subroutine spbstf	(	character	UPLO,
		integer	N,
		integer	KD,
		real, dimension( ldab, * )	AB,
		integer	LDAB,
		integer	INFO
	)

SPBSTF

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Purpose:

 SPBSTF computes a split Cholesky factorization of a real
 symmetric positive definite band matrix A.

 This routine is designed to be used in conjunction with SSBGST.

 The factorization has the form  A = S**T*S  where S is a band matrix
 of the same bandwidth as A and the following structure:

   S = ( U    )
       ( M  L )

 where U is upper triangular of order m = (n+kd)/2, and L is lower
 triangular of order n-m.

Parameters

[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 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,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. 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). On exit, if INFO = 0, the factor S from the split Cholesky factorization A = S*TS. See Further Details.
[in]	LDAB	LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.
[out]	INFO	INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.

Author: Univ. of Tennessee; Univ. of California Berkeley; Univ. of Colorado Denver; NAG Ltd.

Date: November 2011

Further Details:

  The band storage scheme is illustrated by the following example, when
  N = 7, KD = 2:

  S = ( s11  s12  s13                     )
      (      s22  s23  s24                )
      (           s33  s34                )
      (                s44                )
      (           s53  s54  s55           )
      (                s64  s65  s66      )
      (                     s75  s76  s77 )

  If UPLO = 'U', the array AB holds:

  on entry:                          on exit:

   *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75
   *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76
  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77

  If UPLO = 'L', the array AB holds:

  on entry:                          on exit:

  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
  a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
  a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *

  Array elements marked * are not used by the routine.

Definition at line 154 of file spbstf.f.

 *
 *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2011
 *
 *     .. Scalar Arguments ..
       CHARACTER          uplo
       INTEGER            info, kd, ldab, n
 *     ..
 *     .. Array Arguments ..
       REAL               ab( ldab, * )
 *     ..
 *
 *  =====================================================================
 *
 *     .. Parameters ..
       REAL               one, zero
       parameter                ( one = 1.0e+0, zero = 0.0e+0 )
 *     ..
 *     .. Local Scalars ..
       LOGICAL            upper
       INTEGER            j, kld, km, m
       REAL               ajj
 *     ..
 *     .. External Functions ..
       LOGICAL            lsame
       EXTERNAL           lsame
 *     ..
 *     .. External Subroutines ..
       EXTERNAL           sscal, ssyr, xerbla
 *     ..
 *     .. Intrinsic Functions ..
       INTRINSIC          max, min, sqrt
 *     ..
 *     .. Executable Statements ..
 *
 *     Test the input parameters.
 *
       info = 0
       upper = lsame( uplo, 'U' )
       IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
          info = -1
       ELSE IF( n.LT.0 ) THEN
          info = -2
       ELSE IF( kd.LT.0 ) THEN
          info = -3
       ELSE IF( ldab.LT.kd+1 ) THEN
          info = -5
       END IF
       IF( info.NE.0 ) THEN
          CALL xerbla( 'SPBSTF', -info )
          RETURN
       END IF
 *
 *     Quick return if possible
 *
       IF( n.EQ.0 )
      $   RETURN
 *
       kld = max( 1, ldab-1 )
 *
 *     Set the splitting point m.
 *
       m = ( n+kd ) / 2
 *
       IF( upper ) THEN
 *
 *        Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
 *
          DO 10 j = n, m + 1, -1
 *
 *           Compute s(j,j) and test for non-positive-definiteness.
 *
             ajj = ab( kd+1, j )
             IF( ajj.LE.zero )
      $         GO TO 50
             ajj = sqrt( ajj )
             ab( kd+1, j ) = ajj
             km = min( j-1, kd )
 *
 *           Compute elements j-km:j-1 of the j-th column and update the
 *           the leading submatrix within the band.
 *
             CALL sscal( km, one / ajj, ab( kd+1-km, j ), 1 )
             CALL ssyr( 'Upper', km, -one, ab( kd+1-km, j ), 1,
      $                 ab( kd+1, j-km ), kld )
    10    CONTINUE
 *
 *        Factorize the updated submatrix A(1:m,1:m) as U**T*U.
 *
          DO 20 j = 1, m
 *
 *           Compute s(j,j) and test for non-positive-definiteness.
 *
             ajj = ab( kd+1, j )
             IF( ajj.LE.zero )
      $         GO TO 50
             ajj = sqrt( ajj )
             ab( kd+1, j ) = ajj
             km = min( kd, m-j )
 *
 *           Compute elements j+1:j+km of the j-th row and update the
 *           trailing submatrix within the band.
 *
             IF( km.GT.0 ) THEN
                CALL sscal( km, one / ajj, ab( kd, j+1 ), kld )
                CALL ssyr( 'Upper', km, -one, ab( kd, j+1 ), kld,
      $                    ab( kd+1, j+1 ), kld )
             END IF
    20    CONTINUE
       ELSE
 *
 *        Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
 *
          DO 30 j = n, m + 1, -1
 *
 *           Compute s(j,j) and test for non-positive-definiteness.
 *
             ajj = ab( 1, j )
             IF( ajj.LE.zero )
      $         GO TO 50
             ajj = sqrt( ajj )
             ab( 1, j ) = ajj
             km = min( j-1, kd )
 *
 *           Compute elements j-km:j-1 of the j-th row and update the
 *           trailing submatrix within the band.
 *
             CALL sscal( km, one / ajj, ab( km+1, j-km ), kld )
             CALL ssyr( 'Lower', km, -one, ab( km+1, j-km ), kld,
      $                 ab( 1, j-km ), kld )
    30    CONTINUE
 *
 *        Factorize the updated submatrix A(1:m,1:m) as U**T*U.
 *
          DO 40 j = 1, m
 *
 *           Compute s(j,j) and test for non-positive-definiteness.
 *
             ajj = ab( 1, j )
             IF( ajj.LE.zero )
      $         GO TO 50
             ajj = sqrt( ajj )
             ab( 1, j ) = ajj
             km = min( kd, m-j )
 *
 *           Compute elements j+1:j+km of the j-th column and update the
 *           trailing submatrix within the band.
 *
             IF( km.GT.0 ) THEN
                CALL sscal( km, one / ajj, ab( 2, j ), 1 )
                CALL ssyr( 'Lower', km, -one, ab( 2, j ), 1,
      $                    ab( 1, j+1 ), kld )
             END IF
    40    CONTINUE
       END IF
       RETURN
 *
    50 CONTINUE
       info = j
       RETURN
 *
 *     End of SPBSTF
 *

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