#!/usr/bin/env python
# coding: utf-8

# # Chapter 5: Equation solving

# Robert Johansson
# 
# Source code listings for [Numerical Python - Scientific Computing and Data Science Applications with Numpy, SciPy and Matplotlib](https://www.apress.com/us/book/9781484242452) (ISBN 978-1-484242-45-2).

# In[1]:


get_ipython().run_line_magic('matplotlib', 'inline')
import matplotlib.pyplot as plt
import matplotlib
matplotlib.rcParams['mathtext.fontset'] = 'stix'
matplotlib.rcParams['font.family'] = 'serif'
matplotlib.rcParams['font.sans-serif'] = 'stix'


# In[2]:


from scipy import linalg as la


# In[3]:


from scipy import optimize


# In[4]:


import sympy


# In[5]:


sympy.init_printing()


# In[6]:


import numpy as np


# In[7]:


import matplotlib.pyplot as plt
get_ipython().run_line_magic('matplotlib', 'inline')

#import matplotlib as mpl
#mpl.rcParams["font.family"] = "serif"
#mpl.rcParams["font.size"] = "12"


# In[8]:


from __future__ import division


# ## Linear Algebra - Linear Equation Systems

# $$
# 2 x_1 + 3 x_2 = 4
# $$
# 
# $$
# 5 x_1 + 4 x_2 = 3
# $$

# In[9]:


fig, ax = plt.subplots(figsize=(8, 4))

x1 = np.linspace(-4, 2, 100)

x2_1 = (4 - 2 * x1)/3
x2_2 = (3 - 5 * x1)/4

ax.plot(x1, x2_1, 'r', lw=2, label=r"$2x_1+3x_2-4=0$")
ax.plot(x1, x2_2, 'b', lw=2, label=r"$5x_1+4x_2-3=0$")

A = np.array([[2, 3], [5, 4]])
b = np.array([4, 3])
x = la.solve(A, b)

ax.plot(x[0], x[1], 'ko', lw=2)
ax.annotate("The intersection point of\nthe two lines is the solution\nto the equation system",
            xy=(x[0], x[1]), xycoords='data',
            xytext=(-120, -75), textcoords='offset points', 
            arrowprops=dict(arrowstyle="->", connectionstyle="arc3, rad=-.3"))

ax.set_xlabel(r"$x_1$", fontsize=18)
ax.set_ylabel(r"$x_2$", fontsize=18)
ax.legend();

fig.tight_layout()
fig.savefig('ch5-linear-systems-simple.pdf')


# ### Symbolic approach

# In[10]:


A = sympy.Matrix([[2, 3], [5, 4]])
b = sympy.Matrix([4, 3])


# In[11]:


A.rank()


# In[12]:


A.condition_number()


# In[13]:


sympy.N(_)


# In[14]:


A.norm()


# In[15]:


L, U, P = A.LUdecomposition()


# In[16]:


L


# In[17]:


U


# In[18]:


L * U


# In[19]:


x = A.solve(b)


# In[20]:


x


# ### Numerical approach

# In[21]:


A = np.array([[2, 3], [5, 4]])
b = np.array([4, 3])


# In[22]:


np.linalg.matrix_rank(A)


# In[23]:


np.linalg.cond(A)


# In[24]:


np.linalg.norm(A)


# In[25]:


P, L, U = la.lu(A)


# In[26]:


L


# In[27]:


U


# In[28]:


np.dot(L, U)


# In[29]:


np.dot(P, np.dot(L, U))


# In[30]:


P.dot(L.dot(U))


# In[31]:


la.solve(A, b)


# ### Example : rank and condition numbers -> numerical errors

# In[32]:


p = sympy.symbols("p", positive=True)


# In[33]:


A = sympy.Matrix([[1, sympy.sqrt(p)], [1, 1/sympy.sqrt(p)]])


# In[34]:


b = sympy.Matrix([1, 2])


# In[35]:


sympy.simplify(A.solve(b))


# In[36]:


# Symbolic problem specification
p = sympy.symbols("p", positive=True)
A = sympy.Matrix([[1, sympy.sqrt(p)], [1, 1/sympy.sqrt(p)]])
b = sympy.Matrix([1, 2])

