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Chapter 8

Chapter 8. Numerical Integration Lecture (I) 1. 1 Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2 nd ed., Ch. 17 , McGraw Hill, 2008. Outline. Introduction What is integration? When do we need numerical integration?

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Chapter 8

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  1. Chapter 8 Numerical Integration Lecture (I)1 1 Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 17, McGraw Hill, 2008. Dr. Jie Zou PHY3320

  2. Outline • Introduction • What is integration? • When do we need numerical integration? • Applications of integration in engineering and science • Newton-cotes formulas • (1) The trapezoidal rule • Error of the Trapezoidal rule • The composite trapezoidal rule • Implementation in MATLAB Dr. Jie Zou PHY3320

  3. Introduction • What is integration? • Mathematically: A definite integration is represented by . • It means: The total value, or summation, of f(x)dx over the range x = a to b. • Graphical representation: For functions lying above the x axis, the integral corresponds to the area under the curve of f(x) between x = a and b. • When do we need numerical integration (also referred to as quadrature)? • Functions that are difficult to or cannot be integrated analytically. • Only a table of discrete data are available. Ref. Fig. 17.1 Graphical representation of the integral Dr. Jie Zou PHY3320

  4. Applications of integration in engineering and science • Examples related to “the integral as the area under a curve”: • Examples related to the analogy between integration and summation: • An example: To determine the mean of a continuous function Ref. Fig. 17.3

  5. Newton-cotes formulas • Basic strategy: Replacing a complicated function or tabulated data with a polynomial that is easy to integrate. • fn(x) = a0 + a1x + … + an-1xn-1+anxn • n: The order of the polynomial. Ref. Fig. 17.4 The approx. of an integral by the area under (a) a straight line and (b) a parabola Ref. Fig. 17.5 The approx. of an integral by the area under three straight-line segments Dr. Jie Zou PHY3320

  6. Newton-cotes formulas: (1) The trapezoidal rule • Basic idea: Replacing the complicated function or tabulated data with a polynomial or a series of polynomials of the first order (linear). • Single and Composite applications • Single application formula: • Composite application formula: Ref. Fig. 17.7 Single application Ref. Fig. 17.9 Composite application

  7. Error of the trapezoidal rule • For single applications, an estimate for the error: • If the function being integrated is linear, Et = 0; otherwise, Et 0. • For composite applications, an estimate for the error: • If the number of segments is doubled, Et is approximately quartered. • Here, Ref. Fig. 17.8 Truncation error for a single application of the trapezoidal rule

  8. Example: Composite application of the trapezoidal rule • Example 17.2 (Ref.): Use the two-segment trapezoidal ruleto estimate the integral of f(x) = 0.2 + 25x – 200x2 + 675x3 – 900x4 + 400x5 from a = 0 to b = 0.8. Also, find the true error Et and the approximate error, Ea. • (1) By hand. • (2) Implement on a computer-write an M-file. x0 = a x1 x2 = b Two segments n = 2; Dr. Jie Zou PHY3320

  9. Results Dr. Jie Zou PHY3320

  10. Implementation of composite trapezoidal rule on a computer • Write an M-file called My_Trapezoidal_Rule.m to do Example 17.2. • A copy of the code will be handed out later. Dr. Jie Zou PHY3320

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