Explain with suitable example that in computer arithmatics (i.e., numbers represented in computer, with +, -, , / as implemented in a computer) the multiplication operation() may not be distributive over plus (+), i.e. may not be true for some computer numbers a, b and c
In computer arithmetic, due to the finite precision of floating-point representations, certain mathematical properties such as distributivity may not always hold. The distributive property states that for any real numbers $a$, $b$, and $c$, the equation $a \times (b + c) = a \times b + a \times c$ should always be true. However, in computer arithmetic, due to rounding errors and limited precision, this equality might not hold in all ______ _______ __________ ________ ________.
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Perform four iterations (rounded to four decimal places) using
(i) Jacobi Method and
(ii) Gauss-Seidel method,
for the following system of equations
. The exact solution is (1, 2, 3)T. Which method gives better approximation to the exact solution?
Regula-falsi method (ii) Newton Raphson method (iii) Bisection method (iv) Secant method
Obtain Approximate the value of (3.7)-1 using first three terms of Taylor's series expansion.
Calculate a bound for the truncation error in approximating
where
and (fact n) denotes factorial of n
By decinnial census, the population of a town was given below
Year (x) : 1971 1981 1991 2001 2011
Population (y): 112 132 158 189 226 (in thousands)
(i) Using Stirling's central difference formula, estimate the population for the year 2006
(ii) Using Newton's forward formula, estimate the population for the year 1992.
(iii) Using Newton's backward formula, estimate the population for the year 1980.
Find the values of the first and second derivatives of f(x) at x = 76 from the following table. Use 0(h2 ) forward difference method. Also, find Truncation Error (TE) and actual errors.
| x: | 76 | 81 | 86 | 91 |
| f(x) : | 5.3147 | 5.4346 | 5.5637 | 5.6629 |
If values of the function
find the Lagrange's interpolation polynomial of f(x). Also, find f