IGNOU BCS-54 - Computer Oriented Numerical Techniques

Q1. (a) Find floating point representation, if possible normalized, in the 4-digit mantissa, two digit exponent, if necessary use approximation for each of the following numbers: (i) 27.94 (ii) -0.00943 (iii) -6781014 (iv) 0.0644321 Also, find absolute error, if any, in each (b) Convert the decimal integer -465 to binary using both the methods (as shown in Pg No:16 of Block-1). Show all the steps. (c) Convert the number given as binary fraction –(0.101110101)2 to decimal. (d) Find the sum of the two floating numbers x1=0.1364X101 and x2=0.7342X10-1. Further express the result in normal form, using (i) Chopping (ii) Rounding. Also, find the absolute error. Q2. (a) Solve the system of equations 2 x + y + z = 3 x + 3y + 3z = 4 x – 4y + 2z = 9 using Gauss elimination method with partial pivoting. Show all the steps. (b) Perform four iterations (rounded to four decimal places) using (i) Jacobi Method and (ii) Gauss-Seidel method , for the following system of equations Which method gives better approximation to the exact solution? Q3. Determine the smallest positive root of the following equation: f(x) ≡ x3 – 9x2 - x + 9 = 0 to three significant digits using (a) Regula-falsi method (b) Newton-Raphson method (c) Bisectionmethod (d) Secant method Q4. (a) Find Lagrange’s interpolating polynomial for the following data. Hence obtain the value of f(4). (b) Using the inverse Lagrange’s interpolation, find the value of x when y=3 for the following data: Q5. (a) The population of a country for the last 25 years is given in the following table: (i) Using Stirling's central difference formula, estimate the populationfor the year 2007 (ii) Using Newton’s forward formula, estimate the population for theyear 1998. (iii) Using Newton’s backward formula, estimate the population for theyear 2013. (b) Derive the relationship for the operators δ in terms of E. Q6. (a) Find the values of the first and second derivatives of y = f(x) for x=2.1 using the following table. Use forward difference method. Also, find Truncation Error (TE) and actual errors. (b) Find the values of the first and second derivatives of y = f(x) for x=2.1 from the following table using Lagrange’s interpolation formula. Compare the results with (a) part above. Q7. Compute the value of the integral By taking 8 equal subintervals using (a) Trapezoidal Rule and then (b) Simpson's 1/3 Rule. Compare the result with the actual value.

Q1. (a) Use the eight-decimal digit floating-point representation as given in Block 1, Unit 1, Section 1.3.1 page 29 to perform the following operations: (i) Represent 0.09091919 and 2134650987 as floating-point numbers in normalized form using rounding for first number and chopping for second number. (ii) Find the absolute and relative error in the representation of the two numbers given above. (iii) Using the floating-point representation as above, perform the addition and multiplication of the two numbers given in part (i). Find the absolute error in the results of the two operations. (iv)Using the floating-point representation as above, divide the second number by the first number. Convert the result into normalized form. (v) Using the given eight-decimal digit floating-point representation and taking the first number as 5432198765012343210; demonstrate the concept of overflow or underflow. You may assume any second number for demonstrating the concept. (vi)What is the meaning of bias, which in used in exponent of floating-point representation? Explain with the help of an example. (b) What is Subtractive Cancellation in the context of floating-point numbers? Explain with the help of an example. Is there any concept called additive cancellation? (c) Find the Maclaurin series for f(x) =e 5x at x=0. Use first four terms of this series to compute the value of the function at any value of x. Also find the bounds of truncation error. Q2. (a) Solve the system of equations x –7y + 3z = 18 5x –2y–z= 8 3x +4y +5z = 0 using Gauss elimination method with partial pivoting. Show all the steps. (b) Perform four iterations (rounded to four decimal places) using (i) Jacobi Method and (ii) Gauss-Seidel method for the following system of equations. Which method gives better approximation to the exact solution? Q3. Determine the smallest positive root of the following equation: f(x) = 2x4–3x3+5x –7 =0 The root should be correct up to two decimal places, using (a) Regula-falsi method (b) Newton-Raphson method (c) Bisection method (d) Secant method Q4. (a) Find Lagrange’s interpolating polynomial that fits the following data. Hence obtain the value of f(5).Also make the Newtons Divided difference table for the following data. (b) Using the Lagrange's inverse interpolation method, find the value of x when y is 5. Q5. (a) The Expenses of a University for 5 different years are given in the following table: (i) Using Stirling's central difference formula estimate the Expense for the year 2017 (ii) Using Newton’s forward formula estimate the Expense for the year 2015. (iii) Using Newton’s backward formula estimate the Expense for the year 2021. (b) Derive an expression for Shift operator and forward difference operator in terms of δ Q6. (a) Find the values of the first and second derivatives of y = 4x2+10x-9 for x=1.25 using the following table. Use forward difference method. Also, find Truncation Error (TE) and actual errors. (b) Find the values of the first and second derivatives of y = 4x2+10x-9for x=1.25 from the following table using Lagrange’s interpolation formula. Compare the results with (a) part above Q7. Compute the value of the integral By taking 10 equal subintervals using (a) Trapezoidal Rule and then (b) Simpson's 1/3 Rule. Compare the result with the actual value. Q8. (a) Solve the Initial Value Problem, using Euler’s Method for the differential Equation: y' = 1+4x3y, given that y(0) = 1. Find y(1.0) taking (i) h = 0.20 and then (ii) h = 0.5 (b) Solve the following Initial Value Problem using (i) R-K method of O(h2) and (ii) R-Kmethod of O(h4) y' = x3y + x2and y(0) = 1. Find y(0.4) taking h = 0.2, where y' means dy/dx