Non-Linear Equations

x−Ay−f(A)=B−Af(B)−f(A)

At x=x₁ => y=0

x1−A−f(A)=B−Af(B)−f(A)

x1=A−f(A)×f(B)−f(A)B−A

x1=f(B)−f(A)Af(B)−Af(A)−f(B)−f(A)Bf(A)−Af(A)

x1=f(B)−f(A)Af(B)=Bf(A)

x−x0y−f(x0)=f′(x0)

At x=x₁ => y=0

x1−x0−f(x0)=f′(x0)

x1=x0−f′(x0)f(x0)

rearrange the function f(x)=x−ϕ(x)

let α be a root of f(x) that satisfies α=ϕ(α)

let x0 be a point at a horizontal distance h from α

and ϕ0 would be at a vertical distance v from ϕ(α)

if v<h that means, with each iteration we are closing in on the root α

if v>h that means, with each iteration we are moving away from the root α

so, the convergence condition is v<h or ∣ϕ′(x0)∣<1