Find the number of positive and negative roots of the polynomial
P( x) = x3 − 3x2 + 4x − 5 . Find P(2) and P′(2) using synthetic division method.
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Using of x0 = −2 as an initial approximation find an approximation to one of the zeros p(x) = 2x4 − 3x2 + 3x − 4 by using Birge-Vieta method. Perform two iterations.
Using the classical R-K method of o(h4) calculate approximate solution of the IVP, y' = 1− x + 4y, y(0) = 1 at x = 0.6, taking h = 0.1 and 0.2 . Use extrapolation technique to improve the accuracy.
Use the polynomial in part (i) to approximate
.
For the linear system of equations
set up the Gauss-Jacobi and Gauss-Seidal iteration schemes in matrix form. Also check the convergence of the two schemes.
Using the following table of values, find approximately by Simpson’s rule, the arc length of the graph between the points (1 , 1) and
| x | 1 | 2 | 3 | 4 | 5 |
| 1.414 | 1.031 | 1.007 | 1.002 | 1.001 |
Using finite differences, show that the data
| x | − 3 | − 2 | − 1 | 0 | 1 | 2 | 3 |
| f ( x) | 13 | 7 | 3 | 1 | 1 | 3 | 7 |
Solve the system of equations
0.6x + 0.8y + 0.1z = 1
1.1x + 0.4y + 0.3z = 0.2
x + y + 2z = 0.5
by LU decomposition method and find the inverse of the coefficient matrix