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A textbook of Computer Based Numerical and Statiscal Techniques part 9

A textbook of Computer Based Numerical and Statiscal Techniques part 9. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 66 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Putting i 2 in 1 x3 0.02439 Therefore reciprocal of 41 is 0.0244. Example 8. Find the square root of 20 correct to 3 decimal places by using recursion formula 1 1 20 Xi 1 2 1 Xi xi Sol. Put 1 20 i 0 x1 -1 X 1 2 Xo Let x0 4.5 X 1 4.5 1 4.47 1 2 4.5 Put i 1 x 1 4.47 4.472 2 2 4.47 Put i 2 x 1 4.472 -20- 4.4721. 3 21 4.472 Therefore - 20 4.472 correct to three decimal places. Example 9. Show that the following rearrangement of equation x3 6x2 10x - 20 0 does not yield a convergent sequence of successive approximations by iteration method near x 1 x 20 - 6x2 - x3 10. Sol. Here x 2 -- X f x 12x - 3x2 Hence f x -----1 ---- Clearly f x - 1 in nbd of x 1 Hence f x 1 and therefore the method and hence the sequence xn does not converge. Example 10. Find the smallest root of the equation x2 x3 x4 x5 1 x --z -----z ---z -----z . 0. I 2 3 4 5 Sol. Written the given equation as x2 x3 x4 x5 X 1 . 0 x 2 2 3 2 4 2 5 2 Omitting x2 and higher powers of x we get x 1 approximately. Taking x0 1 we obtain x1 0 XO 1 1111 -----n T -----------ö T 2 2 3 2 4 2 5 2 . 1.2239 1.2239 2 x2 x1 1 v 2 2 1.2239 3 1.2239 4 1.2239 5 3 2 4 5 2 . 1.3263 67 ALGEBRAIC AND TRANSCENDENTAL EQUATION Similarly x3 0 x2 1.38 x4 0 x3 1.409 x5 0 x4 1.425 x6 0 x5 1.434 x7 0 x6 1.439 x8 0 x7 1.442 Values of x7 and x8 indicate that the root is 1.44 correct to two decimal places. PROBLEM SET 2.3 1. Use the method of Iteration to find a positive root between 0 and 1 of the equation xex - 1. Ans. 0.5671477 2. Find the Iterative method the real root of the equation 3x - log10 x 6 correct to four significant figures. Ans. 2.108 3. Solve by Iteration method a x3 x 1 0 Ans. -0.682327803 x 1 x-1 b sin x Ans. -0.420365 c x3 - 2x2 - 4 0 Ans. 2.5943 4. By Iteration method find 30 . Ans. 5.477225575 5. If f x is sufficiently differentiable and the iteration xn 1 F xn converges prove that the order of convergence is a positive integer. x2 x3 x4 6. The equation f x 0 where f x .