cpbequ.f

          SUBROUTINE CPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, KD, LDAB, N
          REAL               AMAX, SCOND
    *     ..
    *     .. Array Arguments ..
          REAL               S( * )
          COMPLEX            AB( LDAB, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CPBEQU computes row and column scalings intended to equilibrate a
    *  Hermitian positive definite band matrix A and reduce its condition
    *  number (with respect to the two-norm).  S contains the scale factors,
    *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
    *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
    *  choice of S puts the condition number of B within a factor N of the
    *  smallest possible condition number over all possible diagonal
    *  scalings.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  Upper triangular of A is stored;
    *          = 'L':  Lower triangular of A is stored.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  KD      (input) INTEGER
    *          The number of superdiagonals of the matrix A if UPLO = 'U',
    *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    *
    *  AB      (input) COMPLEX array, dimension (LDAB,N)
    *          The upper or lower triangle of the Hermitian 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).
    *
    *  LDAB     (input) INTEGER
    *          The leading dimension of the array A.  LDAB >= KD+1.
    *
    *  S       (output) REAL array, dimension (N)
    *          If INFO = 0, S contains the scale factors for A.
    *
    *  SCOND   (output) REAL
    *          If INFO = 0, S contains the ratio of the smallest S(i) to
    *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
    *          large nor too small, it is not worth scaling by S.
    *
    *  AMAX    (output) REAL
    *          Absolute value of largest matrix element.  If AMAX is very
    *          close to overflow or very close to underflow, the matrix
    *          should be scaled.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            UPPER
          INTEGER            I, J
          REAL               SMIN
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN, REAL, 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( 'CPBEQU', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 ) THEN
             SCOND = ONE
             AMAX = ZERO
             RETURN
          END IF
    *
          IF( UPPER ) THEN
             J = KD + 1
          ELSE
             J = 1
          END IF
    *
    *     Initialize SMIN and AMAX.
    *
          S( 1 ) = REAL( AB( J, 1 ) )
          SMIN = S( 1 )
          AMAX = S( 1 )
    *
    *     Find the minimum and maximum diagonal elements.
    *
          DO 10 I = 2, N
             S( I ) = REAL( AB( J, I ) )
             SMIN = MIN( SMIN, S( I ) )
             AMAX = MAX( AMAX, S( I ) )
       10 CONTINUE
    *
          IF( SMIN.LE.ZERO ) THEN
    *
    *        Find the first non-positive diagonal element and return.
    *
             DO 20 I = 1, N
                IF( S( I ).LE.ZERO ) THEN
                   INFO = I
                   RETURN
                END IF
       20    CONTINUE
          ELSE
    *
    *        Set the scale factors to the reciprocals
    *        of the diagonal elements.
    *
             DO 30 I = 1, N
                S( I ) = ONE / SQRT( S( I ) )
       30    CONTINUE
    *
    *        Compute SCOND = min(S(I)) / max(S(I))
    *
             SCOND = SQRT( SMIN ) / SQRT( AMAX )
          END IF
          RETURN
    *
    *     End of CPBEQU
    *
          END