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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Solve the system of equations:
x + 2y + z = 3
3x − 2y − z4 = −2
2x + 3y − z = −6
using Gauss-Elimination method.
Using the data sin (0.1) = 0.09983 and sin (0.2) = 0.19867, find an approximate value of sin (0.15) by Lagrange’s interpolation. Also obtain a bound on the truncation error.
) Use the Euler’s method to solve numerically the initial value problem:
with h = 2.0 on the interval [0, 1].
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.
If
f(x) = eax ,
show that
Find the inverse of the matrix:
using LU decomposition method with u11 = u22 = u33 = 1.
From the following data, calculate the population in the year 1985:
| Year | Population (in , 000) |
| 1971 | 12 |
| 1981 | 15 |
| 1991 | 20 |
| 2001 | 27 |
| 2011 | 49 |