fsolve with the option secant

10.4.5 fsolve with the option secant_solver

The secant method is a simplified version of Newton’s method. The computation of x₁ is done using Newton’s method, but then the computation of f′(x_n), n>1 is done approximately. This method is used when the computation of the derivative is expensive:

x_i+1=x_i−

f(x_i)

f′_est

, f′_est=

f(x_i)−f(x_i−1)

x_i−x_i−1

The convergence for roots of multiplicity 1 is of order (1 + √5)/2 ≈ 1.62… .

Examples.

Input:
fsolve((cos(x))=x,x,-1..1,secant_solver)
Output:
⎡
⎣ 0.739085133215 ⎤
⎦
Input:
fsolve((cos(x))=x,x,0,secant_solver)
Output:
0.739085133215