## Description

- In 1685 John Wallis published a book called Algebra in which he described a method devised by Newton for solving equations. In a slightly modi ed form, this method was also published by Joseph Raphson in 1690. This form is the one commonly called Newton’s method or the Newton-Raphson method.

Newton himself discussed the method in 1669 and illustrated it with the equation x^{3} 2x 5 = 0. Wallis used the same example. Find a root of this equation using Newton’s method (either by hand or by coding), thus continuing the tradition that every numerical analysis student should solve this venerable equation.

- Consdier f(x) = x
^{3}2.

- Show that f(x) = 0 has a solution in [1; 2] (use a theorem from class).

- Compute by hand (feel free to use a calculator) the rst three iterations of the following method (pick the appropriate initial value as needed).

- Bisection method.

- Newton’s method

- Secant method

- Linear interpolation

- Write a Matlab code for each of the method mentioned in 1(b). Use your code to nd the approximation of the solution of f(x) = 0. Create an error table for each method.

- Use your code to nd the rst ten positive solutions of equation tan x = x.

p

- Investigate the behavior of the secant method on the function f(x) =jx aj. Explain your nding and discuss the cause.

- Each of the functions

f_{1} |
(x) = sin(x) x 1 | |

f_{2} |
(x) = x(1 | cos(x)) |

f_{3}(x) = e^{x} |
x^{2} + 3x 2 |

have a root in the interval [ 2; 1]. Use all four of the above root nding methods to approximate the roots within absolute error tolerance 10 ^{6} for each function. Limit the number of iterations to 500. For Newton’s Method, use starting value x_{0} = 1; for the Secant Method use x_{0} = 1 and x_{1} = 0:9. Summarize the results of the analysis for each method in table form.

Function Number of Iterations Approximate Root

f_{1}(x)

f_{2}(x)

f_{3}(x)

- Why did the Bisection Method require approximately the same number of iterations to converge to the approximate root for all three test problems?

- Newton’s method should have experienced di culty approximating the root of one of the test functions. Identify which function presented a problem and explain why the di culty occurred.

1

(c) Above, the Bisection Method was used to nd the root of the function f_{1}(x) = sin(x) |
x 1. |

Consider the function g_{1}(x) = (sin(x) x 1)^{2}. Clearly f_{1} and g_{1} have the same root in [ |
2; 1]. |

Could the Bisection Method be used to approximate the root in g_{1}? Why or why not? |

- Let f(x) = sin(x) and g(x) = sin
^{2}(x) and note that both have x = as a root.

- Will Newton’s method converge to root for each of these functions? Why? What rate of convergence do you expect?

- Use Newton’s method to approximate as roots of f and g accurate to machine error. Use Matlab to check the rate of convergence at each step. List your results.

- Plot the number of iterations for Newton’s method in part (b) against the absolute error at each step. Plot the results for f and g in the same graph. What does each curve resemble?

- If the root of f(x) = 0 is a double root (a root with multiplicity 2), then Newton’s method can be accelerated by using

f(x_{n})

^{x}^{n+1} ^{=} ^{x}^{n} ^{2}_{f}0_{(x}_{n}_{)}

_{ }

Create two examples by your own and compare the convergence of this scheme with regular Newton’s method. Explain why this algorithm will work.

- Prove the convergence result of the secant method using a similar argument as in Newton’s method.

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