tridiagonal systems

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1: 3.2 Linear Algebra

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§3.2(ii) Gaussian Elimination for a Tridiagonal Matrix

… ►For more information on solving tridiagonal systems see Golub and Van Loan (1996, pp. 152–160). …

2: 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). …

3: 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. …

4: 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). … ►Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). …

5: 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. … ►If $det(𝐀)\ne 0$ the system of $n$ linear equations in $n$ unknowns, … ►and for the corresponding eigenvectors one has to solve the linear system …

6: 18.39 Applications in the Physical Sciences

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Introduction and One-Dimensional (1D) Systems

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1D Quantum Systems with Analytically Known Stationary States

… ►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: …

7: 18.2 General Orthogonal Polynomials

… ►whereas in the latter case the system $\{p_n(x)\}$ is finite: $n=0,1,\mathrm{\dots },N$. … ►The matrix on the left-hand side is an (infinite tridiagonal) Jacobi matrix. … ►Between the systems $\{p_n(x)\}$ and $\{q_n(x)\}$ there are the contiguous relations … ►A system of OP’s with unique orthogonality measure is always complete, see Shohat and Tamarkin (1970, Theorem 2.14). In particular, a system of OP’s on a bounded interval is always complete. …