Secant Iterative method

Secant iterative method is an opened iterative method which can be considered as an extension of Newton Raphson Method. It is used for finding roots of Non-linear Equations.

1. Overview of Secant Method
Numerical Analysis
Isaac Amornortey Yowetu
NIMS-Ghana
June 27, 2020

2. Background of Secant Iteration
1
Background of Secant Iteration
Graphical Example
2
Derivation of Secant Method
3
Performing Secant Iteration to nd approximate Solution
4
Application of Secant Iteration
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3. Background of Secant Iteration
Background of Secant Iteration
Secant is one of the opened-iterative method for nding root of
a given functionf(x) =0.
This method assumes a function should be approximately linear
to the area under consideration.
It requires two initial approximationsx0andx1to start its
iterations.
It retains only the most recent approximations in its iterative
process.
Isaac Amornortey Yowetu (NIMS-Ghana) Overview of Secant Method June 27, 2020 2 / 12

4. Background of Secant Iteration Graphical Example
Graphical Example
Figure:
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5. Derivation of Secant Method
Steps in Deriving Secant Method
Givenf(x) =0 and Newton Raphson Method as this:
xn+1=xn
f(xn)
f
0
(xn)
(1)
But from the eqn(1), when we can consider expressingf
0
(x)as:
f
0
(xn) =
f(xn)f(xn1)
xnxn1
(2)
Then,
xn+1=xn
f(xn)[xnxn1]
f(xn)f(xn1)
(3)
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6. Derivation of Secant Method
xn+1=
xn[f(xn)f(xn1)]xnf(xn) +xn1f(xn)
f(xn)f(xn1)
(4)
xn+1=
xnf(xn)xnf(xn1)xnf(xn) +xn1f(xn)
f(xn)f(xn1)
(5)
xn+1=
xn1f(xn)xnf(xn1)
f(xn)f(xn1)
(6)
Remarks
Secant Method ignores nding derivatives of functions by falling
on Backward Divide Dierence.
Thexn+1value at which there is converges is the solution of
interest.
It takes 2 approximate values to start its iteration process.
Secant method uses the 2 most current approximations in its
iterative process.
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7. Performing Secant Iteration to nd approximate Solution
Performing Secant Iteration
We start our iteration with initial pointsx0andx1
x2=
x0f(x1)x1f(x0)
f(x1)f(x0)
(7)
x3=
x1f(x2)x2f(x1)
f(x2)f(x1)
(8)
x4=
x2f(x3)x3f(x2)
f(x3)f(x2)
(9)
.
.
. (10)
xn+1=
xn1f(xn)xnf(xn1)
f(xn)f(xn1)
(11)
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8. Performing Secant Iteration to nd approximate Solution
Stopping Criteria
Error Formula
Absolute Error=jxxn+1j
Relative Error=
jxxn+1j
jxj
Wherex;xn+1are true-value and approximated-value respectively.
These error formula is not of direct use as the true valuexis not
known.
Commonly Use Stopping Criteria
jxn+1xnj< "
jxn+1xnj
jxn+1j
< "or
jxnxn+1j
jxnj
< "
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9. Application of Secant Iteration
Application of Secant Iteration
Example 1
Find a root ofxe
x
=1 using Secant iteration. Take
x0=0:45 andx1=0:5
Considering ourf(x) =xe
x
1=0
Solution
Checking the convergence atx0=0:45 andx1=0:5
f(0:45) =0:45(e
0:45
)1=0:2943 (12)
f(0:50) =0:50(e
0:50
)1=0:1756 (13)
x2=
0:45(0:1756)0:50(0:2943)
(0:1756)(0:2943)
=0:5740 (14)
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10. Application of Secant Iteration
Solution Continue...
We ignorex0=0:45
f(0:50) =0:50(e
0:50
)1=0:1756 (15)
f(0:5740) =0:5740(e
0:5740
)1=0:0191 (16)
x3=
0:50(0:0191)0:5740(0:1756)
(0:0191)(0:1756)
=0:5668 (17)
f(0:5740) =0:5740(e
0:5740
)1=0:0191 (18)
f(0:5668) =0:5668(e
0:5668
)1=0:0011 (19)
x4=
0:5740(0:0011)0:5668(0:0191)
(0:0011)(0:0191)
=0:5671 (20)
After successive iterations, the approximated value converges
x

=0:5671
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11. Application of Secant Iteration
Example 2
Consider the nonlinear equationx
3
=2x+1 with a solution with the
intervalI= [1:5;2:0]using Newton Raphson iteration with initial
approximationsx0=1:5 andx1=1:6, nd the approximated root.
Considering ourf(x) =x
3
2x1=0
Solution
Checking the convergence atx0=1:5 andx1=1:6
f(1:5) = (1:5)
3
2(1:5)1=0:625 (21)
f(1:6) = (1:6)
3
2(1:6)1=0:104 (22)
x2=
1:5(0:104)1:6(0:625)
(0:104)(0:625)
=1:6200 (23)
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12. Application of Secant Iteration
Solution Continue...
We ignorex0=1:5
f(1:6) = (1:6)
3
2(1:6)1=0:104 (24)
f(1:62) = (1:62)
3
2(1:62)1=0:0113 (25)
x3=
1:6(0:0113)1:62(0:104)
(0:0113)(0:104)
=1:6180 (26)
f(1:62) = (1:62)
3
2(1:62)1=0:0113 (27)
f(1:618) = (1:618)
3
2(1:618)1=0:0002 (28)
x4=
1:62(0:0002)1:618(0:0113)
(0:0002)(0:0113)
=1:6180 (29)
After successive iterations, the approximated value converges
x

=1:6180
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13. Application of Secant Iteration
End
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