Chapter 3InterpolationInterpolation is the process of defining a function that takes on specified values at specified points. This chapter concentrates on two closely related interpolants, the piecewise cubic spline and the shape-preserving piecewise cubic named “pchip”.3.1 The Interpolating PolynomialWe all know that two points determine a straight line. More precisely, any two points in the plane, and, with , determine a unique first degree polynomial in whose graph passes through the two points. There are many different formulas for the polynomial, but they all lead to the same straight line graph.This generalizes to more than two points. Given points in the plane, ，, with distinct s, there is a unique polynomial in of degree less than whose graph passes through the points. It is easiest to remember that , the number of data points, is also the number of coefficients, although some of the leading coefficients might be zero, so the degree might actually be less than . Again, there are many different formulas for the polynomial, but they all define the same function.This polynomial is called the interpolating polynomial because it exactly re- produces the given data.，Later, we examine other polynomials, of lower degree, that only approximate the data. They are not interpolating polynomials.The most compact representation of the interpolating polynomial is the La- grange form.There are terms in the sum and terms in each product, so this expression defines a polynomial of degree at most . Ifis evaluated at , all the products except the th are zero. Furthermore, the th product is equal to one, so the sum is equal to and the interpolation conditions are satisfied.For example, consider the following data set:x=0:3;y=-5 -6 -1 16;The commanddisp(x;y)displays 0123-5-6-116The Lagrangian form of the polynomial interpolating this data isWe can see that each term is of degree three, so the entire sum has degree at most three. Because the leading term does not vanish, the degree is actually three. Moreover, if we plug in or 3, three of the terms vanish and the fourth produces the corresponding value from the data set.Polynomials are usually not represented in their Lagrangian form. More fre- quently, they are written as something likeThe simple powers of x are called monomials and this form of a polynomial is said to be using the power form.The coefficients of an interpolating polynomial using its power form,can, in principle, be computed by solving a system of simultaneous linear equationsThe matrix of this linear system is known as a Vandermonde matrix. Its elements areThe columns of a Vandermonde matrix are sometimes written in the opposite order, but polynomial coefficient vectors in Matlab always have the highest power first.The Matlab function vander generates Vandermonde matrices. For our ex- ample data set,V = vander(x)generatesV =00011111842127931Thenc = Vycomputes the coefficientsc =1.00000.0000-2.0000-5.0000In fact, the example data was generated from the polynomial .One of the exercises asks you to show that Vandermonde matrices are nonsin- gular if the points are distinct. But another one of the exercises asks you to show that a Vandermonde matrix can be very badly conditioned. Consequently, using the power form and the Vandermonde matrix is a satisfactory technique for problems involving a few well-spaced and well-scaled data points. But as a general-purpose approach, it is dangerous.In this chapter, we describe several Matlab functions that implement various interpolation algorithms. All of them have the calling sequencev = interp(x,y,u)The first two input arguments, and , are vectors of the same length that define the interpolating points. The third input argument, , is a vector of points where the function is to be evaluated. The output, v, is the same length as u and has elements Our first such interpolation function, polyinterp, is based on the Lagrange form. The code uses Matlab array operations to evaluate the polynomial at all the components of u simultaneously.function v = polyinterp(x,y,u)n = length(x);v = zeros(size(u);for k = 1:nw = ones(size(u);for j = 1:k-1 k+1:nw = (u-x(j)./(x(k)-x(j).*w;endendv = v + w*y(k); To illustrate polyinterp, create a vector of densely spaced evaluation points.u = -.25:.01:3.25;Thenv = polyinterp(x,y,u);plot(x,y,o,u,v,-)creates figure 3.1.Figure 3.1. polyinterpThe polyinterp function also works correctly with symbolic variables. For example, createsymx = sym(x)Then evaluate and display the symbolic form of the interpolating polynomial withP = polyinterp(x,y,symx)pretty(P)produces-5 (-1/3 x + 1)(-1/2 x + 1)(-x + 1) - 6 (-1/2 x + 3/2)(-x + 2)x-1/2 (-x + 3)(x - 1)x