Secant Method for Engineering explained with examples

1. Secant Method
(Numerical Analysis)
Presented by: Arindam Goswami(11500219015)
Ayush Jha (11500219023)
Aniket Singh (11500219036)
Avirup Pal (11500219041)

2. INTRODUCTION
In this method roots are found using an algorithm,
that uses succession of root of secant to better
approximate a root of a function. This method can
thought of as finite difference of Newton’s Method
OVERVIEW
“To find the roots of non linear
equation with the help of secant
lines”

3. METHODOLOGY
A secant is defined by using two points on graph of a function f(x). It is
necessary to choose these initial points as xi
and xi-1
. Then next point xi-1
is obtained by computing x-value at which the secant line passing
through the points (xi
, f(xi
)) and (xi+1
, f(xi+1
)) has a y-coordinate of zero.

4. SECANT METHOD
(Derivation)
The secant method can also be derived from geometry :
The Geometric Similar Triangles:
AB/AE = DC/DE
can be written as:
f(xi
) / (xi
- xi+1
) = f(xi-1
) / (xi-1
- xi+1
)
On rearranging, the secant method is
given as:
xi+1
= xi
- (f(xi
)(xi
- xi-1
) / f(xi
) -f(xi-1
))

5. ALGORITHM
Step 1: Find points x0
and x1
such that x0
<x1
and
f(x0
).f(x1
) < 0.
Step 2: find next value using formulae
➔ x2
= x0
- f(x0
).(x1
-x0
) / { f(x1
)-f(x0
) }
➔ x2
= { x0
.f(x1
)- x1
-f(x0
) } / { f(x1
)-f(x0
) }
➔ x2
= x1
- f(x1
). (x1
- x0
) / { f(x1
) - f(x0
) }
(Using any of the formulae, you will get same x2
value)
Step 3: If f(x2
) = 0 then x2
is an exact root, else x0
= x1
and x1
= x2
Step 4: Repeat steps 2 & 3 until f(xi
)=0 or |f(xi
)| ≤ Accuracy

6. EXAMPLE
Find a root of an equation f(x) = x3
- x - 1 using secant
method
Solution:
Here x3
- x - 1 = 0
Let f(x) = x3
- x - 1
Here:
x 0 1 2
f(x) -1 -1 5

7. EXAMPLE contd.
1st Iteration:
x0
= 1 and x1
= 2
f(x0
) = f(1) = -1 and f(x1
) = f(2) = 5
∴ x2
= x1
- f(x1
). (x1
- x0
) / { f(x1
) - f(x0
) }
x2
= 1.16667
∴ f(x2
) = f(1.16667) = -0.5787

8. EXAMPLE contd.
2nd Iteration:
x1
= 2 and x2
= 1.16667
f(x1
) = f(2) = 5 and f(x2
) = f(1.16667) = -0.5787
∴ x3
= x1
- f(x1
). (x2
- x1
) / { f(x2
) - f(x1
) }
x3
= 1.25311
∴ f(x3
) = f(1.25311) = -0.28536

9. EXAMPLE contd.
3rd Iteration:
x2
= 1.16667 and x3
= 1.25311
f(x2
) = f(1.16667) = -0.5787 and f(x3
) = f(1.25311) = -0.28536
∴ x4
= x2
- f(x2
). (x3
- x2
) / { f(x3
) - f(x2
) }
x4
= 1.33721
∴ f(x4
) = f(1.33721) = -0.05388

10. EXAMPLE contd.
4th Iteration:
x3
= 1.25311 and x4
= 1.33721
f(x3
) = f(1.25311) = -0.28536 and f(x4
) = f(1.33721) = -0.28536
∴ x5
= x3
- f(x3
). (x4
- x3
) / { f(x4
) - f(x3
) }
x5
= 1.32385
∴ f(x2
) = f(1.32385) = -0.0037

11. EXAMPLE contd.
5th Iteration:
x4
= 1.33721 and x5
= 1.32385
f(x4
) = f(1.33721) = 0.05388 and f(x5
) = f(1.32385) = -0.0037
∴ x6
= x4
- f(x4
). (x5
- x4
) / { f(x5
) - f(x4
) }
x6
= 1.32471
∴ f(x2
) = f(1.32471) = -0.00004

12. EXAMPLE contd.
Approximate root of the equation f(x) = x3
- x - 1 using Secant method is 1.32471
n x0
f(x0
) x1
f(x1
) x2
f(x2
) update
1 1 -1 2 5 1.16667 -0.5787 x0
=x1
x1
=x2
2 2 5 1.1667 -0.5787 1.25311 -0.28536 x0
=x1
x1
=x2
3 1.6667 -0.5787 1.25311 -0.28536 1.33721 0.05388 x0
=x1
x1
=x2
4 1.25311 -0.28536 1.33721 0.05388 1.32385 -0.0037 x0
=x1
x1
=x2
5 1.33721 0.05388 1.32385 -0.0037 1.32471 -0.00004 x0
=x1
x1
=x2

13. APPLICATIONS
Secant method is one of the analytical procedure available to
earthquake engineers for predicting earthquake and structures
Secant method is used to develop linear dynamic analysis
model to have the potential to influence the behaviour of the
structure in non-linear range.
It is used for nonlinear pushover analysis, which defines the
force-displacement relationship of the walls in the building
under lateral load.

14. ADVANTAGES
It converges faster than a linear rate so it is more rapidly
convergent.
Requires two guesses that do not need to bracket the root.
It doesn't require use of derivative of a given function because
in some practical cases, derivatives become very hard to find.
It requires only one function evaluation per iteration as
compared to Newton’s method which requires two

15. REFERENCES
➔ AtoZmath.com
➔ Wikipedia
➔ IIT Research Material
➔ ScienceDirect.com
➔ OpenGenius.org