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NUMERICAL ANALYSIS

NUMERICAL ANALYSIS. Course for 2nd year Architecture Engineering. and. Course for 3rd year Electrical Engineering. By:. Dr Rania Bahgat Mohamed Amer. Department of Engineering Mathematics and Physics, Faculty of Engineering Zagazig University 2009-2010. CONTENTS OF THE COURSE:.

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NUMERICAL ANALYSIS

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  1. NUMERICAL ANALYSIS Course for 2nd year Architecture Engineering and Course for 3rd year Electrical Engineering

  2. By: Dr\ Rania Bahgat Mohamed Amer Department of Engineering Mathematics and Physics, Faculty of Engineering Zagazig University 2009-2010

  3. CONTENTS OF THE COURSE: Numerical solution of linear equations. Numerical solution of non linear equations. Numerical integrations. Curve fitting. Numerical solution of ordinary differential equations. Numerical solution of partial differential equations.

  4. Next The Main Menu اPrevious Curve Fitting 1- The definitions 2- The Least Squares method

  5. Next The Main Menu اPrevious 1- The definitions Curve fitting A procedure in which the basic problem is to pass a curve through a set of points, representing experimental data, in such a way that the curve shows as well as possible the relationship between the two quantities plotted. i.e. It is a mathematical representation on an xy Cartesian graphing plane which describes the relationship between variables x and y.

  6. Next The Main Menu اPrevious 2- The Least Squares method It is a mathematical procedure for finding the best fitting curve to a given set of points by minimizing the sum of the squares of the offsets of the points from the curve. Given a set of data points (xi, f(xi)), find a curve that bestcaptures the general trend Where g(x) is approximation function

  7. Next The Main Menu اPrevious The steps for solution: Let Error: Error is a function of a0, a1 . For E to have extreme value: 2 equations of two unknowns, solve to get a0, a1. We can put in the matrix form:

  8. Next The Main Menu اPrevious In general, fit polynomial of degree n Error: Error is a function of a0, a1, a2,…, an. Require Solve this equation to get unknowns a0,a1,…..,an. The general matrix form:

  9. Next The Main Menu اPrevious Example (1) Using the method of least squares to fit a straight line to the given points tabulated below: Solution Let The equations of the least squares method are: We get: The required equation is

  10. Next The Main Menu اPrevious Example (2) Using the method of least squares to fit a second degree polynomial to the following points (0, 3), (1, 1), (2, 0), (4, 1), (6, 4). Solution Let The equations of the least squares method are:

  11. Next The Main Menu اPrevious We get: The required equation is

  12. Next The Main Menu اPrevious Example (3) Fit the curve y(x)= a + bx2 + c ln x, to the following points (1, 28.2), (2, 18.9), (3, 11.1), (4, 1.7), (5, -13.8). Solution Let ln x = t Let The equations of the least squares method are:

  13. Next The Main Menu اPrevious We get: The required equation is

  14. Next The Main Menu اPrevious Example (4) Fit the curve y(x)= a + bx2, to the following points (1, 4), (2, 6), (3, 8), (4, 10), (5, 12). Solution Let X = x2. Let Where a = a0, b = a1. The equations of the least squares method are:

  15. Next The Main Menu اPrevious We get: The required equation is

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