LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ 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

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

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 i-by-i principal minor is not positive definite,
    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 minor of order i of A is
                       not positive definite, 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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,
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 *
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
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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
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 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 scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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