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Today’s class
• Numerical Integration• Newton-Cotes
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
1
Discuss Mid-term Exam 1
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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0 10 20 30 40 50 60 70 80 90 1000
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Average = 74.5, Standard dev = 17.6
Max = 98 Min = 35
Histogram of scores
Discuss Mid-term Exam 1
1 (30) Use Taylor series to approximate at x=0.5 until t 1%, using a base point x=0. (Use 4 significant digits with rounding, and the true value is 0.4055)
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
3
Discuss Mid-term Exam 1
2 (30) Use the false-position method with initial guesses of 50 and 70 to determine x to a level of a 0.5%. (Use 4 significant digits with rounding)
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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351653.0135
xex
Discuss Mid-term Exam 1
3 (20) Use the Newton-Raphson method with an initial guess of 6 to determine a root of the following function to a level of a 0.01% (use 4 significant digits with rounding)
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
5
05.045.51 32 xxx
Discuss Mid-term Exam 1
4 (20) Use the Gauss-Seidel method to solve the following linear equation system to a tolerance of s = 5% with a starting point (0, 0, 0).(Use 4 significant digits with rounding)
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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2028
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3862
321
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321
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Numerical Integration
• Numerical technique to solve definite integrals
• Find the area under the curve f(x) from a to b
• Another way to put it is that we need to solve the following differential equation
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
7
Numerical Integration• Where do we use numerical techniques?
• When you can’t integrate directly• When you have a sampling of points
representing f(x) as with the experiments results
• Graphical techniques• Grid approximation• Strip approximation
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Graphical Techniques
• Grid approximation
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Graphical Techniques
• Strip approximation
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Numerical Integration
• Numerical techniques• Treat integral as a summation
• Basic idea is to convert a continuous function into discrete function
• The smaller the interval, the more accurate the solution but also at more computational expense
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
11
Numerical Integration
• Newton-Cotes formulas• Approximate function with a series of
polynomials that are easy to integrate
• Zero-order approximation is equivalent to strip approximation
• First order approximation is equivalent to trapezoidal approximation
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Newton-Cotes Formulas
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Newton-Cotes Formulas
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
• Evaluate integral• Analytical solution
• Trapezoidal solution
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
• Trapezoidal solution• To increase accuracy, shorten up the interval
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
21
Trapezoidal Rule
• If you split the interval into n sub-intervals, each sub-interval has a width of h = (b – a)/n
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
• More accurate as you increase n• Double n and reduce the error by a factor of
four
• However, if n is too large, you will start encountering round-off error and the integral solution can diverge
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
23
Simpson’s Rules
• Second and third-order polynomial forms of the Newton-Cotes formulas
• Simpson’s 1/3 Rule• Use three points on the curve to form a second
order polynomial
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s Rules
• Approximate the function by a parabola
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s Rules
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s Rules
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s 1/3 Rule
• Error
• Multiple Application
• n must be even
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s 3/8-Rule
• Approximate with a cubic polynomial
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s 3/8-Rule
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s 3/8-Rule
• Error
• Multiple Application
• n must be a multiple of 3
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s 3/8-Rule
• Both the cubic and quadratic approximations are of the same order
• The cubic approximation is slightly more accurate then the quadratic approximation
• Usually not worth the extra work required• Only use the 3/8 rule if you need odd
number of segments• Can combine with 1/3 rule on some segments
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Example
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Integration with unequal segments• We have until now assumed that each
segment is equal• If we are integrating based on
experimental or tabular data, this assumption may not be true
• Newton-Cotes formulas can easily be adapted to accommodate unequal segments
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoid Rule with Unequal Segments
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Trapezoidal Rule
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Simpson’s Rule with Unequal Segments
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
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Next class
• Numerical Integration
• HW5 due Tuesday Oct 21
Numerical MethodsLecture 12
Prof. Jinbo BiCSE, UConn
38