Today’s class Numerical Integration Newton-Cotes Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn 1

Today’s class

• Numerical Integration• Newton-Cotes

Numerical MethodsLecture 12

Prof. Jinbo BiCSE, UConn

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Discuss Mid-term Exam 1



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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


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)



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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)



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351653.0135

xex


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)



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05.045.51 32 xxx


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)



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2028

3473

3862

321

321

321

xxx

xxx

xxx

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



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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



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Graphical Techniques

• Grid approximation



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Graphical Techniques

• Strip approximation



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• 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



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• 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



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Newton-Cotes Formulas



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Newton-Cotes Formulas



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Trapezoidal Rule



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Trapezoidal Rule



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Trapezoidal Rule



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Trapezoidal Rule

• Evaluate integral• Analytical solution

• Trapezoidal solution



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Trapezoidal Rule



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Trapezoidal Rule

• Trapezoidal solution• To increase accuracy, shorten up the interval



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Trapezoidal Rule



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Trapezoidal Rule

• If you split the interval into n sub-intervals, each sub-interval has a width of h = (b – a)/n



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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



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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



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Simpson’s Rules

• Approximate the function by a parabola



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Simpson’s Rules



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Simpson’s Rules



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Simpson’s 1/3 Rule

• Error

• Multiple Application

• n must be even



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Simpson’s 3/8-Rule

• Approximate with a cubic polynomial



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• Error

• Multiple Application

• n must be a multiple of 3



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• 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



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Example



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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



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Trapezoid Rule with Unequal Segments



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Trapezoidal Rule



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Simpson’s Rule with Unequal Segments



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Next class

• Numerical Integration

• HW5 due Tuesday Oct 21



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Documents

Today’s class Numerical Integration Newton-Cotes Numerical Methods Lecture 12 Prof. Jinbo Bi CSE, UConn 1