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~ Roots of Equations ~ Open Methods Chapter 6

~ Roots of Equations ~ Open Methods Chapter 6. Credit: Prof. Lale Yurttas, Chemical Eng., Texas A&M University. Open Methods Generally use a single starting value or two starting values that do not need to bracket the root. Open Method (convergent). (divergent).

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~ Roots of Equations ~ Open Methods Chapter 6

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  1. ~ Roots of Equations ~Open MethodsChapter 6 Credit: Prof. Lale Yurttas, Chemical Eng., Texas A&M University

  2. Open Methods • Generally use a singlestartingvalue or two starting values that do not need to bracket the root. Open Method (convergent) (divergent)

  3. Simple Fixed-point Iteration • Rearrange the function so that x is on the left-hand side of the equation: EXAMPLE: • 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.

  4. 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

  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 Ei+1=O(Ei2) (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)

  6. Example:Slow Convergence W:\228\MATLAB\6\newtraph.m & func2

  7. 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

  8. The Secant Method • Requires two initial estimates xo, x1. However, it is not a “bracketing” method. • The Secant Method has the same properties as Newton’s method. Convergence is not guaranteed for all xo, f(x).

  9. Modified Secant Method

  10. Systems of Nonlinear Equations Systems of Nonlinear Equations • Locate the roots of a set of simultaneous nonlinear equations:

  11. First-order Taylor series expansion of a function with more than one variable: • The root of the equation occurs at the value of x1 and x2 where f1(i+1) =0 and f2(i+1) =0 Rearrange to solve for x1(i+1)and x2(i+1) • Since x1(i), x2(i), f1(i), and f2(i) are all known at the ith iteration, this represents a set of two linear equations with two unknowns, x1(i+1) and x2(i+1) • You may use several techniques to solve these equations

  12. General Representation for the solution of Nonlinear Systems of Equations  Jacobian Matrix Solve this set of linear equations at each iteration:

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