1. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical IntegrationIntegration  Definition –Total area within a region –In mathematical terms, it is the total value, or summation, of f(x) dx over the range from a to b:

2. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Newton-Cotes Formulas  General Idea –replace a complicated function or tabulated data with a polynomial that is easy to integrate: –where f n (x) is an n th order interpolating polynomial.

3. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Newton-Cotes Illustrations  The integrating function can be polynomials for any order - for example, (a) straight lines or (b) parabolas.  The integral can be approximated in one step or in a series of steps to improve accuracy.

4. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration The Trapezoidal Rule  Uses straight-line approximation for the function  Uses linear interpolation

5. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Error of the Trapezoidal Rule  The error is dependent upon the curvature of the actual function as well as the distance between the points.  Error can thus be reduced by: –breaking the curve into parts or –using a higher order function

6. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Composite Trapezoidal Rule  Assuming n+1 data points are evenly spaced, there will be n intervals over which to integrate.  The total integral can be calculated by integrating each subinterval and then adding them together:

7. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Trapezoid Functions Trapezoid Functions  For inline functions, use the ‘trap functions  For tabulated data, use trapz(x,y) –Matlab built-in function for numerical integration based on trapezoidal rule –y(x) should be in a tabulated form –can handle unequally spaced data as long as x in ascending order –example: >> x=[0.12.22.32.4.44.54.64.7.8]; >> y=0.2+25*x-200*x.^2+675*x.^3-900*x.^4+400*x.^5; >> trapz(x,y) 1.5948

8. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Trapezoid Rule Examples Trapezoid Rule Examples  Tabulated  Inline Function % f(x)=cos(x)+sin(2x) on [0 pi/2] h=(pi/2-0)/10; >> x=0:h:pi/2; >> y=cos(x)+sin(2*x); >> I2=trapz(x,y) 1.9897 p=inline('cos(x)+sin(2*x)'); >> I3=trap(p,0,pi/2,10) 1.9897 % plot f(x) and I(x) >> for k=1:11 I4(k)=trap(p,0,x(k),20); end; >> plot(x,y,x,I4,'r')

9. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Simpson’s Rules  Increasing the approximation order results in better integration accuracy –Simpson’s 1/3 rule based on taking 2 nd order polynomial integrations use two panels (three points) every integral only for even number of panels –Simpson’s 3/8 rule is based on taking 3 rd order polynomial integrations use three panels (four points) every integral only for three-multiple number of panels

10. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Simpson’s 1/3 Rule  Using the Lagrange form for a quadratic fit of three points: –Integration over the three points simplifies to:  Composite

11. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Simpson’s 3/8 Rule  Basic  Composite

12. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Combined Simpson’s Rule  Combined Simpson’s rule –If n (number of panels) is even, use Simpson’s 1/3 rule –If n is odd, use Simpson’s 3/8 rule once at beginning or end and use Simpson’s 1/3 rule for the rest of the panels

13. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Error of Simpson’s 1/3 Rule  If f(x) is a polynomial function of degree 3 or less, Simpson’s rule provides no error.  Use smaller spacing (h decreases) or more panels to reduce the error.  In general, Simpson’s rule is accurate enough for the most of functions f(x) with much less panels compared to that with the trapezoidal rule.

14. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical Integration Simpson’s Rule Example Simpson’s Rule Example  By Hand x=[0.1 0.2 0.3 0.4 0.5 0.6 0.7]; y=[2.1 1.7 1.6 2.3 2.8 1.7 2.5]; 0=(0.1/3)*(y(1)+4*y(2)+2*y(3)+ 4*y(4)+2*y(5)+4*y(6)+y(7)) >>1.2067 % f(x)=cos(x)+sin(2x) on [0 pi/2] n=10; h=(pi/2-0)/10; % 10panels x=0:h:pi/2; y=cos(x)+sin(2*x); I2=0; for k=1:2:(n-1) I2=I2+h/3*(y(k)+4*y(k+1)+y(k+2)); end 2.0001   Fucntions % define function f(x) p=inline('cos(x)+sin(2.*x)'); I3=simps(p,0,pi/2,10) 2.0001 Try different number of panels Compare with trapezoid rule

15. ES 240: Scientific and Engineering Computation. Chapter 17: Numerical IntegrationLab  Ex 17.3