Lecture Note of Secant Method

University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)








Secant Method
Undergraduate Leve, 3th Stage



Mr. Waleed Ali Tameemi
College of Engineering/ Babylon University
M.Sc. Civil Engineering/ the University of Kansas/ USA



2016-2017
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3.2 –Secant Method
The following steps are required in estimating the root.
Assume two inertial values for the root for ?x_(i-1) and x?_i.
Evaluate f(x_i ).
Calculate the estimated root (x_(i+1)) as following:
x_(i+1)=x_i-(f(x_i)×(x_i-x_(i-1)))/(f(x_i )-f(x_(i-1) ) )
Calculate the error value, f(x_(i+1)), which represents the corresponding value of x_(i+1).
Compare the error value, |f(x_(i+1))|, with the desire accuracy (?).
If |f(x_(i+1))|? ? then x_(i+1) is the required root (accurate enough).
Otherwise go to step 3.

Ex1: Find a root for f(x)=x^3-0.165x^2+3.993×?10?^(-4) with accuracy equal to ?=0.000001.
Solution:
Step 1:
Assume x_(-1)=-0.02 x_0=0.05
Step 2:
f(x_(i-1) )= f(-0.02)=?-0.02?^3-0.165×?-0.02?^2+3.993×?10?^(-4)=0.000325
f(x_i )= f(0.05)=?0.05?^3-0.165×?0.05?^2+3.993×?10?^(-4)=0.000112

Step 3:
x_(i+1)=x_i-(f(x_i )×(x_i-x_(i-1) ))/(f(x_i )-f(x_(i-1) ) )=0.05-(0.000112×(0.05-(-0.02)))/((0.000112-0.000325) )=0.08666

Step 4:

f(x_(i+1))=f(0.08666)=x^3-0.165x^2+3.993×?10?^(-4)=-0.00019

Step 5:
|f(x_1)| > ? ?(not enough accurat) go to step 3
Step # x_i f(x_1 ) f(x_(i-1) ) x_(i+1) f(x_(i+1) ) |f(x_r)|- ?
0 -0.02
1 0.05 0.000112 0.000325 0.08666 -0.00019 -0.000179004
2 0.086656 -0.00019 0.000112 0.06362 -1.1E-05 -1.07014E-06
3 0.063624 -1.1E-05 -0.00019 0.06219 1.66E-06 8.33586E-06
4 0.062191 1.66E-06 -1.1E-05 0.06238 -5.4E-09 9.99459E-06
The required root is equal to x=0.06238.
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