Question

The secant method also begins with two initial points, but without the constraint that the function values are of opposite signs. We use the secant line to extrapolate the next candidate point.

def f(x):
    return (x**3-2*x+7)/(x**4+2)

x = np.arange(-3,5, 0.1);
y = f(x)

p1=plt.plot(x, y)
plt.xlim(-3, 4)
plt.ylim(-.5, 4)
plt.xlabel('x')
plt.axhline(0)
t = np.arange(-10, 5., 0.1)

x0=-1.2
x1=-0.5
xvals = []
xvals.append(x0)
xvals.append(x1)
notconverge = 1
count = 0
cols=['r--','b--','g--','y--']
while (notconverge==1 and count <  3):
    slope=(f(xvals[count+1])-f(xvals[count]))/(xvals[count+1]-xvals[count])
    intercept=-slope*xvals[count+1]+f(xvals[count+1])
    plt.plot(t, slope*t + intercept, cols[count])
    nextval = -intercept/slope
    if abs(f(nextval)) < 0.001:
        notconverge=0
    else:
        xvals.append(nextval)
    count = count+1

plt.show()

The secant method has the advantage of fast convergence. While the bisection method has a linear convergence rate (i.e. error goes to zero at the rate that \(h(x) = x\) goes to zero, the secant method has a convergence rate that is faster than linear, but not quite quadratic (i.e. \(\sim x^\alpha\), where \(\alpha = \frac{1+\sqrt{5}}2 \approx 1.6\))