zppequ(3S)

NAME
     ZPPEQU - compute row and column scalings intended to equilibrate a
     Hermitian positive definite matrix A in packed storage and reduce its
     condition number (with respect to the two-norm)

SYNOPSIS
     SUBROUTINE ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )

         CHARACTER      UPLO

         INTEGER        INFO, N

         DOUBLE         PRECISION AMAX, SCOND

         DOUBLE         PRECISION S( * )

         COMPLEX*16     AP( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     ZPPEQU computes row and column scalings intended to equilibrate a
     Hermitian positive definite matrix A in packed storage 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 triangle of A is stored;
             = 'L':  Lower triangle of A is stored.

     N       (input) INTEGER
             The order of the matrix A.  N >= 0.

     AP      (input) COMPLEX*16 array, dimension (N*(N+1)/2)
             The upper or lower triangle of the Hermitian matrix A, packed
             columnwise in a linear array.  The j-th column of A is stored in
             the array AP as follows:  if UPLO = 'U', AP(i + (j-1)*j/2) =
             A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) =
             A(i,j) for j<=i<=n.

     S       (output) DOUBLE PRECISION array, dimension (N)
             If INFO = 0, S contains the scale factors for A.

     SCOND   (output) DOUBLE PRECISION
             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) DOUBLE PRECISION
             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.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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