NUMERICAL METHODS NMDIT398AR
 div.scroll { background-color: #ffffff; width: 550px; height: 1000px; padding: 8px; overflow: scroll; } NUMERICAL METHODS 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 ************************** 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 = 5 Take 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 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: 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 and diverges. 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 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.   ------------------END OF ITERATION ---------------------- .u-star-rating-16 { list-style:none;margin:0px;padding:0px;width:80px;height:16px;position:relative;background: url('/.s/t/1706/rating.png') top left repeat-x } .u-star-rating-16 li{ padding:0px;margin:0px;float:left } .u-star-rating-16 li a { display:block;width:16px;height: 16px;line-height:16px;text-decoration:none;text-indent:-9000px;z-index:20;position:absolute;padding: 0px;overflow:hidden } .u-star-rating-16 li a:hover { background: url('/.s/t/1706/rating.png') left center;z-index:2;left:0px;border:none } .u-star-rating-16 a.u-one-star { left:0px } .u-star-rating-16 a.u-one-star:hover { width:16px } .u-star-rating-16 a.u-two-stars { left:16px } .u-star-rating-16 a.u-two-stars:hover { width:32px } .u-star-rating-16 a.u-three-stars { left:32px } .u-star-rating-16 a.u-three-stars:hover { width:48px } .u-star-rating-16 a.u-four-stars { left:48px } .u-star-rating-16 a.u-four-stars:hover { width:64px } .u-star-rating-16 a.u-five-stars { left:64px } .u-star-rating-16 a.u-five-stars:hover { width:80px } .u-star-rating-16 li.u-current-rating { top:0 !important; left:0 !important;margin:0 !important;padding:0 !important;outline:none;background: url('/.s/t/1706/rating.png') left bottom;position: absolute;height:16px !important;line-height:16px !important;display:block;text-indent:-9000px;z-index:1 } Category: High School level | Added by: Admin (07/Aug/2016) | Author: Yahya Mohamed E W Views: 496 | | Rating: 0.0/0