DPBEQU(3)      LAPACK routine of NEC Numeric Library Collection      DPBEQU(3)



NAME
       DPBEQU

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



PURPOSE
            DPBEQU computes row and column scalings intended to equilibrate a
            symmetric 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)
                     UPLO is CHARACTER*1
                     = 'U':  Upper triangular of A is stored;
                     = 'L':  Lower triangular of A is stored.

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

           KD        (input)
                     KD is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

           AB        (input)
                     AB is DOUBLE PRECISION array, dimension (LDAB,N)
                     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).

           LDAB      (input)
                     LDAB is INTEGER
                     The leading dimension of the array A.  LDAB >= KD+1.

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

           SCOND     (output)
                     SCOND is 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)
                     AMAX is 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)
                     INFO is 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.



LAPACK routine                  31 October 2017                      DPBEQU(3)