Description

• Write your name, ID#, and Section number clearly in the very front page.
• Write all answers sequentially.
• Start answering a question (not the pat of the question) from the top of a new page.
• Write legibly and in orderly fashion maintaining all mathematical norms and rules. Prepare a single solution file.

1. Consider A function f(x) = x3 − 7×2 + 4x + 12.
(a) (2 marks) Construct two different fixed point functions g(x) such that f(x) = 0.
(b) (3 marks) Compute the convergence rate of each fixed point function g(x) obtained in the previous part, and state which root it is converging to or diverging.
(c) (3 marks) How many iterations will be required to find the root if the machine epsilon is 1.4 × 10−18.
(d) (4 marks) Show 4 iterations using the Bisection Method to find the root of the above function within the interval [4.25, 8.95].
(e) (4 marks) Starting from x0 = 2.26 find the approximate root of f(x) up to four iterations by applying Aitken acceleration appropriately. Express your result up to five decimal places.
2. A linear system is described by the following equations.

(a) (2 marks) From the given linear equations, identify the matrices A, x and b such the the linear system can be expressed as a matrix equation.
(b) (3 marks) Does this system have any unique solution? Explain.
(c) (3 marks) Evaluate the upper triangular matrix U. Note that you have to show the row multipliers mij for each step as necessary.
(d) (4 marks) Using the upper triangular matrix found in the previous question, compute the solution of the given linear system by Gaussian elimination method.
3. A linear system is described by the following equations.
3x + 3y + 4z = 1
=
= 6
4
(a) (2 marks) From the given linear equations, identify the matrices A. Examine if the matrix A has any pivoting problem? Explain why or why not?
(b) (3 marks) Compute the Frobenius matrices F(1) and F(2) for this system.
(c) (3 marks) Evaluate the unit lower triangular matrix L, and the upper triangular matrix U.
(d) (4 marks) Now compute the solution of the given linear system using LU-decomposition method. Use the matrices L and U found in the previous question. Show your works.
Motto: Mathematics is NOT difficult, but what is difficult is to believe that mathematics is NOT difficult.