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Home » Articles » LESSON NOTES (Mathematics) » High School level


NUMERICAL METHODS NMDIT398AR

NUMERICAL METHODS

The following will be discussed in this Topic

  1. Iteration
  2. linear interpolation
  3. Newton-Raphson method
  4. Bisection method
  5. Secant method
  6. Regular falsi method
  7. Numerical integration
  1. Trapezium rule
  2. Simpson rule

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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:

  1. X1= 4 and X2 = 5
  2. Take the average say X3 = (x1 + x2) / 2 that is (4 + 5)/2 = 4.5   then X3 = 4.5.
  3. 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 the square root of 18 is 4.25.

In general the square root of positive number N can be calculated from the iterative formula:

where Xr is the first approximation square 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

 

HOW TO 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:

  1. 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)
  2. 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 = x3/53/5………. (II)
  3. 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 and diverges.

  1. Find the average of the two roots given above i.e. x = 2 and x = 3 then the average is 2.5
  2. 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 Xn + 1 = 5/Xn + 3/X2n from 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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Category: High School level | Added by: Admin (07/Aug/2016) | Author: Yahya Mohamed E W
Views: 436 | Tags: mathematics, Approximation, numerical methods, NMD398ARTC | Rating: 0.0/0
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