Categories: Order Term Paper

Engineering COMPUTATIONAL

You have the option to calculate the answers to the problems below by hand or by writing your own M-file. Show as much work as possible to get as much credit as possible. Please submit a pdf file of your work.

 

  1. Use a first through third order Taylor series expansion with starting point, xi = 0 and h = 1 to estimate the each of the following functions at xi+1 = 1. Evaluate the error between the true value and the approximate at xi+1 = 1 for each expansion.
  • 3x3 + 2x2 + x
  • 5x5 + 3x3 + 2x2 + x

 

  1. Using the Taylor series derive the forward, centered, and backward fist order finite difference.

 

  1. Given the approximate values ̃ = 10, ̃ = 5, ℎ̃ = 15, with errors ̃ = 0.1, ̃ = 0.5, ℎ̃ = 0.2, estimate the resulting error in the function, (, , ℎ) = 2 + ℎ, where

= 9.81.

 

  1. Solve for a root of the equation x3 + x2 – 16x = 16 with a stopping criteria of εs = 5% using:
  • The bisection method with initial bracket xl = -1.5 and xu = 2
  • The fixed point iteration method with initial guess x0 = 3.5
  • The Newton-Raphson method with initial guess x0 = -2.5

 

  1. Solve the following linear system using (a) the graphical method, (b) Cramer’s rule, and (c) elimination of unknowns. Check your answers by confirming that [A]{x} = {b}.

 

5    3    1                    2

[3    7] [2] = [−4]

 

 

 

 

  1. Solve the following linear system using (a) naïve Gauss elimination and (b) Gauss elimination with scaling and pivoting. Check your answers by confirming that [A]{x} = {b}.

 

9    7    8    1                 11

  • 2 3] [2] = [12]

5    6    4    3                 10

 

  1. For the following linear system, (a) compute the LU decomposition and confirm that [A] = [L][U], (b) use the decomposition calculated in (a) to solve the system and check your answers by confirming that [A]{x} = {b}.

 

4    5    7    1                 11

  • 2 3] [2] = [10]

6    8    9    3                 12

 

  1. Using the LU decomposition calculated for the linear system in problem 7 determine the matrix inverse [A]-1 of that system. Check your answers by confirming that [A]-1[A] = [ I ].

 

  1. Solve the following linear system using the Gauss-Seidel method with a stopping criteria of εs = 5%. Check your answers by confirming that [A]{x} ≈ {b}.

 

10     4      5     1                 2

[ 4     10      5 ] [2] = [2]

4      5     10    3                 2

 

  1. Solve the following nonlinear system using the Newton-Raphson method with a stopping criteria of εs = 5%. Check your answers.

 

3x2y + xy = 2x – 3

2x2y + y2 = 3y + 2