Numerical Methods
Solutions of Equations and Eigenvalue problems
(1) Explain Regula-Falsi method of getting a root.
Let f(x)=0 be the given equation
• Find two numbers a and b such that f(a) and f(b) are of different signs then
the roots lies between a and b .
• The first approximation to the root in given by x1=[af(b) − bf(a) ]/[f(b) − f(a)]
• If f(x1) and f(a) are of opposite signs then the actual root lies between x1
and a . Now replacing b by x1 and keeping a as it is we get the next closer
approximation x2 to the actual root.
• This procedure is repeated till the root is found to the desired degree of
accuracy.
(2) What is the condition for convergence of Gauss-Jacobi method of
iteration.
The coefficient matrix should be diagonally dominated.
(3) What is the order of convergence in Newton-Raphson method.
The order of convergence in Newton-Raphson method is two.
(4) Gauss-Seidal method is better then Gauss-Jacobi method,Why.
In Gauss-Seidal method the latest values of unknowns at each stage of iteration are
used in proceeding to the next stage of iteration.Hence the convergence in Gauss-
Seidal method is more rapid than Gauss-Jacobi method.
(5) State the formula to find the root of the equation f(x) = 0 which lies
between x = a and x = b by Regula-Falsi method.
x1 =[af(b) − bf(a)]/[f(b) − f(a)]
where
x1 the first approximate value.
(6) Write the sufficient condition for convergence of gauss seidal method.
Let the given set of equation be
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
then the sufficient condition is
|a1| > |b1| + |c1|
|b2| > |a2| + |c2|
|c3| > |a3| + |b3|
(7) Explain Gauss-Elimination method to solve AX = B .
In this mehod the given system is transformed into an equivalent system with upper-
triangular coefficient matrix in which all element below the diagonal elements are
zero which can be solved by back substitution.
(8) Explain briefly Gauss-Jordan interation to solve simultaneous equation.
In this method the Coefficient matrix is reduced to a diagonal matrix (or a unit
matrix) rather than triangular matrix as in the Gaussian method. Here the
elimination of the unknowns is done not only in the eqauation below, but also in the
equations above the leading diagonal. Here we get the solution without using the
back substitution method.
(9) If g(x) is continuous in [a, b] ,then under what condition the Iteration
method x = g(x) has a unique solution in [a, b] .
Let x = r be a root of x = g(x) . Let I be an internal combining the point x = r
if |g′(x)| < 1 for all x in I, the sequence of approximation x0, x1 . . . . . . , xn will
converge to the root r provided that the initial approximation x0 is chosen in I.
(10) What is the order of convergence for fixed point iteration.
The convergence in linear and the convergence in of order one. The convergence is
quadratic, convergence is of order two.
(11) State the principle used in Gauss - Jordan method.
Coefficient matrix is transformed into diagonal matrix.
(12) For solving a Linear system, compare Gaussian elimination method and
Gauss-Jordan method.
Gaussian elimination method
- Coefficient matrix transformed into upper triangular matrix
- Direct method
- We obtain the solution by back substituion method
- We get exact solution
- Simple take less time
Gauss-Jordan method
1. Coefficient matrix transformed into diagonal
2. Direct method
3. No need of back substituion method
(13) Compare Gauss-Jacobi and Gauss-Seidal method.
Gaus-Jacobi method
1. Convergence rate is slow
2. Indirect method
3. Condition for convergence is the coefficient matrix is diagonally dominant
Gauss-Seidal method
1. The rate of convergence of Gauss-seidal is roughly twice that of Gauss-Jacobi.
2. Indirect method
3. Condition for convergence is the coefficient matrix is diagonally dominant
(14) The convergence in the Gauss-Seidal method is as fast as in Jacobi’s
method say true or false.
Infact the rate of convergence of Gayss-Seidal method is roughly twice that of Gauss-
Jacobi.
(15) Distinguish between direct and indirect method of solving simultaneous
equation.
Direct method
Indirect method
1.Approximate solution
2. Time consuming labours
(16) What is the criterion for the convergence in Newton-Raphson method.
The criterion for the convergence is
|f(x)f′′(x)| < |f′(x)|2
(17) What type of eigen value can be obtained using power mehtod.
The largest eigen value can be obtained by power method.
(18) Give two indirect method to solve a systems linear method.
• Gauss-Jacobi method
• Gauss-seidel method
(19) Give Newton-Raphson iteration formula.
xn+1 = xn –[f(xn)]/[f′(xn)]
where n = 0, 1, 2 . . . . . .
(20) Mention the methods to solve the equation which is algebric (or)
transcendental.
• Regula-Falsi method
• Newton-Raphson method
• Iteration method
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