The purpose of this exercise is to understand when Newton's method works and fails. To this end, solve \( \tanh x=0 \) by Newton's method and study the intermediate details of the algorithm. Start with \( x_0=1.08 \). Plot the tangent in each iteration of Newton's method. Then repeat the calculations and the plotting when \( x_0=1.09 \). Explain what you observe.
Filename: Newton_failure.*.
Does the secant method behave better than Newton's method in the problem described in Exercise 6.1: Understand why Newton's method can fail? Try the initial guesses
secant_failure.*.
Solve the same problem as in Exercise 6.1: Understand why Newton's method can fail, using the bisection method, but let the initial interval be \( [-5,3] \). Report how the interval containing the solution evolves during the iterations.
Filename: bisection_nonfailure.*.
An attractive idea is to combine the reliability of the bisection method with the speed of Newton's method. Such a combination is implemented by running the bisection method until we have a narrow interval, and then switch to Newton's method for speed.
Write a function that implements this idea. Start with an interval \( [a,b] \) and switch to Newton's method when the current interval in the bisection method is a fraction \( s \) of the initial interval (i.e., when the interval has length \( s(b-a) \)). Potential divergence of Newton's method is still an issue, so if the approximate root jumps out of the narrowed interval (where the solution is known to lie), one can switch back to the bisection method. The value of \( s \) must be given as an argument to the function, but it may have a default value of 0.1.
Try the new method on \( \tanh(x)=0 \) with an initial interval \( [-10,15] \).
Filename: bisection_Newton.py.
The purpose of this function is to verify the implementation of Newton's
method in the Newton function in the file
nonlinear_solvers.py.
Construct an algebraic equation and perform two iterations of Newton's
method by
hand or with the aid of SymPy.
Find the corresponding size of \( |f(x)| \) and use this
as value for eps when calling Newton. The function should then
also perform two iterations and return the same approximation to
the root as you calculated manually. Implement this idea for a unit test
as a test function test_Newton().
Filename: test_Newton.py.
An important engineering problem that arises in a lot of applications is the vibrations of a clamped beam where the other end is free. This problem can be analyzed analytically, but the calculations boil down to solving the following nonlinear algebraic equation: $$ \cosh\beta \cos\beta = -1,$$ where \( \beta \) is related to important beam parameters through $$ \beta^4 = \omega^2 \frac{\varrho A}{EI},$$ where \( \varrho \) is the density of the beam, \( A \) is the area of the cross section, \( E \) is Young's modulus, and \( I \) is the moment of the inertia of the cross section. The most important parameter of interest is \( \omega \), which is the frequency of the beam. We want to compute the frequencies of a vibrating steel beam with a rectangular cross section having width \( b=25 \) mm and height \( h=8 \) mm. The density of steel is \( 7850 \mbox{ kg/m}^3 \), and \( E= 2\cdot 10^{11} \) Pa. The moment of inertia of a rectangular cross section is \( I=bh^3/12 \).
a) Plot the equation to be solved so that one can inspect where the zero crossings occur.
Hint. When writing the equation as \( f(\beta)=0 \), the \( f \) function increases its amplitude dramatically with \( \beta \). It is therefore wise to look at an equation with damped amplitude, \( g(\beta) = e^{-\beta}f(\beta) = 0 \). Plot \( g \) instead.
b) Compute the first three frequencies.
Filename: beam_vib.py.