(SEM VI) THEORY EXAMINATION 2021-22 COMPUTER BASED NUMERICAL TECHNIQUES

COMPUTER BASED NUMERICAL TECHNIQUES (KOE-065)

B.Tech Semester VI – Theory Examination (2021–22) 

Computer Based Numerical Techniques is a fundamental applied mathematics subject that deals with numerical methods for solving mathematical problems using computational approaches. Many real-life engineering problems cannot be solved analytically using exact mathematical formulas, and hence numerical methods are employed to obtain approximate but accurate solutions. This subject focuses on the numerical solution of algebraic equations, interpolation, numerical differentiation and integration, solution of ordinary and partial differential equations, error analysis, and eigenvalue problems.

The uploaded question paper clearly shows that the examination emphasizes root-finding techniques, interpolation formulas, numerical integration rules, predictor-corrector methods, stability concepts, finite difference methods, numerical solution of differential equations, error analysis, and eigenvalue computation. To score well, answers must be written in clear, continuous, logically connected paragraphs, showing complete mathematical reasoning and step-by-step explanation wherever required.

SECTION A – BASIC DEFINITIONS AND CONCEPTS

(Based on Section A on Page-1) 

The rate of convergence of the Bisection method is defined as the speed at which the sequence of approximations approaches the exact root. The bisection method converges linearly, which means that the error decreases in a proportional manner at each iteration.
 

When performing addition and subtraction of floating-point numbers, normalization and alignment of exponents are essential to avoid rounding errors and loss of significance, which are common in computer arithmetic.
 

The evaluation of finite difference operators such as Δⁿ(e³ˣ⁺⁵) requires understanding of forward difference tables and properties of exponential functions.
 

The relationship between divided differences and ordinary differences shows that divided differences generalize ordinary finite differences and are particularly useful when data points are not equally spaced.

The generalized Simpson’s 1/3 rule is a numerical integration formula used to approximate definite integrals by fitting parabolic arcs over subintervals and summing their areas.
 

The differentiation of Newton’s forward difference formula provides a numerical method to approximate derivatives using tabulated data values.
 

A Predictor–Corrector method is a numerical technique for solving ordinary differential equations in which an initial approximate value is predicted and then refined using a correction formula to improve accuracy.
 

The stability of a solution refers to the behavior of numerical errors during computation. A stable method ensures that small errors do not grow uncontrollably during successive iterations.

The classification of the partial differential equation
uₓₓ + 3uₓᵧ + uᵧᵧ = 0
is performed by comparing coefficients, and it is classified as a hyperbolic equation, which commonly arises in wave propagation problems.
 

An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, results in a scalar multiple of itself. Eigenvectors are fundamental in vibration analysis, stability analysis, and numerical methods.
 

SECTION B – NUMERICAL METHODS AND APPLICATIONS

(Based on Section B on Page-1) 

The Regula Falsi method is a root-finding technique that combines the concepts of bisection and linear interpolation. When applied to the equation x³ − 4x − 9 = 0, the method iteratively refines the interval containing the root by replacing one endpoint with the point where the secant line intersects the x-axis. Performing three iterations yields a progressively accurate approximation of the real root.

The Lagrange interpolation formula constructs a polynomial that exactly passes through all given data points. Using the given table of x and f(x) values, the polynomial is formed and then evaluated at x = 3 to obtain the required function value.

The Simpson’s 3/8 rule is a numerical integration method used when the number of subintervals is a multiple of three. In the given velocity–time data of a car, Simpson’s 3/8 rule is applied to compute the approximate distance traveled by integrating velocity over time.

The fourth-order Runge–Kutta method is a powerful numerical technique for solving ordinary differential equations. It provides high accuracy by computing weighted slopes at intermediate points. Using step size h = 0.05, the value of y(1.1) is obtained for the given differential equation by systematically calculating k₁, k₂, k₃, and k₄.

The finite difference method for boundary value problems converts differential equations into algebraic equations by replacing derivatives with difference quotients. This method is widely used in engineering problems involving heat conduction and structural analysis.

SECTION C – ERROR ANALYSIS AND ROOT APPROXIMATION

(Based on Section C on Page-1) 
 

When given a function such as
u = (4x²y³) / z⁴
and known errors in x, y, and z, the relative maximum error in u is computed using differential error analysis. By substituting x = y = z = 1 and applying logarithmic differentiation, the combined effect of individual errors on the final result is obtained.

The Newton–Raphson method is an iterative root-finding technique with quadratic convergence. To calculate √12, the equation x² − 12 = 0 is formulated and successive approximations are obtained until the desired accuracy is achieved.

INTERPOLATION, DIFFERENTIATION AND INTEGRATION

(Based on Questions 4 and 5 on Page-2) 
 

The proof of
Δ log f(x) = log [1 + Δf(x)/f(x)]
is obtained using properties of logarithms and finite differences, showing the relationship between incremental changes in a function and its logarithm.
 

The Newton forward interpolation polynomial is constructed using equally spaced data points. Once the polynomial is obtained, it is used to estimate the function value at x = 5.

The numerical differentiation problem involving tabulated values of √x requires application of forward or central difference formulas to compute f′(16) accurately.
 

The trapezoidal rule is a numerical integration technique that approximates the area under a curve by dividing it into trapezoids. Using step size h = 0.25, the integral of 1/(1 + x²) is evaluated numerically.

DIFFERENTIAL EQUATIONS AND EIGENVALUE PROBLEMS

(Based on Questions 6 and 7 on Page-2) 

Picard’s method is an iterative technique for solving initial value problems. By repeatedly integrating the given differential equation, successive approximations of y(0.1) and y(0.2) are obtained.

The Euler’s method algorithm provides a simple numerical approach for solving ordinary differential equations by advancing the solution in small step sizes using slope information.

The explicit finite difference method for solving the one-dimensional parabolic heat equation involves discretizing both time and space variables, leading to a stable numerical solution under specific conditions.

The Power method is an iterative technique used to compute the dominant eigenvalue and corresponding eigenvector of a matrix. Applying this method to the given matrix yields approximate eigenvalues and eigenvectors through repeated matrix multiplication and normalization.

HOW TO WRITE COMPUTER BASED NUMERICAL TECHNIQUES ANSWERS IN THE EXAM

In Computer Based Numerical Techniques, never write answers in short bullet points. Always start with the definition or principle, followed by derivation or algorithm, and then step-by-step numerical computation. Clearly mention formulas, assumptions, and intermediate steps. Examiners give high weightage to logical flow, correctness of method, and clarity of numerical working.