UNIT II SOLUTION OF NON-LINEAR AND SIMULTANEOUS LINEAR EQUATION. If g x is continuous in a, b, then under what condition the iterative method x = g x has a unique solution in a, b? g x < in a, b. State the iterative formula for Regula Falsi method to solve f x =. The iteration formula to find the root of the equation f x =. which lies between x = a and x = b is x =. Explain Regula Falsi method of getting a root. a f b b f(a) f b f(a). Find two numbers a and b such that f a and f(b) are of opposite signs. Then a root lies between a and b.. The first approximation to the root is given by x = a f b b f(a) f b f(a). If f x and f(a) are opposite signs, then the actual root lies between x and a. Now replacing b by x and keeping a as it is we get the closer approximation x to the actual root. 4. This procedure is repeated till the root is found to the desired degree of accuracy. 4. How to reduce the number of iterations while finding the root of an equation by Regula Falsi method. The number of iterations to get a good approximation to the real root can be reduced, if we start with a smaller interval for the root. 5. What is the order of convergence of Newton- Raphson method if the multiplicity of the root is one. The order of convergence of Newton-Raphson method is 6. What is the rate of convergence in Newton-Raphson method? The rate of convergence in Newton-Raphson method is. 7. What is the criterion for convergence of Newton- Raphson method?.
The Criterion for convergence of Newton- Raphson -method is f x f x < f x in te interval considered. 8. Write the iterative formula for Newton- Raphson method. The Newton- Raphson formula is + = f f 9. What is the condition of convergence of a fixed point iteration method? Let f x = be the given equation whose actual root is r. The equation f x = be written as x = g x. Let I be the interval containing the root x = r. If g x < for all x in I, then the sequence of approximations x, x, x,. will converge to r, if the initial starting value x is chosen in I.. If g x is continuous in a, b, then under what condition the iterative method x = g x has a unique solution in a, b? g x < in a, b. What is the order of convergence of fixed point iteration method. The order of convergence of Newton-Raphson method is.. In what form is the coefficient matrix transformed into when A X = B is solved by Gauss elimination method? Upper triangular matrix.. In what form is the coefficient matrix transformed into when A X = B is solved by Gauss Jordan method? Diagonal matrix. 4. Explain briefly Gauss Jordan iteration to solve simultaneous equation? Consider the system of equations A X = B. If A is a square matrix the given system reduces to a a a nn x x = b b b n This system is reduces to the following n equations. a x = b, a x = b, a nn = b n x = b a, x = b a, = b n a nn
The method of obtaining the solution of the system of equations by reducing the matrix A to diagonal matrix is known as Gauss Jordan elimination method. 5. For solving a linear system, compare Gauss - elimination method and Gauss Jordan method. Gauss - elimination method Coefficient matrix transformed into upper triangular matrix Gauss Jordan method Coefficient matrix transformed into diagonal matrix Direct method Direct method We obtain the solution by Backward substitution method No need of Backward substitution method 6. State the principle used in Gauss Jordan method. Coefficient matrix transformed into upper triangular matrix 7. Write the sufficient condition for Gauss - Siedal method to converge. The coefficient of matrix should be diagonally dominant. a > b + c, b > a + c, c > a + b 8. State the sufficient condition for Gauss - Jacobi method to converge. The coefficient of matrix should be diagonally dominant. a > b + c, b > a + c, c > a + b 9. Give two indirect methods to solve a system of linear equations. (). Gauss Jacobi method (). Gauss Siedal method. Compare Gauss Jacobi method and Gauss Siedal method Gauss Jacobi method Convergence rate is slow Indirect method Indirect method Condition for the convergence is the coefficient matrix is diagonally dominant Gauss Siedal method The rate of convergence of Gauss Siedal method is roughly twice that of Gauss - Jacobi Condition for the convergence is the coefficient matrix is diagonally dominant
. Find the first approximation to the root lying between and of x + x = by Newton s method. f x = x + x and f x = x + x = f x f x =... Find an iteration formula for finding the square root of N by Newton method. f x = x N and f x = x + = f f N = x n + N, n =,,,. How do you express the equation x + x = for the positive root by iteration method? x + x = x x + = x x + = x = x + x = = g x x + 4. Can we write iteration method to find the root of the equation x cos x = in, π?. x cos x = x = + cos x x = + cos x = φ x φ x = + cos x φ x = sin x φ = sin < and φ π = sin π =.5 <. Hence φ x = sin x < for all I =, π. Therefore we can use iteration method. 5. Is the system of equations x + 9y z =, 4x + y + z =, 4x y + z = Diagonally dominant? If not make it diagonally dominant. Let x + 9y z =, 4x + y + z =, 4x y + z = a > b + c, b > a + c and c > a + b 4
Hence the given system is not diagonally dominant. Hence we rearrange the system as follows 4x y + z = x + 9y z =, 4x + y + z = 6. Write Newton s formula for finding cube root of N. x N = f x = x N and f x = x By newton smetod, we ave 7. Write Newton s formula for finding reciprocal of a positive N. + = N, n =,, N = f x = N and x x f x = x By newton x N smetod + = n = x n N, n =,, 8. What is the basic principle involved in triangularisation method? Ans. The method of triangularisation is based on the fact that the coefficient matrix can be expressed as the product of a unit lower triangular matrix and an upper triangular matrix. 5