# Solve symbolically
x_sym_sol = A.solve(b)
x_sym_sol.simplify()
x_sym_sol
Acond = A.condition_number().simplify()

# Function for solving numerically
AA = lambda p: np.array([[1, np.sqrt(p)], [1, 1/np.sqrt(p)]])
bb = np.array([1, 2])
x_num_sol = lambda p: np.linalg.solve(AA(p), bb)

# Graph the difference between the symbolic (exact) and numerical results.
p_vec = np.linspace(0.9, 1.1, 200)

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

for n in range(2):
    x_sym = np.array([x_sym_sol[n].subs(p, pp).evalf() for pp in p_vec])
    x_num = np.array([x_num_sol(pp)[n] for pp in p_vec])
    axes[0].plot(p_vec, (x_num - x_sym)/x_sym, 'k')
axes[0].set_title("Error in solution\n(numerical - symbolic)/symbolic")
axes[0].set_xlabel(r'$p$', fontsize=18)

axes[1].plot(p_vec, [Acond.subs(p, pp).evalf() for pp in p_vec])
axes[1].set_title("Condition number")
axes[1].set_xlabel(r'$p$', fontsize=18)

fig.tight_layout()
fig.savefig('ch5-linear-systems-condition-number.pdf')


# ### Rectangular systems

# ### Underdetermined

# In[37]:


unknown = sympy.symbols("x, y, z")


# In[38]:


A = sympy.Matrix([[1, 2, 3], [4, 5, 6]])


# In[39]:


x = sympy.Matrix(unknown)


# In[40]:


b = sympy.Matrix([7, 8])


# In[41]:


AA = A * x - b


# In[42]:


sympy.solve(A*x - b, unknown)


# ### Overdetermined: least squares

# In[43]:


np.random.seed(1234)

# define true model parameters
x = np.linspace(-1, 1, 100)
a, b, c = 1, 2, 3
y_exact = a + b * x + c * x**2

# simulate noisy data points
m = 100
X = 1 - 2 * np.random.rand(m)
Y = a + b * X + c * X**2 + np.random.randn(m)

# fit the data to the model using linear least square
A = np.vstack([X**0, X**1, X**2])  # see np.vander for alternative
sol, r, rank, sv = la.lstsq(A.T, Y)
y_fit = sol[0] + sol[1] * x + sol[2] * x**2
fig, ax = plt.subplots(figsize=(12, 4))

ax.plot(X, Y, 'go', alpha=0.5, label='Simulated data')
ax.plot(x, y_exact, 'k', lw=2, label='True value $y = 1 + 2x + 3x^2$')
ax.plot(x, y_fit, 'b', lw=2, label='Least square fit')
ax.set_xlabel(r"$x$", fontsize=18)
ax.set_ylabel(r"$y$", fontsize=18)
ax.legend(loc=2);

fig.savefig('ch5-linear-systems-least-square.pdf')


# In[44]:


# fit the data to the model using linear least square: 
# 1st order polynomial
A = np.vstack([X**n for n in range(2)])
sol, r, rank, sv = la.lstsq(A.T, Y)
y_fit1 = sum([s * x**n for n, s in enumerate(sol)])

# 15th order polynomial
A = np.vstack([X**n for n in range(16)])
sol, r, rank, sv = la.lstsq(A.T, Y)
y_fit15 = sum([s * x**n for n, s in enumerate(sol)])

fig, ax = plt.subplots(figsize=(12, 4))
ax.plot(X, Y, 'go', alpha=0.5, label='Simulated data')
ax.plot(x, y_exact, 'k', lw=2, label='True value $y = 1 + 2x + 3x^2$')
ax.plot(x, y_fit1, 'b', lw=2, label='Least square fit [1st order]')
ax.plot(x, y_fit15, 'm', lw=2, label='Least square fit [15th order]')
ax.set_xlabel(r"$x$", fontsize=18)
ax.set_ylabel(r"$y$", fontsize=18)
ax.legend(loc=2);

fig.savefig('ch5-linear-systems-least-square-2.pdf')


