conjugate_gradient.py
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	#!/usr/bin/env python3
	# -- coding: utf-8 --
	"""
	Created on Wed Oct 10 10:29:34 2018

	@author: sajjadazimi
	"""

	import numpy as np
	import optimizer as opt


	# part 1: implementation of linear conjugate gradient method
	def conjugateGradient(A, b, x, tol):
	r = b - np.einsum ('ij,j->i', A, x)
	p = r
	x_steps = x # x_steps is a matrix whose rows are the value of x at different steps
	while np.sqrt(np.einsum('k,k->', r, r)) > tol:
	alpha = np.einsum('k,k->', r, r) / np.einsum('k,k', p, np.einsum('ij,j->i', A, p))
	x = x + alpha * p
	x_steps = np.vstack([x_steps,x])
	rnew = r - alpha * np.einsum('ij,j->i', A, p)
	beta = np.einsum('k,k->', rnew, rnew) / np.einsum('k,k->', r, r)
	p = rnew + beta * p
	r = rnew
	return x_steps


	# part 2: solve for the function S given in exercise 1
	# using the matrix form definition of the minimizer, 1/2 x'Ax - x'b = S(x), so
	A = [[4,1],[1,3]] #matrix A for S
	b = [1,2] #vector b for S
	x = [3., 1.]
	x_steps = conjugateGradient(A,b,x,1e-7)
	n_steps = np.shape(x_steps)[0]-1 #number of steps
	print("Number of Steps :", n_steps)
	print("The converged solution :", x_steps[n_steps,:])


	# part 3: plot
	opt.plotFunc(opt.S, x_steps)