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04/18/23http://
numericalmethods.eng.usf.edu 1
Secant Method
Industrial Engineering Majors
Authors: Autar Kaw, Jai Paul
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
Secant Method
http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu3
Secant Method – Derivation
)(xf
)f(x - = xx
i
iii 1
f ( x )
f ( x i )
f ( x i - 1 )
x i + 2 x i + 1 x i X
ii xfx ,
1
1 )()()(
ii
iii xx
xfxfxf
)()(
))((
1
11
ii
iiiii xfxf
xxxfxx
Newton’s Method
Approximate the derivative
Substituting Equation (2) into Equation (1) gives the Secant method
(1)
(2)
Figure 1 Geometrical illustration of the Newton-Raphson method.
http://numericalmethods.eng.usf.edu4
Secant Method – Derivation
)()(
))((
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
xx
xf
xx
xf
DE
DC
AE
AB
Figure 2 Geometrical representation of the Secant method.
The secant method can also be derived from geometry:
can be written as
On rearranging, the secant method is given as
http://numericalmethods.eng.usf.edu6
Step 1
0101
1 x
- xx =
i
iia
Calculate the next estimate of the root from two initial guesses
Find the absolute relative approximate error
)()(
))((
1
11
ii
iiiii xfxf
xxxfxx
http://numericalmethods.eng.usf.edu7
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.
http://numericalmethods.eng.usf.edu8
ExampleYou are working for a start-up computer assembly company and have been asked to determine the minimum number of computers that the shop will have to sell to make a profit.
The equation that gives the minimum number of computers ‘x’ to be sold after considering the total costs and the total sales is:
03500087540)f( 5.1 xxx
http://numericalmethods.eng.usf.edu9
Solution
Use the Secant method of finding roots of equations to find
The minimum number of computers that need to be sold to make a profit. Conduct three iterations to estimate the root of the above equation.
Find the absolute relative approximate error at the end of each iteration, and
The number of significant digits at least correct at the end of each iteration.
http://numericalmethods.eng.usf.edu10
Graph of function f(x) 03500087540 51 xxxf .
0 20 40 60 80 1002 10
4
1 104
0
1 104
2 104
3 104
4 104
f(x)
3.5 104
1.25 104
0
f x( )
1000 x
Graph of function f(x)
http://numericalmethods.eng.usf.edu11
Iteration #1
%474.17
60.587
181255392
2550539250
50,25
1
10
10001
01
a
x
xfxf
xxxfxx
xx
0 20 40 60 80 1003 10
4
2 104
1 104
0
1 104
2 104
3 104
4 104
f(x)x'1, (first guess)x0, (previous guess)Secant linex1, (new guess)
3.5 104
2.007 104
0
f x( )
f x( )
f x( )
secant x( )
f x( )
1000 x x 0 x 1' x x 1
The number of significant digits at least correct is 0.
http://numericalmethods.eng.usf.edu12
Iteration #2
%1672.3
62.569
5392133.850
50587.60133.8505871.60
587.60,50
2
01
01112
10
a
x
xfxf
xxxfxx
xx
0 20 40 60 80 1002 10
4
1 104
0
1 104
2 104
3 104
4 104
f(x)x1 (guess)x0 (previous guess)Secant linex2 (new guess)
3.5 104
1.606 104
0
f x( )
f x( )
f x( )
secant x( )
f x( )
1000 x x 1 x 0 x x 2 The number of significant digits at least correct is 1.
http://numericalmethods.eng.usf.edu13
Iteration #3
%19425.0
62.690
133.850219.49
587.60569.62219.49
569.62
569.62,587.60
a
3
12
12223
21
x
xfxf
xxxfxx
xx
Entered function along given interval with current and next root and the tangent line of the curve at the current root
0 20 40 60 80 1002 10
4
1 104
0
1 104
2 104
3 104
4 104
f(x)x2 (guess)x1 (previous guess)Secant linex3 (new guess)
3.5 104
1.508 104
0
f x( )
f x( )
f x( )
secant x( )
f x( )
1000 x x 2 x 1 x x 3
The number of significant digits at least correct is 2.
http://numericalmethods.eng.usf.edu14
Advantages
Converges fast, if it converges Requires two guesses that do not need
to bracket the root
http://numericalmethods.eng.usf.edu15
Drawbacks
Division by zero
10 5 0 5 102
1
0
1
2
f(x)prev. guessnew guess
2
2
0
f x( )
f x( )
f x( )
1010 x x guess1 x guess2
0 xSinxf
http://numericalmethods.eng.usf.edu16
Drawbacks (continued)
Root Jumping
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
0Sinxxf
Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit
http://numericalmethods.eng.usf.edu/topics/secant_method.html