secant method
TRANSCRIPT
04/08/23http://
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
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.edu5
Algorithm for Secant Method
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
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.
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
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).
http://numericalmethods.eng.usf.edu11
Example 1 Cont.Let us assume the initial guesses of the root of as and
0xf 02.0 1 x
Iteration 1The estimate of the root is
06461.0
10993.302.0165.002.010993.305.0165.005.0
02.005.010993.305.0165.005.005.0
423423
423
10
10001
xfxf
xxxfxx
.05.00 x
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.
http://numericalmethods.eng.usf.edu13
Example 1 Cont.
Figure 5 Graph of results of Iteration 1.
http://numericalmethods.eng.usf.edu14
Example 1 Cont.
Iteration 2The estimate of the root is
06241.0
10993.305.0165.005.010993.306461.0165.006461.0
05.006461.010993.306461.0165.006461.006461.0
423423
423
01
01112
xfxf
xxxfxx
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.
http://numericalmethods.eng.usf.edu16
Example 1 Cont.
Figure 6 Graph of results of Iteration 2.
http://numericalmethods.eng.usf.edu17
Example 1 Cont.
Iteration 3The estimate of the root is
06238.0
10993.306461.0165.005.010993.306241.0165.006241.0
06461.006241.010993.306241.0165.006241.006241.0
423423
423
12
12223
xfxf
xxxfxx
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.
http://numericalmethods.eng.usf.edu19
Iteration #3
Figure 7 Graph of results of Iteration 3.
http://numericalmethods.eng.usf.edu20
Advantages
Converges fast, if it converges Requires two guesses that do not need
to bracket the root
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
0
f x( )
f x( )
f x( )
1010 x x guess1 x guess2
0 xSinxf
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
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