Chapter 5: Equation solving

Robert Johansson

Source code listings for Numerical Python - A Practical Techniques Approach for Industry (ISBN 978-1-484205-54-9).

The source code listings can be downloaded from http://www.apress.com/9781484205549

In [1]:
from scipy import linalg as la
In [2]:
from scipy import optimize
In [3]:
import sympy
In [4]:
sympy.init_printing()
In [5]:
import numpy as np
In [6]:
import matplotlib.pyplot as plt
%matplotlib inline

import matplotlib as mpl
mpl.rcParams["font.family"] = "serif"
mpl.rcParams["font.size"] = "12"
In [7]:
from __future__ import division

Linear Algebra - Linear Equation Systems

$$ 2 x_1 + 3 x_2 = 4 $$

$$ 5 x_1 + 4 x_2 = 3 $$

In [8]:
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 [9]:
A = sympy.Matrix([[2, 3], [5, 4]])
b = sympy.Matrix([4, 3])
In [10]:
A.rank()
Out[10]:
$$2$$
In [11]:
A.condition_number()
Out[11]:
$$\frac{\sqrt{2 \sqrt{170} + 27}}{\sqrt{- 2 \sqrt{170} + 27}}$$
In [12]:
sympy.N(_)
Out[12]:
$$7.58240137440151$$
In [13]:
A.norm()
Out[13]:
$$3 \sqrt{6}$$
In [14]:
L, U, _ = A.LUdecomposition()
In [15]:
L
Out[15]:
$$\left[\begin{matrix}1 & 0\\\frac{5}{2} & 1\end{matrix}\right]$$
In [16]:
U
Out[16]:
$$\left[\begin{matrix}2 & 3\\0 & - \frac{7}{2}\end{matrix}\right]$$
In [17]:
L * U
Out[17]:
$$\left[\begin{matrix}2 & 3\\5 & 4\end{matrix}\right]$$
In [18]:
x = A.solve(b)
In [19]:
x
Out[19]:
$$\left[\begin{matrix}-1\\2\end{matrix}\right]$$

Numerical approach

In [20]:
A = np.array([[2, 3], [5, 4]])
b = np.array([4, 3])
In [21]:
np.linalg.matrix_rank(A)
Out[21]:
$$2$$
In [22]:
np.linalg.cond(A)
Out[22]:
$$7.5824013744$$
In [23]:
np.linalg.norm(A)
Out[23]:
$$7.34846922835$$
In [24]:
P, L, U = la.lu(A)
In [25]:
L
Out[25]:
array([[ 1. ,  0. ],
       [ 0.4,  1. ]])
In [26]:
U
Out[26]:
array([[ 5. ,  4. ],
       [ 0. ,  1.4]])
In [27]:
L*U
Out[27]:
array([[ 5. ,  0. ],
       [ 0. ,  1.4]])
In [28]:
la.solve(A, b)
Out[28]:
array([-1.,  2.])

Example : rank and condition numbers -> numerical errors

In [29]:
p = sympy.symbols("p", positive=True)
In [30]:
A = sympy.Matrix([[1, sympy.sqrt(p)], [1, 1/sympy.sqrt(p)]])
In [31]:
b = sympy.Matrix([1, 2])
In [32]:
sympy.simplify(A.solve(b))
Out[32]:
$$\left[\begin{matrix}\frac{2 p - 1}{p - 1}\\- \frac{\sqrt{p}}{p - 1}\end{matrix}\right]$$
In [33]:
# 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 [34]:
unknown = sympy.symbols("x, y, z")
In [35]:
A = sympy.Matrix([[1, 2, 3], [4, 5, 6]])
In [36]:
x = sympy.Matrix(unknown)
In [37]:
b = sympy.Matrix([7, 8])
In [38]:
AA = A * x - b
In [39]:
sympy.solve(A*x - b, unknown)
Out[39]:
$$\left \{ x : z - \frac{19}{3}, \quad y : - 2 z + \frac{20}{3}\right \}$$

Overdetermined: least squares

In [40]:
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 [41]:
# 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 [42]:
eps, delta = sympy.symbols("epsilon, delta")
In [43]:
H = sympy.Matrix([[eps, delta], [delta, -eps]])
H
Out[43]:
$$\left[\begin{matrix}\epsilon & \delta\\\delta & - \epsilon\end{matrix}\right]$$
In [44]:
eval1, eval2 = H.eigenvals()
In [45]:
eval1, eval2
Out[45]:
$$\left ( - \sqrt{\delta^{2} + \epsilon^{2}}, \quad \sqrt{\delta^{2} + \epsilon^{2}}\right )$$
In [46]:
H.eigenvects()
Out[46]:
$$\left [ \left ( - \sqrt{\delta^{2} + \epsilon^{2}}, \quad 1, \quad \left [ \left[\begin{matrix}- \frac{\delta}{\epsilon + \sqrt{\delta^{2} + \epsilon^{2}}}\\1\end{matrix}\right]\right ]\right ), \quad \left ( \sqrt{\delta^{2} + \epsilon^{2}}, \quad 1, \quad \left [ \left[\begin{matrix}- \frac{\delta}{\epsilon - \sqrt{\delta^{2} + \epsilon^{2}}}\\1\end{matrix}\right]\right ]\right )\right ]$$
In [47]:
(eval1, _, evec1), (eval2, _, evec2) = H.eigenvects()
In [48]:
sympy.simplify(evec1[0].T * evec2[0])
Out[48]:
$$\left[\begin{matrix}0\end{matrix}\right]$$
In [49]:
A = np.array([[1, 3, 5], [3, 5, 3], [5, 3, 9]])
A
Out[49]:
array([[1, 3, 5],
       [3, 5, 3],
       [5, 3, 9]])
In [50]:
evals, evecs = la.eig(A)
In [51]:
evals
Out[51]:
array([ 13.35310908+0.j,  -1.75902942+0.j,   3.40592034+0.j])
In [52]:
evecs
Out[52]:
array([[ 0.42663918,  0.90353276, -0.04009445],
       [ 0.43751227, -0.24498225, -0.8651975 ],
       [ 0.79155671, -0.35158534,  0.49982569]])
In [53]:
la.eigvalsh(A)
Out[53]:
array([ -1.75902942,   3.40592034,  13.35310908])