# ## Eigenvalue problems

# In[45]:


eps, delta = sympy.symbols("epsilon, delta")


# In[46]:


H = sympy.Matrix([[eps, delta], [delta, -eps]])
H


# In[47]:


eval1, eval2 = H.eigenvals()


# In[48]:


eval1, eval2


# In[49]:


H.eigenvects()


# In[50]:


(eval1, _, evec1), (eval2, _, evec2) = H.eigenvects()


# In[51]:


sympy.simplify(evec1[0].T * evec2[0])


# In[52]:


A = np.array([[1, 3, 5], [3, 5, 3], [5, 3, 9]])
A


# In[53]:


evals, evecs = la.eig(A)


# In[54]:


evals


# In[55]:


evecs


# In[56]:


la.eigvalsh(A)


# ## Nonlinear equations

# ### Univariate

# In[57]:


x = np.linspace(-2, 2, 1000)

# four examples of nonlinear functions
f1 = x**2 - x - 1
f2 = x**3 - 3 * np.sin(x)
f3 = np.exp(x) - 2
f4 = 1 - x**2 + np.sin(50 / (1 + x**2))

# plot each function
fig, axes = plt.subplots(1, 4, figsize=(12, 3), sharey=True)

for n, f in enumerate([f1, f2, f3, f4]):
    axes[n].plot(x, f, lw=1.5)
    axes[n].axhline(0, ls=':', color='k')
    axes[n].set_ylim(-5, 5)
    axes[n].set_xticks([-2, -1, 0, 1, 2])
    axes[n].set_xlabel(r'$x$', fontsize=18)

axes[0].set_ylabel(r'$f(x)$', fontsize=18)

titles = [r'$f(x)=x^2-x-1$', r'$f(x)=x^3-3\sin(x)$',
          r'$f(x)=\exp(x)-2$', r'$f(x)=\sin\left(50/(1+x^2)\right)+1-x^2$']
for n, title in enumerate(titles):
    axes[n].set_title(title)
    
fig.tight_layout()
fig.savefig('ch5-nonlinear-plot-equations.pdf')


# ### Symbolic

# In[58]:


x, a, b, c = sympy.symbols("x, a, b, c")


# In[59]:


e = a + b*x + c*x**2


# In[60]:


sol1, sol2 = sympy.solve(e, x)

sol1, sol2


# In[61]:


e.subs(x, sol1).expand()


# In[62]:


e.subs(x, sol2).expand()


# In[63]:


e = a * sympy.cos(x) - b * sympy.sin(x)


# In[64]:


sol1, sol2 = sympy.solve(e, x)

sol1, sol2


# In[65]:


e.subs(x, sympy.atan(a/b))


# In[66]:


e.subs(x, sol1).simplify()


# In[67]:


e.subs(x, sol2).simplify()


# In[68]:


sympy.solve(sympy.sin(x)-x, x)


# ### Bisection method

# In[ ]:


# define a function, desired tolerance and starting interval [a, b]
f = lambda x: np.exp(x) - 2
tol = 0.1
a, b = -2, 2
x = np.linspace(-2.1, 2.1, 1000)

# graph the function f
fig, ax = plt.subplots(1, 1, figsize=(12, 4))

ax.plot(x, f(x), lw=1.5)
ax.axhline(0, ls=':', color='k')
ax.set_xticks([-2, -1, 0, 1, 2])
ax.set_xlabel(r'$x$', fontsize=18)
ax.set_ylabel(r'$f(x)$', fontsize=18)

# find the root using the bisection method and visualize
# the steps in the method in the graph
fa, fb = f(a), f(b)

ax.plot(a, fa, 'ko')
ax.plot(b, fb, 'ko')
ax.text(a, fa + 0.5, r"$a$", ha='center', fontsize=18)
ax.text(b, fb + 0.5, r"$b$", ha='center', fontsize=18)

n = 1
while b - a > tol:
    m = a + (b - a)/2
    fm = f(m)

    ax.plot(m, fm, 'ko')
    ax.text(m, fm - 0.5, r"$m_%d$" % n, ha='center')
    n += 1
    
    if np.sign(fa) == np.sign(fm):
        a, fa = m, fm
    else:
        b, fb = m, fm

ax.plot(m, fm, 'r*', markersize=10)
ax.annotate("Root approximately at %.3f" % m,
            fontsize=14, family="serif",
            xy=(a, fm), xycoords='data',
            xytext=(-150, +50), textcoords='offset points', 
            arrowprops=dict(arrowstyle="->", connectionstyle="arc3, rad=-.5"))

ax.set_title("Bisection method")

fig.tight_layout()
fig.savefig('ch5-nonlinear-bisection.pdf')


