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tridiagonal systems

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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 = λ , where λ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns w 1 ( N ) , w 2 ( N ) , , w N 1 ( N ) ; see §3.2(ii). …
3: 3.7 Ordinary Differential Equations
If, for example, β 0 = β 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 ) 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 , , 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. …