The secant method is a root-finding procedure in numerical analysis that uses a series of roots of secant lines to better approximate a root of a function f. Let us learn more about the second method, its formula, advantages and limitations, and secant method solved example with detailed explanations in this article.
Table of Contents:
What is a Secant Method?
The tangent line to the curve of y = f(x) with the point of tangency (x0, f(x0) was used in Newton’s approach. The graph of the tangent line about x = α is essentially the same as the graph of y = f(x) when x0 ≈ α. The root of the tangent line was used to approximate α.
Consider employing an approximating line based on ‘interpolation’. Let’s pretend we have two root estimations of root α, say, x0 and x1. Then, we have a linear function
q(x) = a0 + a1x
using q(x0) = f (x0), q(x1) = f (x1).
This line is also known as a secant line. Its formula is as follows:
The linear equation q(x) = 0 is now solved, with the root denoted by x2. This results in
Let the above form be equation (1)
The procedure can now be repeated. Employ x1 and x2 to create a new secant line, and then use the root of that line to approximate α;…
Secant Method Steps
The secant method procedures are given below using equation (1).
Step 1: Initialization
x0 and x1 of α are taken as initial guesses.
Step 2: Iteration
In the case of n = 1, 2, 3, …,
until a specific criterion for termination has been met (i.e., The desired accuracy of the answer or the maximum number of iterations has been attained).
Secant Method Convergence
If the initial values x0 and x1 are close enough to the root, the secant method iterates xn and converges to a root of function f. The order of convergence is given by φ, where
Which is the golden ratio.
The convergence is particularly superlinear, but not really quadratic. This solution is only valid under certain technical requirements, such as f being two times continuously differentiable and the root being simple in the question (i.e., having multiplicity 1).
There is no certainty that the secant method will converge if the beginning values are not close enough to the root. For instance, if the function f is differentiable on the interval [x0, x1], and there is a point on the interval where f’ =0, the algorithm may not converge.
Secant Method Advantages and Disadvantages
The secant method has the following advantages:
- It converges quicker than a linear rate, making it more convergent than the bisection method.
- It does not necessitate the usage of the function’s derivative, which is not available in a number of applications.
- Unlike Newton’s technique, which requires two function evaluations in every iteration, it only requires one.
The secant method has the following drawbacks:
- The secant method may not converge.
- The computed iterates have no guaranteed error bounds.
- If f0(α) = 0, it is likely to be challenging. This means that when x = α, the x-axis is tangent to the graph of y = f(x).
- Newton’s approach is more easily generalized to new ways for solving nonlinear simultaneous systems of equations.
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Secant Method Solved Example
Example:
Compute two iterations for the function f(x) = x3 – 5x + 1 = 0 using the secant method, in which the real roots of the equation f(x) lies in the interval (0, 1).
Solution:
Using the given data, we have,
x0 = 0, x1 = 1, and
f(x0) = 1, f(x1) = -3
Using the secant method formula, we can write
x2 = x1 – [(x0 – x1) / (f(x0) – f(x1))]f(x1)
Now, substitute the known values in the formula,
= 1 – [(0 – 1) / ((1-(-3))](-3)
= 0.25.
Therefore, f(x2) = – 0.234375
Performing the second approximation, ,
x3 = x2 – [( x1 – x2) / (f(x1) – f(x2))]f(x2)
=(- 0.234375) – [(1 – 0.25)/(-3 – (- 0.234375))](- 0.234375)
= 0.186441
Hence, f(x3) = 0.074276
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Frequently Asked Questions on Secant Method
What is secant method in numerical analysis?
The secant method is a root-finding procedure in numerical analysis that uses a series of roots of secant lines to better approximate a function’s root.
Mention the advantages of the secant method.
- It is more convergent than the bisection approach since it converges faster than a linear rate.
- It does not demand the use of the derivative of the function, which is not available in many applications.
- Unlike Newton’s method, which necessitates two function evaluations every iteration, this method just necessitates one.
Is the secant method converging faster than the bisection method?
Yes, the secant approach is faster than the bisection method in terms of convergence.
What is the other name of the secant method?
The 2-point method is also known as the Secant Method.
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