numerical methods
CHENNAI INSTITUTE OF TECHNOLOGY
Sarathy Nagar, Kundrathur, Pudupedu, Chennai– 600 069.
Department of Mathematics
Subject Name: Numerical Methods
Subject Code: MA1251
Unit I
1) Write the Descartes rule of signs
Sol:
1) An equation f ( x) = 0 cannot have more number of positive roots than there are changes of sign in the terms of the polynomial f ( x) .
2)An equation f ( x) = 0 cannot have more number of positive roots than there are changes of sign in the terms of the polynomial f ( x) .
2) What is the order of convergence of Newton Raphson method if the multiplicity of the root is one.
Sol: Order of convergence of N.R method is 2.
3) Newton Raphson method is also known as the method of ………..
Sol: Iteration ( Newton’s iteration method)
Derive newtons algorithm to derive p th root of a number N.
1
p
Sol: If x = N then x p N 0 is the equation to be solved.
Let f ( x) = x p − N , f ′ ( x) = px p 1
=
−
−
By NR rule ,if xr is the rth iterate f ( xr ) xr 1 = x r − f ′ (rx )
+
xr p − N ( p − 1) xr p + N
=xr−
= p 1 p 1 pxr pxr
−
−
4) When would we not use N-R method .
Sol: If x1 is the exact root and x0 is its approximate value of the equation f ( x0 ) f ′ ( x0 ) f ( x0 )
If f ′ ( x0 ) is small,the will be large and the computation of the root by f ′ ( x0 error ) this,method will be a slow process or may even be impossible.
Hence the method should not be used in cases where the graph of the function when it crosses the x axis is nearly horizontal. f ( x) = 0.we know that x1 = x0 -
5) What is the convergence in NR method?
Sol: x n +1 = x n -
f ( xn ) f ′( x n )
7) State the order of convergence and convergence condition for NR method?
Sol: The order of convergence is 2
2
Condition of convergence is f ( x ) f ′′( x ) < f ′( x )
8) 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].
Sol: Let x = r be a root of x =
Sarathy Nagar, Kundrathur, Pudupedu, Chennai– 600 069.
Department of Mathematics
Subject Name: Numerical Methods
Subject Code: MA1251
Unit I
1) Write the Descartes rule of signs
Sol:
1) An equation f ( x) = 0 cannot have more number of positive roots than there are changes of sign in the terms of the polynomial f ( x) .
2)An equation f ( x) = 0 cannot have more number of positive roots than there are changes of sign in the terms of the polynomial f ( x) .
2) What is the order of convergence of Newton Raphson method if the multiplicity of the root is one.
Sol: Order of convergence of N.R method is 2.
3) Newton Raphson method is also known as the method of ………..
Sol: Iteration ( Newton’s iteration method)
Derive newtons algorithm to derive p th root of a number N.
1
p
Sol: If x = N then x p N 0 is the equation to be solved.
Let f ( x) = x p − N , f ′ ( x) = px p 1
=
−
−
By NR rule ,if xr is the rth iterate f ( xr ) xr 1 = x r − f ′ (rx )
+
xr p − N ( p − 1) xr p + N
=xr−
= p 1 p 1 pxr pxr
−
−
4) When would we not use N-R method .
Sol: If x1 is the exact root and x0 is its approximate value of the equation f ( x0 ) f ′ ( x0 ) f ( x0 )
If f ′ ( x0 ) is small,the will be large and the computation of the root by f ′ ( x0 error ) this,method will be a slow process or may even be impossible.
Hence the method should not be used in cases where the graph of the function when it crosses the x axis is nearly horizontal. f ( x) = 0.we know that x1 = x0 -
5) What is the convergence in NR method?
Sol: x n +1 = x n -
f ( xn ) f ′( x n )
7) State the order of convergence and convergence condition for NR method?
Sol: The order of convergence is 2
2
Condition of convergence is f ( x ) f ′′( x ) < f ′( x )
8) 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].
Sol: Let x = r be a root of x =