# tridiagonal systems

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## 7 matching pages

##### 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

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►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).
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##### 3: 3.7 Ordinary Differential Equations

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►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.
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##### 4: 29.20 Methods of Computation

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►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).
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►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).
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##### 5: 1.2 Elementary Algebra

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*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 $det(\mathbf{A})\ne 0$ 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

… ►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

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►whereas in the latter case the system
$\{{p}_{n}(x)\}$ is finite: $n=0,1,\mathrm{\dots},N$.
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►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. …