Numerical Analysis Assignment 3 — Systems of Equations
 Question #1
Consider the following system of equations: $$-3 x_1 + 2 x_2 + 4 x_3 + 5 x_4 = 1 \\ - x_1 + 3 x_2 - x_3 + 5 x_4 = 2 \\ -4 x_1 + 4 x_3 + 2 x_4 = 3 \\ 6 x_2 + 4 x_3 - 5 x_4 = 4$$ Find $x_1$, $x_2$, $x_3$, $x_4$ using Gaussian elimination in two different ways:
(a)  By hand.
(b)  By writing a C program that starts as follows:
 09.02.16
 Question #2
Consider a system of equations expressed as: $$A X = B$$ with $$A=\left[\begin{array}{cccc} -2 & 0 & 0 & 0 \\ 0 & 3 & -1 & 5 \\ -4 & 0 & 4 & 2 \\ 0 & 6 & -2 & -5 \end{array} \right]$$ Find the matrices $L$ and $U$ by hand such that $A=LU$ and $L$ is a lower-triangular matrix and $U$ is a upper-triangular matrix.
 Question #3
Consider the system of equations $$AX=B$$ with $$B=\left[ \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right]$$ and with $A$ as outlined in Question #2. Using the $L$ and $U$ matrices found by hand in Question #2 solve for $X$ using lower-upper decomposition in two ways:
(a)  By hand
(b)  Using a C program that starts as follows:
 Question #4
For the following system of equations $$AX=B$$ do the following:
 (a) Prove that the work needed in making the Gaussian elimination augmented matrix $M$ upper triangular is $$W_{\rm ut}=\sum_{n=1}^{N-1} \sum_{m=n+1}^N (2(N-n)+3)$$ Note that $M$ has a size of $(N+1) \times N$ because it is composed of $A$ and $B$. Explain carefully all the logical steps involved in proving the latter. (b) Should $A=LU$ with $L$ lower triangular and $U$ upper triangular and for $L$ and $U$ given, prove that the work needed in finding $X$ corresponds to: $$W_{LU}=\sum_{n=1}^{N}(4(N-n)+2)$$ Explain carefully all the logical steps involved in proving the latter.
 08.29.17
 Due on Monday October 29th at 16:30. Do all 4 problems. Note that this assignment is more difficult than average and will be given 6 points bonus instead of 3.
 10.18.18
 $\pi$