HP MLIB User's Guide Vol. 2 7th Ed.

1004 HP MLIB User’s Guide
Associated documentation
Associated documentation
The following documents provide supplemental material for this chapter:
Grimes, R.G., J.G. Lewis and H.D. Simon. “Eigenvalue Problems and
Algorithms in Structural Engineering.” Large Scale Eigenvalue Problems.
North-Holland. 1986. pp. 81-95.
Grimes, R.G., J.G. Lewis and H.D. Simon. “The Implementation of a Block
Shifted and Inverted Lanczos Algorithm for Eigenvalue Problems in Structural
Engineering.” Boeing Computer Services Technical Report, ETA-TR-39. August,
1986.
Nour-Omid, B., B.N. Parlett, T. Ericsson and P.S. Jensen. “How to Implement
the Spectral Transformation,” Mathematics of Computation. 1987. 48, 178, pp.
663-673.
Parlett, B.N. The Symmetric Eigenvalue Problem. Englewood Cliffs, NJ:
Prentice-Hall, Inc. 1980.
What you need to know to use these subprograms
Generalized symmetric eigenproblems
This package is capable of solving ordinary symmetric eigenvalue problems
Ax=λx and certain generalized symmetric eigenvalue problems Ax=λBx. In the
latter case, both A and B are symmetric. The ordinary symmetric eigenproblem
has the property that all eigenvalues are real and that there are n real
orthogonal real eigenvectors. This latter condition is equivalent to x
i
T
x
i
= 1 and
x
i
T
x
j
= 0 for all eigenvectors x
i
and x
j
with i ≠ j.
To have similar properties in the generalized case, more conditions are needed
than just symmetry. When B is singular, there are fewer than n finite
eigenvalues. If, however, there exists a positive definite matrix C=αA+βB for
some scalars α and β, then all finite eigenvalues are real and the corresponding
eigenvectors are real and C-orthogonal. That is, x
i
T
Cx
i
= 1 and x
i
T
Cx
j
= 0 for
all eigenvectors x
i
and x
j
with i ≠ j.