# Miller algorithm

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##### 1: 3.6 Linear Difference Equations
###### §3.6(iii) Miller’s Algorithm
For further information on Miller’s algorithm, including examples, convergence proofs, and error analyses, see Wimp (1984, Chapter 4), Gautschi (1967, 1997b), and Olver (1964a). See also Gautschi (1967) and Gil et al. (2007a, Chapter 4) for the computation of recessive solutions via continued fractions. … 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)). …
##### 2: 6.18 Methods of Computation
$A_{0}$, $B_{0}$, and $C_{0}$ can be computed by Miller’s algorithm3.6(iii)), starting with initial values $(A_{N},B_{N},C_{N})=(1,0,0)$, say, where $N$ is an arbitrary large integer, and normalizing via $C_{0}=1/z$. …
##### 3: Bibliography O
• F. W. J. Olver (1964a) Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.
• ##### 4: Bibliography C
• C. W. Clenshaw, G. F. Miller, and M. Woodger (1962) Algorithms for special functions. I. Numer. Math. 4, pp. 403–419.
• ##### 5: 3.11 Approximation Techniques
For details and examples of these methods, see Clenshaw (1957, 1962) and Miller (1966). …
###### Summation of Chebyshev Series: Clenshaw’s Algorithm
A widely implemented and used algorithm for calculating the coefficients $p_{j}$ and $q_{j}$ in (3.11.16) is Remez’s second algorithm. … is of fundamental importance in the FFT algorithm. …For further details and algorithms, see Van Loan (1992). …
##### 6: Bibliography P
• K. A. Paciorek (1970) Algorithm 385: Exponential integral $\mathrm{Ei}(x)$ . Comm. ACM 13 (7), pp. 446–447.
• V. I. Pagurova (1961) Tables of the Exponential Integral ${E}_{\nu}(x)=\int_{1}^{\infty}e^{-xu}u^{-\nu}du$ . Pergamon Press, New York.
• R. Piessens and M. Branders (1984) Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.
• G. P. M. Poppe and C. M. J. Wijers (1990) Algorithm 680: Evaluation of the complex error function. ACM Trans. Math. Software 16 (1), pp. 47.
• P. J. Prince (1975) Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.
• ##### 7: 8.25 Methods of Computation
DiDonato and Morris (1986) describes an algorithm for computing $P\left(a,x\right)$ and $Q\left(a,x\right)$ for $a\geq 0$, $x\geq 0$, and $a+x\neq 0$ from the uniform expansions in §8.12. The algorithm supplies 14S accuracy. … Stable recursive schemes for the computation of $E_{p}\left(x\right)$ are described in Miller (1960) for $x>0$ and integer $p$. …
##### 8: Bibliography L
• J. C. Lagarias, V. S. Miller, and A. M. Odlyzko (1985) Computing $\pi(x)$: The Meissel-Lehmer method. Math. Comp. 44 (170), pp. 537–560.
• W. R. Leeb (1979) Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.
• H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.
• Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.
• Y. L. Luke (1977b) Algorithms for the Computation of Mathematical Functions. Academic Press, New York.
• ##### 9: Bibliography G
• B. Gabutti and G. Allasia (2008) Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms. Numer. Algorithms 49 (1-4), pp. 159–168.
• W. Gautschi (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.
• W. Gautschi (1973) Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.
• W. Gautschi (1979a) Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.
• H. Gupta, C. E. Gwyther, and J. C. P. Miller (1958) Tables of Partitions. Royal Society Math. Tables, Vol. 4, Cambridge University Press.
• ##### 10: 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.
• C. Flammer (1957) Spheroidal Wave Functions. Stanford University Press, Stanford, CA.
• A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie (1962) An Index of Mathematical Tables. Vols. I, II. 2nd edition, Published for Scientific Computing Service Ltd., London, by Addison-Wesley Publishing Co., Inc., Reading, MA.