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R9490/homework1/conjugate_gradient.py
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conjugate_gradient.py
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# EXERCISE 2, PART 1.
import numpy as np
def conjugate_gradient(A,b,x0,tol,maxIter):
# This function returns the minimizer and all the iterations performed
# until convergence, of a quadratic function of N-dimensions as defined
# in exercise 2, by using the conjugate gradient (CG) method
# The inputs are:
# A: matrix of size N x N --> array
# b: vector of size N --> array
# x0: initial guess, vector of size N --> array
# maxIter: maximum number of iterations in case method does not converge --> float
r = b - np.einsum('ij,j->i',A,x0) # initial residual
p = r # initial direction
x = x0 # initial guess
rdot0 = np.einsum('i,i->', r, r) # initial dot product of residual
# Save initial guess in the lists iterations_x and iterations_y. These
# lists will change size depending on the number of iterations
iterations_x = [x0[0]]
iterations_y = [x0[1]]
j = 0 # iteration counter
# CG method - Loop
while True:
j+=1
# Compute new approximatted solution
Ap = np.einsum('ij,j->i',A,p) # A * p
pAp = np.einsum('i,i->', p, Ap) # pT * A * p
alpha = rdot0/pAp # alpha value
x = x + alpha * p # new value of x
# Save result of current iteration
iterations_x.append(x[0])
iterations_y.append(x[1])
r = r - alpha * Ap; # current value of residual
rdot1 = np.einsum('i,i->', r, r) # current dot product of residual
# test if loop is finished
if j >= maxIter:
break
if np.sqrt(rdot1) < tol:
break
p = r + (rdot1/rdot0) * p; # current direction
rdot0 = rdot1 # update dor product of residual for next iteration
# Return the optimizer "x", all the iteration points "iterations_x" and
# "iterations_y", the number of iterations "j", and the norm of the residual
return x, iterations_x, iterations_y, j, np.sqrt(rdot0)
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