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esenummet.py
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esenummet.py
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import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import scipy.optimize as scop
import scipy.interpolate as scip
from mpl_toolkits.axes_grid1.inset_locator import zoomed_inset_axes
from mpl_toolkits.axes_grid1.inset_locator import mark_inset
# ================
# NONLINER SESSION
# ================
def plot_picard_convergence_pattern(x_p, fp_p, max_labels=6):
fig, ax = plt.subplots()
# the iteration results
ax.scatter(x_p, fp_p, marker='x', color='blue', s=30)
# the convergence pattern
x_pattern = [x_p[0]]
fp_pattern = [fp_p[0]]
for i in range(1,len(x_p)):
x_pattern.append(x_p[i])
fp_pattern.append(fp_p[i-1])
x_pattern.append(x_p[i])
fp_pattern.append(fp_p[i])
ax.plot(x_pattern, fp_pattern, color='green', ls='--')
# the function
idx_sort = np.argsort(x_p)
ax.plot(np.array(x_p)[idx_sort], np.array(fp_p)[idx_sort], color='red', alpha=0.5)
# initial guess
dxt = (np.max(x_pattern)-np.min(x_pattern))/35.
dyt = (np.max(fp_pattern)-np.min(fp_pattern))/35.
ax.text(x_p[0]+dxt, fp_p[0]+dyt, '$x_0$')
for i in range(1, min(max_labels+1, len(x_p))):
label = ''.join(['$x_', str(i), '$'])
ax.text(x_p[i]+dxt, fp_p[i]+dyt, label)
ax.set_xlabel('$x$')
ax.set_ylabel('$f_p(x)$')
plt.title('Picard')
plt.show()
def plot_newton_convergence_pattern(fn, dx, x_0, tol, max_labels0=1, max_labels1=2, inset=True, ixmin=3.0, ixmax=3.2, iymin=-0.1, iymax=0.1, zoom=8, loc0=1, loc1=3, loc2=2, maxiter=100):
fig, ax = plt.subplots()
plt.title('Newton')
x_n = [x_0]
y_n = [fn(x_0)]
fx_n = [x_0]
fy_n = [fn(x_0)]
# solving
i = 0
while 1:
dfdx = (fn(x_n[-1]+dx) - fn(x_n[-1])) / dx
a = fn(x_n[-1])-(dfdx*x_n[-1])
x_zero = -a/dfdx
x_n.append(x_zero)
y_n.append(0.)
fx_n.append(x_zero)
fy_n.append(fn(x_zero))
if abs(x_n[-1]-x_n[-2]) < tol:
break
x_n.append(x_zero)
y_n.append(fn(x_zero))
i = i+1
if i >= maxiter:
break
# the iteration results
ax.scatter(fx_n, fy_n, marker='x', color='blue', s=30)
# the convergence pattern
ax.plot(x_n, y_n, color='green', ls='--')
# the function
x = np.linspace( np.min(x_n), np.max(x_n), 50)
f = fn(x)
ax.plot(x, f, color='red', alpha=0.5)
# initial guess
dxt = (np.max(x_n)-np.min(x_n))/35.
dyt = (np.max(y_n)-np.min(y_n))/35.
ax.text(fx_n[0]+dxt, fy_n[0]+dyt, '$x_0$')
for i in range(1, min(max_labels0+1, len(fx_n))):
label = ''.join(['$x_', str(i), '$'])
ax.text(fx_n[i]+dxt, fy_n[i]+dyt, label)
# zero line
xlim = ax.get_xlim()
ax.plot([xlim[0],xlim[1]],[0.,0.],color='gray',ls='-.',alpha=0.75)
ax.set_xlim(xlim)
# zoomed inset
if inset:
axins = zoomed_inset_axes(ax, zoom, loc=loc0)
axins.scatter(fx_n, fy_n, marker='x', color='blue', s=30)
axins.plot(x_n, y_n, color='green', ls='--')
axins.plot(x, f, color='red', alpha=0.5)
for i in range(max_labels0+1, min(max_labels0+1+max_labels1,len(fx_n))):
label = ''.join(['$x_', str(i), '$'])
axins.text(fx_n[i]+dxt/8., fy_n[i]+dyt/8., label)
axins.plot([ixmin,ixmax],[0.,0.],color='gray',ls='-.',alpha=0.75)
axins.set_xlim(ixmin, ixmax)
axins.set_ylim(iymin, iymax)
axins.get_xaxis().set_visible(False)
axins.get_yaxis().set_visible(False)
mark_inset(ax, axins, loc1=loc1, loc2=loc2, fc="none", ec="0.5")
ax.set_xlabel('$x$')
