# Interpolation and polynomial approximation pdf

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## Polynomial interpolation

In numerical analysis , polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset. Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography , [ citation needed ] given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions : pick a few known data points, create a lookup table , and interpolate between those data points.

This results in significantly faster computations. Polynomial interpolation is also essential to perform sub-quadratic multiplication and squaring such as Karatsuba multiplication and Toom—Cook multiplication , where an interpolation through points on a polynomial which defines the product yields the product itself. Finding points along W x by substituting x for small values in f x and g x yields points on the curve.

Interpolation based on those points will yield the terms of W x and subsequently the product ab. In the case of Karatsuba multiplication this technique is substantially faster than quadratic multiplication, even for modest-sized inputs.

This is especially true when implemented in parallel hardware. It follows that the linear combination. The unisolvence theorem states that such a polynomial p exists and is unique, and can be proved by the Vandermonde matrix , as described below. If we substitute equation 1 in here, we get a system of linear equations in the coefficients a k.

The system in matrix-vector form reads the following multiplication :. We have to solve this system for a k to construct the interpolant p x.

The matrix on the left is commonly referred to as a Vandermonde matrix. The condition number of the Vandermonde matrix may be large, [4] causing large errors when computing the coefficients a i if the system of equations is solved using Gaussian elimination.

Several authors have therefore proposed algorithms which exploit the structure of the Vandermonde matrix to compute numerically stable solutions in O n 2 operations instead of the O n 3 required by Gaussian elimination. Alternatively, we may write down the polynomial immediately in terms of Lagrange polynomials :. For matrix arguments, this formula is called Sylvester's formula and the matrix-valued Lagrange polynomials are the Frobenius covariants.

We know,. It has one root too many. So q x which could be any polynomial, so long as it interpolates the points is identical with p x , and q x is unique.

To prove that V is nonsingular we use the Vandermonde determinant formula:. Either way this means that no matter what method we use to do our interpolation: direct, Lagrange etc. This can be a very costly operation as counted in clock cycles of a computer trying to do the job. One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients, e.

Neville's algorithm. The cost is O n 2 operations, while Gaussian elimination costs O n 3 operations. Furthermore, you only need to do O n extra work if an extra point is added to the data set, while for the other methods, you have to redo the whole computation.

Another method is to use the Lagrange form of the interpolation polynomial. The resulting formula immediately shows that the interpolation polynomial exists under the conditions stated in the above theorem. Lagrange formula is to be preferred to Vandermonde formula when we are not interested in computing the coefficients of the polynomial, but in computing the value of p x in a given x not in the original data set. In this case, we can reduce complexity to O n 2. The Lagrange form of the interpolating polynomial is a linear combination of the given values.

In many scenarios, an efficient and convenient polynomial interpolation is a linear combination of the given values , using previously known coefficients. The interpolation polynomial in the Lagrange form is the linear combination. This quadratic interpolation is valid for any position x , near or far from the given positions. This is a quadratic interpolation typically used in the Multigrid method.

In the above polynomial interpolations using a linear combination of the given values, the coefficients were determined using the Lagrange method.

In some scenarios, the coefficients can be more easily determined using other methods. Examples follow. This area is surveyed here. The triangle of binomial transform coefficients is like Pascal's triangle. The BTC triangle can also be used to derive other polynomial interpolations. For example, the above quadratic interpolation. Second, the unwanted data point y 0. Similar to other uses of linear equations, the above derivation scales and adds vectors of coefficients. In polynomial interpolation as a linear combination of values, the elements of a vector correspond to a contiguous sequence of regularly spaced positions.

The p non-zero elements of a vector are the p coefficients in a linear equation obeyed by any sequence of p data points from any degree d polynomial on any regularly spaced grid, where d is noted by the subscript of the vector. When adding vectors with various subscript values, the lowest subscript applies for the resulting vector.

Similarly, the cubic interpolation typical in the Multigrid method ,. When interpolating a given function f by a polynomial of degree n at the nodes x 0 , The Chebyshev nodes achieve this. Thus the remainder term in the Lagrange form of the Taylor theorem is a special case of interpolation error when all interpolation nodes x i are identical.

Thus, the maximum error will occur at some point in the interval between two successive nodes. That question is treated in the section Convergence properties. We fix the interpolation nodes x 0 , The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C [ a , b ] of all continuous functions on [ a , b ] to itself. The Lebesgue constant L is defined as the operator norm of X.

