Answer in python:

The banded, upper triangular matrix A E RnXn can be written as 0 a2 b2 c2 0 3 3 an-2 bn-2 Cn-2 where (, , an-2,an-1, an) E R", 6- (bi,b2,...,bn-2,bn,-1) R-1, n-2 The entries of , b, and are all assumed to be nonzero (a) Write a pseudocode for computing the matrix-vector product of A with a vector x. That is, given vectors a, b, and c that define A as in definition (1), and given a vector x E Rn, your algorithm should compute the matrix-vector product y- Aa Your pseudocode should have the following form Input: vector re Rn and vectors a E R", be Rn-i and Rn-2 Insert pseudocode here Output: vector fE R" such that y- At (b) Analyse the complexity of the algorithm from part (a). That is, determine how many flops are required to compute the product y - Ai with your algorithm. In terms of "Big-Oh" notation, what is the asymptotic behaviour of your algorithm as n increases? (c) Implement your pseudo-code in as a function called tridag matvec.py. Also, write a script called compare.py to generate a tridiagonal test matrix and a test vector for a n -10*, k- 2' .. , 7, compute the product and measure the time your code takes to complete. Produce a plot of the time taken versus n on a logarithmic scale, along with your prediction. (d) Repeat the test, but now using the built-in matrix-vector product (scipy.dot/scipy.matmul) and the matrix A defined as an n n matrix with mostly zeros. Plot the time taken in the same plot as for (c). Which algorithm is faster? Is the difference as great as you expectedi