK is REAL array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is <= N: the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.```
Further Details:
```  The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

a11  a12  a13
a22  a23  a24
a33  a34  a35
a44  a45  a46
a55  a56
(aij=conjg(aji))         a66

Band storage of the upper triangle of A:

*    *   a13  a24  a35  a46
*   a12  a23  a34  a45  a56
a11  a22  a33  a44  a55  a66

Similarly, if UPLO = 'L' the format of A is as follows:

a11  a22  a33  a44  a55  a66
a21  a32  a43  a54  a65   *
a31  a42  a53  a64   *    *

Array elements marked * are not used by the routine.```

Definition at line 339 of file cpbsvx.f.

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,
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*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
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
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
real function clanhb(norm, uplo, n, k, ab, ldab, work)
CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition clanhb.f:132
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
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine cpbcon(uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info)
CPBCON
Definition cpbcon.f:133
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 cpbtrf(uplo, n, kd, ab, ldab, info)
CPBTRF
Definition cpbtrf.f:142
subroutine cpbtrs(uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
CPBTRS
Definition cpbtrs.f:121
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