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    1:       SUBROUTINE ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       CHARACTER          UPLO
   10:       INTEGER            INFO, N
   11:       DOUBLE PRECISION   AMAX, SCOND
   12: *     ..
   13: *     .. Array Arguments ..
   14:       DOUBLE PRECISION   S( * )
   15:       COMPLEX*16         AP( * )
   16: *     ..
   17: *
   18: *  Purpose
   19: *  =======
   20: *
   21: *  ZPPEQU computes row and column scalings intended to equilibrate a
   22: *  Hermitian positive definite matrix A in packed storage and reduce
   23: *  its condition number (with respect to the two-norm).  S contains the
   24: *  scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
   25: *  B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
   26: *  This choice of S puts the condition number of B within a factor N of
   27: *  the smallest possible condition number over all possible diagonal
   28: *  scalings.
   29: *
   30: *  Arguments
   31: *  =========
   32: *
   33: *  UPLO    (input) CHARACTER*1
   34: *          = 'U':  Upper triangle of A is stored;
   35: *          = 'L':  Lower triangle of A is stored.
   36: *
   37: *  N       (input) INTEGER
   38: *          The order of the matrix A.  N >= 0.
   39: *
   40: *  AP      (input) COMPLEX*16 array, dimension (N*(N+1)/2)
   41: *          The upper or lower triangle of the Hermitian matrix A, packed
   42: *          columnwise in a linear array.  The j-th column of A is stored
   43: *          in the array AP as follows:
   44: *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   45: *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
   46: *
   47: *  S       (output) DOUBLE PRECISION array, dimension (N)
   48: *          If INFO = 0, S contains the scale factors for A.
   49: *
   50: *  SCOND   (output) DOUBLE PRECISION
   51: *          If INFO = 0, S contains the ratio of the smallest S(i) to
   52: *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
   53: *          large nor too small, it is not worth scaling by S.
   54: *
   55: *  AMAX    (output) DOUBLE PRECISION
   56: *          Absolute value of largest matrix element.  If AMAX is very
   57: *          close to overflow or very close to underflow, the matrix
   58: *          should be scaled.
   59: *
   60: *  INFO    (output) INTEGER
   61: *          = 0:  successful exit
   62: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   63: *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
   64: *
   65: *  =====================================================================
   66: *
   67: *     .. Parameters ..
   68:       DOUBLE PRECISION   ONE, ZERO
   69:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
   70: *     ..
   71: *     .. Local Scalars ..
   72:       LOGICAL            UPPER
   73:       INTEGER            I, JJ
   74:       DOUBLE PRECISION   SMIN
   75: *     ..
   76: *     .. External Functions ..
   77:       LOGICAL            LSAME
   78:       EXTERNAL           LSAME
   79: *     ..
   80: *     .. External Subroutines ..
   81:       EXTERNAL           XERBLA
   82: *     ..
   83: *     .. Intrinsic Functions ..
   84:       INTRINSIC          DBLE, MAX, MIN, SQRT
   85: *     ..
   86: *     .. Executable Statements ..
   87: *
   88: *     Test the input parameters.
   89: *
   90:       INFO = 0
   91:       UPPER = LSAME( UPLO, 'U' )
   92:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
   93:          INFO = -1
   94:       ELSE IF( N.LT.0 ) THEN
   95:          INFO = -2
   96:       END IF
   97:       IF( INFO.NE.0 ) THEN
   98:          CALL XERBLA( 'ZPPEQU', -INFO )
   99:          RETURN
  100:       END IF
  101: *
  102: *     Quick return if possible
  103: *
  104:       IF( N.EQ.0 ) THEN
  105:          SCOND = ONE
  106:          AMAX = ZERO
  107:          RETURN
  108:       END IF
  109: *
  110: *     Initialize SMIN and AMAX.
  111: *
  112:       S( 1 ) = DBLE( AP( 1 ) )
  113:       SMIN = S( 1 )
  114:       AMAX = S( 1 )
  115: *
  116:       IF( UPPER ) THEN
  117: *
  118: *        UPLO = 'U':  Upper triangle of A is stored.
  119: *        Find the minimum and maximum diagonal elements.
  120: *
  121:          JJ = 1
  122:          DO 10 I = 2, N
  123:             JJ = JJ + I
  124:             S( I ) = DBLE( AP( JJ ) )
  125:             SMIN = MIN( SMIN, S( I ) )
  126:             AMAX = MAX( AMAX, S( I ) )
  127:    10    CONTINUE
  128: *
  129:       ELSE
  130: *
  131: *        UPLO = 'L':  Lower triangle of A is stored.
  132: *        Find the minimum and maximum diagonal elements.
  133: *
  134:          JJ = 1
  135:          DO 20 I = 2, N
  136:             JJ = JJ + N - I + 2
  137:             S( I ) = DBLE( AP( JJ ) )
  138:             SMIN = MIN( SMIN, S( I ) )
  139:             AMAX = MAX( AMAX, S( I ) )
  140:    20    CONTINUE
  141:       END IF
  142: *
  143:       IF( SMIN.LE.ZERO ) THEN
  144: *
  145: *        Find the first non-positive diagonal element and return.
  146: *
  147:          DO 30 I = 1, N
  148:             IF( S( I ).LE.ZERO ) THEN
  149:                INFO = I
  150:                RETURN
  151:             END IF
  152:    30    CONTINUE
  153:       ELSE
  154: *
  155: *        Set the scale factors to the reciprocals
  156: *        of the diagonal elements.
  157: *
  158:          DO 40 I = 1, N
  159:             S( I ) = ONE / SQRT( S( I ) )
  160:    40    CONTINUE
  161: *
  162: *        Compute SCOND = min(S(I)) / max(S(I))
  163: *
  164:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
  165:       END IF
  166:       RETURN
  167: *
  168: *     End of ZPPEQU
  169: *
  170:       END