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Numerical Computation Lecture 4: Root Finding Methods - II United International College

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Review During our Last Class we covered: – Algorithm Convergence – Root Finding: Bisection Method, Newton’s Method

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Today We will cover: – Root Finding: More on Newton’s Method, Secant Method – Section 4.3 in Pav – Sections 4.4 and 4.6 in Moler

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Uses of Newton’s Method Newton's Method can be used to program more complex functions using only simple functions. Problem: A computer only has addition, subtraction, multiplication, division, and we need to compute some complex function g(x). Solution: Use Newton's Method to solve some equation equivalent to g(z) - x = 0, where z is the input to the subroutine. Then, x is the numerical value for g(z).

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Uses of Newton’s Method Example: Find Solution: Use Newton's Method for f(x) = z - x 2. – Iterations : – Simplify : Practice: Write a Matlab M-function program to find the sqrt(z) to a specified accuracy eps.

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Secant Method Newton’s Method is fast (quadratic convergence). However, Newton’s Method requires knowledge of the derivative of f(x). This is hard to do programmatically. Needed: An algorithm that is (hopefully) as fast as Newton’s Method, but does not require f’(x). Solution: Secant Method

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Secant Method Recall: Newton’s method Problem is f’(x k ). We know that the derivative is a limit of the difference quotient We can approximate this by using x=x k-1, as we assume that the iterates {x k } are close to one another.

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Secant Method In other words, instead of following the tangent line to get the next iterate (Newton’s Method), we follow the secant line.

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Secant Method The secant line has equation: We want to find where it crosses the x-axis, i.e. where y = 0. So,

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Secant Method Secant Iterates: Note: We will have to supply two initial guesses: x 0 and x 1.

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Secant Method Matlab Implementation: function v = secant( f, x1, x0, eps) %Secant method % Assumes f is differentiable k = 0; xprev = x0; xnext = x1; while abs(xnext - xprev) > eps*abs(xnext) xtmp = xnext; xnext = xnext - ((xnext-xprev)/(f(xnext)-f(xprev)))*f(xnext); xprev = xtmp; k = k + 1; v = [xnext k]; end

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Secant Method Practice: Run this program to find the sqrt(2) to an accuracy of 0.01. How does the speed of the Secant Method compare to Newton’s Method?

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Secant Method Convergence: Pav (Section 4.3) shows that the error in the Secant method has the following order of growth: where This called super-linear convergence. It is slower than quadratic, but faster than linear. So, somewhere between the speed of bisection and Newton.

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Inverse Quadratic Interpolation Method (IQI) Secant Method uses a line to approximate f(x) and then finds where that line crosses the x- axis. Idea: Use a parabola (quadratic) to approximate f(x) and find where parabola crosses x-axis. Problem: Parabola might have complex roots! It might not cross the x-axis.

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Inverse Quadratic Interpolation Method (IQI) Solution: Inverse approximation (interpolation) – we think of x k as a function of y k =f(x k ) We get a quadratic p(y) and the next approximation x k+1 is just p(0).

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Inverse Quadratic Interpolation Method (IQI) f(x)

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Inverse Quadratic Interpolation Method (IQI) Convergence: It can be shown (Michael T Heath’s book Scientific Computing) that where α ≈ 1.893. So this method is almost as fast as Newton’s method, and does not require derivatives.

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Matlab Root-finding fzero algorithm: Matlab has a function called fzero that will find the root of a function, starting from an initial guess. It uses a combination of the bisection, secant, and inverse quadratic methods. Idea: We use bisection and secant to get a good approximation, then use IQI to rapidly close in on the root. Algorithm is listed as zeroin in section 4.6 of Moler text.

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Practice (If time) Try Exercise 4.9. in Moler Find the first ten positive values of x for which x = tan x. How can we do this in one Matlab M-function file?

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