iterative methods

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1: 3.8 Nonlinear Equations

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Bisection Method

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Secant Method

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Steffensen’s Method

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Eigenvalue Methods

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2: 29.20 Methods of Computation

… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …

3: 3.9 Acceleration of Convergence

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§3.9(v) Levin’s and Weniger’s Transformations

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4: Bibliography T

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Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..

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External Links:

MathReview (W. C. Rheinboldt), ZentralBlatt

Referenced by:

§3.8(iii).

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5: Bibliography S

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Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

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Referenced by:

§18.38(i).

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6: 3.11 Approximation Techniques

… ►to $f(x)$, then the coefficients $a_k$ can be computed iteratively. … ►The iterative process converges locally and quadratically (§3.8(i)). ►A method for obtaining a sufficiently accurate first approximation is described in the next subsection. … ►For details and examples of these methods, see Clenshaw (1957, 1962) and Miller (1966). … ►The method of the fast Fourier transform (FFT) exploits the structure of the matrix $𝛀$ with elements ${\omega }_n^{jk}$, $j,k=0,1,\mathrm{\dots },n-1$. …

7: 19.36 Methods of Computation

§19.36 Methods of Computation

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§19.36(i) Duplication Method

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►

§19.36(iv) Other Methods

►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. …

8: 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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External Links:

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

Referenced by:

§17.18, §17.18, §5.21, §5.21.

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On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.

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ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt

Referenced by:

§10.74(vii).

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On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.

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ISSN 0771-050X, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

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ISSN 0044-2275, Document, MathReview

Referenced by:

§10.71(ii).

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Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.

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External Links:

ISBN 981-02-3333-7, MathReview (K. Moszyński), ZentralBlatt

Referenced by:

§18.38(i).

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9: 22.20 Methods of Computation

§22.20 Methods of Computation

… ►Four iterations of (22.20.1) lead to $c_4=\mathrm{6.5\times 10⁻¹²}$. … ►If $k=k^{\prime }=1/\sqrt{2}$, then three iterations of (22.20.1) give $M=\mathrm{0.84721\hspace{0.33em}30848}$, and from (22.20.6) $K=\pi /(2M)=\mathrm{1.85407\hspace{0.33em}46773}$ — in agreement with the value of ${\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2/\left(4\sqrt{\pi }\right)$; compare (23.17.3) and (23.22.2). … ►If $k^{\prime }=1-\mathrm{i}$, then four iterations of (22.20.1) give $K=\mathrm{1.23969\hspace{0.33em}74481}+\mathrm{i}\mathrm{0.56499\hspace{0.33em}30988}$. … ►For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947). …

10: Bibliography O

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Asymptotic Enumeration Methods. In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.), pp. 1063–1229.

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External Links:

ISBN 0-444-82351-4, Pdf, MathReview (Konrad Engel), ZentralBlatt

Referenced by:

Ch.2.

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Computing $\pi (x)$: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Notes:

Online fulltext contains postpublication annotations.

External Links:

FullText

Referenced by:

§27.18.

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations. Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.

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External Links:

Document, MathReview (J. Todd), ZentralBlatt

Referenced by:

§10.21(ii).

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Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

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Notes:

Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania

External Links:

MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))

Referenced by:

§3.8(vii).

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