Math 230 Assignment 4 Due date: May 19 5pm



Math 230 Assignment 5 Submission is NOT REQUIRED. The answer will be provided on May 16.

1: Euler’s Method

(a) Given the initial-value problem

[pic]

With exact solution [pic]

Use Euler’s method with h=0.25 to approximate the solution, and compare it with the

actual value of y.

b) Use answers generated in part (a) and linear interpolation to approximation the following value y(2.85) , and compare it with the actual value.

c) Compute the value of h necessary for [pic]using equation

[pic] in Theorem 5.9 (see page 261 of text book).

2: Linear System of Equations

2.1: (a)Use Gaussian Elimination method 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.

2.2: Find the row interchanges that are required to solve the following linear system using (a) Gaussian Elimination with Backward Substitution; (b) Gaussian Elimination with Partial Pivoting; (c) Gaussian Elimination with Scaled Partial Pivoting and write its Algorithm.

5x1 +x2 -6x3 = 7,

2x1 +x2 -x3 = 8,

6x1 +12x2 +x3 = 9.

2.3: Solve the following linear system using matrix factorization technique.

[pic]

2.4 Factor the matrices into LU decomposition using the LU Factorization Algorithm with lii=1 for all i.

[pic]

2.5. Using (a) Jacobi method and (b) Gauss-Seidel method to solve for the following linear system with TOL=10-3 in the [pic] norm (i.e. [pic], K+1 is iterative step required )

4x1 +x2 -x3 +x4 = -2,

x1 +4x2 -x3 -x4 = -1,

-x1 -x2 +5x3 +x4 = 0,

x1 -x2 +x3 +3x4 =-1.

2.6. Find the first two iterations of the SOR method with [pic] for the linear system, using x(0)=0

-2x1 +x2 +0.5x3 = 4,

x1 -2x2 -0.5x3 = -4,

x2 +2x3 = 0.

3: Discrete Least Squares

Find the least squares polynomials of degree 1 and 2 for the data in the following table. Compute the error E2 in each case. Graph the data and the polynomials.

xi 0 0.15 0.31 0.5 0.6 0.75

yi 1.0 1.0004 1.031 1.117 1.223 1.422

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