Tridiagonalization by Permutations

Tridiagonalizing a matrix by similarity transformations
is an important computational tool 
in numerical linear algebra. Consider the class of sparse
matrices which can be tridiagonalized using 
only row and corresponding column permutations.  The
advantages of using such a transformation include 
the absence of round-off errors and improved computation time
when compared with standard transformations. 
 A graph theoretic algorithm which examines an arbitrary
n x n matrix and determines whether or not it 
can be permuted into tridiagonal form is given.  The
algorithm requires no arithmetic while the number 
of comparisons, the number of assignments, and the number
of increments are linear in n.  This compares 
very favorably with standard transformation methods.
 If the matrix is permutable into tridiagonal form, 
the algorithm gives the explicit tridiagonal form.
 Otherwise, early rejection will occur.

CACM January, 1974

Gibbs, N. E.
Poole, W. G. Jr.

tridiagonal matrix, permutation, algorithm,
eigenvalues, graph, bandwidth, sparse matrix

5.14 5.32

CA740104 JB January 18, 1978  2:31 PM

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