1 numerical integration dr. asaf varol [email protected]
TRANSCRIPT
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Numerical Integration
• Numerical integration is a primary tool used for definite integrals that cannot be solved analytically. A numerical integration rule has the form
• we investigate several basic quadrature formulas that use function values at equally spaced points; these methods are known as Newton-Cotes formulas. There are two types of Newton –Cotes formulas, depending on whether or not the function values at the ends of the interval of integration are used. The trapezoid and Simpson rules are examples of “closed” formulas, in which the endpoint values are used. The midpoint rule is the simplest example of an “open” formula, in which the endpoints are not used [2].
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Newton-Cotes Closed FormulasTrapezoid Rule
• One of the simplest ways to approximate the area under a curve is to approximate the curve by a straight line. The trapezoid rule approximates the curve by the straight line that passes through the points [a, f(a) and b, f(b)], the two ends of the interval of interest. We have x0=a, x1=b, and h=b-a, and then
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Example
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Matlab Program
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Diagram
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Newton-Cotes Closed FormulasSimpson’s Rule
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Newton-Cotes Closed FormulasSimpson’s Rule (Cont’d)
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Matlab Programdx
x
1
021
1
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Diagram dxx
1
021
1
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Newton-Cotes Closed FormulasMidpoint Rule
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Figure
• Figure given on the right side compares the actual value of the area with that found by using the midpoint rule. The area given by the integral S (hatched) and the approximation using the midpoint rule (shaded) [2].
dxx
xS
0
)sin(
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Matlab Program dxx
xS
0
)sin(
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Diagram dxx
xS
0
)sin(
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Gaussian quadrature
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Gaussian quadrature
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Example 2xe
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End of Chapter 5
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References• Celik, Ismail, B., “Introductory Numerical Methods for Engineering Applications”,
Ararat Books & Publishing, LCC., Morgantown, 2001 • Fausett, Laurene, V. “Numerical Methods, Algorithms and Applications”, Prentice
Hall, 2003 by Pearson Education, Inc., Upper Saddle River, NJ 07458
• Rao, Singiresu, S., “Applied Numerical Methods for Engineers and Scientists, 2002 Prentice Hall, Upper Saddle River, NJ 07458
• Mathews, John, H.; Fink, Kurtis, D., “Numerical Methods Using MATLAB” Fourth Edition, 2004 Prentice Hall, Upper Saddle River, NJ 07458
• Varol, A., “Sayisal Analiz (Numerical Analysis), in Turkish, Course notes, Firat University, 2001