1. The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 22 Integration of Equations

2. Gauss Quadrature Gauss quadrature implements a strategy of positioning any two points on a curve to define a straight line that would balance the positive and negative errors. Hence, the area evaluated under this straight line provides an improved estimate of the integral.

4. Two points Gauss-Legendre Formula The objective of Gauss quadrature is to determine the equations of the form: c 0 and c 1 are constants, the function arguments x 0 and x 1 are unknowns…….(4 unknowns)

6. Two points Gauss-Legendre Formula Thus, four unknowns to be evaluated require four conditions. If this integration is exact for a constant, 1 st order, 2 nd order, and 3 rd order functions:

7. Two points Gauss-Legendre Formula Solving these 4 equations, we can determine c 1, c 2, x 1 and x 2.

8. Two points Gauss-Legendre Formula Since we used limits for the previous integration from –1 to 1 and the actual limits are usually from a to b, then we need first to transform both the function and the integration from the x-system to the x d -system f(x) x ab f(x o ) f(x 1 ) xoxo x1x1  1 f(  )

10. Higher-Points Gauss-Legendre Formula Higher points version of Gauss Legender can be developed in the form: Where n is the number of points, c’s and x’s up to the six points are tabulated.

11. Multiple Points Gauss-Legendre Points Weighting factor Function argument Exact for 21.0-0.577350269up to 3 rd 1.0 0.577350269degree 30.5555556-0.774596669up to 5 th 0.88888890.0degree 0.55555560.774596669 40.3478548-0.861136312up to 7 th 0.6521452-0.339981044degree 0.6521452 0.339981044 0.3478548 0.861136312 60.1713245-0.932469514up to 11 th 0.3607616-0.661209386degree 0.4679139-0.238619186 0.4679139 0.238619186 0.3607616 0.661209386 0.1713245 0.932469514

12. Gauss Quadrature - Example Note that f(x) corresponds to the transformed function

13. Improper Integral Improper integrals can be evaluated by making a change of variable that transforms the infinite range to one that is finite, Can be evaluated by Newton-Cotes or Gauss quadrature closed formula Can be evaluated by Extended Midpoint rule

14. Improper Integral Extended Midpoint Rule In the extended midpoint rule, the function is evaluated at points that are h/2 after and before the interval limits. Where h is the interval width and x i/2 is the midpoint of the interval between x i-1 and x i If the integral is divided into m intervals, h=(b-a)/m.

15. Extended Midpoint Rule A common practice is to take m=4 intervals dividing the integral limits. For example, if the integral limits are 0 and ½, then h=(½- 0 )/4 = 1/8, and the integral is evaluated as

16. Improper Integral - Examples.

17. Example (Improper Integral) Use Numerical Integration to evaluate the following integral: I = The integral is first decomposed to the form: The first term is evaluated by means of closed form integration (e.g. Gauss quadrature) while the second is evaluated by extended midpoint rule.

18. Example (Improper Integral) Using Gauss Quadrature with three points, we have:

19. Example (Improper Integral) Where f(1/16)= [1/(1/16) 3 ]*e- 1/(1/16) and so on….