cpbequ(3S)

NAME
     CPBEQU - compute 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)

SYNOPSIS
     SUBROUTINE CPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )

         CHARACTER      UPLO

         INTEGER        INFO, KD, LDAB, N

         REAL           AMAX, SCOND

         REAL           S( * )

         COMPLEX        AB( LDAB, * )

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
     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.

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

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