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

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

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

Purpose:
``` SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
compute the solution to a real system of linear equations
A * X = B,
where A is an N-by-N symmetric 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**T * U,  if UPLO = 'U', or
A = L * L**T,  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 REAL array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric 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 REAL 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**T*U or A = L*L**T 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**T*U or A = L*L**T. 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**T*U or A = L*L**T 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 REAL 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 REAL 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 REAL array, dimension (3*N)` [out] IWORK ` IWORK is INTEGER 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 symmetric 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 340 of file spbsvx.f.

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,
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*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
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
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
real function slansb(norm, uplo, n, k, ab, ldab, work)
SLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition slansb.f:129
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
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine spbcon(uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info)
SPBCON
Definition spbcon.f:132
subroutine spbequ(uplo, n, kd, ab, ldab, s, scond, amax, info)
SPBEQU
Definition spbequ.f:129
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 spbtrf(uplo, n, kd, ab, ldab, info)
SPBTRF
Definition spbtrf.f:142
subroutine spbtrs(uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
SPBTRS
Definition spbtrs.f:121
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