Nonlinear equations

Univariate

In [54]:
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 [55]:
import sympy as s
In [56]:
x, a, b, c = sympy.symbols("x, a, b, c")
In [57]:
sympy.solve(a + b*x + c*x**2, x)
Out[57]:
$$\left [ \frac{1}{2 c} \left(- b + \sqrt{- 4 a c + b^{2}}\right), \quad - \frac{1}{2 c} \left(b + \sqrt{- 4 a c + b^{2}}\right)\right ]$$
In [58]:
sympy.solve(a * sympy.cos(x) - b * sympy.sin(x), x)
Out[58]:
$$\left [ - 2 \operatorname{atan}{\left (\frac{1}{a} \left(b - \sqrt{a^{2} + b^{2}}\right) \right )}, \quad - 2 \operatorname{atan}{\left (\frac{1}{a} \left(b + \sqrt{a^{2} + b^{2}}\right) \right )}\right ]$$
In [59]:
sympy.solve(sympy.sin(x)-x, x)
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-59-5e1a5dc1567d> in <module>()
----> 1 sympy.solve(sympy.sin(x)-x, x)

/Users/rob/miniconda/envs/py27-npm/lib/python2.7/site-packages/sympy/solvers/solvers.pyc in solve(f, *symbols, **flags)
    907     ###########################################################################
    908     if bare_f:
--> 909         solution = _solve(f[0], *symbols, **flags)
    910     else:
    911         solution = _solve_system(f, symbols, **flags)

/Users/rob/miniconda/envs/py27-npm/lib/python2.7/site-packages/sympy/solvers/solvers.pyc in _solve(f, *symbols, **flags)
   1412     if result is False:
   1413         raise NotImplementedError(msg +
-> 1414         "\nNo algorithms are implemented to solve equation %s" % f)
   1415 
   1416     if flags.get('simplify', True):

NotImplementedError: multiple generators [x, sin(x)]
No algorithms are implemented to solve equation -x + sin(x)

Bisection method

In [60]:
# 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 [61]:
# 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 [62]:
optimize.bisect(lambda x: np.exp(x) - 2, -2, 2)
Out[62]:
$$0.693147180559$$
In [63]:
optimize.newton(lambda x: np.exp(x) - 2, 2)
Out[63]:
$$0.69314718056$$
In [64]:
x_root_guess = 2
In [65]:
f = lambda x: np.exp(x) - 2
In [66]:
fprime = lambda x: np.exp(x)
In [67]:
optimize.newton(f, x_root_guess)
Out[67]:
$$0.69314718056$$
In [68]:
optimize.newton(f, x_root_guess, fprime=fprime)
Out[68]:
$$0.69314718056$$
In [69]:
optimize.brentq(lambda x: np.exp(x) - 2, -2, 2)
Out[69]:
$$0.69314718056$$
In [70]:
optimize.brenth(lambda x: np.exp(x) - 2, -2, 2)
Out[70]:
$$0.69314718056$$
In [71]:
optimize.ridder(lambda x: np.exp(x) - 2, -2, 2)
Out[71]:
$$0.69314718056$$

Multivariate

In [72]:
def f(x):
    return [x[1] - x[0]**3 - 2 * x[0]**2 + 1, x[1] + x[0]**2 - 1]
In [73]:
optimize.fsolve(f, [1, 1])
Out[73]:
array([ 0.73205081,  0.46410162])
In [74]:
def f_jacobian(x):
    return [[-3*x[0]**2-4*x[0], 1], [2*x[0], 1]]
In [75]:
optimize.fsolve(f, [1, 1], fprime=f_jacobian)
Out[75]:
array([ 0.73205081,  0.46410162])
In [76]:
#import sympy as s
In [77]:
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]))
Out[77]:
$$\left[\begin{matrix}- 3 x^{2} - 4 x & 1\\2 x & 1\end{matrix}\right]$$
In [78]:
#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 [79]:
optimize.broyden2(f, x_guesses[1])
Out[79]:
array([ 0.73205079,  0.46410162])
In [80]:
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])

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')
/Users/rob/miniconda/envs/py27-npm/lib/python2.7/site-packages/scipy/optimize/minpack.py:236: RuntimeWarning: The iteration is not making good progress, as measured by the 
  improvement from the last ten iterations.
  warnings.warn(msg, RuntimeWarning)

Versions

In [81]:
%reload_ext version_information
In [82]:
%version_information sympy, scipy, numpy, matplotlib
Out[82]:
SoftwareVersion
Python2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]
IPython3.2.1
OSDarwin 14.1.0 x86_64 i386 64bit
sympy0.7.6
scipy0.16.0
numpy1.9.2
matplotlib1.4.3