Homework # 3 Due Feb 19, 1999 Numerical Linear Algebra Math 4267 Spring 1999

Use pencil and paper solutions throughout. I want DETAILS. No computer software is allowed. Of course, you can check your results using such software. Write on one side of the paper only. Place in order assigned and staple results. Put all papers in a manila folder. 10 points per day excluding weekends for late penalty. 10 points penalty over the weekend.

1. p 375; 5a. Be sure to show steps such as m₂₁ = a₂₁/a₁₁, m₃₁ = a₃₁/a₁₁,...., a₂₂ = a₂₂-m₂₁*a₁₂, etc. which was discussed in algorithm 6.1. Careful: Be sure to chop after each computation.

2.p 375, 7a Be sure to show steps such as m₂₁ = a₂₁/a₁₁, m₃₁ = a₃₁/a₁₁,...., a₂₂ = a₂₂-m₂₁*a₁₂, etc. which was discussed in algorithm 6.1. Careful: Be sure to chop after each computation.

3. Consider the following matrices.

A = and b =

a. Proceed as in example 1 on p 397 to find L and U. Be sure to show steps such as

m₂₁ = a₂₁/a₁₁, m₃₁ = a₃₁/a₁₁,...., a₂₂ = a₂₂-m₂₁*a₁₂, etc. which was discussed in algorithm 6.1 to determine U. Note that L contains the multipliers m₂₁, etc.

b. From a. with A = LU, we wish to solve Ax = b. With LUx = b, let y = Ux. Then solve Ly = b by hand .

c. Having found y from b, solve Ux = y for x by hand.

4. Find the matrix P so that PA can be factored into the product LU, where L is lower triangular and U is upper triangular. (Hint: Perform Gaussian elimination and determine which rows must be swapped to avoid zero pivots, etc.)

A =

5. p 416, 3b

6. p 457, 1a

7. p 457, 2a