cpbequ man page on Scientific

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CPBEQU(1)		 LAPACK routine (version 3.2)		     CPBEQU(1)

NAME
       CPBEQU  -  computes  row	 and column scalings intended to equilibrate a
       Hermitian positive definite band matrix A and reduce its condition num‐
       ber (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, * )

PURPOSE
       CPBEQU  computes row and column scalings intended to equilibrate a Her‐
       mitian 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 condi‐
       tion 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.

 LAPACK routine (version 3.2)	 November 2008			     CPBEQU(1)
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