Section 5.5 Piecewise Interpolation

Transcription

1 Section 5.5 Piecewise Interpolation Key terms Runge phenomena polynomial wiggle problem Piecewise polynomial interpolation

2 We have considered polynomial interpolation to sets of distinct data like {( ) 012 } S = x, y i =,,,...,n i i 1.5 The result was a unique polynomial of 1 degree n or less. Unfortunately for n large the 0.5 resulting polynomial may contain undesired 0 oscillations. A classic example is to generate -0.5 polynomial interpolants of higher and higher -1 degree to the data sets at equispaced points -1.5 in the interval [-5, 5] determined by function f(x) = 1+ x 2 2 To demonstrate this we have MATLAB routine rungedem. For successive experiments choose spacing h for data in [-5, 5] using h= 1,h=1/2,h=1/4,h=1/5.

3 h = 1 h = 1/2 h = 1/4 For this function 1 f(x) = 1+ x 2 it is possible to show that the interpolation error goes to infinity as the spacing h of the points goes to zero (or equivalently the degree of the interpolant becomes larger and larger).

4 1 Note that f(x) = 1+ x 2 is infinitely differentiable. But this doesn t always happen. Let g(x) = sin(x) over [0, 2π]. The interpolant to the equispaced points with h = 0.1is shown as red dots on the sine function The behavior described for is called the Runge phenomena or the polynomial wiggle problem. Runge, Carl (1901), "Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten", Zeitschrift für Mathematik und Physik 46:

5 The wiggle problem suggests that we use low order polynomial interpolation over subsets of the data set. This is an alternative to polynomial interpolation through an entire data set is to construct polynomial interpolants to subsets of the data. Thus different polynomials can be used over subintervals of the data points involved. This is referred to as piecewise polynomial interpolation. We briefly consider linear, quadratic, and cubic piecewise interpolation. Piecewise linear interpolation amounts to connecting successive data points with straight lines ( connect-the-dots ). The interpolant is continuous, but usually not differentiable because sharp points occur at the data points. Piecewise linear interpolation was used to develop the trapezoidal rule. Piecewise quadratic interpolation uses pairs of adjoining subintervals over which a quadratic interpolant is constructed through three points. Again the interpolant is continuous, but not guaranteed to be differentiable at endpoints where two different quadratic pieces meet. Simpson's 1/3 rule really uses piecewise quadratic interpolation and integrates the quadratic pieces. Piecewise cubic interpolation uses a set of 3 adjoining subintervals over which a cubic interpolant is constructed through four points. Again the interpolant is continuous, but not guaranteed to be differentiable at endpoints where two different cubic pieces meet.

6 To experiment with piecewise polynomial interpolation of this type use routine MATLAB pwinterp. PWINTERP Piecewise polynomial interpolation using: LINEAR QUADRATIC or CUBIC pieces. INPUT: Vectors x and y must contain the x and y coordinates of the data points respectively. OUTPUT: The coefficients of the pieces are in arrays lf qf and cf respectively. Various interpolants can be superimposed graphically. => pwinterp(x,y) or [lf,qf,cf]=pwinterp(x,y) <= Example: Use the following commands to generate data for the curve f(x)= 1/(1 + x^2) over[-4, 4] and investigate the piecewise interpolants available in pwinterp for constructing interpolating functions. In pwinterp set the graphics window so 4 x 4 and 0 y 2 and superimpose the piecewise interpolants. x=[-4:0.5:4]'; y=(1)./(1+x.^2); pwinterp(x,y)

7 1 Piecewise Linear Interpolant Piecewise Polynomial Interpolation magenta=linear, blue=quadratic, black=cubic;**press ENTER to Continue**

8 1 Piecewise Quadratic Interpolant Piecewise Polynomial Interpolation magenta=linear, blue=quadratic, black=cubic;**press ENTER to Continue**

9 1 Piecewise Cubic Interpolant Piecewise Polynomial Interpolation magenta=linear, blue=quadratic, black=cubic;**press ENTER to Continue**

10 Compared to polynomial interpolation to the entire data set (see below for a comparison of the interpolant with the original function), the piecewise interpolations provide a better model over the interval as a whole. PW Lnear PW Quadratic PW Cubic

11 Piecewise Polynomial Interpolation Error Inspect how the error terms change.