Use secant method to determine the root of the equation cos x - xex = 0. Take the initial approximation as x0 = 0, x1 = 1 and perform two iterations of the method.
The given equation is $f(x) = \cos x - x e^{x} = 0$. The secant method uses the formula $x_{n+1} = x_{n} - f(x_{n}) \frac{(x_{n} - x_{n-1})}{f(x_{n}) - f(x_{n-1})}$. Starting __________ _________ _______ ____ ________ _______ _________ __________ __________ __________ ________ ___.
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Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:
i) The equation x3 4x 16 = 0 has a root in the interval [3, 4].
ii) The order of convergence of the secant method is 0.62.
iii) For the system of linear equations:
5x + y + 2z = 34
4y − 3z = 12
10x − 2y + z = −4
the matrix is diagonally dominant.
iv) The numerical method:
is relatively stable.
v) The method:
converges to 1.5 for any choice of initial approximation.
If
f(x) = eax ,
show that
Solve the system of equations:
x + 2y + z = 3
3x − 2y − z4 = −2
2x + 3y − z = −6
using Gauss-Elimination method.
Using Runge-Kutta fourth order method with h = 0.1, find an approximation value of y(0.1) for the initial value problem:
b) Evaluate the integral:
using composite trapezoidal rule with h = ,2.0 compare with the exact value.
Find the first term of the series whose second and subsequent terms are ,0,3,8 − 0,1 using difference table.
Calculate the nth divided difference of 1/x, on the nodal points X0 , X1 , ......., Xn .
) Use the Euler’s method to solve numerically the initial value problem:
with h = 2.0 on the interval [0, 1].