Two working algorithms for the eigenvalues of a symmetric tridiagonal matrix
Two working algorithms for the eigenvalues of a symmetric tridiagonal matrix
Algorithm 685: a program for solving separable elliptic equations
ACM Transactions on Mathematical Software (TOMS)
Banded Eigenvalue Solvers on Vector Machines
ACM Transactions on Mathematical Software (TOMS)
Band reduction algorithms revisited
ACM Transactions on Mathematical Software (TOMS)
Spectral Chebyshev Collocation for the Poisson and Biharmonic Equations
SIAM Journal on Scientific Computing
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An algorithm is described for reducing the generalized eigenvalue problem Ax = &lgr;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 of C requires order n2m 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.