SGEBAL balances a general real matrix A.  This involves, first,


 permuting A by a similarity transformation to isolate eigenvalues


 in the first 1 to ILO-1 and last IHI+1 to N elements on the


 diagonal; and second, applying a diagonal similarity transformation


 to rows and columns ILO to IHI to make the rows and columns as


 close in norm as possible.  Both steps are optional.


 Balancing may reduce the 1-norm of the matrix, and improve the


 accuracy of the computed eigenvalues and/or eigenvectors.

JOB

          JOB is CHARACTER*1


          Specifies the operations to be performed on A:


          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(I) = 1.0


                  for i = 1,...,N;


          = 'P':  permute only;


          = 'S':  scale only;


          = 'B':  both permute and scale.

N

          N is INTEGER


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

A

          A is REAL array, dimension (LDA,N)


          On entry, the input matrix A.


          On exit,  A is overwritten by the balanced matrix.


          If JOB = 'N', A is not referenced.


          See Further Details.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


          ILO and IHI are set to integers such that on exit


          A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.


          If JOB = 'N' or 'S', ILO = 1 and IHI = N.

SCALE

          SCALE is REAL array, dimension (N)


          Details of the permutations and scaling factors applied to


          A.  If P(j) is the index of the row and column interchanged


          with row and column j and D(j) is the scaling factor


          applied to row and column j, then


          SCALE(j) = P(j)    for j = 1,...,ILO-1


                   = D(j)    for j = ILO,...,IHI


                   = P(j)    for j = IHI+1,...,N.


          The order in which the interchanges are made is N to IHI+1,


          then 1 to ILO-1.

INFO

          INFO is INTEGER


          = 0:  successful exit.


          < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  The permutations consist of row and column interchanges which put


  the matrix in the form


             ( T1   X   Y  )


     P A P = (  0   B   Z  )


             (  0   0   T2 )


  where T1 and T2 are upper triangular matrices whose eigenvalues lie


  along the diagonal.  The column indices ILO and IHI mark the starting


  and ending columns of the submatrix B. Balancing consists of applying


  a diagonal similarity transformation inv(D) * B * D to make the


  1-norms of each row of B and its corresponding column nearly equal.


  The output matrix is


     ( T1     X*D          Y    )


     (  0  inv(D)*B*D  inv(D)*Z ).


     (  0      0           T2   )


  Information about the permutations P and the diagonal matrix D is


  returned in the vector SCALE.


  This subroutine is based on the EISPACK routine BALANC.


  Modified by Tzu-Yi Chen, Computer Science Division, University of


    California at Berkeley, USA