ax.set_ylabel('$f_n(x)$')
plt.show()
def plot_root_bracketing_pattern(f, a, b, dx, xbounds=(-0.1,1.4), ybounds=(-5,6)):
x = np.linspace(a, b, int((b-a)/dx)+1)
y = f(x)
fig, ax = plt.subplots()
ax.plot(x, y, marker='x', color='r')
switched = False
for i, xt in enumerate(x):
ft = str('$f_' + str(i) + '=$' + '${:4.2f}$'.format(y[i]))
ax.text(xt, y[i], ft)
if i > 0:
if not switched and np.sign(y[i]) != np.sign(y[i-1]):
pass
ax.plot([x[i],x[i-1]], [y[i],y[i-1]], color='b')
ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
ax.plot([xbounds[0],xbounds[1]],[0.,0.],color='gray',ls='-.',alpha=0.75)
ax.set_xlim(xbounds[0],xbounds[1])
ax.set_ylim(ybounds[0],ybounds[1])
plt.title('Bracketing')
plt.show()
def plot_bisection_pattern(f, x1, x2, tol=1.0e-5, inset=True, ixmin=0.715, ixmax=0.755, iymin=-0.25, iymax=0.2, zoom=3, loc0=3, loc1=3, loc2=2):
fig, ax = plt.subplots()
plt.title('Bisection')
x = np.linspace(x1, x2, 100)
y = f(x)
ax.plot(x, y, color='r')
f1 = f(x1)
f2 = f(x2)
x1s = x1
x2s = x2
ax.scatter( [x1,x2], [f1,f2], marker='x', s=50)
virgin = True
while abs(x1-x2) > tol:
x3 = 0.5*(x1 + x2)
f3 = f(x3)
ax.scatter( [x3], [f3], marker='x', s=50)
if f2*f3 < 0.0:
x1 = x3
f1 = f3
else:
x2 = x3
f2 = f3
# zero line
xlim = ax.get_xlim()
ax.plot([xlim[0],xlim[1]],[0.,0.],color='gray',ls='-.',alpha=0.75)
ax.set_xlim(xlim)
# zoomed inset
if inset:
x1, x2 = x1s, x2s
axins = zoomed_inset_axes(ax, zoom, loc=loc0)
x = np.linspace(x1, x2, 100)
y = f(x)
axins.plot(x, y, color='r')
f1 = f(x1)
f2 = f(x2)
axins.scatter( [x1,x2], [f1,f2], marker='x', s=50)
virgin = True
while abs(x1-x2) > tol:
x3 = 0.5*(x1 + x2)
f3 = f(x3)
axins.scatter( [x3], [f3], marker='x', s=50)
if f2*f3 < 0.0:
x1 = x3
f1 = f3
else:
x2 = x3
f2 = f3
axins.plot([ixmin,ixmax],[0.,0.],color='gray',ls='-.',alpha=0.75)
axins.set_xlim(ixmin, ixmax)
axins.set_ylim(iymin, iymax)
axins.get_xaxis().set_visible(False)
axins.get_yaxis().set_visible(False)
mark_inset(ax, axins, loc1=loc1, loc2=loc2, fc="none", ec="0.5")
xf = (x1 + x2)/2.0
ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
plt.show()
# =============================
# INTERPOLATION & CURVE FITTING
# =============================
def interpolation_vs_curve_fitting():
matplotlib.rcParams['figure.figsize'] = 16,6
plt.subplot(1, 2, 1)
x = np.linspace(0, 10, 10)
y = np.cos(-x**2/8.0)
f = scip.interp1d(x, y)
f2 = scip.interp1d(x, y, kind='cubic')
xnew = np.linspace(0, 10, 40)
plt.scatter(x,y, marker='x', s=45)
plt.plot(x,y,'-', color='green')
plt.plot(xnew, f2(xnew),'--', color='red')
plt.title('Interpolation')
plt.subplot(1, 2, 2)
def foo(x, a, b, c):
return a*np.exp(-b*x) + c
t = np.linspace(0,4,50)
T = foo(t, 5.0, 1.5, 0.5)
d = T + 0.25*np.random.normal(size=len(T))
fp, fc = scop.curve_fit(foo, t, d)
plt.scatter(t, d, marker='x', s=30)
plt.plot(t, foo(t, fp[0], fp[1], fp[2]), color='green')
plt.xlim(0,4.5)
plt.title('Curve Fitting')
plt.show()
def least_squares():
def foo(x):
return x**1.5
x_sample = np.linspace(0,5,7)
y_sample = foo(x_sample) + 2.*np.random.normal(size=len(x_sample))
x_fit = np.linspace(0,5,100)
y_fit = foo(x_fit)
for i, x in enumerate(x_sample):
plt.plot([x,x], [foo(x), y_sample[i]], color='red')
plt.plot(x_sample, y_sample, 'x', mew=2, ms=5)
plt.plot(x_fit,y_fit)
plt.xlim(-0.5, 5.5)
plt.xlabel('x')
plt.ylabel('y')
plt.show()