![Page 1: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/1.jpg)
1/10/2010 http://numericalmethods.eng.usf.edu 1
Secant Method
Major: All Engineering Majors
Authors: Autar Kaw, Jai Paul
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
![Page 3: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/3.jpg)
http://numericalmethods.eng.usf.edu3
Secant Method – Derivation
)(xf)f(x - = xx
i
iii ′+1
f(x)
f(xi)
f(xi-1)
xi+2 xi+1 xi X θ
( )[ ]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.
![Page 4: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/4.jpg)
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
xxxf
xxxf
DEDC
AEAB
=
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
![Page 5: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/5.jpg)
http://numericalmethods.eng.usf.edu5
Algorithm for Secant Method
![Page 6: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/6.jpg)
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
![Page 7: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/7.jpg)
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.
![Page 8: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/8.jpg)
http://numericalmethods.eng.usf.edu8
Example 1You are working for ‘DOWN THE TOILET COMPANY’ that makes floats for ABC commodes. The floating ball has a specific gravity of 0.6 and has a radius of 5.5 cm. You are asked to find the depth to which the ball is submerged when floating in water.
Figure 3 Floating Ball Problem.
![Page 9: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/9.jpg)
http://numericalmethods.eng.usf.edu9
Example 1 Cont.
Use the Secant method of finding roots of equations to find the depth x to which the ball is submerged under water. • Conduct three iterations to estimate the root of the above equation. • Find the absolute relative approximate error and the number of significant digits at least correct at the end of each iteration.
The equation that gives the depth x to which the ball is submerged under water is given by
( ) 423 1099331650 -.+x.-xxf ×=
![Page 10: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/10.jpg)
http://numericalmethods.eng.usf.edu10
Example 1 Cont.
( ) 423 1099331650 -.+x.-xxf ×=
To aid in the understanding of how this method works to find the root of an equation, the graph of f(x) is shown to the right,
where
Solution
Figure 4 Graph of the function f(x).
![Page 11: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/11.jpg)
http://numericalmethods.eng.usf.edu11
Example 1 Cont.Let us assume the initial guesses of the root of as and
( ) 0=xf 02.0 1 =−x
Iteration 1The estimate of the root is
( )( )( ) ( )
( )( )( )( )( ) ( )( )
06461.010993.302.0165.002.010993.305.0165.005.0
02.005.010993.305.0165.005.005.0423423
423
10
10001
=×+−−×+−
−×+−−=
−−
−=
−−
−
−
−
xfxfxxxfxx
.05.00 =x
![Page 12: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/12.jpg)
http://numericalmethods.eng.usf.edu12
Example 1 Cont.The absolute relative approximate error at the end of Iteration 1 is
%62.22
10006461.0
05.006461.0
1001
01
=
×−
=
×−
=∈x
xxa
a∈
The number of significant digits at least correct is 0, as you need an absolute relative approximate error of 5% or less for one significant digits to be correct in your result.
![Page 13: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/13.jpg)
http://numericalmethods.eng.usf.edu13
Example 1 Cont.
Figure 5 Graph of results of Iteration 1.
![Page 14: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/14.jpg)
http://numericalmethods.eng.usf.edu14
Example 1 Cont.
Iteration 2The estimate of the root is
( )( )( ) ( )
( )( )( )( )( ) ( )( )
06241.010993.305.0165.005.010993.306461.0165.006461.0
05.006461.010993.306461.0165.006461.006461.0423423
423
01
01112
=×+−−×+−
−×+−−=
−−
−=
−−
−
xfxfxxxfxx
![Page 15: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/15.jpg)
http://numericalmethods.eng.usf.edu15
Example 1 Cont.The absolute relative approximate error at the end of Iteration 2 is
%525.3
10006241.0
06461.006241.0
1002
12
=
×−
=
×−
=∈x
xxa
a∈
The number of significant digits at least correct is 1, as you need an absolute relative approximate error of 5% or less.
![Page 16: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/16.jpg)
http://numericalmethods.eng.usf.edu16
Example 1 Cont.
Figure 6 Graph of results of Iteration 2.
![Page 17: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/17.jpg)
http://numericalmethods.eng.usf.edu17
Example 1 Cont.
Iteration 3The estimate of the root is
( )( )( ) ( )
( )( )( )( )( ) ( )( )
06238.010993.306461.0165.005.010993.306241.0165.006241.0
06461.006241.010993.306241.0165.006241.006241.0423423
423
12
12223
=×+−−×+−
−×+−−=
−−
−=
−−
−
xfxfxxxfxx
![Page 18: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/18.jpg)
http://numericalmethods.eng.usf.edu18
Example 1 Cont.The absolute relative approximate error at the end of Iteration 3 is
%0595.0
10006238.0
06241.006238.0
1003
23
=
×−
=
×−
=∈x
xxa
a∈
The number of significant digits at least correct is 5, as you need an absolute relative approximate error of 0.5% or less.
![Page 19: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/19.jpg)
http://numericalmethods.eng.usf.edu19
Iteration #3
Figure 7 Graph of results of Iteration 3.
![Page 20: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/20.jpg)
http://numericalmethods.eng.usf.edu20
Advantages
Converges fast, if it converges Requires two guesses that do not need to
bracket the root
![Page 21: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/21.jpg)
http://numericalmethods.eng.usf.edu21
Drawbacks
Division by zero
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,
( ) ( ) 0== xSinxf
![Page 22: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/22.jpg)
http://numericalmethods.eng.usf.edu22
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,
( ) 0== Sinxxf
![Page 23: Secant Method Nonlinear Equations - MATH FOR COLLEGEmathforcollege.com/nm/mws/gen/03nle/mws_gen_nle_ppt_secant.pdf · Secant Method – Derivation f (x) f(x ) x = x - i i i ... For](https://reader031.vdocuments.mx/reader031/viewer/2022030420/5aa7bf487f8b9a424f8cb4e0/html5/thumbnails/23.jpg)
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