b) Find by Newton’s method the roots of the following equations correct to three places of decimals
i) near
ii)
To solve these equations numerically using Newton's method, we follow the iterative formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] Where \( f(x_n) \) is the function and \( f'(x_n) \) is its derivative. ### i) Equation: \( x \log_{10} x ___ _______ __________ _________ ____ ___ _____ ____ ___ _________ ________.
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b) Estimate the eigenvalues of the matrix
using the Gershgorin bounds. Draw a rough sketch of the region where the eigenvalues lie.
b) The position of a particle moving in a line at various times
is given in the following table. Estimate the velocity and acceleration of the particle at
| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 |
| f(x) | 2.72 | 3.32 | 4.06 | 4.96 | 6.05 | 7.39 | 9.02 |
a) Using as an initial approximation find an approximation to one of the zeros of
by using Birge-Vieta method. Perform two iterations.
b) Find by Newton’s method the roots of the following equations correct to three places of decimals
i) near
ii)
b) Solve the system of equations
with partial pivoting. Store the multipliers and also write the pivoting vectors.
a) The equation has two real roots p and q such that
If we use the fixed point iteration
to find a root then to which root does it converge?
c) Find the inverse of the matrix
using Gauss Jordan method.