Today’s class Romberg integration Gauss quadrature Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn 1

Today’s class

• Romberg integration• Gauss quadrature

Numerical MethodsLecture 13

Prof. Jinbo BiCSE, UConn

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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



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Numerical Integration



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a=0, b = 0.8

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



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Richardson’s Extrapolation



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• 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)



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• Special case where you always halve the interval - i.e. h2=h1/2



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Romberg Integration• Accelerated Trapezoid Rule

+1

k: level of integration

j: level of accurancy



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Romberg Integration• Accelerated Trapezoid Rule



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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



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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



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Gauss Quadratures

• Newton-Cotes Formulas• use evenly-spaced functional values

• Gauss Quadratures• select functional values at non-uniformly

distributed points to achieve higher accuracy• Gauss-Legendre formulas



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Gauss Quadratures

• Trapezoidal Method



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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



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Gauss Quadratures

• Method of Undetermined Coefficients

• Trapezoidal method should be exact for



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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



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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



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Gauss Quadratures

0.4dxd



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Gauss Quadratures

• Equivalent accuracy to Simpson’s 1/3 rule (O(n3))

• Fewer function evaluations• Can be extended to higher-point versions



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Gauss Quadratures

Numerical Methods Lecture 13


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Gauss Quadratures on [-1, 1]



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Gauss Quadratures on [-1, 1]



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Example: Gauss Quadratures



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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



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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



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Improper integrals

• Example: Normal distribution• Split the integral at a point where the function

starts to approach zero faster than 1/x2



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Summary• Integration Techniques

– 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 integralsNumerical MethodsLecture 13


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Next class

• Numerical Differentiation• Read Chapter 23



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Documents

Today’s class Romberg integration Gauss quadrature Numerical Methods Lecture 13 Prof. Jinbo Bi CSE, UConn 1