Numerical Analysis Assignment #4 — Partial Pivoting and Non-Linear Systems  
Question #1
Consider the system of equations $AX=B$ with $A$ equal to: $$ A=\left[ \begin{array}{ccc} -2 & 2 & -1 \\ 6 & -6 & 7 \\ 3 & -8 & 4 \end{array} \right] $$ and $B$ equal to: $$ B=\left[ \begin{array}{c} -1 \\ -7 \\ -6 \end{array} \right] $$ Find $X$ using partial pivoting (by hand).
09.02.16
Question #2
Consider the following non-linear system of equations: $$ x_1^4 + x_2 = 5 $$ $$ x_1 x_2 + x_2^{1.5} =8 $$ Solve the latter using Newton's method using the initial conditions $x_1=1$ and $x_2=1$. Do it in two ways:
(a)  by hand and solve the first 3 iterations only.
(b)  By writing a C code that starts as follows:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <assert.h>

double f(double x1, double x2){
  return(x1*x1*x1*x1+x2-5.0);   
}

double dfdx1(double x1, double x2){
Question #3
Consider the following non-linear system of equations: $$ x_2 x_1 x_3 = 5 $$ $$ \frac{1}{2}x_1^2 + \frac{1}{2}x_2^2 = 100 $$ $$ x_2 + x_3 =0 $$ Do the following:
(a)  Find $x_1$, $x_2$, and $x_3$ using Newton's method using the initial conditions $x_1=0$, $x_2=1$, and $x_3=1$. Do so by hand and solve the first 2 iterations only.
(b)  Using the results obtained in (a) estimate the order of convergence of the method.
08.29.17
Question #4
Consider the following non-linear system of equations: $$ \sin^2(x_1) \cos(x_2)=0.5 $$ $$ \sqrt{x_1}-x_2=0.3 $$ Do the following:
(a)  Find the root of the system starting from the guess $x_1=x_2=0.6$ and make sure the root is correct to at least 6 significant digits. Hint: first substitute one equation in the other to obtain 1 equation for 1 unknown, and then find the root for such unknown through an iterative root solver of your choice.
(b)  Starting from the guess $x_1=x_2=0.6$ and with the first steps $\Delta x_1=\Delta x_2=0.00001$, use the secant method in system form to obtain the root of the system. Solve by hand the first 2 iterations only. Outline clearly all the steps including the expressions used to compute the Jacobians.
Answers
1.  2, $\frac{1}{4}$, $-\frac{5}{2}$.
2.  
3.  4.91244, 51.125, -51.125, 0.1, -1.1, -1.18.
4.  0.919332, 0.658818, 0.9197, 0.6586.
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