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Transcript of 2/2/2016 1 Secant Method Electrical Engineering Majors Authors: Autar Kaw, Jai Paul
05/03/23http://
numericalmethods.eng.usf.edu 1
Secant Method
Electrical 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
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
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 2Find 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.
3843 ln10775468.8ln10341077.210129241.11 RRT
http://numericalmethods.eng.usf.edu8
Example 1Thermistors are temperature-measuring devices based on the principle that the thermistor material exhibits a change in electrical resistance with a change in temperature. By measuring the resistance of the thermistor material, one can then determine the temperature.
Figure 3 A typical thermistor.
For a 10K3A Betatherm thermistor, the relationship between the resistance, R, of the thermistor and the temperature is given by
where T is in Kelvin and R is in ohms.
Thermally conductive epoxy coating
Tin plated copper alloy lead wires
http://numericalmethods.eng.usf.edu9
Example 1 Cont.For the thermistor, error of no more than ±0.01oC is acceptable. To find the range of the resistance that is within this acceptable limit at 19oC, we need to solve
and
Use the Newton-Raphson method of finding roots of equations to find the resistance R at 18.99oC.
a) Conduct three iterations to estimate the root of the above equation.
b) 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.
3843 ln10775468.8ln10341077.210129241.115.27301.19
1 RR
3843 ln10775468.8ln10341077.210129241.115.27399.18
1 RR
http://numericalmethods.eng.usf.edu10
Example 1 Cont.
3384 10293775.2ln10775468.8ln10341077.2)( RRRf
11000 12000 13000 14000 15000
0.00006
0.00004
0.00002
0.00002
Entered function on given interval
Figure 4 Graph of the function f(R).
http://numericalmethods.eng.usf.edu11
13014 107563.1105383.3
1400015000105383.315000
55
5
1
10
10001
R
RfRfRRRfRR
12000 14000 16000 18000 20000
0.00005
0.00005
0.0001
Entered function on given interval with two initial guesses and estimated root
Example 1 Cont.Initial guesses: 15000,14000 01 RR
Iteration 1The estimate of the root is
Figure 5 Graph of the estimate of the root after Iteration 1.
%257.15
10013014
1500013014a
The absolute relative approximate error is
The number of significant digits at least correct is 0.
http://numericalmethods.eng.usf.edu12
13083 105383.3102658.1
1500013014102658.113014
R
56
6
2
01
01112
R
RffRRRfRR
12000 14000 16000 18000 20000
0.00005
0.00005
0.0001
Entered function on given interval with two initial guesses and estimated root
Example 1 Cont.Iteration 2The estimate of the root is
Figure 6 Graph of the estimate of the root after Iteration 2.
The absolute relative approximate error is
The number of significant digits at least correct is 1.
%52422.0
10013083
1301413083a
12000 14000 16000 18000 20000
0.00005
0.00005
0.0001
Entered function on given interval with two initial guesses and estimated root
http://numericalmethods.eng.usf.edu13
13078 102658.1108911.8
1301413083108911.813083 R
R
68
8
3
12
12223
RffRRRfRR
%034415.0
10013078
1308313078a
Example 1 Cont.Iteration 3The estimate of the root is
Figure 7 Graph of the estimate of the root after Iteration 3.
The absolute relative approximate error is
The number of significant digits at least correct is 3.
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
0f 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