One has a special case of Lebesgue's lemma :. This suggests that we look for a set of interpolation nodes that makes L small. In particular, we have for Chebyshev nodes :. We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation, as the growth in n is exponential for equidistant nodes.

However, those nodes are not optimal. Convergence may be understood in different ways, e. The situation is rather bad for equidistant nodes, in that uniform convergence is not even guaranteed for infinitely differentiable functions. One might think that better convergence properties may be obtained by choosing different interpolation nodes.

The following result seems to give a rather encouraging answer:. But this is true due to a special property of polynomials of best approximation known from the equioscillation theorem. Choosing the points of intersection as interpolation nodes we obtain the interpolating polynomial coinciding with the best approximation polynomial. The defect of this method, however, is that interpolation nodes should be calculated anew for each new function f x , but the algorithm is hard to be implemented numerically.

Does there exist a single table of nodes for which the sequence of interpolating polynomials converge to any continuous function f x?

The answer is unfortunately negative:. Now we seek a table of nodes for which. Due to the Banach—Steinhaus theorem , this is only possible when norms of X n are uniformly bounded, which cannot be true since we know that. For example, if equidistant points are chosen as interpolation nodes, the function from Runge's phenomenon demonstrates divergence of such interpolation.

For better Chebyshev nodes , however, such an example is much harder to find due to the following result:. Runge's phenomenon shows that for high values of n , the interpolation polynomial may oscillate wildly between the data points. This problem is commonly resolved by the use of spline interpolation. Here, the interpolant is not a polynomial but a spline : a chain of several polynomials of a lower degree. Interpolation of periodic functions by harmonic functions is accomplished by Fourier transform.

This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial. Hermite interpolation problems are those where not only the values of the polynomial p at the nodes are given, but also all derivatives up to a given order. This turns out to be equivalent to a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials.

Birkhoff interpolation is a further generalization where only derivatives of some orders are prescribed, not necessarily all orders from 0 to a k. Collocation methods for the solution of differential and integral equations are based on polynomial interpolation. The technique of rational function modeling is a generalization that considers ratios of polynomial functions. At last, multivariate interpolation for higher dimensions.

June Learn how and when to remove this template message. May—June Retrieved 3 November Society for Industrial and Applied Mathematics. Numerische Mathematik.

## Numerical Analysis : Approximation, Interpolation, Integration

Due to the COVID crisis, the information below is subject to change, in particular that concerning the teaching mode presential, distance or in a comodal or hybrid format. Teacher s. Absil Pierre-Antoine ;. Interpolation Function approximation Numerical integration. At the end of this learning unit, the student is able to : 1 AA1. Transversal learning outcomes : Use a reference book in English; Use programming languages for scientific computing. Polynomial interpolation: Lagrange's interpolation formula, Neville's algorithm, Newton's interpolation formula, divided differences, Hermite interpolation.

In numerical analysis , polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset. Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography , [ citation needed ] given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions : pick a few known data points, create a lookup table , and interpolate between those data points. This results in significantly faster computations. Polynomial interpolation is also essential to perform sub-quadratic multiplication and squaring such as Karatsuba multiplication and Toom—Cook multiplication , where an interpolation through points on a polynomial which defines the product yields the product itself. Finding points along W x by substituting x for small values in f x and g x yields points on the curve. Interpolation based on those points will yield the terms of W x and subsequently the product ab.

through given data is called polynomial interpolation. ▻ Polynomials are often used because they have the property of approximating any continuous function.

## Interpolation and Approximation

Principles and Procedures of Numerical Analysis pp Cite as. Often functions arising in economics and engineering are not specified by explicit formulas, but rather by values at distinct points. Engineering and mathematical analysis cannot easily be applied to problems having partially specified functions, and so interpolation functions of explicit and uncomplicated form are sought that agree with the known function values.

The Lagrange interpolating polynomial is the polynomial of degree that passes through the points , , , , and is given by. Lagrange Interpolation Formula. Lagrange polynomials are used for polynomial interpolation.

In numerical analysis , Lagrange polynomials are used for polynomial interpolation. Although named after Joseph-Louis Lagrange , who published it in , the method was first discovered in by Edward Waring. Uses of Lagrange polynomials include the Newton—Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation.