def newtonRaphson(x, tol=1.0e-9):
def f(x):
return x * x - 1
def jacobian(f, x):
h = 1.0e-4
n = len(x)
jac = np.zeros((n, n))
f0 = f(x)
for i in range(n):
temp = x[i]
x[i] = temp + h
f1 = f(x)
x[i] = temp
jac[:, i] = (f1 - f0) / h
return jac, f0
for i in range(30):
jac, f0 = jacobian(f, x)
if math.sqrt(np.dot(f0, f0) / len(x)) < tol:
print(x)
dx = gaussPivot(jac, -f0)
x = x + dx
if math.sqrt(np.dot(dx, dx)) < tol * max(max(abs(x)), 1.0): return x
print('Too many iterations')
def newtonRaphson2(f,x,tol=1.0e-9,s=None):
def jacobian(f,x):
h = 1.0e-4
n = len(x)
jac = np.zeros((n,n))
f0 = f(x)
for i in range(n):
temp = x[i]
x[i] = temp + h
f1 = f(x)
x[i] = temp
jac[:,i] = (f1 - f0)/h
return jac,f0
for i in range(30):
jac,f0 = jacobian(f,x)
if math.sqrt(np.dot(f0,f0)/len(x)) < tol:
return x
dx = gaussPivot(jac,-f0)
x = x + dx
if s != None:
s.append(x)
if math.sqrt(np.dot(dx,dx)) < tol*max(max(abs(x)),1.0): return x
print('Too many iterations')
def polyFit(xData,yData,m):
a = zeros((m+1,m+1))
b = zeros(m+1)
s = zeros(2*m+1)
for i in range(len(xData)):
temp = yData[i]
for j in range(m+1):
b[j] = b[j] + temp
temp = temp*xData[i]
temp = 1.0
for j in range(2*m+1):
s[j] = s[j] + temp
temp = temp*xData[i]
for i in range(m+1):
for j in range(m+1):
a[i,j] = s[i+j]
return gaussPivot(a,b)
def polyFit(xData, yData, m):
a = zeros((m + 1, m + 1))
b = zeros(m + 1)
s = zeros(2 * m + 1)
for i in range(len(xData)):
temp = yData[i]
for j in range(m + 1):
b[j] = b[j] + temp
temp = temp * xData[i]
temp = 1.0
for j in range(2 * m + 1):
s[j] = s[j] + temp
temp = temp * xData[i]
for i in range(m + 1):
for j in range(m + 1):
a[i, j] = s[i + j]
return gaussPivot(a, b)
def newtonRaphsonSEnoL(f, x, tol=1.0e-9):
def jacobian(f, x):
h = 1.0e-4
n = len(x)
jac = zeros((n, n), dtype=float)
f0 = f(x)
for i in range(n):
temp = x[i]
x[i] = temp + h
f1 = f(x)
x[i] = temp
jac[:, i] = (f1 - f0) / h
return jac, f0
for i in range(30):
jac, f0 = jacobian(f, x)
if sqrt(dot(f0, f0) / len(x)) < tol: return x
dx = gaussPivot(jac, -f0)
x = x + dx
if sqrt(dot(dx, dx)) < tol * max(max(abs(x)), 1.0): return x
print 'Too many iterations'
def newtonRaphson2(f,x,tol=1.0e-9):
def jacobian(f,x):
h = 1.0e-4
n = len(x)
jac = zeros((n,n),type=Float64)
f0 = f(x)
for i in range(n):
temp = x[i]
x[i] = temp + h
f1 = f(x)
x[i] = temp
jac[:,i] = (f1 - f0)/h
return jac,f0
for i in range(30):
jac,f0 = jacobian(f,x)
if sqrt(dot(f0,f0)/len(x)) < tol: return x
dx = gaussPivot(jac,-f0)
x = x + dx
if sqrt(dot(dx,dx)) < tol*max(max(abs(x)),1.0): return x
print 'Too many iterations'
def newtonRaphson2(f,q,tol=1.0e-15): ##1.0e-9 may be too large
for i in range(50):
h = 1.0e-4 ##1.0e-4 may be too large.
n = len(q)
jac = zeros((n,n))
if 1.0+q[0]-x<0.0:
#print "Problem occurs i=",i
#print "PROBLEM OCCURS.......... 1.0+q[0]-x=",1.0+q[0]-x
q[0] = x-1.0 + 0.0001
#print "NOW problem is fixed as 1.0+q[0]-x=",1.0+q[0]-x
f0 = f(q)
for i in range(n):
temp = q[i]
q[i] = temp + h
f1 = f(q)
q[i] = temp
jac[:,i] = (f1 - f0)/h
if sqrt(dot(f0,f0)/len(q)) < tol: return q
dq = gaussPivot(jac,-f0)
q = q + dq
if sqrt(dot(dq,dq)) < tol*max(max(abs(q)),1.0): return q
print('Too many iterations')
def newtonRaphson2(f,q,tol=1.0e-15): ##1.0e-9 may be too large
for i in range(50):
h = 1.0e-4 ##1.0e-4 may be too large.