# In[ ]:


# define a function, desired tolerance and starting point xk
tol = 0.01
xk = 2

s_x = sympy.symbols("x")
s_f = sympy.exp(s_x) - 2

f = lambda x: sympy.lambdify(s_x, s_f, 'numpy')(x)
fp = lambda x: sympy.lambdify(s_x, sympy.diff(s_f, s_x), 'numpy')(x)

x = np.linspace(-1, 2.1, 1000)

# setup a graph for visualizing the root finding steps
fig, ax = plt.subplots(1, 1, figsize=(12,4))

ax.plot(x, f(x))
ax.axhline(0, ls=':', color='k')

# repeat Newton's method until convergence to the desired tolerance has been reached
n = 0
while f(xk) > tol:
    xk_new = xk - f(xk) / fp(xk)

    ax.plot([xk, xk], [0, f(xk)], color='k', ls=':')
    ax.plot(xk, f(xk), 'ko')
    ax.text(xk, -.5, r'$x_%d$' % n, ha='center')
    ax.plot([xk, xk_new], [f(xk), 0], 'k-')

    xk = xk_new
    n += 1

ax.plot(xk, f(xk), 'r*', markersize=15)
ax.annotate("Root approximately at %.3f" % xk,
            fontsize=14, family="serif",
            xy=(xk, f(xk)), xycoords='data',
            xytext=(-150, +50), textcoords='offset points', 
            arrowprops=dict(arrowstyle="->", connectionstyle="arc3, rad=-.5"))

ax.set_title("Newton's method")
ax.set_xticks([-1, 0, 1, 2])
fig.tight_layout()
fig.savefig('ch5-nonlinear-newton.pdf')


# ### `scipy.optimize` functions for root-finding

# In[69]:


optimize.bisect(lambda x: np.exp(x) - 2, -2, 2)


# In[70]:


optimize.newton(lambda x: np.exp(x) - 2, 2)


# In[71]:


x_root_guess = 2


# In[72]:


f = lambda x: np.exp(x) - 2


# In[73]:


fprime = lambda x: np.exp(x)


# In[74]:


optimize.newton(f, x_root_guess)


# In[75]:


optimize.newton(f, x_root_guess, fprime=fprime)


# In[76]:


optimize.brentq(lambda x: np.exp(x) - 2, -2, 2)


# In[77]:


optimize.brenth(lambda x: np.exp(x) - 2, -2, 2)


# In[78]:


optimize.ridder(lambda x: np.exp(x) - 2, -2, 2)


# ### Multivariate

# In[79]:


def f(x):
    return [x[1] - x[0]**3 - 2 * x[0]**2 + 1, x[1] + x[0]**2 - 1]


# In[80]:


optimize.fsolve(f, [1, 1])


# In[81]:


def f_jacobian(x):
    return [[-3*x[0]**2-4*x[0], 1], [2*x[0], 1]]


# In[82]:


optimize.fsolve(f, [1, 1], fprime=f_jacobian)


# In[83]:


x, y = sympy.symbols("x, y")

f_mat = sympy.Matrix([y - x**3 -2*x**2 + 1, y + x**2 - 1])
f_mat.jacobian(sympy.Matrix([x, y]))


