use excel spreadsheet , help please Solve the following problems using Microsoft
ID: 675193 • Letter: U
Question
use excel spreadsheet , help please Solve the following problems using Microsoft Excel Consider the cubic polynomial function f(x) = x^3 + 4x^2 -10 = 0 Propose one iterative scheme using the Fixed-Point Iteration Method that will converge to a root of the function, starting with the initial guess x-0 = 1. Prove that it converges by running enough iterations to determine that it does. Use the Halley's Rational Method to find one root, starting with the same initial guess. Continue your iterations until the absolute value of the function is smaller that 0.0001Explanation / Answer
Let us find location of the + ive roots.
x
0
1
2
> 2
f(x)
- 10
- 5
14
Sign f(x)
-
-
+
+
There is only one + ive root and it lies between 1 and 2. Let x1 = 1 and x2 = 2; at x = 1, f(x) is
– ive and at x = 2, f(x) is + ive. We examine the sign of f(x) at and check whether the root lies in the interval (1, 1.5) or (1.5, 2). Let us show the computations in the table below :
Iteration No.
Sign f(x)
Sign f(x) ´ f(x2)
x1
x2
1
1.5
+ 2.375
+
1
1.5
2
1.25
- 1.797
-
1.25
1.5
3
1.375
+ 0.162
+
1.25
1.375
4
1.3125
- 0.8484
-
1.3125
1.375
5
1.3438
- 0.3502
-
1.3438
1.375
6
1.3594
- 0.0960
-
1.3594
1.375
7
1.367
- 0.0471
-
1.367
1.375
8
1.371
+ 0.0956
+
1.367
1.371
We see that .
We can choose the root as .
Regula-Falsi Method (or Method of False Position)
In this method also we find two values of x say x1 and x2 where function f(x) has opposite signs and there is only one root in the interval (x1, x2). Let us express the function of y = f(x) and we are interested in finding the value of x where curve y = f(x) intersects x-axis i.e. y = 0. We identify two points (x1, y1) and (x2, y2) on the curve. Then we approximate the curve by a straight line joining these two points. We find the point on the x-axis where this line cuts the x-axis. The equation of the straight line passing through (x1, y1) and (x2, y2) is given by
The point on x-axis where y = 0 is given by
Now we check the sign of f(x) and proceed like in the bisection method. That is, if f(x) has same sign as f(x2) then root lies in the interval (x1, x) and if they have opposite signs, then it lies in the interval (x, x2). See Figure 1.
Figure 1 : Regula-Falsi Method, Superscript Shows Iteration Number
Example
Find positive root of by Regula-Falsi method. Compute upto the two decimal places only.
Solution
It is the same problem as given in the previous example. We start by taking x1 = 1 and x2 = 2. We have ; y1 = - 5 and y2 = 14. The point on the curve are (1, – 5) and
(2, 14). The points on the x-axis where the line joining these two pints cuts it, is given by
I-Iteration
II-Iteration
Take points (1.26, – 1.65) and (2, 14)
III-Iteration
Take two points (1.34, – 0.41) and (2, 14)
IV-Iteration
Take two points (1.36, – 0.086) and (2, 14)
Since value of x repeats we take the root as x = 1.36.
x
0
1
2
> 2
f(x)
- 10
- 5
14
Sign f(x)
-
-
+
+
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