Next: Criterion for Determining the Up: GUPTRI Algorithm Previous: GUPTRI Algorithm   Contents   Index

A Singular Pencil in GUPTRI Form.

For completeness, we consider a pencil with all different types of structure blocks in its KCF:

Consequently, the GUPTRI form of this pencil includes all types of diagonal blocks in (8.34) and (8.35). Since and the example have the same right singular structure and Jordan structure for the zero eigenvalue, they also have the same and (see above). The GUPTRI form of the Jordan structure of the infinity eigenvalue ( ) and the left singular structure ( ) are as follows:

Since these forms are computed using the -staircase reduction, the block indices and start to count from the southeast corner. Now, superdiagonal blocks of and have full row rank and diagonal blocks of and have full column rank. In the following table, the structure indices for the -, -, and -staircase forms are summarized.

 1 2 3 4 7 4 3 1 5 3 2 0
 1 2 3 4 5 3 2 1 3 2 1 0
 1 2 3 4 2 1 1 0 2 1 1 0
Finally, the block in the GUPTRI form corresponding to the finite and nonzero eigenvalues , and has the following appearance:

The diagonal elements in and are the eigenvalue pairs whose ratios are , and (in any order).

So far the description for computing the GUPTRI form has relied on infinite precision arithmetic. In the presence of roundoff the problem is regularized by allowing a deflation criterion for range/null space separations and thereby makes it possible to compute the GUPTRI form of a nearby matrix pencil.

This GUPTRI form is computed by a sequence of unitary equivalence transformations. The equivalence transformations are built from rank-revealing factorizations used to find orthonormal bases for different null spaces associated with the matrix pair.

Next: Criterion for Determining the Up: GUPTRI Algorithm Previous: GUPTRI Algorithm   Contents   Index
Susan Blackford 2000-11-20