# In[84]:


#def f(x):
#    return [x[1] - x[0]**3 - 2 * x[0]**2 + 1, x[1] + x[0]**2 - 1]

x = np.linspace(-3, 2, 5000)
y1 = x**3 + 2 * x**2 -1
y2 = -x**2 + 1

fig, ax = plt.subplots(figsize=(8, 4))

ax.plot(x, y1, 'b', lw=1.5, label=r'$y = x^3 + 2x^2 - 1$')
ax.plot(x, y2, 'g', lw=1.5, label=r'$y = -x^2 + 1$')

x_guesses = [[-2, 2], [1, -1], [-2, -5]]
for x_guess in x_guesses:
    sol = optimize.fsolve(f, x_guess)
    ax.plot(sol[0], sol[1], 'r*', markersize=15)

    ax.plot(x_guess[0], x_guess[1], 'ko')
    ax.annotate("", xy=(sol[0], sol[1]), xytext=(x_guess[0], x_guess[1]),
                arrowprops=dict(arrowstyle="->", linewidth=2.5))
    
ax.legend(loc=0)
ax.set_xlabel(r'$x$', fontsize=18)
fig.tight_layout()
fig.savefig('ch5-nonlinear-system.pdf')


# In[85]:


optimize.broyden2(f, x_guesses[1])


# In[86]:


def f(x):
    return [x[1] - x[0]**3 - 2 * x[0]**2 + 1,
            x[1] + x[0]**2 - 1]

x = np.linspace(-3, 2, 5000)
y1 = x**3 + 2 * x**2 -1
y2 = -x**2 + 1

fig, ax = plt.subplots(figsize=(8, 4))

ax.plot(x, y1, 'k', lw=1.5, label=r'$y = x^3 + 2x^2 - 1$')
ax.plot(x, y2, 'k', lw=1.5, label=r'$y = -x^2 + 1$')

sol1 = optimize.fsolve(f, [-2,  2])
sol2 = optimize.fsolve(f, [ 1, -1])
sol3 = optimize.fsolve(f, [-2, -5])
sols = [sol1, sol2, sol3]
colors = ['r', 'b', 'g']
for idx, s in enumerate(sols):
    ax.plot(s[0], s[1], colors[idx]+'*', markersize=15)

for m in np.linspace(-4, 3, 80):
    for n in np.linspace(-15, 15, 40):
        x_guess = [m, n]
        sol = optimize.fsolve(f, x_guess)
        idx = (abs(sols - sol)**2).sum(axis=1).argmin()
        ax.plot(x_guess[0], x_guess[1], colors[idx]+'.')
    
ax.set_xlabel(r'$x$', fontsize=18)
ax.set_xlim(-4, 3)
ax.set_ylim(-15, 15)
fig.tight_layout()
fig.savefig('ch5-nonlinear-system-map.pdf')


# In[87]:


def f(x):
    return [x[1] - x[0]**3 - 2 * x[0]**2 + 1,
            x[1] + x[0]**2 - 1]

x = np.linspace(-3, 2, 5000)
y1 = x**3 + 2 * x**2 -1
y2 = -x**2 + 1

fig, ax = plt.subplots(figsize=(8, 4))

ax.plot(x, y1, 'k', lw=1.5, label=r'$y = x^3 + 2x^2 - 1$')
ax.plot(x, y2, 'k', lw=1.5, label=r'$y = -x^2 + 1$')

sol1 = optimize.fsolve(f, [-2,  2])
sol2 = optimize.fsolve(f, [ 1, -1])
sol3 = optimize.fsolve(f, [-2, -5])

for idx, s in enumerate([sol1, sol2, sol3]):
    ax.plot(s[0], s[1], colors[idx]+'*', markersize=15)

colors = ['r', 'b', 'g']
for m in np.linspace(-4, 3, 80):
    for n in np.linspace(-15, 15, 40):
        x_guess = [m, n]
        sol = optimize.fsolve(f, x_guess)

        for idx, s in enumerate([sol1, sol2, sol3]):
            if abs(s-sol).max() < 1e-8:
                # ax.plot(sol[0], sol[1], colors[idx]+'*', markersize=15)
                ax.plot(x_guess[0], x_guess[1], colors[idx]+'.')
    
ax.set_xlabel(r'$x$', fontsize=18)
fig.tight_layout()
fig.savefig('ch5-nonlinear-system-map.pdf')


# ## Versions

# In[88]:


get_ipython().run_line_magic('reload_ext', 'version_information')


# In[91]:


get_ipython().run_line_magic('version_information', 'sympy, scipy, numpy, matplotlib')