#### Publication Date

12-17-1976

#### Abstract

Numerous routines are available to find the eigenvalues of a real symmetric tridiagonal matrix. Since it is known to converge in exact arithmetic, the tridiagonal QL algorithm with origin shift is widely used. Here we analyze the algorithm in floating-point arithmetic. This analysis suggests two modifications to the EISPACK implementation TQLl that enable one to prove correctness and hence convergence of the routine.

Also, it is known that the implicit and explicit versions of the QL algorithm produce the same results in exact arithmetic. A counter-example to the floating-point analog of this theorem is presented.

#### Degree Name

Mathematics

#### Level of Degree

Doctoral

#### Department Name

Mathematics & Statistics

#### First Committee Member (Chair)

Cleve B. Moler

#### Second Committee Member

Steven A. Pruess

#### Third Committee Member

Donald R. Morrison

#### Language

English

#### Document Type

Dissertation

#### Recommended Citation

Sanderson, James George. "A Proof of Convergence for the Tridiagonal QL Algorithm in Floating-Point Arithmetic." (1976). https://digitalrepository.unm.edu/math_etds/137