Lagrange interpolation

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1: 3.3 Interpolation

§3.3 Interpolation

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§3.3(i) Lagrange Interpolation

… ► ►With an error term the Lagrange interpolation formula for $f$ is given by … ►

§3.3(ii) Lagrange Interpolation with Equally-Spaced Nodes

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2: 18.40 Methods of Computation

… ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: … ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. … ►

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Figure 18.40.2: Derivative Rule inversions for $w^{\mathrm{RCP}}(x)$ carried out via Lagrange and PWCF interpolations. …For the derivative rule Lagrange interpolation (red points) gives $\sim 15$ digits in the central region, while PWCF interpolation (blue points) gives $\sim 25$. Magnify

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3: 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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4: 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 $[a,b]$. …

5: 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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6: 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 $w(x)$. …

7: 3.4 Differentiation

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§3.4(i) Equally-Spaced Nodes

►The Lagrange $(n+1)$-point formula is …The $B_k^n$ are the differentiated Lagrangian interpolation coefficients: ►

3.4.2 $$B_k^n=d{A}_k^n/dt,$$

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Defines:

$B_k^n$: differentiated Lagrangian interpolation coefficients (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $A_k^n$: Lagrangian interpolation coefficients

Referenced by:

§3.4(i)

Permalink:

http://dlmf.nist.gov/3.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4 and Ch.3

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