TAILIEUCHUNG - A textbook of Computer Based Numerical and Statiscal Techniques part 40

A textbook of Computer Based Numerical and Statiscal Techniques part 40. 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. | 376 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES largest coefficient of y. We continue this process till last equation. This procedure is known as partial pivoting. In general the rearrangement of equation is done even if pivot element is non-zero to improve the accuracy of solution by reducing the round off errors involved in elimination process by getting a larger determinant which is done by finding a largest element of the row as the pivotal element. Complete Pivoting If the order of elimination of xv xv xv is not important then we may choose at each stage the largest coefficient of the whole matrix of coefficients. We may search the largest value not only in rows but also in columns. After searching largest value we bring at the diagonal position. This method of elimination is known as complete pivoting. The superiority of this method is that it gives the solution of a system provided its determinant does not vanish in finite number of steps. ILL-CONDITIONED SYSTEM OF EQUATIONS A system of equations A X B is said to be ill-conditioned or unstable if it is highly sensitive to small changes in A and B . small change in A or B causes a large change in the solution of the system. On the other hand if small changes in A and B give small changes in the solution the system is said to be stable or well conditioned. Thus in a ill-conditioned system even the small round off errors effect the solutions very badly. Unfortunately it is quite difficult to recognize an ill-conditioned system. For example consider the following two almost identical systems. x1 - x2 1 x1 - x2 1 x1 x2 0 and x1 - x2 0 Respective solutions are 100001 100000 and -99999 -100000 obviously the two solutions differ very widely. Therefore the system is ill conditioned. Example 3. Show that the following system of linear equations is ill-conditioned. 7x - 10y 1 5x 7y Sol. On solving the given equations we get x 0 and y . Now we make slight changes in the given .

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