Today’s class Romberg integration Gauss quadrature Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Numerical Integration Multiple application of the Newton-Cotes Formulas will improve the accuracy of the approximation As you increase the number of segments, you reduce the error However, if you increase the number of segments too much, round-off errors begin to dominate and the error will start to increase Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Numerical Integration a=0, b = 0.8 Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Romberg integration Richardson’s extrapolation Perform a numerical algorithm using multiple values of a parameter h and then extrapolate that result to the limit h=0 With numerical integration use two estimates of the integral to come up with a third more accurate approximation Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Richardson’s Extrapolation The estimated integral is a function of subinterval size. With smaller h, usually I(h), the integral estimate, is more accurate, which means the error caused by choosing this interval size is smaller. If we have two integral esitmates, each one is obtained by using a different segment size, or equivalently different number of segments. We can use estimate the error for each integral estimate using our early analysis. If we are able to estimate this error, we can use this error to update the estimate so the new one is closer to the true integral. However, we cannot estimate this one because we don’t know this item related to second order derivative. So with two e Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Richardson’s Extrapolation We’ve used the error calculation to come up with a new estimate It can also be shown that this new estimate is O(h4), whereas trapezoidal rule is only O(h2) In 1978, two researchers show that this new esitmate is much more accuract with an error term in the forth order of the subinterval size. Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Richardson’s Extrapolation Special case where you always halve the interval - i.e. h2=h1/2 Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Richardson’s Extrapolation Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Richardson’s Extrapolation Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Romberg Integration Accelerated Trapezoid Rule k: level of integration +1 k: level of integration j: level of accurancy Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Romberg Integration Accelerated Trapezoid Rule Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Romberg Integration Termination Criteria Trapezoidal method 9 iterations before hitting precision limit (n=256) 511 function evaluations Romberg integration 3 iterations before hitting precision limit 15 function evaluations Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Romberg Integration very good convergence properties less susceptible to round-off error than Trapezoidal or Simpson’s rule extra levels of extrapolation require very little computational work Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures Newton-Cotes Formulas Gauss Quadratures use evenly-spaced functional values Gauss Quadratures select functional values at non-uniformly distributed points to achieve higher accuracy Gauss-Legendre formulas Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures Trapezoidal Method Numerical Methods Prof. Jinbo Bi Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures Find interior points so that the trapezoidal area outside the curve is equal to the area below the curve and above the trapezoid Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures Method of Undetermined Coefficients Trapezoidal method should be exact for Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures Two-Point Gauss Legendre Derivation Find a solution to following equation over range [-1:1] Using similar reasoning as before, solve for the four unknowns using the following equations Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures This formula gives an integral estimate that is third-order accurate To be usable, the bounds of the definite integral have to be from -1 to 1 It is easy to convert by using a new variable xd Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures 0.4dxd Numerical Methods Prof. Jinbo Bi Lecture 13 CSE, UConn

Gauss Quadratures Equivalent accuracy to Simpson’s 1/3 rule (O(n3)) Fewer function evaluations Can be extended to higher-point versions Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures Numerical Methods Prof. Jinbo Bi Lecture 13 CSE, UConn

Gauss Quadratures on [-1, 1] Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures on [-1, 1] Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Example: Gauss Quadratures Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Gauss Quadratures High accuracy with few function evaluations Error is proportional to the (2n+2)th derivative Function must be known - not appropriate for tabular data Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Improper integrals How do you handle integrals where one of the bounds of the integral is ±∞? Do a translation of the bounds into a proper integral Works as long as a is -∞ and b is negative or a is positive and b is ∞. The function f(x) must also asymptotically approach zero at least as fast as 1/x2 Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Improper integrals Example: Normal distribution Split the integral at a point where the function starts to approach zero faster than 1/x2 Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Summary Integration Techniques Gaussian Quadrature Improper integrals Trapezoidal Rule : Linear Simpson’s 1/3-Rule : Quadratic Simpson’s 3/8-Rule : Cubic Improvement techniques Multiple application or composite methods Romberg integration Gaussian Quadrature Improper integrals Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn

Next class Numerical Differentiation Read Chapter 23 Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn