Further Details: Error Bounds for the Generalized Symmetric Definite Eigenproblem

The error analysis of the driver routine xSYGV, or xHEGV in the complex case      (see subsection 2.2.5.1), goes as follows. In all cases is the absolute gap   between and the nearest other eigenvalue.  

1. . The computed eigenvalues can differ from true eigenvalues by at most about

   

       The angular difference between the computed eigenvector and a true eigenvector is

   

2. or . The computed eigenvalues can differ from true eigenvalues by at most about

   

      

   The angular difference between the computed eigenvector and a true eigenvector is

   

The code fragments above replace p(n) by 1, and makes sure neither RCONDB nor RCONDZ is so small as to cause overflow when used as divisors in the expressions for error bounds.

These error bounds are large when B is ill-conditioned with respect to inversion ( is large). It is often the case that the eigenvalues and eigenvectors are much better conditioned than indicated here. We mention three ways to get tighter bounds. The first way is effective when the diagonal entries of B differ widely in magnitude:

1. . Let be a diagonal matrix. Then replace B by DBD and A by DAD in the above bounds.
2. or . Let be a diagonal matrix. Then replace B by DBD and A by in the above bounds.

The second way to get tighter bounds does not actually supply guaranteed bounds, but its estimates are often better in practice. It is not guaranteed because it assumes the algorithm is backward stable, which is not necessarily true when B is ill-conditioned.     It estimates the chordal distance between a true eigenvalue and a computed eigenvalue :  

To interpret this measure we write and . Then . In other words, if represents the one-dimensional subspace consisting of the line through the origin with slope , and represents the analogous subspace S, then is the sine of the acute angle between these subspaces.     Thus X is bounded by one, and is small when both arguments are large. It applies only to the first problem, :

Suppose a computed eigenvalue of is the exact eigenvalue of a perturbed problem . Let be the unit eigenvector () for the exact eigenvalue . Then if ||E|| is small compared to |A|, and if ||F|| is small compared to ||B||, we have

Thus is a condition number for eigenvalue .  

The third way applies only to the first problem , and only when A is positive definite. We use a different algorithm:

1. Compute the Cholesky factorization of , using xPOTRF.
2. Compute the Cholesky factorization of , using xPOTRF.
3. Compute the generalized singular value decomposition of the pair , using xTGSJA. The squares of the generalized singular      values are the desired eigenvalues.

Other yet more refined algorithms and error bounds are discussed in [78][73][13], and will be available in future releases.