SPBEQU 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangular of A is stored;


          = 'L':  Lower triangular of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

KD

          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

          AB is REAL 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

          LDAB is INTEGER


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

S

          S is REAL array, dimension (N)


          If INFO = 0, S contains the scale factors for A.

SCOND

          SCOND is 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

          AMAX is 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

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016