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Therefore, after inserting the above expressions for [U.sub.N/2-2] and [U.sub.N/2+2] in (4.1), we transform the system of equations in (3.2) into a tridiagonal system of equations and obtain the following form:
A linear functional quasi-definite on [P.sub.n] determines a family of irreducible tridiagonal matrices that are diagonally similar, where this diagonal similarity is equivalent to a rescaling of the sequence of orthogonal polynomials.
We consider the tridiagonal matrix A = [W.sub.n]/25, where [W.sub.n] = diag(1,2,1).
The first nontrivial tridiagonal matrix (12) with N = 4 may represent, for example, a schematic quantum system with Hermitian-matrix interaction
The vectors [[psi].sup.(ev).sub.m] (j), j = 1, ..., M, are the eigenstates of the tridiagonal M x M matrix [H.sup.(ev)]:
For pressure field in the gap between the piston and the bore the tridiagonal matrix method was applied.
Zhang, "Synchronizing chaotic systems using control based on tridiagonal structure," Chaos, Solitons and Fractals, vol.
In this case [[theta].sub.t] is the second largest eigenvalue of the tridiagonal matrix U = U(C) and standard results about Sturm sequences (see, e.g., [Mar94, Mar00]) guarantee that Etx has [rho] + 1 distinct entries in all such cases.
The tridiagonal equations about [m.sub.j] can be obtained from the above conditions.
We note here that the whole linear system has a block diagonal matrix on the left hand side, and a block tridiagonal matrix on the right, and this characteristic will be exploited to improve the time integration, as explained in the following section.
(30); H is a tridiagonal matrix with < u, Hv > = < Hu, v > .