Day 6: Root-finding in 1D, cont'd

HW hints

Newton can fail

import numpy as np
import matplotlib.pyplot as plt

def newton_step(f,fp,x):
    return -f(x)/fp(x)

def draw_newton_iteration(f,fp,x0s,a,b,nits):
    fig, (ax1,ax2) = plt.subplots(2,1,figsize=(3,6))
    x = np.linspace(a,b,200)
    # Newton construction
    ax1.plot(x,0*x ,color='k',lw=2,alpha=0.5) # horizontal axis
    ax1.plot(x,f(x),color='m',lw=3,alpha=0.5)

    def g(x):
        return x - f(x)/fp(x)
    ax2.plot(x,x,color='k',lw=2,alpha=0.5)
    ax2.plot(x,g(x),color='g',lw=2,alpha=0.5)

    ax1.text(a,1,'f',color='m',fontsize=30,alpha=0.5)
    ax2.text(a,b,'g',color='g',fontsize=30,alpha=0.5)

    for x0 in x0s:
        x = x0
        print(0,f'{x:20.15f}')
        for k in range(nits):
            xnew = x + newton_step(f,fp,x)
            ax1.plot([x,x,xnew],[0,f(x),0],lw=1.5,alpha=0.5,color='b')
            ax2.plot([x,x,xnew],[x,xnew,xnew],lw=1.5,alpha=0.5,color='r')
            x = xnew
            print(k+1,f'{x:20.15f}')
    ax1.set_xlim(a,b)
    ax1.set_ylim(-1,1)
    ax2.set_xlim(a,b)
    ax2.set_ylim(a,b)

def f(x):
    return np.tanh(x)
def fp(x):
    return 1 - np.tanh(x)**2
draw_newton_iteration(f,fp,[1.06],-1.8,1.8,6)

draw_newton_iteration(f,fp,[1.1 ],-1.8,1.8,6)