Functions

This module contains the algorithms for the nonlinear equations lab.

The materials follow Miranda and Fackler (2004, [MF04]) (Chapter 3). The python code draws on Romero-Aguilar (2020, [RA20]).

labs.nonlinear_equations.nonlinear_algorithms.bisect(f, a, b, tolerance=1.5e-08)[source]

Apply bisect method to root finding problem.

Iterative procedure to find the root of a continuous real-values function $f(x)$ defined on a bounded interval of the real line. Define interval $[a, b]$ that is known to contain or bracket the root of $f$ (i.e. the signs of $f(a)$ and $f(b)$ must differ). The given interval $[a, b]$ is then repeatedly bisected into subintervals of equal length. Each iteration, one of the two subintervals has endpoints of different signs (thus containing the root of $f$ ) and is again bisected until the size of the subinterval containing the root reaches a specified convergence tolerance.

Parameters

f (callable) – Continuous, real-valued, univariate function $f(x)$ .
a (int or float) – Lower bound $a$ for $x \in [a,b]$ .
b (int or float) – Upper bound $a$ for $x \in [a,b]$ . Select $a$ and $b$ so that $f(b)$ has different sign than $f(a)$ .
tolerance (float) – Convergence tolerance.

Returns

x – Solution to the root finding problem within specified tolerance.

Return type

float

Examples

>>> x = bisect(f=lambda x : x ** 3 - 2, a=1, b=2)[0]
>>> round(x, 4)
1.2599

labs.nonlinear_equations.nonlinear_algorithms.fischer(u, v, sign)[source]

Define Fischer’s function.

$\phi_{i}^{\pm}(u, v) = u_{i} + v_{i} \pm \sqrt{u_{i}^{2} + v_{i}^{2}}$

Parameters

u (float) –
v (float) –
sign (float or int) – Gives sign of equation. Should be either 1 or -1.

Returns

Return type

callable

labs.nonlinear_equations.nonlinear_algorithms.fixpoint(f, x0, tolerance=0.0001)[source]

Compute fixed point using function iteration.

Parameters

f (callable) – Function $f(x)$ .
x0 (float) – Initial guess for fixed point (starting value for function iteration).
tolerance (float) – Convergence tolerance (tolerance < 1).

Returns

x – Solution of function iteration.

Return type

float

Examples

>>> import numpy as np
>>> x = fixpoint(f=lambda x : x**0.5, x0=0.4, tolerance=1e-10)[0]
>>> np.allclose(x, 1)
True

labs.nonlinear_equations.nonlinear_algorithms.funcit(f, x0=2)[source]: Apply function iteration using the fixpoint method.

labs.nonlinear_equations.nonlinear_algorithms.newton_method(f, x0, tolerance=1.5e-08)[source]

Apply Newton’s method to solving nonlinear equation.

Solve equation using successive linearization, which replaces the nonlinear problem by a sequence of linear problems whose solutions converge to the solution of the nonlinear problem.

Parameters

f (callable) – (Univariate) function $f(x)$ .
x0 (float) – Initial guess for the root of $f$ .
tolerance (float) – Convergence tolerance.

Returns

xn – Solution of function iteration.

Return type

float