Rate of convergence of numerical methods like Secant method ,Regula Falsi method,Newton Raphson method.

Question

Is there any good resource which has a good introduction to rate of convergence of numerical methods . I searched but unable to find one.

Any kind of material,video lectures,books,links is great.

Answer 1

Normally the speed of convergence is given whenever the method is introduced with an in-depth discussion of convergence.

Secant method: golden ratio as order of convergence in the case that the iteration converges at all.

Regula Falsi: Convergence is assured by bracketing, convergence speed is linear in most cases due to stalling of one of the bracketing interval ends. With anti-stalling variations the number of steps is roughly double that of the secant method for the same level of error reduction.

Newton: Order 2 convergence close to a simple root. However, the evaluation of the derivative has the cost of 1 to 2 function evaluations, so in that measure, function evaluations rather than iteration steps, it may be slower than the secant method. See Ostrowski index.