Numerical Analysis Solution of Algebraic Equation (non-linear equation) 1- Trial and Error. 2- Fixed point

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1 Numerical Analysis Solution of Algebraic Equation (non-linear equation) 1- Trial and Error In this method we assume initial value of x, and substitute in the equation. Then modify x and continue till we get the solution. Ex:- solve the equation 2x 2 x 1(by using trial and error) 2x 2 x 1 0 Let x 1 F(0) -1 Let x 0.5 F(0.5) -1 Let x 1 F(1) 0 (first root) Let x F(-0.5) 0 (second root) 2- Fixed point In this method, we try to put x g(x) than assuming an initial value for x and evaluating the next value of x, as follows:- X n+1 g(x n ) And continue in this iteration till we get X n+1 g(x n ) Ex:- resolve the last example using fixed point method 2x 2 x 1 x 2x 2 1 Let x o 0 n x n X n (first root) Let x o 2 n x n X n Divergence Let x o -2 n x n X n Divergence Let x Let x o 0

2 n x n x n (second root) 3- Newton-Raphson method x 0 x 1 Generally x n+1 x n Ex:- resolve the last example using Newton Raphson method F(x) 2x 2 x 1 0 x) 4x-1 x n+1 x n Let x o 0 n x n X n (first root) Let x o 1 n x n X n (second root) Curve Fitting 1- Leaner Fitting y(x) ax+b E + + +

3 Multiply equ.2 by and subtract it from equ.1 Ex:- Find the best line passing though the following points (0,1), (1.3), (2,4), (3,8), (4,10) Sol: i Average Polynomial Fitting y(x) m : degree of polynomial For m2 y(x) 1

4 Generally..2.3 Ex:- Find the best parabola representing the following points (0,1), (1.3), (2,4), (3,11), (4,15) Sol: y i average Then find by using Gauss elimination method or by using Gauss-Jordon forward b 3- Exponential Fitting Taking Ex:- Find the best exponential function passing though the following points (1,3), (2,8), (3,27), (4,80) Sol:

5 i Y Average Power Fitting 1- Taking 2- Perform linear fitting ; Ex:- Find the best power function passing though the following points (1,3), (2,8), (3,27), (4,80) Sol: i Y X Average

6 Newton Interpolation Formula a- Forward Interpolation Formula b- Backward Interpolation Form Where r and h Ex:- give the following data (x n, f n ) find f(0.25), f(1.3) and f(-0.35) x F Using linear, quadratic interpolation and 4-trem Newton interpolation formula Sol:- n X F(0.25) h 0.2 let 0.2, 3.43 and 2.72 linear f (0.25) quadratic f (0.25) trem Newton interpolation formula r 0.25 F(0.25) F(1.3) h 0.2 r 2.5

7 let 0.8, 25.7 and 29.9 linear f (1.3) quadratic let 0.6; ; ; h 0.2 r 3.5 f (1.3) trem Newton interpolation formula r 4.5 F(1.3) F(-0.35) h 0.2 let 0.0, 1.2 and 2.23 linear f (-0.35) quadratic f ( 0.35) trem Newton interpolation formula r f ( 0.35) Note:- Newton interpolation formula can be used only when points are equally spaced ( h constant) Lagrange Interpolation Formula where j k for example when N2 f(x)

8 Note:- Lagrange interpolation formula can be used for equally or non-equally spaced points. Ex:-for the same data in the last example find f(0.25) using 3-trem Lagrange interpolation formula f(0.25) Hw. Complete the example and find f(1.3) and f(-0.35) Finite Difference, Interpolation and Extrapolation 1- Finite Difference a- Forward Finite Difference. first forward difference ) ). second forward difference. third forward difference Ex:- construct the forward finite difference for the following data x F Sol:- n x b- Backward Finite Difference. first backward difference ) ). second backward difference. third forward difference Ex:- construct the backward finite difference for the same data in the last example

9 Sol:- x F n x c- Central Finite Difference (first central difference) (second central difference) Ex:- construct the central finite difference for the same data in the last example Sol:- x F n x Interpolation and Extrapolation 1- Linear interpolation F 1 F F o x o x x 1 let h (step) or let r

10 2- Quadratic interpolation Solution of Ordinary Differential of Equation 1- Euler Methods Ex:- solve the equation for from 0 to 1 given that Let n Modified Euler Method Ex:- re-solve last example using modify Euler methods Let

11 n Numerical Analysis Numerical Integration and Differential Integration 1- Trapezoidal Method N number of division (strip)

12 Ex:- evaluated the following integral using trapezoidal method Sol:- x F I Simpson's Rule N number of division (must be even) Ex:- :- evaluated the last integral using Simpson's rule Sol:- x F

13 I Gauss Quadrature method m number of term Ex:- evaluated the last integral using 4- term Gauss Quadrature methods Sol : k I Differentiation F 1 F F o

14 r Solution of Linear Simultaneous Equation b b b n Where m number of variables n number of equations If n m; then there is one exact solution If n > m; then there is no solution If n < m; then there are many solution 1- Iteration (Gauss-Siedel) Method Extract the value of each variable from one of equation and substituted the obtained values in the next iteration process till we obtain convergence i j To guarantee convergence Ex:- solve the following set equations (1) (2) (3) + 2 (1) (2)

15 + 5..(3) Where is very small j 1,2,,m i column of iteration Note: Gauss Elimination Methods(Forward Elimination & Backward Substitution) For example Step1:-eliminate coefficient of from equations (2) and (3), multiplying equation (1) by and add it to equation (2) then multiplying equation (1) by and add it to equation (3). Step2:- eliminate by multiplying equation (2) by and add it to equation (3). Step 3:- back substitution

16 Ex:- Resolve the last example using Gauss Elimination Methods + 2 (1) (2) + 5..(3) Step1:-eliminate coeff. of from equ. (2) and (3), multiplying equ. (1) by and add it to equ. (2) then multiplying equ. (1) by and add it to equ. (3). + 2 (1) (2) (3) Step2:-eliminate coeff. of from equ. (3), multiplying equ. (2) by and add it to equ. (3). + 2 (1) (2) (3) Back substitution Gauss-Jordon Methods (Forward &Backward Elimination) For example Step1 and step2 same as in Gauss Elimination method

17 Step3:- eliminate coefficient of from equations (1) and (2), multiplying equation (3) by and add it to equation (1) then multiplying equation (3) by and add it to equation (2). Step4:- eliminate coefficient of and add it to equation (1) from equations (1), multiplying equation (2) by Ex:- Ex:- Resolve the last example using Gauss- Jordon Methods + 2 (1) (2) + 5..(3) Step1:-eliminate coeff. of from equ. (2) and (3), multiplying equ. (1) by and add it to equ. (2) then multiplying equ. (1) by and add it to equ. (3). + 2 (1) (2) (3) Step2:-eliminate coeff. of from equ. (3), multiplying equ. (2) by and add it to equ. (3). + 2 (1) (2) (3) Step3 Multiplying equ. (3) by and add it to equ. (2) and add it to equ. (1), and then multiplying equ. (3) by

18 (1) (2) (3) Step4:- Multiplying equ. (2) by and add it to equ. (1) (1) (2) (3) So