NationMaster - Encyclopedia: Lagrange polynomial

In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form. It was first discovered by Edward Waring in 1779 and later rediscovered by Leonhard Euler in 1783. Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Joseph-Louis Lagrange Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... Edward Waring (1736 - August 15, 1798) was British mathematician who was born in Old Heath (near Shrewsbury) Shropshire England and died in Pontesbury Shropshire England He was Lucasian professor of mathematics at Cambridge University from 1760 until his death. ... Euler redirects here. ...

As there is only one interpolation polynomial for a given set of data points it is a bit misleading to call the polynomial Lagrange interpolation polynomial. The more precise name is interpolation polynomial in the Lagrange form.

This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x), which is the sum of the scaled basis polynomials y0l0(x), y1l1(x), y2l2(x) and y3l3(x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.

This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x), which is the sum of the scaled basis polynomials y₀l₀(x), y₁l₁(x), y₂l₂(x) and y₃l₃(x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.

1 Definition
2 Proof
3 Main idea
4 Usage
- 4.1 Example
- 4.2 Notes
5 See also

Lagrange interpolation polynomials // #include <stdio. ...

Definition

Given a set of k + 1 data points

$(x_0, y_0),ldots,(x_k, y_k)$

where no two x_j are the same, the interpolation polynomial in the Lagrange form is a linear combination In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

$L(x) := sum_{j=0}^{k} y_j ell_j(x)$

of Lagrange basis polynomials

$ell_j(x) := prod_{i=0,, ineq j}^{k} frac{x-x_i}{x_j-x_i} = frac{x-x_0}{x_j-x_0} cdots frac{x-x_{j-1}}{x_j-x_{j-1}} frac{x-x_{j+1}}{x_j-x_{j+1}} cdots frac{x-x_{k}}{x_j-x_{k}}.$

Proof

The function we are looking for has to be a polynomial function L(x) of degree k with

$L(x_j) = y_j qquad j=0,ldots,k$

The Lagrange polynomial is a solution to the interpolation problem:

As can be easily seen

$ell_j(x)$ is a polynomial and has degree k.
$ell_i(x_j) = delta_{ij},quad 0 leq i,j leq k.,$

Thus the function L(x) is a polynomial with degree k and

$L(x_i) = sum_{j=0}^{k} y_j ell_j(x_i) = y_i.$

There can be only one solution to the interpolation problem since the difference of two such solutions would be a polynomial with degree k and k+1 zeros.

Therefore L(x) is our unique interpolation polynomial.

Main idea

Solving an interpolation problem leads to a problem in linear algebra where we have to solve a matrix. Using a standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix. By choosing another basis, the Lagrange basis we get the much simpler identity matrix = δ_i,j which we can solve instantly. this article has been removed from wikipedia because it was useless. ... In linear algebra, a Vandermonde matrix, named after Alexandre-ThÃ©ophile Vandermonde, is a matrix with a geometric progression in each row, i. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

Usage

Example

The tangent function and its interpolant

We wish to interpolate $f (x) = tan(x)$ at the points Image File history File links Download high resolution version (1200x900, 9 KB) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (1200x900, 9 KB) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

$x 0 = - 1.5$	$f (x 0) = - 14.1014$
$x 1 = - 0.75$	$f (x 1) = - 0.931596$
$x 2 = 0$	$f (x 2) = 0$
$x 3 = 0.75$	$f (x 3) = 0.931596$
$x 4 = 1.5$	$f (x 4) = 14.1014$

Now, the basis polynomials are:

$ell_0(x)={x - x_1 over x_0 - x_1}cdot{x - x_2 over x_0 - x_2}cdot{x - x_3 over x_0 - x_3}cdot{x - x_4 over x_0 - x_4} ={1over 243} x (2x-3)(4x-3)(4x+3)$

$ell_1(x)={x - x_0 over x_1 - x_0}cdot{x - x_2 over x_1 - x_2}cdot{x - x_3 over x_1 - x_3}cdot{x - x_4 over x_1 - x_4} =-{8over 243} x (2x-3)(2x+3)(4x-3)$

$ell_2(x)={x - x_0 over x_2 - x_0}cdot{x - x_1 over x_2 - x_1}cdot{x - x_3 over x_2 - x_3}cdot{x - x_4 over x_2 - x_4} ={1over 243} (243-540x^2+192x^4)$

$ell_3(x)={x - x_0 over x_3 - x_0}cdot{x - x_1 over x_3 - x_1}cdot{x - x_2 over x_3 - x_2}cdot{x - x_4 over x_3 - x_4} =-{8over 243} x (2x-3)(2x+3)(4x+3)$

$ell_4(x)={x - x_0 over x_4 - x_0}cdot{x - x_1 over x_4 - x_1}cdot{x - x_2 over x_4 - x_2}cdot{x - x_3 over x_4 - x_3} ={1over 243} x (2x+3)(4x-3)(4x+3)$

Thus the interpolating polynomial then is

${1over 243}Big(f(x_0)x (2x-3)(4x-3)(4x+3)-8f(x_1)x (2x-3)(2x+3)(4x-3)$

$+f(x_2)(243-540x^2+192x^4)-8f(x_3)x (2x-3)(2x+3)(4x+3) ,$

$+f(x_4)x (2x+3)(4x-3)(4x+3)Big),$

$=-1.47748x+4.83456x^3.,$

Notes

The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments. But, as can be seen from the construction, each time a node x_k changes, all Lagrange basis polynomials have to be recalculated. A better form of the interpolation polynomial for practical (or computational) purposes is the Newton polynomial. Using nested multiplication amounts to the same idea. In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. ... In the mathematical subfield of numerical analysis the Horner scheme or Horner algorithm, named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. ...

Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This oscillation is lessened by choosing interpolation points at Chebyshev nodes. In the mathematical subfield of numerical analysis Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind. ...

The Lagrange basis polynomials are used in numerical integration to derive the Newton-Cotes formulas. Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... In numerical analysis, the Newton-Cotes formulas, also called the Newton-Cotes rules, are a group of formulas for numerical integration (also called quadrature) based on evaluating the integrand at n+1 equally-spaced points. ...

Lagrange interpolation is often used in digital signal processing of audio for the implementation of fractional delay FIR filters (e.g., to precisely tune digital waveguides in physical modelling synthesis). Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... A finite impulse response (FIR) filter is a type of a digital filter. ... Digital waveguide synthesis is the synthesis of audio using a digital waveguide. ... Physical modelling synthesis is the synthesis of sound by using a set of equations and algorithms to simulate a physical source of sound. ...

Categories: Interpolation | Polynomials In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ... In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. ... In the mathematical subfield of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials. ... In numerical analysis, the Newton-Cotes formulas, also called the Newton-Cotes rules, are a group of formulas for numerical integration (also called quadrature) based on evaluating the integrand at n+1 equally-spaced points. ...

Results from FactBites:

Lab for Lagrange Polynomial Approximation (242 words)
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Lagrange polynomial - Wikipedia, the free encyclopedia (457 words)
	In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form.
	The Lagrange basis polynomials are used in numerical integration to derive the Newton-Cotes formulas.
	Lagrange interpolation is often used in digital signal processing of audio for the implementation of fractional delay FIR filters (e.g., to precisely tune digital waveguides in physical modelling synthesis).

More results at FactBites »

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