- (a) (3 points) Solve the following system using GEPP (Gaussian elimination with partial pivoting):
A = 42
88 = 168
Show intermediate matrices, vectors and multipliers at each step.
(b) (2 points) Compute the LU factorization of the matrix in the previous question with partial pivoting: PA = LU. You have to show intermediate results at each step. This question and the previous one can be answered together.
Note: Do the computations by your hands and don’t consider any rounding errors.
- (3 points) Suppose we have a complex linear system Ax = b, where A ∈Cn×n is nonsingular and b ∈Cn. Can you solve it in real arithmetic operations? What is the cost?
Hint: Rewrite Ax = b as an equivalent 2n × 2n real linear system.
- Suppose A ∈Rn×n is nonsingular.
- (4 points) Given B ∈Rn×p, show how to use the LU factorization with partial pivoting to solve AX = B. What is the cost of your method?
Hint: AX = B is equivalent to AX(:,j) = B(:,j) for j = 1 : p.
- (5 points) Use m given in the lecture notes to solve AX = B, where 10 × 10 Hilbert matrix A = (aij), aij = 1/(i + j − 1);
10 × 5 B = A ∗randn(10,5).
Here randn is a MATLAB built-in function to generate a random matrix. Denote this randn(10,5) by Xt and your computed solution by Xc.
- Compute kXc − XtkF/kXtkF and ǫkAkFkA−1kF, where ǫ is the machine epsilon. Check MTALAB built-in functions or constants norm, cond and eps, to see how to compute or get related quantities.
- Compute the relative residual kB − AXcompkF/(kAkFkXcompkF).
- Run your code 10 times (you may use a loop). Notice each time you have differentB, since Xtrue is random. Answer the following questions:
- Do you see any rough relation between kXc − XtkF/kXtkF and ǫkAkFkA−1kF? iii. Do you see any rough relation between kB − AXckF/(kAkFkXckF) and ǫ ?
Print out your MATLAB code and the results.
(c) Computing A−1
- (3 points) Show how to use the LU factorization with partial pivoting to computethe inverse of a general n × n nonsingular matrix A. What is the cost of your method? (Hint: Think about what matrix equation AX = B you should solve to get A−1).
Compute the inverse of a 5 × 5 Hilbert matrix defined in 3(b). Print out your MATLAB code and the result.
- (3 bonus points) For the sake of simplicity, suppose we use the LU factorizationwith no pivoting to compute A−1. Briefly state an algorithm which costs 2n3 You need to explain why its cost is 2n3 flops.