GEBAL - 5.2 English - 68552

GEBAL It balances a real matrix

Purpose:

 GEBAL 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.

Parameters:

• JOB – [in]

  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 – [in]

  N is INTEGER

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

• A – [inout]

  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 – [in]

  LDA is INTEGER

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

• ILO – [out] ILO is INTEGER

• IHI – [out]

  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 – [out]

  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 – [out]

  INFO is INTEGER

  = 0: successful exit.

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