NON-LINEAR ALGEBRAIC EQUATIONS Lec. 5.1: Nonlinear Equation in Single Variable

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1 NON-LINEAR ALGEBRAIC EQUATIONS Lec. 5.1: Nonlinear Equation in Single Variable Dr. Niket Kaisare Department of Chemical Engineering IIT Madras NPTEL Course: MATLAB Programming for Numerical Computations Week-5 A Simple Example To solve the following equation: 2.0 % + ln % = 0 The solution is the location where the curve intersects the X-axis x + ln(x) It is possible to have multiple solutions (finite) to the problem Computational Techniques: Module-4

2 General Setup Let x be a variable of interest. The objective is to find the value of x which satisfies the following nonlinear equation: * % = 0 Example: Model for a reactor c A0 C kc A 0 A A 2 = τ τ ( 1+ KC ) $!!!! # A f ( C A ) c A C!!!!" 0 General Strategy of Solution Start with initial guess(es) Using a chosen strategy, move in the direction of the solution x + ln(x) Verify if stopping criterion is satisfied Yes Solution!

3 Bisection Method (Single Variable Only) Start with two initial guesses, % + and %, Verify that signs of * % + and * %, are different Repeat the following steps New guess % (./0) is the midpoint of the previous two guesses Calculate * %./0 Replace either % (+) or % (,) with based on sign of * %./0 Related: Computational Techniques Module-4 Part-2: Methods to Solve Nonlinear Equations Bracketing Methods Bisection method Regula Falsi Open Methods Secant method Fixed-point iteration Newton-Raphson MATLAB functions fzero and fsolve

4 End of Lecture 5.1 NON-LINEAR ALGEBRAIC EQUATIONS Lec. 5.2: Using MATLAB Function fzero Dr. Niket Kaisare Department of Chemical Engineering IIT Madras NPTEL Course: MATLAB Programming for Numerical Computations Week-5

5 MATLAB Function fzero Solves nonlinear algebraic equation in single variable Uses bracketing method Usage: xsol = fzero(@(x) funname(x),x0) xsol is the resulting solution funname is a function file that we provide to calculate * % x0=[xl;xh]; is vector of initial guesses Problem to Solve Use fzero to solve the nonlinear equation: 2 % + ln % = 0 Modify bisection method from previous lecture

6 End of Lecture 5.2 NON-LINEAR ALGEBRAIC EQUATIONS Lec. 5.3: Fixed-Point Iteration (Single Variable) Dr. Niket Kaisare Department of Chemical Engineering IIT Madras NPTEL Course: MATLAB Programming for Numerical Computations Week-5

7 Fixed Point Iteration Also known as Method of successive substitution Related: Computational Techniques Module-4 Part-2: Used for solving equations of the type: % = 2 % How are 2 % and * % related è % 2 % 3 4 = 0 Fixed Point Iteration Also known as Method of successive substitution Related: Computational Techniques Module-4 Part-2: % 567 = 2 % 5

8 Example 2 % + ln % = 0 % = 2 + ln (%) % = 9 4:; Example 2 5% 9 4/; = 0 % = /;??? Module-4 Part-3:

9 End of Lecture 5.3 NON-LINEAR ALGEBRAIC EQUATIONS Lec. 5.4: Newton-Raphson (Single Variable) Dr. Niket Kaisare Department of Chemical Engineering IIT Madras NPTEL Course: MATLAB Programming for Numerical Computations Week-5

10 Newton Raphson Popular Method Example: Computational Techniques Module-4 Part-2: % 567 = % 5 * % 5 * > % 5 Example Solve the previous example using Newton-Raphson: * % = 2 % + ln % * > % =

11 Newton Raphson Derivation and Analysis: Computational Techniques Module-4 Part-4: % 565 = % 5 * % 5 * > % 5 Has quadratic rate of convergence ; Fixed point iteration has linear rate of convergence End of Lecture 5.4

12 NON-LINEAR ALGEBRAIC EQUATIONS Lec. 5.5: Using MATLAB function fsolve Dr. Niket Kaisare Department of Chemical Engineering IIT Madras NPTEL Course: MATLAB Programming for Numerical Computations Week-5 Recap and Next Steps Solved nonlinear equation 2 % + ln % = 0 1. Bisection Method 2. MATLAB function fzero 3. Fixed-Point Iteration 4. Newton-Raphson Next: MATLAB function fsolve Multivariate problems using fsolve and Newton-Raphson

13 MATLAB Function fsolve Solves nonlinear algebraic equation Usage: xsol = fsolve(@(x) funname(x),x0) xsol funname x0 is the resulting solution is a function file that we provide to calculate A B initial guess (same dimension as number of variables) Problem to Solve Use fsolve to solve the nonlinear equation: 2 % + ln % = 0

14 Multivariate Example: Lorenz Equation Wikipedia: First example to demonstrate Chaos Observed by Edward Lorenz for atmospheric convection Example problem in this Module: Find steady-state solution: % C = 0 2% %D C = 0 %C 3D = 0 End of Lecture 5.5

15 NON-LINEAR ALGEBRAIC EQUATIONS Lec. 5.6: Multivariable Newton Raphson Dr. Niket Kaisare Department of Chemical Engineering IIT Madras NPTEL Course: MATLAB Programming for Numerical Computations Week-5 Multivariate Example: Lorenz Equation Steady-state Lorenz Equation: % C = 0 2% %D C = 0 %C 3D = 0 B 567 = B 5 G 5 :7 A 5 Compute the Jacobian: F = D 1 % C % 3

16 End of Lecture 5.6