Reduction of a Band-Symmetric Generalized Eigenvalue Problem

An algorithm is described for reducing the
generalized eigenvalue problem Ax = lambda Bx to 
an ordinary problem, in case A and B are symmetric band
matrices with B positive definite.  If n is the 
order of the matrix and m the bandwidth, the matrices
A and B are partitioned into m-by-m blocks; and 
the algorithm is described in terms of these blocks.
 The algorithm reduces the generalized problem to 
an ordinary eigenvalue problem for a symmetric band
matrix C whose bandwidth is the same as A and B. 
 The algorithm is similar to those of Rutishauser and
Schwartz for the reduction of symmetric matrices 
to band form.  The calculation C requires order mn^2
operation.  The round-off error in the calculation 
of C is of the same order as the sum of the errors at
each of the n/m steps of the algorithm, the latter 
errors being largely determined by the condition of B with respect to inversion.

CACM January, 1973

Crawford, C. R.

generalized eigenvalues, symmetric band matrices

5.14

CA730107 JB January 24, 1978  4:26 PM

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