#!/usr/bin/env python
# coding: utf-8
# # Jacobi vs Conjugate Gradient
#
# Copyright (C) 2010-2020 Luke Olson
# Copyright (C) 2020 Andreas Kloeckner
#
#
# MIT License
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
#
# The above copyright notice and this permission notice shall be included in
# all copies or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
# THE SOFTWARE.
#
#
# Let's consider solving the discretized 2D Poisson equation with iterative methods.
# In[1]:
import numpy as np
import numpy.linalg as la
from matplotlib import pyplot as pt
# In[2]:
n=16
T = 2*np.eye(n)-np.diag(np.ones(n-1),1)-np.diag(np.ones(n-1),-1)
A = np.kron(np.eye(n),T)+np.kron(T,np.eye(n))
pt.spy(A)
# Define a right-hand side and solve the resulting system directly.
# In[3]:
h = 1/(n-1)
b = h*np.arange(0,n*n)
x = la.solve(A,b)
# Split the matrix into its diagonal and strictly lower/upper triangular parts.
# In[4]:
d = np.diag(A)
D = np.diag(d)
L = np.tril(A,-1)
U = np.triu(A,1)
la.norm(A-(D+L+U))
# Jacobi iteration proceeds using
# $$\boldsymbol x^{(i+1)} = \boldsymbol D^{-1}(\boldsymbol b- (\boldsymbol L+\boldsymbol U)\boldsymbol x^{(i)}).$$
# In[10]:
def jacobi(niter,x0):
xi = x0
for i in range(niter):
xi = np.diag(1./d)@(b-(L+U)@xi)
return xi
niters = np.asarray(2**np.arange(4,12),dtype=int)
x0 = np.random.random(n*n)
jacobi_results = []
err = []
for niter in niters:
jacobi_results.append(jacobi(niter,x0.copy()))
err.append(la.norm(jacobi_results[-1]-x))
pt.plot(niters,err)
pt.yscale('log')
pt.xscale('log')
# In[16]:
def cg(A,b,niter,x0):
rk = b - A @ x0
sk = rk
xk = x0
for i in range(niter):
alpha = np.inner(rk,rk)/np.inner(sk, A @ sk)
xk1 = xk + alpha * sk
rk1 = rk - alpha * A @ sk
beta = np.inner(rk1,rk1)/np.inner(rk,rk)
sk1 = rk1 + beta*sk
rk = rk1
xk = xk1
sk = sk1
return xk
iters = np.asarray(2**np.arange(2,7),dtype=int)
x0 = np.random.random(n*n)
cg_results = []
err = []
for niter in iters:
cg_results.append(cg(A,b,niter,x0.copy()))
err.append(la.norm(cg_results[-1]-x))
pt.plot(iters,err)
pt.yscale('log')
pt.xscale('log')
# In[ ]: