# tridiagonal systems

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## 7 matching pages

##### 1: 3.2 Linear Algebra
###### §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 $w_{1}^{(N)},w_{2}^{(N)},\dots,w_{N-1}^{(N)}$; 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 $\mathbf{A}_{P}$ 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 $\mathbf{M}$ 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 $\det(\mathbf{A})\neq 0$ 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
###### Introduction and One-Dimensional (1D) Systems
As in classical dynamics this sum is the total energy of the one particle system. …
###### 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. …
##### 7: 18.2 General Orthogonal Polynomials
whereas in the latter case the system $\{p_{n}(x)\}$ is finite: $n=0,1,\ldots,N$. … 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. …