triangular%20matrices
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11: 3.8 Nonlinear Equations
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►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).
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3.8.15
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►Consider and .
We have and .
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12: Bibliography D
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Computing Riemann matrices of algebraic curves.
Phys. D 152/153, pp. 28–46.
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Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach.
Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical
Sciences, New York.
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D gravity and random matrices.
Phys. Rep. 254 (1-2), pp. 1–133.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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13: 35.1 Special Notation
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►All matrices are of order , unless specified otherwise.
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complex variables. | |
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space of all real symmetric matrices. | |
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space of positive-definite real symmetric matrices. | |
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is positive definite. Similarly, is equivalent. | |
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space of orthogonal matrices. | |
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14: 8 Incomplete Gamma and Related
Functions
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15: 28 Mathieu Functions and Hill’s Equation
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16: 29.20 Methods of Computation
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►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv).
These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that has to be chosen sufficiently large.
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►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices
given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998).
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17: Bibliography G
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Riemann surfaces, plane algebraic curves and their period matrices.
J. Symbolic Comput. 26 (6), pp. 789–803.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Matrices, moments and quadrature with applications.
Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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18: 35.9 Applications
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►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument , with and .
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►In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).
19: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
20: 23 Weierstrass Elliptic and Modular
Functions
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