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1: 3.2 Linear Algebra

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§3.2(ii) Gaussian Elimination for a Tridiagonal Matrix

►Tridiagonal matrices are ones in which the only nonzero elements occur on the main diagonal and two adjacent diagonals. … ►For more information on solving tridiagonal systems see Golub and Van Loan (1996, pp. 152–160). … ►

§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix

… ►The tridiagonal matrix …

2: 30.16 Methods of Computation

… ►For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $𝐀=[A_{j,k}]$ with nonzero elements …

3: 29.20 Methods of Computation

… ►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices $𝐌$ given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …

4: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►Let $\alpha =-n$, $n=0,1,2,\mathrm{\dots }$, and $q_{n,m}$, $m=0,1,\mathrm{\dots },n$, be the eigenvalues of the tridiagonal matrix …

5: 3.7 Ordinary Differential Equations

… ►If, for example, ${\beta }_0={\beta }_1=0$, then on moving the contributions of $w(z_0)$ and $w(z_P)$ to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of ${𝐀}_P$ that lie below the main diagonal and its two adjacent diagonals. … ►This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in $\lambda$; compare §3.2(vi). …

6: 3.8 Nonlinear Equations

… ►For the computation of zeros of orthogonal polynomials as eigenvalues of finite tridiagonal matrices (§3.5(vi)), see Gil et al. (2007a, pp. 205–207). For the computation of zeros of Bessel functions, Coulomb functions, and conical functions as eigenvalues of finite parts of infinite tridiagonal matrices, see Grad and Zakrajšek (1973), Ikebe (1975), Ikebe et al. (1991), Ball (2000), and Gil et al. (2007a, pp. 205–213). …

7: 29.15 Fourier Series and Chebyshev Series

… ►be the tridiagonal matrix with ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$ as in (29.3.11), (29.3.12). … ►The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an $(n+1)\times (n+1)$ tridiagonal matrix; see Arscott and Khabaza (1962). …

8: 1.2 Elementary Algebra

… ►a tridiagonal matrix if … ►Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix. …

9: 3.6 Linear Difference Equations

… ►Let us assume the normalizing condition is of the form $w_0=\lambda$, where $\lambda$ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns $w_1^{(N)},w_2^{(N)},\mathrm{\dots },w_{N-1}^{(N)}$; see §3.2(ii). …

10: 18.39 Applications in the Physical Sciences

… ►The technique to accomplish this follows the DVR idea, in which methods are based on finding tridiagonal representations of the co-ordinate, $x$. Here tridiagonal representations of simple Schrödinger operators play a similar role. …is tridiagonalized in the complete $L^2$ non-orthogonal (with measure $dr$, $r\in [0,\mathrm{\infty })$) basis of Laguerre functions: …