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The following will be discussed in this Topic - Iteration
- linear interpolation
- Newton-Raphson method
- Bisection method
- Secant method
- Regular falsi method
- Numerical integration
- Trapezium rule
- Simpson rule
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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 4 Then x - X
_{1}= 4 and X_{2}= 5 - Take the average say X
_{3}=^{(x1 + x2)}/ 2 that is (4 + 5)/2 = 4.5**then**X_{3}= 4.5. - Take the average again say X
_{4 }=^{(x1 + x3) }/2 that is (4 + 4.5)/2 = 4.25 then X_{4}= 4.25
In general the square root of positive number N can be calculated from the iterative formula:
where X
a). 145, b). 47 c).65 d).82
Establish the formula to solve the equation x
- Make x the subject of the formula using x
^{3}that will be x^{3 }= 5x + 3, and by dividing by x^{2}the formula will be x =^{5}/_{x}+^{3}/x^{2}……….. (I) - Also make x the subject of the formula using 5x that will 5x = x
^{3}– 3 and dividing by 5 both sides the formula will become x = x^{3}/_{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.
- Find the average of the two roots given above i.e. x = 2 and x = 3 then the average is 2.5
- find 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 To find the roots we start by x
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