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:
 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. Hint: the root should be found from the iterative procedure as the solution to the next iteration (valid as we are converging). Thus, the approximate root here would be 4.573738E+00, 2.731503E+01, -2.731503E+01.
 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.
 Question #5
Consider the system of equations $AX=B$ with $A$ equal to: $$A=\left[ \begin{array}{cccc} -2 & 0 & 1 &1 \\ 2 & 1 & 0 &0 \\ 0 & 1 & 1 &2 \\ 0 & 0 & 2 &1 \\ \end{array} \right]$$ and $B$ equal to: $$B=\left[ \begin{array}{c} -1 \\ -7 \\ 3\\ -6 \end{array} \right]$$ Using partial pivoting only when the pivot is zero, find the lower and upper triangular matrices associated with matrix $A$. Outline all the steps needed to obtain the matrix $L$, the matrix $U$, and the permutation matrices. Also, indicate clearly how $A$ can be written as a function of $L$, $U$, and the permutation matrices.
 11.07.18
 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. 5. $P_{34}LU$.
 $\pi$