Implement a solver for the stiff system of differential

Question

In this problem set, you will implement a solver for the stiff system of differential equations

y' = -Ay,

y(0) = y_o,

where A is a symmetric positive definite n x n matrix, y is a function from [0, ∞) to Rⁿ, and y_o is a vector in Rⁿ. You can choose one of the following two algorithms to implement.

Method 1. Solve the differential equation using the backward Euler method. Then, apply one step of Richardson extrapolation.

Method 2. Diagonalize the matrix A. Then, find an explicit formula for the solution. In C, your calling sequence should be

void stiff_solve(double* a, int n, double* yO, double t, int m, double* y)

Input arguments:
a - The n x n symmetric positive definite matrix of doubles A
n - The dimension n of the matrix A
y0 - An array of doubles of length n containing the initial value yo
t - The final time t
m - The number of steps used in backward Euler (the step size should be t/m). This argument should be ignored if you are using method 2.
Output arguments:
y - An array of doubles of length n containing the solution y(t) at time t