Explain with example that rate of convergence of false position method is faster than that of the bisection method.
Introduction
False position method
In numerical analysis, the false position method or regula falsi method is a root-finding algorithm that combines features from the bisection method and the secant method.
The method:
The first two iterations of the false position method. The red curve shows the function f and the blue lines are the secants.
Like the bisection method, the false position method starts with two points a0 and b0 such that f(a0) and f(b0) are of opposite signs, which implies by the intermediate value theorem that the function f has a root in the interval [a0, b0], assuming continuity of the function f. The method proceeds by producing a sequence of shrinking intervals [ak, bk] that all contain a root of f.
At iteration number k, the number
is computed. As explained below, ck is the root of the secant line through (ak, f(ak)) and (bk, f(bk)). If f(ak) and f(ck) have the same sign, then we set ak+1 = ck and bk+1 = bk, otherwise we set ak+1 = ak and bk+1 = ck. This process is repeated until the root is approximated sufficiently well.
The above formula is also used in the secant method, but the secant method always retains the last two computed points, while the false position method retains two points which certainly bracket a root. On the other hand, the only difference between the false position method and the bisection method is that the latter uses ck = (ak + bk) / 2.
Bisection method
In mathematics, the bisection method is a root-finding algorithm which repeatedly bisects an interval then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow.
The method is applicable when we wish to solve the equation for the scalar variable x, where f is a continuous function.
The bisection method requires two initial points a and b such that f(a) and f(b) have opposite signs. This is called a bracket of a root, for by the intermediate value theorem the continuous function f must have at least one root in the interval (a, b). The method now divides the interval in two by computing the midpoint c = (a+b) / 2 of the interval. Unless c is itself a root–which is very unlikely, but possible–there are now two possibilities: either f(a) and f(c) have opposite signs and bracket a root, or f(c) and f(b) have opposite signs and bracket a root. We select the subinterval that is a bracket, and apply the same bisection step to it. In this way the interval that might contain a zero of f is reduced in width by 50% at each step. We continue until we have a bracket sufficiently small for our purposes. This is similar to the computer science Binary Search, where the range of possible solutions is halved each iteration.
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Explicitly, if f(a) f(c) < 0, then the method sets c as the new value for b, and if f(b) f(c) < 0, then the method sets c as the new a. In both cases, the new f(a) and f(b) have opposite signs, so the method is applicable to this smaller interval. A practical implementation of this method must guard against the uncommon occurrence that the midpoint is indeed a solution.
Advantages and drawbacks of the bisection method
Advantages of Bisection Method
The bisection method is always convergent. Since the method brackets the root, the method is guaranteed to converge.
As iterations are conducted, the interval gets halved. So one can guarantee the decrease in the error in the solution of the equation.
Drawbacks of Bisection Method
The convergence of bisection method is slow as it is simply based on halving the interval.
If one of the initial guesses is closer to the root, it will take larger number of iterations to reach the root.
If a function is such that it just touches the x-axis (Figure 3.8) such as
it will be unable to find the lower guess, , and upper guess, , such that
For functions where there is a singularity and it reverses sign at the singularity, bisection method may converge on the singularity (Figure 3.9).
An example include
and, are valid initial guesses which satisfy
.
However, the function is not continuous and the theorem that a root exists is also not applicable.
Figure.3.8. Function has a single root at that cannot be bracketed.
Figure.3.9. Function has no root but changes sign.
Explanation
Source code for False position method:
Example code of False-position method
C code was written for clarity instead of efficiency. It was designed to solve the same problem as solved by the Newton’s method and secant method code: to find the positive number x where cos(x) = x3. This problem is transformed into a root-finding problem of the form f(x) = cos(x) – x3 = 0.
#include #include double f(double x)
return cos(x) – x*x*x;
double FalsiMethod(double s, double t, double e, int m)
int n,side=0;
double r,fr,fs = f(s),ft = f(t);
for (n = 1; n <= m; n++)
r = (fs*t – ft*s) / (fs – ft);
if (fabs(t-s) < e*fabs(t+s)) break;
fr = f(r);
if (fr * ft > 0)
t = r; ft = fr;
if (side==-1) fs /= 2;
side = -1;
else if (fs * fr > 0)
s = r; fs = fr;
if (side==+1) ft /= 2;
side = +1;
return r;
int main(void)
printf(“%0.15fn”, FalsiMethod(0, 1, 5E-15, 100));
return 0;
After running this code, the final answer is approximately 0.865474033101614
Consider finding the root of f(x) = x2 – 3. Let εstep = 0.01, εabs = 0.01 and start with the interval [1, 2].
Table 1. False-position method applied to f(x) = x2 – 3.
1.0
2.0
-2.00
1.00
1.6667
-0.2221
a = c
0.6667
1.6667
2.0
-0.2221
1.0
1.7273
-0.0164
a = c
0.0606
1.7273
2.0
-0.0164
1.0
1.7317
0.0012
a = c
0.0044
Thus, with the third iteration, we note that the last step 1.7273 → 1.7317 is less than 0.01 and |f(1.7317)| < 0.01, and therefore we chose b = 1.7317 to be our approximation of the root.
Note that after three iterations of the false-position method, we have an acceptable answer (1.7317 where f(1.7317) = -0.0044) whereas with the bisection method, it took seven iterations to find a (notable less accurate) acceptable answer (1.71344 where f(1.73144) = 0.0082)
Consider finding the root of f(x) = e-x(3.2 sin(x) – 0.5 cos(x)) on the interval [3, 4], this time with εstep = 0.001, εabs = 0.001.
