Chapter 3 Interpolation - MathWorks

Chapter 3 InterpolationInterpolation is the process of defining a function that takes on specified values atspecified points. This Chapter concentrates on two closely related interpolants: thepiecewise cubic spline and the shape-preserving piecewise cubic named pchip. The Interpolating PolynomialWe all know that two points determine a straight line. More precisely, any twopoints in the plane, (x1,y1) and (x2,y2), withx1 =x2, determine a unique first-degree polynomial inxwhose graph passes through the two points. There aremany different formulas for the polynomial, but they all lead to the same straightline generalizes to more than two points. Givennpoints in the plane,(xk,yk),k= 1,..,n, with distinctxk s, there is a unique polynomial inxof degreeless thannwhose graph passes through the points. It is easiest to remember thatn,the number of data points, is also the number of coefficients, although some of theleading coefficients might be zero, so the degree might actually be less thann , there are many different formulas for the polynomial, but they all define thesame polynomial is called theinterpolatingpolynomial because it exactly re-produces the given data:P(xk) =yk,k= 1.

2 Chapter 3. Interpolation There are n terms in the sum and n − 1 terms in each product, so this expression defines a polynomial of degree at most n−1.If P(x) is evaluated at x = xk, all the products except the kth are zero.Furthermore, the kth product is equal to one, so the sum is equal to yk and the interpolation conditions are satisfied. For example, consider the following data set.

Loading..

Tags:

  Chapter, Chapter 3

Information

Related search queries