By Michael A. Malcolm, John Palmer
Communications of the ACM,
January 1974,
Vol. 17 No. 1, Pages 14-17
10.1145/360767.360777
Comments
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 LU decomposition of the coefficient matrix rapidly converge to full floating-point precision. It is also proved that the computed LU decomposition converges when floating-point arithmetic is used and that the limits of the LU 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 8n - 7 to 5n + 2k - 3, where k is much less than n, the order of the matrix. If the elements of the subdiagnals and superdiagonals are 1, then only 4n + 2k - 3 operations are needed. The entire LU decomposition takes k words of storage, and considerable savings in array subscripting are achieved. Upper and lower bounds on k 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.
The full text of this article is premium content
No entries found
Log in to Read the Full Article
Need Access?
Please select one of the options below for access to premium content and features.
Create a Web Account
If you are already an ACM member, Communications subscriber, or Digital Library subscriber, please set up a web account to access premium content on this site.
Join the ACM
Become a member to take full advantage of ACM's outstanding computing information resources, networking opportunities, and other benefits.
Subscribe to Communications of the ACM Magazine
Get full access to 50+ years of CACM content and receive the print version of the magazine monthly.
Purchase the Article
Non-members can purchase this article or a copy of the magazine in which it appears.