What is the meaning of the term "Subtractive Cancellation"? Explain with the help of an example. How is subtractive cancellation related to or different from an Unstable Algorithm? Explain with the help of an example.
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What is the meaning of the term "Subtractive Cancellation"? Explain with the help of an example. How is subtractive cancellation related to or different from an Unstable Algorithm? Explain with the help of an example.
(a) Use the eight-decimal digit floating point representation as given in your Block 1, Unit 1, Section 1.3.1 page 29 to perform the following operations:
(i) Represent 0.000001235432 and 257890000012 as floating point numbers in normalised form using chopping for first number and rounding for second number.
(ii) What is the absolute and relative error in the representation of the two numbers given above?
(iii) Using the floating point representation, perform an addition of the two numbers given above. What is the error in the resulting number?
(iv) Using the floating point representation, multiply the first number and second number. Convert the result into normalized form in the given format.
(v) Take the first number as 0.00000000000003234 and assume any second number to demonstrate the concept of overflow or underflow for the given representation. (You may assume any second number to demonstrate overflow or underflow).
(vi) What is use of bias in binary floating point representation. Explain the concept of bias with the help of an example for binary floating point numbers.
(b) Perform four iterations (rounded to four decimal places) using
(i) Jacobi Method and
(ii) Gauss-Seidel method for the following system of equations.
With x (0) = (0, 0, 0)T . The exact solution is (3, 2, 1)T
Which method gives better approximation to the exact solution?
Solve the following Initial Value Problem using (i)R-K method of O(h2 ) and (ii) R-K method of O(h4 )
Find y(0.4) taking h = 0.2, where y' means dy/dx
(a) Find Lagrange’s interpolating polynomial that fits the following data. Hence obtain the value of f(4).
| X | 1 | 3 | 6 | 10 |
| f(x) | 1 | 5 | 26 | 82 |
Determine the smallest positive root of the following equation:
f(x) = 3x4 – 5x2 – 11x - 13 = 0
The root should be correct up to 2 decimal places, using (a) Regula-falsi method (b) Newton-Raphson method (c) Bisection method
(d) Secant method
The depositions in a Bank for 5 different years are given in the following table:
| Year (x) | 2011 | 2013 | 2015 | 2017 | 2019 |
| Deposits(y) (Crores (INR) ) : | 5 | 15 | 45 | 102 | 2019 |