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Numerical Methods Algorithm and C Program
The document outlines algorithms and C programs for three numerical methods: the bisection method, power method, and Lagrange's interpolation method. It provides step-by-step algorithms for each method, followed by corresponding C program implementations to find roots of nonlinear equations, largest eigenvalues and eigenvectors, and to perform interpolation, respectively. The content is aimed at students and practitioners in the field of numerical analysis.
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Numerical Methods Algorithm and C Program
- 1. Numerical Methods Algorithms &C Program Dr. Nirav Vyas – Associate Professor, Dept. of Science & Humanities, FoET Atmiya University, INDIA
- 2. 1. Bisection Method 2.Power Method 3. Lagrange’s Interpolation Method
- 3. Bisection Method Algorithm &C Program
- 4. Bisection Method -Algorithm 1. Start 2. Define function f(x) 3. Choose initial guesses x0 and x1 such that f(x0)f(x1) < 0 4. Choose pre-specified tolerable error e 5. Calculate new approximated root as x2 = (x0 + x1)/2 6. Calculate f(x0)f(x2) a) if f(x0)f(x2) < 0 then x0 = x0 and x1 = x2 b) if f(x0)f(x2) > 0 then x0 = x2 and x1 = x1 c) if f(x0)f(x2) = 0 then goto (8) 7. if |f(x2)| > e then goto (5) otherwise goto (8) 8. Display x2 as root. 9. Stop
- 5. Bisection Method –C Program ▪ Program: Finding real roots of nonlinear equation using Bisection Method ▪ Header files ▪ User define function for nonlinear equation: x^3 – 9x + 1 = 0
- 6. ▪ Declaration ofvariables ▪ Input from the user
- 7. ▪ Checking ofthe input values ▪ Input from the user for tolerable error ▪ Setting the output screen format
- 8. ▪ Start thelooping structure until fc gets less then tolerable error ▪ Inside loop first find c value using bisection method & function of c ▪ Print on screen all values of step, a, b, c and fc
- 9. ▪ Now tocheck whether c will replace a or b ▪ Increase the step size by 1 ▪ Print the root of the equation
- 10. Power Method Algorithm &C Program
- 11. Power Method -Algorithm ▪ 1. Start ▪ 2. Read Order of Matrix (n) and Tolerable Error (e) ▪ 3. Read Matrix A of Size n x n ▪ 4. Read Initial Guess Vector X of Size n x 1 ▪ 5. Initialize: Lambda_Old = 1 ▪ 6. Multiply: X_NEW = A * X ▪ 7. Replace X by X_NEW ▪ 8. Find Largest Element (Lamda_New) by Magnitude from X_NEW ▪ 9. Normalize or Divide X by Lamda_New ▪ 10. Display Lamda_New and X ▪ 11. If |Lambda_Old - Lamda_New| > e then set Lambda_Old = Lamda_New and goto step (6) otherwise goto step (12) ▪ 12. Stop
- 12. Power Method –C Program ▪ Program: Write a C program to find Larges Eigen Value and Eigen Vector using Power Method. ▪ Header files ▪ Universal Constant (maximum size of matrix) ▪ Declaration of variables
- 13. ▪ Title andinput from the user ▪ Reading Matrix from user (order n x n)
- 14. ▪ Reading initialGuess Vector from user (order n x 1) ▪ Initializing Lambda_old ▪ Multiplication
- 15. ▪ Replacing oldeigen vector with new ▪ Finding largest numerical value from the vector ▪ Normalization – dividing each element of vector with the largest value
- 16. ▪ Display step,eigen value and eigen vector ▪ Checking the accuracy (either repeat the process or stop )
- 17.
- 18. Lagrange’s Interpolation Method- Algorithm ▪ 1. Start ▪ 2. Read number of data (n) ( how many pairs of (x,y) ) ▪ 3. Read data and for ▪ 4. Read the value of whose corresponding value of is to be determined. ▪ 5. Initialize: = 0 ▪ 6. Formula for Lagrange’s Interpolation: y= where ▪ 7. Display the value of as interpolated value. ▪ 8. Stop
- 19. Lagrange’s Interpolation Method– C Program ▪ Program: Write a C program to interpolate value of y for the given value of x from the set of data given by the user using Lagrange’s interpolation formula. ▪ Header files ▪ Declaration of variables
- 20. ▪ Read numberof data (n) and read data and for ▪ Read the value of whose corresponding value of is to be determined
- 21. ▪ Implementing Lagrange’sformula ▪ Display interpolated value y