n = len(q)
jac = zeros((n,n))
if 1.0+q[0]-x<0.0:
#print "Problem occurs i=",i
#print "PROBLEM OCCURS.......... 1.0+q[0]-x=",1.0+q[0]-x
q[0] = x-1.0 + 0.0001
#print "NOW problem is fixed as 1.0+q[0]-x=",1.0+q[0]-x
f0 = f(q)
for i in range(n):
temp = q[i]
q[i] = temp + h
f1 = f(q)
q[i] = temp
jac[:,i] = (f1 - f0)/h
if sqrt(dot(f0,f0)/len(q)) < tol: return q
dq = gaussPivot(jac,-f0)
q = q + dq
if sqrt(dot(dq,dq)) < tol*max(max(abs(q)),1.0): return q
print 'Too many iterations'
import numpy
from numpy import *
import scipy
from scipy import linalg
from gaussPivot import *
from gaussElimin import *
from numpy.linalg import solve
from LUdecomp import *
import matplotlib.pylab as plt
from scipy import *
G = array(
[[2.0, -1.0, 0.0, 0.0], [0.0, 0.0, -1.0, 1.0], [0.0, -1.0, 2.0, -1.0],
[-1.0, 2.0, -1.0, 1.0]]
) #setting up a matrix for G in order to solve system of simultaneous equations
print "G=", G
H = array([[1.0], [0.0], [0.0], [1.0]])
print "H=", H
I = gaussPivot(G, H)
print "Gauss Pivot solutions are", I
print "x1=1, x2=1, x3=1, x4=1"
J = gaussElimin(
G, H
) # this gives different solutions. gauss pivot is designed to avoid dividing by zero or dividing by small numbers
print "Gaussian Elimination solutions are", J
#Problem sheet 2 from numpy import array from gaussPivot import * from gaussElimin import * a= array([[2.0,-1.0,0.0,0.0], [0.0,0.0,-1.0,1.0], [0.0,-1.0,2.0,-1.0], [-1.0,2.0,-1.0,1.0]]) b= array([1.0,0.0,0.0,1.0]) #builds the matrix a and b print 'original matrix' print a print 'matrix b' print b x=gaussPivot(a,b) #pivots the matrix print 'pivot' print x print 'part 2' a= array([[2.0,-1.0,0.0,0.0], [0.0,0.0,-1.0,1.0], [0.0,-1.0,2.0,-1.0], [-1.0,2.0,-1.0,1.0]]) b= array([1.0,0.0,0.0,1.0]) #builds the matrix a and b y=gaussElimin(a,b) #does not give an answer, more a nan error print x print y
print 'Upper matrix: ', '\n',U, '\n', 'Lower matrix: ','\n', np.dot(L,P) #displays the upper and lower matrices
print 'Testing L dot U, this should be equal to a: ', '\n', np.dot(np.dot(L,P),U)
print 'The variables are: ', '\n', x #outputs the result of the gaussElimin
#solves a set of simultanious equations, uses the decomp method from textbook
if in001==2 and in005==2:
d=LUdecomp(c) #LU decomposition using LUdecomp. Arguments:(square coefficients matrix)
e=LUsolve(d,b) #solves the decomposed matrix. Arguments:(result of LUdecomp, dependent variable matrix)
print 'The variables are: ', '\n', e #outputs the result of the LUsolve
###Gaussian Pivot
#solves a set of simultanious equations
if in001==3:
d=gaussPivot(c,b)
print 'The variables are: ' '\n' , d #outputs result of gaussPivot
###Fourier Transform
#finds the coefficients of a Fourier series and plots the result
if in001==4:
FT001=np.fft.rfft(a) #Performs the real Fast Fourier Transform. Arguments:(source data)
for i in range(len(FT001)): #loop removes the requested percentage of data from the array of coefficients, this smooths the graph
if i>(len(FT001))*(1-(in007*0.01)):
FT001[i]=0
iFT001=np.fft.irfft(FT001) #performs the inverse real Fast Fourier Transform. Argument:(Fourier coefficient array)
plt.figure(1)
plt.plot(b,a,'g-',label='Original Data')
plt.plot(b,iFT001,'r-',label='Smoothed Fourier Transform')
plt.xlabel(in008)
