Explain each of the following concepts, along with at least one suitable example for each: (i) Fixed-point number representation (ii) round-off error (iii) representation of zero as floating point number (iv) significant digits in a decimal number representation (v) normalized representation of a floating point number (vi) overflow
(i) Fixed-point number representation: Fixed-point representation is a method of storing real numbers by fixing the number of digits after (and sometimes before) the decimal point. It is useful for applications requiring fast computation and limited precision. For example, in 8-bit fixed-point format with 4 bits for the integer part and 4 bits for the fractional part, the number 5.25 would be represented as 0101.0100. This method is __________ __________ ______ ______ _______ _________ ____ _________ ______ ____ _________.
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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.
If values of the function
find the Lagrange's interpolation polynomial of f(x). Also, find f
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
Solve the system of equations
using Gauss elimination method with partial pivoting.
Determine the smallest roots of the following equation:
to three significant digits using
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 |
Compute the value of the integral
by using
Rectangular Rule (ii) Trapezoidal Rule and then (iii) Simpson's 1/3 Rule