Table 2. False-position method applied to f(x) = e-x(3.2 sin(x) – 0.5 cos(x)).
3.0
4.0
0.047127
-0.038372
3.5513
-0.023411
b = c
0.4487
3.0
3.5513
0.047127
-0.023411
3.3683
-0.0079940
b = c
0.1830
3.0
3.3683
0.047127
-0.0079940
3.3149
-0.0021548
b = c
0.0534
3.0
3.3149
0.047127
-0.0021548
3.3010
-0.00052616
b = c
0.0139
3.0
3.3010
0.047127
-0.00052616
3.2978
-0.00014453
b = c
0.0032
3.0
3.2978
0.047127
-0.00014453
3.2969
-0.000036998
b = c
0.0009
Thus, after the sixth iteration, we note that the final step, 3.2978 → 3.2969 has a size less than 0.001 and |f(3.2969)| < 0.001 and therefore we chose b = 3.2969 to be our approximation of the root.
In this case, the solution we found was not as good as the solution we found using the bisection method (f(3.2963) = 0.000034799) however, we only used six instead of eleven iterations.
#include #include #define epsilon 1e-6
main()
double g1,g2,g,v,v1,v2,dx;
int found,converged,i;
found=0;
printf(” enter the first guessn”);
scanf(“%lf”,&g1);
v1=g1*g1*g1-15;
printf(“value 1 is %lfn”,v1);
while (found==0)
printf(“enter the second guessn”);
scanf(“%lf”,&g2);
v2=g2*g2*g2-15;
printf(” value 2 is %lfn”,v2);
if (v1*v2>0)
{found=0;}
else
found=1;
printf(“right guessn”);
i=1;
while (converged==0)
printf(“n iteration=%dn”,i);
g=(g1+g2)/2;
printf(“new guess is %lfn”,g);
v=g*g*g-15;
printf(“new value is%lfn”,v);
if (v*v1>0)
g1=g;
printf(“the next guess is %lfn”,g);
dx=(g1-g2)/g1;
else
g2=g;
printf(“the next guess is %lfn”,g);
dx=(g1-g2)/g1;
if (fabs(dx)’less than’ epsilon
{converged=1;}
i=i+1;
printf(“nth calculated value is %lfn”,v);
Consider finding the root of f(x) = x2 – 3. Let εstep = 0.01, εabs = 0.01 and start with the interval [1, 2].
Table 1. Bisection method applied to f(x) = x2 – 3.
1.0
2.0
-2.0
1.0
1.5
-0.75
a = c
0.5
1.5
2.0
-0.75
1.0
1.75
0.062
b = c
0.25
1.5
1.75
-0.75
0.0625
1.625
-0.359
a = c
0.125
1.625
1.75
-0.3594
0.0625
1.6875
-0.1523
a = c
0.0625
1.6875
1.75
-0.1523
0.0625
1.7188
-0.0457
a = c
0.0313
1.7188
1.75
-0.0457
0.0625
1.7344
0.0081
b = c
0.0156
1.71988/td>
1.7344
-0.0457
0.0081
1.7266
-0.0189
a = c
0.0078
Thus, with the seventh iteration, we note that the final interval, [1.7266, 1.7344], has a width less than 0.01 and |f(1.7344)| < 0.01, and therefore we chose b = 1.7344 to be our approximation of the root.
Consider finding the root of f(x) = e-x(3.2 sin(x) – 0.5 cos(x)) on the interval [3, 4], this time with εstep = 0.001, εabs = 0.001.
Table 1. Bisection method applied to f(x) = e-x(3.2 sin(x) – 0.5 cos(x)).
3.0
4.0
0.047127
-0.038372
3.5
-0.019757
b = c
0.5
3.0
3.5
0.047127
-0.019757
3.25
0.0058479
a = c
0.25
3.25
3.5
0.0058479
-0.019757
3.375
-0.0086808
b = c
0.125
3.25
3.375
0.0058479
-0.0086808
3.3125
-0.0018773
b = c
0.0625
3.25
3.3125
0.0058479
-0.0018773
3.2812
0.0018739
a = c
0.0313
3.2812
3.3125
0.0018739
-0.0018773
3.2968
-0.000024791
b = c
0.0156
3.2812
3.2968
0.0018739
-0.000024791
3.289
0.00091736
a = c
0.0078
3.289
3.2968
0.00091736
-0.000024791
3.2929
0.00044352
a = c
0.0039
3.2929
3.2968
0.00044352
-0.000024791
3.2948
0.00021466
a = c
0.002
3.2948
3.2968
0.00021466
-0.000024791
3.2958
0.000094077
a = c
0.001
3.2958
3.2968
0.000094077
-0.000024791
3.2963
0.000034799
a = c
0.0005
Thus, after the 11th iteration, we note that the final interval, [3.2958, 3.2968] has a width less than 0.001 and |f(3.2968)| < 0.001 and therefore we chose b = 3.2968 to be our approximation of the root.
Why don’t we always use false position method?
There are times it may converge very, very slowly.
Example:
What other methods can we use?
{
}
{
{
{
}
{
}
else break;
}
}
{
}
Example 1
a
b
f(a)
f(b)
c
f(c)
Update
Step Size
Example 2
a
b
f(a)
f(b)
c
f(c)
Update
Step Size
Source code for Bisection method
{
{
}
{
{
}
{
}
}
}
Example 1
a
b
f(a)
f(b)
c = (a + b)/2
f(c)
Update
new b − a
Example 2
a
b
f(a)
f(b)
c = (a + b)/2
f(c)
Update
new b − a
Convergence Rate
Comparison of rate of convergence for bisection and false-position method
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