Using Google Collab, build and solve question your own matrix to solve certain matrices

Tasks (a) Implement a tridiagonal solver in Python. The function to implement must be called tri_solver, with input/output as described in the template. A tri-diagonal solver is a simplified Gaussian elemination solver that makes use of the banded nature of the matrix to reduce the amount of storage and computation required. We give you the following pseudo-code to get started: % Inpu t : % l ( 0 :N−1) −− a r ra y o f s i z e N c o n t ai ni n g low e r di a g o n al % u ( 0 :N−1) −− a r ra y o f s i z e N c o n t ai ni n g upper di a g o n al % d ( 0 :N) −− a r ra y o f s i z e N+1 c o n t ai ni n g main di a g o n al % b ( 0 :N) −− a r ra y o f s i z e N+1 c o n t ai ni n g the r i g h t hand s i d e . 3 % Output : a r ra y b i s o v e r w ri t t e n with s o l u t i o n . % Forward e l i m i n a t i o n s f o r i from 0 to N−1 do % Compute m u l t i p l i e r mult = l ( i )/ d ( i ) % Per form row r e d u c ti o n d ( i +1) = d ( i +1) −mult∗u ( i ) % Reduce r i g h t hand s i d e b ( i +1) = b ( i +1) − mult∗b ( i ) end f o r % Back s u b s t i t u t i o n s b (N) = b (N)/ d (N) f o r i from N−1 to 0 do b ( i ) = ( b ( i ) − u ( i )∗b ( i +1))/d ( i ) end f o r (b) Implement a function called compute_derivative that computes the derivative using 4th-order compact finite-differences. Refer to the template for details on input/output of this function. This function will use tri_solver coded in part (a) to solve the system (⋆). (c) Consider the test function f(x) = sin(5x) for 0 ≤ x ≤ 3. Compute the derivatives of f using your function from 4b) on a grid with N = 16 segments (x0 = 0, x1 = 3/16, x2 = 2 × 3/16, ..., x16 = 3). On the same figure, plot your numerically obtained derivative alongside the exact derivative of f evaluated at the nodes x0,..., x16. (d) Plot the numerical solution with N = 32, 64, and 128. How does the agreement with the analytical solution change? (e) Conduct a performance study of your function compute_derivative by timing the code execution with N = 10, 100, 1000, 5000, 10000. Plot mesh size versus time. Briefly discuss your results.