zlaqsb.f

          SUBROUTINE ZLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          EQUED, UPLO
          INTEGER            KD, LDAB, N
          DOUBLE PRECISION   AMAX, SCOND
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   S( * )
          COMPLEX*16         AB( LDAB, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZLAQSB equilibrates a symmetric band matrix A using the scaling
    *  factors in the vector S.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          Specifies whether the upper or lower triangular part of the
    *          symmetric matrix A is stored.
    *          = 'U':  Upper triangular
    *          = 'L':  Lower triangular
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  KD      (input) INTEGER
    *          The number of super-diagonals of the matrix A if UPLO = 'U',
    *          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
    *
    *  AB      (input/output) COMPLEX*16 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 triangular factor U or L from the
    *          Cholesky factorization A = U'*U or A = L*L' of the band
    *          matrix A, in the same storage format as A.
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= KD+1.
    *
    *  S       (input) DOUBLE PRECISION array, dimension (N)
    *          The scale factors for A.
    *
    *  SCOND   (input) DOUBLE PRECISION
    *          Ratio of the smallest S(i) to the largest S(i).
    *
    *  AMAX    (input) DOUBLE PRECISION
    *          Absolute value of largest matrix entry.
    *
    *  EQUED   (output) CHARACTER*1
    *          Specifies whether or not equilibration was done.
    *          = 'N':  No equilibration.
    *          = 'Y':  Equilibration was done, i.e., A has been replaced by
    *                  diag(S) * A * diag(S).
    *
    *  Internal Parameters
    *  ===================
    *
    *  THRESH is a threshold value used to decide if scaling should be done
    *  based on the ratio of the scaling factors.  If SCOND < THRESH,
    *  scaling is done.
    *
    *  LARGE and SMALL are threshold values used to decide if scaling should
    *  be done based on the absolute size of the largest matrix element.
    *  If AMAX > LARGE or AMAX < SMALL, scaling is done.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, THRESH
          PARAMETER          ( ONE = 1.0D+0, THRESH = 0.1D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, J
          DOUBLE PRECISION   CJ, LARGE, SMALL
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH
          EXTERNAL           LSAME, DLAMCH
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Quick return if possible
    *
          IF( N.LE.0 ) THEN
             EQUED = 'N'
             RETURN
          END IF
    *
    *     Initialize LARGE and SMALL.
    *
          SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
          LARGE = ONE / SMALL
    *
          IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN
    *
    *        No equilibration
    *
             EQUED = 'N'
          ELSE
    *
    *        Replace A by diag(S) * A * diag(S).
    *
             IF( LSAME( UPLO, 'U' ) ) THEN
    *
    *           Upper triangle of A is stored in band format.
    *
                DO 20 J = 1, N
                   CJ = S( J )
                   DO 10 I = MAX( 1, J-KD ), J
                      AB( KD+1+I-J, J ) = CJ*S( I )*AB( KD+1+I-J, J )
       10          CONTINUE
       20       CONTINUE
             ELSE
    *
    *           Lower triangle of A is stored.
    *
                DO 40 J = 1, N
                   CJ = S( J )
                   DO 30 I = J, MIN( N, J+KD )
                      AB( 1+I-J, J ) = CJ*S( I )*AB( 1+I-J, J )
       30          CONTINUE
       40       CONTINUE
             END IF
             EQUED = 'Y'
          END IF
    *
          RETURN
    *
    *     End of ZLAQSB
    *
          END