University:
Whitman College
Course:
MATH-467 | Numerical Analysis
Academic year:

2022
Views:

358

Pages:

2
Author:

Eden Robbins

Piecewise Polynomials and Matlab First, we see how to define piecewise polynomials and evaluate these polynomials in Matlab. Here are two polynomials: ( P1 (x) = P2 (x) = 3x2 − 5x + 1, 3x − 2  2   x − 5x,   3x − 2 −x2 + 1 if 0 ≤ x <≤ 5 if 5 < x ≤ 10 if − 2 ≤ x ≤ 0 if 0 < x ≤ 2 if 2 < x ≤ 4 In Matlab, we need to build a structure that contains the “break” points, and the polynomial coefficients. To build P1 (x), use mkpp: Coefs=[3,-5,1;0,3,-2]; breaks=[0 5 10]; P1=mkpp(breaks,Coefs); To evaluate this polynomial, say at equally spaced points between -3 and 12 and plot the result, use: xx=linspace(-3,12); yy=ppval(xx,P1); plot(xx,yy) Notice that if you evaluate past the intervals on which the polynomial is defined, Matlab will use the first or last function. Now build P2 (x), and plot it on the interval from −3 to 5. Coefs=[1,-5,0;0,3,-2;-1,0,1]; breaks=[-2 0 2 4]; P1=mkpp(breaks,Coefs); xx=linspace(-3,5); yy=ppval(xx,P1); plot(xx,yy) Cubic Splines and PPVAL In this example, we create a cubic spine approximating sin(x) for −2 ≤ x ≤ 2 on 12 equally spaced points: x=linspace(-2,2,12); y=sin(x); P=spline(x,y); The coefficients are stored in P.coefs and the breaks are stored as P.breaks. The coefficients are stored just as we did previously- That is, if a row of P.coefs looks like: [1 2 3 4], that corresponds to 1 · x3 + 2 · x2 + 3 · x + 4 Now, we would like to plot the derivative of our splines. Note that each spline has the form: ax3 + bx2 + cx + d so the derivative is 3ax2 + 2bx + c, so the coefficients are 3a, 2b, c. We can re-form the coefficient matrix: T=P.coefs; %Temporary storage Coefs=[3*T(:,1),2*T(:,2),T(:,3)]; P=mkpp(P.breaks,Coefs); clear Coefs T To plot the result, we could say: t=linspace(-2,2); newY=ppval(t,P); plot(t,newY);

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