# Olver algorithm

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##### 1: 3.6 Linear Difference Equations
###### §3.6(v) Olver’s Algorithm
The backward recursion can be carried out using independently computed values of $J_{N}\left(1\right)$ and $J_{N+1}\left(1\right)$ or by use of Miller’s algorithm3.6(iii)) or Olver’s algorithm3.6(v)). … Thus the asymptotic behavior of the particular solution $\mathbf{E}_{n}\left(1\right)$ is intermediate to those of the complementary functions $J_{n}\left(1\right)$ and $Y_{n}\left(1\right)$; moreover, the conditions for Olver’s algorithm are satisfied. …
##### 2: 10.74 Methods of Computation
In the case of $J_{n}\left(x\right)$, the need for initial values can be avoided by application of Olver’s algorithm3.6(v)) in conjunction with Equation (10.12.4) used as a normalizing condition, or in the case of noninteger orders, (10.23.15). …
##### 3: 13.29 Methods of Computation
In the following two examples Olver’s algorithm3.6(v)) can be used. …
##### 4: 3.7 Ordinary Differential Equations
The equations can then be solved by the method of §3.2(ii), if the differential equation is homogeneous, or by Olver’s algorithm3.6(v)). …
##### 5: Bibliography O
• F. W. J. Olver and D. J. Sookne (1972) Note on backward recurrence algorithms. Math. Comp. 26 (120), pp. 941–947.
• F. W. J. Olver (1964a) Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.
• ##### 6: Bibliography T
• N. M. Temme (1979a) An algorithm with ALGOL 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives. J. Comput. Phys. 32 (2), pp. 270–279.
• N. M. Temme (1994a) A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.
• N. M. Temme (1997) Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.
• H. C. Thacher Jr. (1963) Algorithm 165: Complete elliptic integrals. Comm. ACM 6 (4), pp. 163–164.
• I. J. Thompson (2004) Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.
• ##### 7: 2.11 Remainder Terms; Stokes Phenomenon
Further details for this example are supplied in Olver (1991a, 1994b). … For further details see Olde Daalhuis and Olver (1994). … For second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997). … Further improvements in accuracy can be realized by making a second application of the Euler transformation; see Olver (1997b, pp. 540–543). Similar improvements are achievable by Aitken’s $\Delta^{2}$-process, Wynn’s $\epsilon$-algorithm, and other acceleration transformations. …
##### 8: Bibliography C
• P. J. Cameron (1994) Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press, Cambridge.
• S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.
• B. C. Carlson and E. M. Notis (1981) Algorithm 577: Algorithm for incomplete elliptic intergrals [S21]. ACM Trans. Math. Software 7 (3), pp. 398–403.
• B. C. Carlson (1972a) An algorithm for computing logarithms and arctangents. Math. Comp. 26 (118), pp. 543–549.
• J. E. Cremona (1997) Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, Cambridge.
• ##### 9: Bibliography F
• B. R. Fabijonas, D. W. Lozier, and J. M. Rappoport (2003) Algorithms and codes for the Macdonald function: Recent progress and comparisons. J. Comput. Appl. Math. 161 (1), pp. 179–192.
• B. R. Fabijonas (2004) Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.
• A. M. S. Filho and G. Schwachheim (1967) Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.
• S. Fillebrown (1992) Faster computation of Bernoulli numbers. J. Algorithms 13 (3), pp. 431–445.
• L. W. Fullerton (1972) Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.
• ##### 10: Philip J. Davis
In 1961, Davis hired Frank W. J. Olver, a founding member of the Mathematics Division and Head of the Numerical Methods Section at the National Physical Laboratory, Teddington, U. …Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. …Decades later, Olver became Editor-in-Chief and Mathematics Editor of the NIST Digital Library of Mathematical Functions (DLMF), a complete revision of A&S that was publicly released in 2010. … After receiving an overview of the project and watching a short demo that included a few preliminary colorful, but static, 3D graphs constructed for the first Chapter, “Airy and Related Functions”, written by Olver, Davis expressed the hope that designing a web-based resource would allow the team to incorporate interesting computer graphics, such as function surfaces that could be rotated and examined. This immediately led to discussions among some of the project members about what might be possible, and the discovery that some interactive graphics work had already been done for the NIST Matrix Market, a publicly available repository of test matrices for comparing the effectiveness of numerical linear algebra algorithms. …