Engineering COMPUTATIONAL

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1. Use a first through third order Taylor series expansion with starting point, x_i = 0 and h = 1 to estimate the each of the following functions at x_i+1 = 1. Evaluate the error between the true value and the approximate at x_i+1 = 1 for each expansion.

• 3x³ + 2x² + x
• 5x⁵ + 3x³ + 2x² + x

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

3. 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.

4. Solve for a root of the equation x³ + x² – 16x = 16 with a stopping criteria of ε_s = 5% using:

• The bisection method with initial bracket x_l = -1.5 and x_u = 2
• The fixed point iteration method with initial guess x₀ = 3.5
• The Newton-Raphson method with initial guess x₀ = -2.5

5. 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]

6. 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

7. 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

8. 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 ].

9. Solve the following linear system using the Gauss-Seidel method with a stopping criteria of ε_s = 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

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

3x²y + xy = 2x – 3

2x²y + y² = 3y + 2

Last Updated on March 23, 2019