Miller algorithm

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1: 3.6 Linear Difference Equations

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§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 algorithm (§3.6(iii)) or Olver’s algorithm (§3.6(v)). …

2: 6.18 Methods of Computation

… ► $A_0$, $B_0$, and $C_0$ can be computed by Miller’s algorithm (§3.6(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

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Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.

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External Links:

FullText, MathReview (J.C.P. Miller), ZentralBlatt

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§3.6(iii).

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4: Bibliography C

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Algorithms for special functions. I. Numer. Math. 4, pp. 403–419.

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ISSN 0029-599X, Document, MathReview (A. S. Householder), ZentralBlatt

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§5.24(ii).

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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

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Algorithm 385: Exponential integral $\mathrm{Ei}(x)$ . Comm. ACM 13 (7), pp. 446–447.

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Notes:

Includes Fortran code, maximum accuracy 20D. For certification and remarks see same journal v. 13 (1970), pp. 448–449 and p. 750; v. 15 (1972), p. 1074.

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ISSN 0001-0782, Document

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§6.21(ii).

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Tables of the Exponential Integral $E_{\nu }(x)={\int }_1^{\mathrm{\infty }}{e}^{-xu}{u}^{-\nu }𝑑u$ . Pergamon Press, New York.

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MathReview (J. C. P. Miller), ZentralBlatt

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3rd item.

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Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.

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ISSN 0377-0427, Document, ZentralBlatt

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§10.77(ix).

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Algorithm 680: Evaluation of the complex error function. ACM Trans. Math. Software 16 (1), pp. 47.

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Notes:

Single-precision Fortran, maximum accuracy 14S.

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ISSN 0098-3500, Document, GAMS (module 680; package TOMS), MathReview, ZentralBlatt

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1st item.

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Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.

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ISSN 0098-3500, Document, GAMS (module 498; package TOMS), ZentralBlatt

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1st item, 4th item.

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7: 8.25 Methods of Computation

… ►DiDonato and Morris (1986) describes an algorithm for computing $P(a,x)$ and $Q(a,x)$ for $a\ge 0$, $x\ge 0$, and $a+x\ne 0$ 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

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Computing $\pi (x)$: The Meissel-Lehmer method. Math. Comp. 44 (170), pp. 537–560.

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ISSN 0025-5718, FullText, MathReview (Kenneth A. Jukes), ZentralBlatt

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§27.18.

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Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.

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ISSN 0098-3500, Document, GAMS (module 537; package TOMS)

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2nd item, 2nd item.

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Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

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Notes:

Includes Algol program, maximum accuracy 12D.

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ISSN 0001-0782, Document

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§7.25(iv).

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Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

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External Links:

MathReview (F. Gotze), ZentralBlatt

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§13.31(iii).

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Algorithms for the Computation of Mathematical Functions. Academic Press, New York.

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MathReview (Gh. Adam)

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§16.26.

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9: Bibliography G

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Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms. Numer. Algorithms 49 (1-4), pp. 159–168.

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ISSN 1017-1398, Document, FullText, MathReview (Necdet Batir), ZentralBlatt

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§17.18, §17.18, §5.21, §5.21.

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Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

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ISSN 0001-0782, Document

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§33.26(ii).

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Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

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ISSN 0001-0782, Document

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§6.21(ii), §8.28(vi).

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Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.

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Notes:

Single-precision Fortran.

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ISSN 0098-3500, Document, GAMS (module 542; package TOMS), ZentralBlatt

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2nd item.

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Tables of Partitions. Royal Society Math. Tables, Vol. 4, Cambridge University Press.

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ZentralBlatt

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§27.21.

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10: Bibliography F

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Algorithms and codes for the Macdonald function: Recent progress and comparisons. J. Comput. Appl. Math. 161 (1), pp. 179–192.

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External Links:

ISSN 0377-0427, MathReview (Boro Döring), ZentralBlatt

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§10.77(vi).

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

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ISSN 0098-3500, Document, Code, MathReview Entry, ZentralBlatt

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3rd item, 2nd item, 1st item.

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Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

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ISSN 0001-0782, Document

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§5.24(ii).

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Spheroidal Wave Functions. Stanford University Press, Stanford, CA.

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External Links:

MathReview (J. C. P. Miller), ZentralBlatt

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§30.1, §30.10, 2nd item, Ch.30.

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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.

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Notes:

Table errata: Math. Comp. v.27 (1973), p. 1009.

External Links:

MathReview

Referenced by:

§10.75(i), §11.14(i), §12.19, §13.30, §14.33, §18.41(iii), §19.37(i), §20.15, §22.21, §23.23, §24.20, 4th item, §28.35(iv), §30.17, §33.24, §4.23(viii), §4.46, §5.22(i), §6.19(i), §7.23(i), §8.26(i), §9.18(i).

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