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A table of values is to be constructed for the function ) f (x given by
x
f x
+
=
1
1
( ) in
the interval ]4,1[ with equal step length. Determine the spacing h such that
quadratic interpolation gives result with accuracy 6
1 10−
× .
x
f x
+
=
1
1
( ) in
the interval ]4,1[ with equal step length. Determine the spacing h such that
quadratic interpolation gives result with accuracy 6
1 10−
× .
Show that
2
1 1
2
2 2 δ
+ µ δ = + where µ and δ are the average and central
differences operators, respectively
2
1 1
2
2 2 δ
+ µ δ = + where µ and δ are the average and central
differences operators, respectively
Derive a suitable numerical differentiation formula of ) (0
2
h to find )4.2( f ′′ with
h = 1.0 given the table
5
x 0.1 1.2 2.4 3.9
f (x) 3.41 2.68 1.37 − 48
2
h to find )4.2( f ′′ with
h = 1.0 given the table
5
x 0.1 1.2 2.4 3.9
f (x) 3.41 2.68 1.37 − 48
Determine the constants α, in the differentiation formula β, γ
y x( ) x(y )h x(y ) x(y )h ′
0 = α 0 − + β 0 + γ 0 +
so that the method is of the highest possible order. Find the order and the error term
of the method.
y x( ) x(y )h x(y ) x(y )h ′
0 = α 0 − + β 0 + γ 0 +
so that the method is of the highest possible order. Find the order and the error term
of the method.
The equation 0 2 5
3 2
x + x − = has a positive root in the interval [2,1] . Write a fixed
point iteration method and show that it converges. Starting with initial approximation
5.1 x0 = find the root of the equation. Perform two iterations.
3 2
x + x − = has a positive root in the interval [2,1] . Write a fixed
point iteration method and show that it converges. Starting with initial approximation
5.1 x0 = find the root of the equation. Perform two iterations.
For the linear system of equations
=
3
7
4
1 1 3
3 5 1
4 1 2
z
y
x
i) Set up the Gauss-Seidel iteration scheme in matrix form. (2)
ii) Show that this iteration scheme converges and find its rate of convergence. (1)
iii) Starting with initial approximation T
44.0[ perform two iterations of 8.0 44.0 ]
this scheme.
=
3
7
4
1 1 3
3 5 1
4 1 2
z
y
x
i) Set up the Gauss-Seidel iteration scheme in matrix form. (2)
ii) Show that this iteration scheme converges and find its rate of convergence. (1)
iii) Starting with initial approximation T
44.0[ perform two iterations of 8.0 44.0 ]
this scheme.
Solve the system of equations
7 3x + 2y + 4z =
7 2x + y + z =
2 x + 3y + 5z =
using Gauss elimination with pivoting. Store the multipliers and also write the
pivoting vector.
7 3x + 2y + 4z =
7 2x + y + z =
2 x + 3y + 5z =
using Gauss elimination with pivoting. Store the multipliers and also write the
pivoting vector.
The polynomial 3 ( ) 2 5
3 2
f x = x − x + is given. Find )5.1( f ),5.1( f ′ ),5.1( f ′′ and
f ′′′ )5.1( using synthetic division.
3 2
f x = x − x + is given. Find )5.1( f ),5.1( f ′ ),5.1( f ′′ and
f ′′′ )5.1( using synthetic division.
Perform three iterations of the inverse power method to obtain the smallest
eigenvalue in magnitude of the matrix
− 3 5
1 1
. Take the initial approximation to
the eigenvector as T
9.0[ . ]0.1 (
eigenvalue in magnitude of the matrix
− 3 5
1 1
. Take the initial approximation to
the eigenvector as T
9.0[ . ]0.1 (
Before leaving to visit mexico,Levant traded 270 american dollars and received 3,000 Mexican pesos.When he returned from mexico, he had 100 pesos left. How much will he receive when he exchanges pesos for dollars?
