Realizations of this method for symmetric tridiagonal matrix
can be found, for instance, in the monograph [9, Section 8.
Above algebraic equations are in tridiagonal matrix
form which is solved by using LU decomposition in the present work.
Then the shifts are introduced and the tridiagonal matrix
When A is symmetric or Hermitian, the Arnoldi process reduces to the Lanczos algorithm, in which the matrix H is a symmetric tridiagonal matrix
m+1,m] is a symmetric tridiagonal matrix
, and if [mu] is an eigenvalue of [B.
with a tridiagonal matrix
B with 2's on the diagonal and -1's on the subdiagonals and C is a matrix with all entries zero except for a 1 in the lower left corner.
The latter offers to compute k eigenpairs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], of a symmetric tridiagonal matrix
T [member of][R.
CHU, Symbolic calculation of the trace of the power of a tridiagonal matrix
, Computing, 35 (1985), pp.
As is well known, this allows us to reformulate a minimum residual problem as an equivalent or approximately equivalent least squares problem in coordinate space, which can be solved by updating the QR decomposition of a Hessenberg or tridiagonal matrix
i], i [member of] I, of a symmetric tridiagonal matrix
T [member of] [R.
ii) a formula for a real normal tridiagonal matrix
Each diagonal block in the matrix corresponds to the restriction of the operator to one of the subdomains, and ordering the nodes in each domain in groups and then the domains consecutively, results in a block tridiagonal matrix