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
(a) Regula-Falsi Method:
The Regula-Falsi method involves bracketing the root and updating the interval based on the function values at the endpoints. Apply the method iteratively until convergence. The smallest positive root of the given equation using ____ __________ _____ _______ _________ _____ __________ ________ ____.
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Compute the value of the integral
By taking 12 equal subintervals using (a) Trapezoidal Rule and then (b) Simpson's 1/3 Rule. Compare the result with the actual value.
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 |
Solve the Initial Value Problem, using Euler’s Method for the differential Equation:
y, = 1+3xy, given that y(0) = 1.
Find y(1.0) taking (i) h = 0.25 and then (ii) h = 0.5
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.
Find the values of the first and second derivatives of y = 2x2+3x-2 for x=1.25 using the following table. Use forward difference method. Also, find Truncation Error (TE) and actual errors.
| x | 1 | 1.5 | 2 | 2.5 |
| y | 3 | 7 | 12 | 18
|
(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.
Find the values of the first and second derivatives of y = 2x2+3x-2 for x=1.25 from the following table using Lagrange’s interpolation formula. Compare the results with (a) part above.
| x | 1 | 1.5 | 2 | 2.5 |
| y | 3 | 7 | 12 | 18 |