Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 ~ Roots of Equations ~ Open Methods Chapter 6 Credit: Prof. Lale Yurttas, Chemical Eng., Texas A&M University

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2 Open Methods Generally use a single starting value or two starting values that do not need to bracket the root. Open Method (convergent) (divergent)

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3 Simple Fixed-point Iteration Bracketing methods are “convergent” if you have a bracket to start with Fixed-point methods may sometimes “converge”, depending on the starting point (initial guess) and how the function behaves. Rearrange the function so that x is on the left-hand side of the equation: EXAMPLE:

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Newton-Raphson Method Most widely used formula for locating roots. Can be derived using Taylor series or the geometric interpretation of the slope in the figure

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 5 Newton-Raphson is a convenient method if f’(x) (the derivative) can be evaluated analytically Rate of convergence is quadratic, i.e. the error is roughly proportional to the square of the previous error E i+1 =O(E i 2 ) (proof is given in the Text) But: it does not always converge  There is no convergence criterion Sometimes, it may converge very very slowly (see next slide)

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 6 Example : Slow Convergence Iteration x ∞ W:\228\MATLAB\6\newtraph.m & func2

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The Secant Method If derivative f’(x) can not be computed analytically then we need to compute it numerically (backward finite divided difference method) RESULT: N-R  becomes  SECANT METHOD

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 8 Requires two initial estimates x o, x 1. However, it is not a “bracketing” method. The Secant Method has the same properties as Newton’s method. Convergence is not guaranteed for all x o, f(x). The Secant Method

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 9 Modified Secant Method

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10 Systems of Nonlinear Equations Locate the roots of a set of simultaneous nonlinear equations: Systems of Nonlinear Equations

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. First-order Taylor series expansion of a function with more than one variable: The root of the equation occurs at the value of x 1 and x 2 where f 1(i+1) =0 and f 2(i+1) =0 Rearrange to solve for x 1(i+1) and x 2(i+1) Since x 1(i), x 2(i), f 1(i), and f 2(i) are all known at the i th iteration, this represents a set of two linear equations with two unknowns, x 1(i+1) and x 2(i+1) You may use several techniques to solve these equations

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 12 General Representation for the solution of Nonlinear Systems of Equations Solve this set of linear equations at each iteration:  Jacobian Matrix