(SEM VI) THEORY EXAMINATION 2023-24 COMPUTER BASED NUMERICAL TECHNIQUES

COMPUTER BASED NUMERICAL TECHNIQUES – KOE065

Section-wise Important Questions & Ready Answers

SECTION A

(Attempt all – 2 marks each)

(a) Concept of Error in Series Approximation

Error in series approximation is the difference between the exact value of a function and its approximate value obtained using a finite number of terms. Since infinite terms cannot be used practically, truncation leads to approximation error, which decreases as more terms are included.

(b) Example of a Simple Iteration Method

The simple iteration method rewrites an equation in the form x=g(x)x = g(x)x=g(x) and repeatedly substitutes values until convergence. For example, solving x=cos⁡xx = \cos xx=cosx using xn+1=cos⁡xnx_{n+1} = \cos x_nxn+1​=cosxn​.

(c) Hermite’s Interpolation

Hermite interpolation is a method of interpolation where both function values and derivative values at given points are used to construct an interpolating polynomial, resulting in higher accuracy than simple interpolation.

(d) Principle of Finite Difference

The principle of finite difference is based on approximating derivatives by differences between function values at discrete points. It replaces continuous derivatives with algebraic difference expressions.

(e) Newton’s Forward vs Backward Formula (Differentiation)

Newton’s forward formula is used when interpolation is required near the beginning of the data set, while Newton’s backward formula is preferred near the end. The choice depends on the location of the required value.

(f) Lagrange Interpolation for Numerical Differentiation

Although Lagrange interpolation is mainly used for interpolation, its polynomial can be differentiated analytically to obtain approximate derivatives at a point.

(g) Types of Errors

Errors are classified as truncation error, round-off error, absolute error, relative error, and percentage error. These arise due to approximation and finite precision arithmetic.

(h) Differential Equation

A differential equation is an equation involving derivatives of a dependent variable with respect to one or more independent variables. It describes physical systems like heat transfer and motion.

(i) Boundary Value Problem (BVP)

A boundary value problem is a differential equation with conditions specified at more than one point of the independent variable, unlike initial value problems.

(j) Calculation of a Distillation Column

Numerical methods are used to calculate temperature profiles, compositions, and stages in a distillation column where analytical solutions are complex or unavailable.

SECTION B

(Attempt any three – 10 marks each)

2(a) Rounding and Truncation Errors and Their Impact

Truncation removes digits beyond a certain decimal place, while rounding adjusts the last retained digit. Both introduce errors that accumulate during computations, affecting accuracy and stability of numerical solutions.

2(b) Gauss vs Central Difference Interpolation Formulas

Gauss forward and backward formulas are suitable for values near the center of the table, while Stirling’s, Bessel’s, and Everett’s formulas provide better accuracy for evenly spaced data. Central difference formulas are preferred when data is symmetric.

2(c) Simpson’s 1/3 Rule – Derivation, Advantages & Limitations

Simpson’s 1/3 rule approximates integration using quadratic polynomials. It offers higher accuracy than trapezoidal rule but requires an even number of intervals and smooth functions.

2(d) Picard’s Method for dy/dx = y + x

Picard’s method generates successive approximations using integration. Starting with an initial guess, repeated integration yields increasingly accurate solutions, up to the fifth approximation as required in the question.

2(e) Finite Difference Method for Eigenvalue Problems

Finite difference method converts differential equations into algebraic equations. Eigenvalue problems are solved by discretizing the domain and solving the resulting matrix equation.

SECTION C

3(a) Importance of Error Analysis and Uncertainty Quantification

Error analysis ensures reliability of numerical results by estimating uncertainty. Techniques include error bounds, sensitivity analysis, and propagation of errors, which are crucial in engineering simulations.

3(b) Iterations Required for Root Accuracy (Numerical)

For the function f(x)=x3−x−1f(x)=x^3-x-1f(x)=x3−x−1 in the interval [1,2], the number of iterations is determined using error tolerance formulas. This ensures accuracy up to two decimal places.

4(a) Everett’s Interpolation Formula (Numerical)

Everett’s formula is a central interpolation method used when the interpolation point lies near the center. Using the given table (page-1), the value of f(1.15)f(1.15)f(1.15) is computed accurately.

4(b) Newton’s Divided Difference Formula

Unlike forward and backward formulas, Newton’s divided difference formula does not require equally spaced data. It is highly flexible and widely used for polynomial interpolation.