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# Homework 7

Please answer the following questions in complete sentences in a typed manuscript and submit the solution on blackboard by on April 18th at noon.

## Problem 0: Homework checklist

• Please identify anyone, whether or not they are in the class, with whom you discussed your homework. This problem is worth 1 point, but on a multiplicative scale.

• Make sure you have included your source-code and prepared your solution according to the most recent Piazza note on homework submissions.

## Problem 1: Gautschi Exercise 4.19

Consider the equation

1. Show graphically, as simply as possible, that in the interval $[(1/2)\pi, \pi]$ there is exactly one root $\alpha$.

2. Does Newton's method converge to a root $\alpha \in [(1/2)\pi, \pi]$ if the initial approximation is taken to be $x_0 = \pi$? Justify your answer.

## Problem 2: Gautschi Exercise 4.25

Consider Newton's method for computing $\alpha = \sqrt{a}$. Let $d_k = x_{k+1} - x_{k}$.

1. Show that

2. Use (1) to show that Discuss the significance of this result with regard to the overall behavior of Newton's iteration.

## Problem 3: Nonlinear systems

Express the Newton iteration for solving these systems of equations and show (analytically or experimentally) what happens if you start from $x_1 = 0, x_2 =0$.

1. $x_1^2 + x_1 x_2^3 = 9$, $3x_1^2 x_2 - x_2^3 = 4$

2. $x_1 + x_2 - 2 x_1 x_2 = 0$, $x_1^2 + x_2^2 - 2x_1 + 2 x_2 = -1$

## Problem 4: Fixed point methods

Can either of the two systems above be solved via the fixed-point method? Two variations on the fixed point method to consider are:

• if $F(\vx) = 0$, then $\phi(\vx) = \vx-F(\vx) = \vx$, so we can look for a fixed point of $\phi$, but the same analysis holds with $\alpha F(\vx) = 0$, which gives, $\phi=x-\alpha F(\vx)$
• if $\phi(\vx) = \vx$, then we can also note that $\gamma \vx + (1-\gamma) \phi(\vx) = \vx$ also has a fixed point at the solution $F(\vx) = 0$ for any $\gamma$.

Do either of those modifications help? (This problem requires you to implement and investigate.)