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A textbook of Computer Based Numerical and Statiscal Techniques part 7

A textbook of Computer Based Numerical and Statiscal Techniques part 7. 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. | 46 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES PROBLEM SET 2.1 1. Find the smallest root lying in the interval 1 2 up to four decimal places for the equation x6 - x4 - x3 - 1 0 by Bisection Method Ans. 1.4036 2. Find the smallest root of x3 - 9x 1 0 using Bisection Method correct to three decimal places. Ans. 0.111 3. Find the real root of ex 3x by Bisection Method. Ans. 1.5121375 4. Find the positive real root of x - cos x 0 by Bisection Method correct to four decimal places between 0 and 1. Ans. 0.7393 5. Find a root of x3 - x - 11 0 using Bisection Method correct to three decimal places which lies between 2 and 3. Ans. 2.374 6. Find the positive root of the equation xex 1 which lies between 0 and 1. Ans. 0.5671433 7. Solve x3 - 9x 1 0 for the root between x 2 and x 4 by the method of Bisection. U.P.T.U. 2005 Ans. 2.94282 8. Compute the root of log x cos x correct to 2 decimal places using Bisection Method. Ans. 1.5121375 9. Find the root of tan x x 0 up to two decimal places which lies between 2 and 2.1 using Bisection Method. Ans. 2.02875625 10. Use the Bisection Method to find out the positive square root of 30 correct to 4 decimal places. Ans. 5.4771 2.5 FALSE POSITION METHOD OR REGULA FALSI METHOD This method is essentially same as the bisection method except that instead of bisecting the interval. In this method we choose two points x0 and x1 such that f x0 and f x1 are of opposite signs. Since the graph of y f x crosses the X-axis between these two points a root must lie in between these points. Consequently f x0 f x1 0. Equation of the chord joining points x0 f x0 and x1 f x1 is v f x -f x1 -f x0 x x y - f x0 ------------ x - x0 x1 - x0 The method consists in replacing the curve AB by means of the chord AB and taking the point of intersection of the chord with X-axis as an approximation to the root. So the abscissa of the point where chord cuts y 0 is given by X2 X0 - X1 - X0 f X1 - f X0 f x0 ALGEBRAIC AND TRANSCENDENTAL EQUATION 47 The value of .