Next: Convergence of secant Method: Up: Main: Previous: False position or Regula
In this method an improvement over Regula Falsi method is
obtained by replacing the secant by straight lines of even-smaller
slope until falls to the otherside of the zero of
. The
various steps in the method are given in the algorithm below:
Algorithm:
Given a function continuous on an
interval
satisfying the criteria
, carry out
the following steps to find the root of
of
in
:
(1)Set
(2) For n=0,1,2...., until convergence criteria is satisfied,
do:
(a) compute
(b) If
then
Set
Also if
Set
Otherwise
Set
Also if
Set
Example:
Solve
for the root in the
interval [1,2] by Modified Regula Falsi method.
Solution: Since
we go ahead with
finding the root of given function f(x) in [1,2]. Setting
and following the above algorithm. Results
are provided in
the table below:
Modified Regula Falsi Method
Iteration no. | ![]() |
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0 | 1.0000000000 | 2.0000000000 | 1.4782608747 | -2.2348976135 |
1 | 1.4782608747 | 2.0000000000 | 1.7010031939 | 0.5908976793 |
2 | 1.4782608747 | 1.7010031939 | 1.6544258595 | -0.0793241411 |
3 | 1.6544258595 | 1.7010031939 | 1.6599385738 | -0.0022699926 |
4 | 1.6599385738 | 1.7010031939 | 1.6602516174 | 0.0021237291 |
5 | 1.6599385738 | 1.6602516174 | 1.6601003408 | 0.0000002435 |
The geometric view of the example is provided in the figure below:
Example: Solve
for the root in the
interval [0.5,1.5] by Modified Regula Falsi Method.
Modified Regula Falsi Method
Iteration no. | ![]() |
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0 | 0.5000000000 | 1.5000000000 |
0.8773435354 | 2.1035263538 |
1 | 0.5000000000 | 0.8773435354 |
0.7222673893 | 0.2828366458 |
2 | 0.5000000000 | 0.7222673893 |
0.6871531010 | -0.1967970580 |
3 | 0.6871531010 | 0.7222673893 |
0.7015607357 | 0.0026464546 |
4 | 0.6871531010 | 0.7015607357 |
0.7013695836 | 0.0000239155 |
5 | 0.6871531010 | 0.7013695836 |
0.7013661265 | -0.0000235377 |
6 | 0.7013661265 | 0.7013695836 |
0.7013678551 | -0.0000003363 |
Secant Method
Like the Regula Falsi method and the
Bisection method this method also requires two initial estimates
of the root of f(x)=0 but unlike those
earlier methods it gives up the demand of bracketing the root.
Like in the Regula Falsi method, this method too retains the use
of secants throughout while tracking the root of f(x)=0. The
secant joining the points
is given by
Example:
Solve
for the root with
by secant method to an accuracy of
.
Solution:
Set
Repeat the
process with
and so on
till you get a
s.t.
These
results are tabulated below:
Secant Method | ||||
---|---|---|---|---|
Iteration no. | ![]() |
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0 |
1.0000000000 | 2.0000000000 | 1.4782608747 | -2.2348976135 |
1 |
2.0000000000 | 1.4782608747 | 1.6198574305 | -0.5488323569 |
2 |
1.4782608747 | 1.6198574305 | 1.6659486294 | 0.0824255496 |
3 | 1.6198574305 | 1.6659486294 | 1.6599303484 | -0.0023854144 |
4 | 1.6659486294 | 1.6599303484 | 1.6600996256 | -0.0000097955 |
Geometrical visualization of the root tracking procedure by Secant method for the above example.
Exercise: Find the solutions accurate to within for the following problems using Secant's Method.
(1)
(2)