today’s class romberg integration gauss quadrature numerical methods lecture 13 prof. jinbo bi...
TRANSCRIPT
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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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 MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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)
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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Richardson’s Extrapolation
• Special case where you always halve the interval - i.e. h2=h1/2
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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Romberg Integration• Accelerated Trapezoid Rule
+1
k: level of integration
j: level of accurancy
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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Romberg Integration• Accelerated Trapezoid Rule
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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Gauss Quadratures
• Method of Undetermined Coefficients
• Trapezoidal method should be exact for
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Numerical MethodsLecture 13
Prof. Jinbo BiCSE, UConn
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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
Prof. Jinbo BiCSE, UConn
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