| Admin | Date: Monday, 28/Jan/2013, 00:07 | Message # 1 |
 Sergeant
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| What is your knowledge today? Just start teaching your friends anything you know HERE. you are most welcome.
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| Admin | Date: Monday, 28/Jan/2013, 00:10 | Message # 2 |
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Sergeant
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| 1. ITERATION
Also known as successive
approximation which is the process of finding the roots of the number.
Suppose we need to find the
square root of 18 i.e. √ (18) but 18 lies between 4 and 5 it means 42
= 16, less 2 from18 and 52 = 25, 7 more than 18 that’s why it’s said
to lie between 4 and 5
Then x1= first
approximation and x2 = second approximation then:
X1= 4 and X2 = 5Take the average say X3 = (x1 + x2) / 2 that is (4 + 5)/2 = 4.5 then X3 = 4.5.Take the average again say X4 = (x1 + x3) /2 that is (4 + 4.5)/2 = 4.25 then X4 = 4.25
Therefore by two iteration thesquare root of 18 is 4.25.
In general the square root of
positive number N can be calculated from the iterative formula Xr+1 =1/2(Xr+ N/xr) where Xr is the first approximationsquare root of the given positive number N. e.g. if given to find √(27) then Xr
is 5.
Qn: find the square roots of the
following using iterative formula.
a). 145, b). 47
c).65 d).82
HOWTO FIND THE FORMULAR FOR FINDING ROOTS OF EQUATION
Example:
Establish the formula to solve
the equation x3 – 5x – 3 = 0 for the roots which lie between x = 2
and x = 3.
Solution:
Make x the subject of the formula using x3 that will be x3 = 5x + 3, and by dividing by x2 the formula will be x = 5/x + 3/x2……….. (I) Also make x the subject of the formula using 5x that will 5x = x3 – 3 and dividing by 5 both sides the formula will become x = x3/5 – 3/5………. (II)Then we test for converges and diverges in each equation above, if f ́(x) < 1 the formula will be converges and if f
́(x) > 1 the formula will be diverges, but we need converges.
How to test for converges anddiverges.
Find the average of the two roots given above i.e. x = 2 and x = 3 then the average is 2.5find the derivative of the equations above i.e. eqn. (I) will be f ́(x) = -5x-2 – 6x-3 then substitute the value of x by average above say x = 2.5 then by substitution f’(2.5) =
0.835, that is less than 1 then the formula is converges and there is no
need for testing another because the converges equation is already
obtained.
There fore the iterative formula
is Xn+ 1 = 5/Xn + 3/X2nfrom the first equation.
To find the roots we start by x1
= 2.5 in order to get X2 i.e. X2 = 5/2.5
+ 3/ (2.5)2 then continue up to 3 or 4 iteration.
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| don | Date: Wednesday, 05/Apr/2017, 20:21 | Message # 3 |
 Private
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| Hallo friends, to me mathematics is my best subject because i love it if you see my results you will see mathematics 100 percent why? Because I love it and following all the rules and regulations concerning mathematics that's all I wanted to say about mathematics to me  byeeeeee friends love you aaalllll!.
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| Admin | Date: Wednesday, 05/Apr/2017, 20:32 | Message # 4 |
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Sergeant
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| Yeah.. I know and hat off to you mr.Don keep it up. Go East and West Mathematics is Best. 
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