# Functions, Roots, Convergence, Fixed Point Method, Aitken's DELTA-method and Accelerating Convergence

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Consider the function

f(x) = 2sinx + e^-x - 1

on the interval r E [?2,2]. If you plot the function, you will see that it has two roots on this interval

(a) Write down a first order fixed point method for finding one of the two roots.

(b) Will this fixed point method converge for both of the roots (Justify)? If it does not converge to the second root, explain how you might modify your method so that it will converge to the other root.

(c) Write a program to implement your fixed point method for this equation and apply your code to find one of the roots. Use |xn ? xn-1| as your stopping criterion with a tolerance of 10^-8.

(d) Numerically confirm that your method does indeed converge linearly

(e) Write a second program which implements your fixed point method and makes use of Aitken's DELTA-Method (Accelerating Convergence - see Section 2.6 and example 2.14). Use the same tolerance and stopping criterion above.

(f) Numerically confirm that your second method converges faster than 1st order.

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Functions, Roots, Convergence, Fixed Point Method, Aitken's DELTA-method and Accelerating Convergence are investigated. The solution is detailed and well presented.

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