Using the third order Taylor’s series method, find the solution of the initial value problem:
at x = 0.1 taking h = 0.1
To solve the given initial value problem using the third-order Taylor series method, we begin with the general formula for Taylor series: $$ y_{n+1} = y_n + h y'_n + \frac{h^2}{2!} y''_n + \frac{h^3}{3!} y'''_n $$ where $y' = f(x, y)$, $y'' = \frac{d}{dx} f(x, y)$, _________ ______ ________ ______ ___ __________.
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Estimate the eigen values of the matrix:
using the Gerschgorin bounds. Also, draw the rough sketch of the region in which the eigen values lie.
Find one root of the equation:
x3 - 2x 5 = 0
in the interval [3,2] using Birge-Vieta method. Perform only one iteration.
Given the data:
f(3) = 168,
f(7) = 120
and f(9) = 72
If f(k) is estimated as 138 using the Newton’s form of the interpolating polynomial, then find the value of .k
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 |
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.
Using Newton-Rapshson method, find an iterative formula to compute the reciprocal of a natural number N.
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.