# Answered Essay: # import necessary packages # import numpy as np The following code imports a matrix from the file mat.npy and stores it as mati. It also imports a vector fro

# import necessary packages # import numpy as np The following code imports a matrix from the file mat.npy and stores it as mati. It also imports a vector from vect.npy and stores it as vect1. For objects like arrays, this can be much simpler than the more general methods in the sample files mat1 = np. load (“mat .npy.) vect1 = np.load(‘vect.npy’) print(‘This is a positive definite matrix An’, mat1, “n and this is a vector bn’, vect1) … Replace this comment with code write a function to perform forward substitution to solve the matrix equation 6x = b assume G is invertible and lower triangular the function should return the solution vector x the function should work for square matrices of any size … return(x)

PYTHON CODE for FORWARD SUBSTITUTION:

import numpy as np

def solve(m, y): #definition of solve function

x = [0 for i in range(len(m))]

for i in range (len(m)):

q = np.array(m[i,:]).flatten() # Flattening ith row of matrix

v = np.dot(v,q) # Dot product of flattened ith row and soln vector v formed till now

x[i] = (y[i] – v )/float(m[i,i]) #This operation gets the ith element in solution vector x

return x

m = np.matrix(‘1 0;1 2’) # m is a 2×2 lower triangular matrix

b = np.array([2,4]); # b is an array with 2 elements

print(“Solve (G,b ) gives soln vector x = “)

x = solve(G,b)

print(x)

OUTPUT:

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