University of Engineering and Technology, LhrDept. of Computer Science and Engg.

Numerical Analysis (Presentation)

Instructed by: SIR Ahmad Awais

Members: Hafiz Hassaan Tariq (2015-CS-67) Muhammad Umair (2015-CS-5) Ahmad Afraz Khan(2015-CS-27)

1

Secant Method

Problem Statement:“To find the roots of a non-linear equation with the help of secant

lines”.

Introduction:

In this method roots are found using an algorithm, that uses

succession of roots of secant lines to better approximate a root of

a function. This method can be thought of as a finite difference of

Newton’s Method.

2

3 Methodology

A secant line is defined by using two points on graph of a function f(x). It is necessary to choose these two 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.

f(x)

f(xi)

f(xi-1)

xi+1 xi-1 xi X

B

C

E D A

Secant Method – Derivation4

)()())((

1

11

ii

iiiii xfxf

xxxfxx

The Geometric Similar Triangles

f(x)

f(xi)

f(xi-1)

xi+1 xi-1 xi X

B

C

E D A

11

1

1

)()(

ii

i

ii

i

xxxf

xxxf

DEDC

AEAB

The secant method can also be derived from geometry:

can be written as

On rearranging, the secant method is given as

Algorithm

Step 1

5

Calculate the next estimate of the root from two initial guesses

)()())((

1

11

ii

iiiii xfxf

xxxfxx

Find the absolute relative approximate error

0101

1 x

- xx = i

iia

Step 2

Find if the absolute relative approximate error is greater than the prespecified relative error tolerance.

If so, go back to step 1, else stop the algorithm.

Also check if the number of iterations has exceeded the maximum number of iterations.

6

7 Applications

• Secant method is one of the analytical procedure available to earthquake engineers for predicting earthquake performance and structures.

• Secant method is used to develop linear dynamic analysis model to have the potential to influence the behavior of the structure in non-linear range.

• It is used for non-linear push over analysis, which defines the force-displacement relationship of the walls in the building under lateral load.

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.

8

Limitations9

10 5 0 5 102

1

0

1

2

f(x)prev. guessnew guess

2

2

0f x( )

f x( )

f x( )

1010 x x guess1 x guess2

Division by zero

0Sinxxf

Root Jumping

10

10 5 0 5 102

1

0

1

2

f(x)x'1, (first guess)x0, (previous guess)Secant linex1, (new guess)

2

2

0

f x( )

f x( )

f x( )

secant x( )

f x( )

1010 x x 0 x 1' x x 1