How long does it take approximately (in years) for Supercomputer Fugaku to complete a LU factorization for a square matrix with n = 108? How about n = 109, n = 1010?

Finish 3 math question

MATH 3510 FALL 2021 HOMEWORK 4
(DUE IN CLASS OCTOBER 6)
Instructions: For proof-based questions, write your answers on your own papers. For pro-gramming questions, submit your codes using MATLAB Grader. Hand in your homework (written part) on Wednesday, October 6 in class.

1. Write a MATLAB function to perform the LU factorization (without pivoting). Your input should be a nonsingular n×n matrix A. Your output should be a lower triangular matrix L and an upper triangular matrix U such that A = LU. See detailed description in MATLAB Grader.

2. We discussed in class that Gaussian Elimination/LU factorization is an O(n3) algorithm. This means that for a n ×n matrix, LU factorization needs approximately n3 flops. The top supercomputer today is Supercomputer Fugaku in Japan. This machine can execute, theoretically, 4.42 ×1017 flops per second.

a. How long does it take approximately (in years) for Supercomputer Fugaku to complete a LU factorization for a square matrix with n = 108? How about n = 109, n = 1010?

b. Suppose we have an O(n) algorithm to solve the linear system Ax = b, how long does it take for Supercomputer Fugaku to solve the system with size n = 108? How about n = 109, n = 1010? Remark 1. From the results you should see why we need fast algorithms.

3. (Pivoting) In class, we find the LUP factorization of the following matrix

In other words, we have PA = LU.
Use this LUP factorization to solve the system Ax = b by hand where b =

2 (DUE IN CLASS OCTOBER 6)
Remark 2. You need to use forward substitution and back substitution.

4. (Vector Norms)
a. Let x =

Compute the norms ‖x‖1 and ‖x‖2.

b. Define a vector ∞-norm ‖x‖∞ for any x ∈Rnas follows,‖x‖∞ = maxk|xk|.
Prove the following properties for ‖·‖∞:
1. ‖x‖∞ ≥0. ‖x‖∞ = 0 if and only if x = 0.
2. ‖αx‖∞ = |α|‖x‖∞, where α ∈R.
3. ‖x + y‖∞ ≤‖x‖∞ + ‖y‖∞.