# A Fast Method for Solving a Class of Tridiagonal Linear Systems

• Published 1974 in Commun. ACM

#### Abstract

The solution of linear systems having real, symmetric, diagonally dominant, tridiagonal coefficient matrices with constant diagonals is considered. It is proved that the diagonals of the <italic>LU</italic> decomposition of the coefficient matrix rapidly converge to full floating-point precision. It is also proved that the computed <italic>LU</italic> decomposition converges when floating-point arithmetic is used and that the limits of the <italic>LU</italic> diagonals using floating point are roughly within machine precision of the limits using real arithmetic. This fact is exploited to reduce the number of floating-point operations required to solve a linear system from 8<italic>n</italic> - 7 to 5<italic>n</italic> + 2<italic>k</italic> - 3, where <italic>k</italic> is much less than <italic>n</italic>, the order of the matrix. If the elements of the subdiagnals and superdiagonals are 1, then only 4<italic>n</italic> + 2<italic>k</italic> - 3 operations are needed. The entire <italic>LU</italic> decomposition takes <italic>k</italic> words of storage, and considerable savings in array subscripting are achieved. Upper and lower bounds on <italic>k</italic> are obtained in terms of the ratio of the coefficient matrix diagonal constants and parameters of the floating-point number system. Various generalizations of these results are discussed.

DOI: 10.1145/360767.360777

### Cite this paper

@article{Malcolm1974AFM, title={A Fast Method for Solving a Class of Tridiagonal Linear Systems}, author={Michael A. Malcolm and John Palmer}, journal={Commun. ACM}, year={1974}, volume={17}, pages={14-17} }