The secant method is a root-finding technique that does not require bracketing or sign changes and is also referred to as the chord method. It uses two initial estimates to iteratively approximate roots of equations without calculating derivatives, though it may not converge in some cases. This method is particularly useful in engineering applications, like predicting earthquake performance in structures.

1. SECANT
METHOD

2. Secant Method
• The word secant comes from the latin, secare means to cut.
• It is also known as Root finding method.
• Root: The root (somethings also called “zeros”) of an equation are the
values of for which the equation is satisfied.
Example:
𝑓 𝑥 = 0
• The secant method is not a bracketing method because it not
required to change signs between estimates.
• It is also known as chord method.

3. Method
• Starting with initial values 𝑥0 and 𝑥1, we construction line through the
points (𝑥0, 𝑓 𝑥0 ) & (𝑥1, 𝑓(𝑥1)).
𝑥2 = 𝑥1 − 𝑓 𝑥1
𝑥1 − 𝑥0
𝑓 𝑥1 − 𝑓 𝑥0
We continue this process, solving for x₃, x₄, etc. Until we reach a
sufficiently high level of precision (a sufficiently small difference
between 𝑥𝑛 and 𝑥𝑛−1).

4. Continue:- Method
• This new value replaces the oldest x value being used in the
calculation.
• 𝑥2 = 𝑥1 − 𝑓 𝑥1
𝑥1−𝑥0
𝑓 𝑥1 −𝑓 𝑥0
• 𝑥𝑛+1= 𝑥𝑛 − 𝑓(𝑥𝑛)
𝑥𝑛−𝑥𝑛−1
𝑓 𝑥𝑛 −𝑓(𝑥𝑛−1)
General Formula

5. Example:-
• Question:- Use the secant method to determine root of equation.
𝑥3
− 2𝑥 − 5 = 0
Solution:- First we have to take initial approximation as 𝑥0 = 2, 𝑥1 = 3
We have for secant method,
𝑓 2 = −1 & 𝑓 3 = 16

6. Approximation to root by Secant Method-
n 𝒙𝒏+𝟏 𝒇(𝒙𝟐)
1 2.058823 - 0.390799
2 2.081263 - 0.147204
3 2.094824 0.003042
4 2.094549 0.003042
Hence, the root is 2.094 correct to three decimal places.

7. Excel Worksheet on Secant
method

8. Advantages:
• It does not require the computation of the first order derivative
• No need to check for sign.
• Sometimes it is good to start findings a root using the bisection
method then once you know you are close to the root you can switch
to the secant method to achieve faster convergence.
• The secant method converges more rapidly near a root.

9. Disadvantages:
• The secant method is not a bracketing method it may not converge.
• Another problem of this method that does not know when to stop. It
must be performed several times until the f of the current guess is
very small.
• If the function is very “flat” the secant method can fail.

10. Regula falsie vs Secant:-
• It is similar to Regula falsie except:-
• Will convergence always, speed can be slow.
• No need to check for sign.
• Begin with a, b, as usual.
• Regula falsie a variant of the secant method which maintains a
bracket around the solution.
• Secant method keeps the most recent two estimates, while the false
position method retains the most recent estimate and the next recent
one the most recent estimate and the next recent one which has an
opposite sign in the function value.

11. Applications:
• The Secant method is one of a number of analytical procedure
available to earthquake engineers today for predicting the earthquake
performance of structures.
• Designing of multi-story building.

12. THANK
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