Balancing

The routine xGEBAL     may be used to balance the matrix A prior to reduction to Hessenberg form . Balancing involves two steps, either of which is optional:

• first, xGEBAL     attempts to permute A by a similarity transformation  to block upper triangular form:

  

  where P is a permutation matrix and and are upper triangular. Thus the matrix is already in Schur form  outside the central diagonal block in rows and columns ILO to IHI.   Subsequent operations by xGEBAL, xGEHRD or xHSEQR need only be applied to these rows and columns; therefore ILO and IHI are passed as arguments to xGEHRD     and xHSEQR    . This can save a significant amount of work if ILO > 1 or IHI < n. If no suitable permutation can be found (as is very often the case), xGEBAL sets ILO = 1 and IHI = n, and is the whole of A.

• secondly, xGEBAL applies a diagonal similarity transformation to to make the rows and columns of as close in norm in possible:

  

  This can improve the accuracy of later processing in some cases; see subsection 4.8.1.2.

If A was balanced by xGEBAL, then eigenvectors computed by subsequent operations are eigenvectors of the balanced matrix ; xGEBAK     must then be called to transform them back to eigenvectors of the original matrix A.