MATH 230: Introduction to Numerical Methods (Spring 2009)

Homework Assignments


 
Assignment 1:
  1. 2.1:  1.
  2. 2.2: 1b, 2b.
  3. 2.3: 1, 3a.

 

Assignment 2:

 

  1. 3.1: 1a
  2. 3.2: 7

 

Assignment 3:

 

  1. 8.1: 1

 

  1. Write a MATLAB program to construct the least squares polynomials of degree 1, 2, and 3 for the data in 8.1.1 and plot the constructed polynomials.

Assignment 4:

 

Q1. Numerical Differentiation: Exercise Set 4.1: 1a, 20.

 

Q2. (Excise Set 4.3: 2a, 6a,10a) Elements of Numerical Integration: Approximate the following integrals using the Trapezoidal, Simpson’s and Midpoint rules.

 

 

 

Q3 (Exercise Set 4.4: 2a, 4a, 6a)  Composite Numerical Integration: Use Composite Trapezoidal, Simpson’s and Midpoint rules to approximate the integral

 

     n=4

 

Q4. (Exercise Set 4.7: 1a) Gaussian Quadrature

Approximate the following integral using Gaussian quadrature with n=3.

 

 

Assignment 5:

 

Q1. (Exercise Set 5.2: 2c and 4c) Euler’s Method: Given the initial-value problem

        ,

       with the exact solution .

 

       Use Euler’s method with h=0.25 to approximate the solution, and compare your computed solution with the exact solution at each step.

 

Assignment 6:

 

Q1: (Exercise Set 6.1: 6d)  Use Gaussian Elimination with Backward Substitution method (Algorithm 6.1) to solve the following linear system, if possible, and determine whether row interchanges are necessary.

 

 x1       +x2                +x4 = 2,

2x1      +x2      -x3     +x4 = 1,

-x1     +2x2   +3x3      -x4 = 4,

3x1       -x2      -x3    +2x4 =-3.

 

Q2: (Exercise Set 6.2: 2c,4c)  Find the row interchanges that are required to solve the following linear system using (a) Algorithm 61: Gaussian Elimination with Backward Substitution; (b) Algorithm 6.2: Gaussian Elimination with Partial Pivoting.

 

5x1       +x2     -6x3    = 7,

2x1       +x2      -x3     = 8,

6x1   +12x2     +x3     = 9.

 

Q3: (Exercise Set 6.5: 2a) Solve the following linear system using matrix factorization technique.

 

 

 

Q4: (Exercise Set 6.5: 6a): Factor the matrices into LU decomposition using the LU Factorization Algorithm with lii=1 for all i.

 

 

 

Assignment 7:
 

Q1: (Exercise Set 7.3: 2c,4c) Find the first two iterations using (a) Jacobi method and (b) Gauss-Seidel method for the following linear system, using x(0)=0

 

4x1       +x2     -x3     +x4 = -2,

 x1      +4x2      -x3    -x4 = -1,

-x1       -x2     +5x3    +x4 = 0,

 x1       -x2      +x3    +3x4 =-1.

 

 

Q2: (Exercise Set 7.3: 9a) Find the first two iterations of the SOR method with  for the linear system, using x(0)=0

 

3x1       -x2     +x3    = 1,

3x1   +6x2     +2x3   = 0,

3x1  +3x2     +7x3   = 0.

 

 

 



Last Revised: 30/8/2004


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