Secant Methods:

In this lecture we introduce two additional methods to search numerical solutions of the equation f(x) = 0. Both of these methods are basis on approximating the function by secant lines now as Newton’s method was based on approximating the function by tangent lines.

The Secant Method:

The secant method requires two initial points x₀ with x₁ which are both reasonably close to the solution x*. Preferably the signs of y₀ = f(x₀) and y₁ = f(x₁) must be different. Once x₀ also x₁ are determined the method proceeds by the following formula:

x_i+1 = x_i– [{x_i− x_i−1}/{y_i− y_i−1}]*[y_i− y_i−1]

Illustration - Suppose f(x) = x⁴ − 5 for which the true solution is x*= 4√ 5. Plotting this function disclose that the solution is at about 1.25. If we let x₀ = 1 as well as x₁ = 2 then we know that the root is in between x₀ as well as x₁. After that we have that y₀ = f(1) = −4 and y₁ = f(2) = 11. We may then compute x₂ from the formula:

Pluggin x₂ = 19/15 into f(x) we acquire y₂ = f(19/15) ≈ −2.425758.... In the subsequent step we would use x₁ = 2 and x₂ = 19/15 in the above formula to find x₃ and so on.

Given Below is a program for the secant method. Notice that it necessitate two input guesses x₀ and x₁, however it does not require the derivative to be input.

function x = mysecant(f,x0,x1,n)
format long % prints more digits
format compact % makes the output more compact
% Solves f(x) = 0 by doing n steps of the secant method starting with x0 and x1.
% Inputs: f -- the function input as an inline function
% x0 -- starting guess, a number
% x1 -- second starting geuss
% n -- the number of steps to do
% Output: x -- the approximate solution
y0 = f(x0);
y1 = f(x1);
for i = 1:n % Do n times
x = x1 - (x1-x0)*y1/(y1-y0) % secant formula.
y=f(x) % y value at the new approximate solution.
% Move numbers to get ready for the next step
x0=x1;
y0=y1;
x1=x;
y1=y;
end

The Regula Falsi Method:

The Regula Falsi method is fairly a combination of the secant method and bisection method. The idea is to utilize secant lines to approximate f(x) but choose how to update using the sign of f(xn). Now as in the bisection method we begin with a and b for which f(a) and f(b) have different signs.

Then let:

x = b –[{b – a}/{f(b) − f(a)}]*[f(b)].

Then check the sign of f(x). If it is the similar as the sign of f(a) then x becomes the new a Otherwise let b = x.

Convergence:

If we can start with a good choice x0 then Newton’s method will converge to x* rapidly. The secant method is a slightly slower than Newton’s method and the Regula Falsi method is slightly slower than that. Both nevertheless are still much faster than the bisection method.

If we don’t have a good starting point or interval, after that the secant method, just similar to Newton’s method can fail altogether. The Regula Falsi method just similar to the bisection method always works for the reason that it keeps the solution inside a definite interval.

Simulations and Experiments:

Although Newton’s method converges quicker than any other method there are contexts when it isn’t convenient or even impossible. One noticeable situation is when it is difficult to calculate a formula for f′(x) even though one knows the formula for f(x). This is habitually the case when f(x) isn’t defined explicitly however implicitly. There are other situations, which are extremely common in engineering and science where even a formula for f(x) isn’t known. This happens when f(x) is the result of experiment or else simulation rather than a formula. In such circumstances the secant method is usually the best choice.

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