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11: 3.11 Approximation Techniques

… ►If $J=n+1$, then $p_n(x)$ is the Lagrange interpolation polynomial for the set $x_1,x_2,\mathrm{\dots },x_J$ (§3.3(i)). … ►For many applications a spline function is a more adaptable approximating tool than the Lagrange interpolation polynomial involving a comparable number of parameters; see §3.3(i), where a single polynomial is used for interpolating $f(x)$ on the complete interval $[a,b]$. …

12: 10.75 Tables

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British Association for the Advancement of Science (1937) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $x=0(.001)16(.01)25$, 10D; $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0.01(.01)25$, 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $Y_0\left(x\right)$, $Y_1\left(x\right)$ for small values of $x$, as well as auxiliary functions to compute all four functions for large values of $x$.

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British Association for the Advancement of Science (1937) tabulates $I_0\left(x\right)$, $I_1\left(x\right)$, $x=0(.001)5$, 7–8D; $K_0\left(x\right)$, $K_1\left(x\right)$, $x=0.01(.01)5$, 7–10D; ${\mathrm{e}}^{-x}{I}_0\left(x\right)$, ${\mathrm{e}}^{-x}{I}_1\left(x\right)$, ${\mathrm{e}}^x{K}_0\left(x\right)$, ${\mathrm{e}}^x{K}_1\left(x\right)$, $x=5(.01)10(.1)20$, 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $K_0\left(x\right)$, $K_1\left(x\right)$ for small values of $x$.

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Young and Kirk (1964) tabulates ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, $n=0,1$, $x=0(.1)10$, 15D; ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, modulus and phase functions $M_n\left(x\right)$, ${\theta }_n\left(x\right)$, $N_n\left(x\right)$, ${\varphi }_n\left(x\right)$, $n=0,1,2$, $x=0(.01)2.5$, 8S, and $n=0(1)10$, $x=0(.1)10$, 7S. Also included are auxiliary functions to facilitate interpolation of the tables for $n=0(1)10$ for small values of $x$. (Concerning the phase functions see §10.68(iv).)

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13: Bibliography B

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Barycentric Lagrange interpolation. SIAM Rev. 46 (3), pp. 501–517.

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External Links:

ISSN 0036-1445, Document, Link, MathReview (Mao Dong Ye), ZentralBlatt

Referenced by:

§3.3(i).

Encodings:

BibTeX

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Interpolation und genäherte Quadratur. In Mathematische Hilfsmittel des Ingenieurs. Teil III, R. Sauer and I. Szabó (Eds.), Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 141, pp. 232–319.

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External Links:

MathReview (E. Kreyszig)

Referenced by:

§3.3(vi).

Encodings:

BibTeX

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14: Bibliography D

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Interpolation and Approximation. Dover Publications Inc., New York.

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Notes:

Republication of the original 1963 edition (Blaisdell Publishing Co.), with minor corrections and a new preface and bibliography

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.3(i), §3.3(iii), §3.3(iv), §3.3(vi).

Encodings:

BibTeX

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15: Bibliography G

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On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Notes:

Orthogonal polynomials and numerical methods

External Links:

ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan), ZentralBlatt

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

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16: Bibliography F

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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

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Notes:

Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp (i\theta )$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii), §19.37(ii).

Encodings:

BibTeX

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17: Bibliography S

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Cardinal Spline Interpolation. Society for Industrial and Applied Mathematics, Philadelphia, PA.

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Notes:

Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12

External Links:

MathReview (H. L. Loeb), ZentralBlatt

Referenced by:

§24.17(ii).

Encodings:

BibTeX

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18: 3.5 Quadrature

… ►The nodes $x_1,x_2,\mathrm{\dots },x_n$ are prescribed, and the weights $w_k$ and error term $E_n(f)$ are found by integrating the product of the Lagrange interpolation polynomial of degree $n-1$ and $w(x)$. …

19: Errata

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These equations, originally added in Other Changes and Other Changes, respectively, have been assigned interpolated numbers.

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