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LAPACK version 3.0
dpbequ(3)

PURPOSE
  DPBEQU - compute 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)

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

      CHARACTER	     UPLO

      INTEGER	     INFO, KD, LDAB, N

      DOUBLE	     PRECISION AMAX, SCOND

      DOUBLE	     PRECISION AB( LDAB, * ), S( * )

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

  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.