Numerical Methods Viva Questions PDF

# Numerical Methods: Concepts and Algorithms ## Error Analysis 1. **Significant Figures:** How do yo determine the number of significant figures in a measurement or calculation? 2. **Accuracy and Precision:** Can you explain the difference between accuracy and precision in the context of numerical...

# Numerical Methods: Concepts and Algorithms ## Error Analysis 1. **Significant Figures:** How do yo determine the number of significant figures in a measurement or calculation? 2. **Accuracy and Precision:** Can you explain the difference between accuracy and precision in the context of numerical computations? 3. **Error Definitions:** What are the various types of errors encountered in numerical calculations, and how are they defined? 4. **Round-Off Errors:** How do round-off errors occur during numerical computations, and what techniques can be used to mitigate them? 5. **Truncation Errors and the Taylor Series:** What is the Taylor series, and how is it used to approximate functions? 6. **Error Propagation:** How does error propagate through mathematical operations, and what methods exist to estimate propagated errors? 7. **Total Numerical Errors:** How do you quantify the total numerical error in a computational task considering borth round-off and truncation errors? 8. **Formulation Errors:** What are formulation errors, and how do they arise in the process of developing mathematical models or algorithms? 9. **Data Uncertainty:** How is data uncertainty accounted for in mathematical modeling, and what impact does it have on the accuracy of numerical solutions? 10. **Error Analysis:** How do you perform error analysis to assess the reliability and validity of numerical results obtained from computational models? ## Root-Finding Methods 1. **Bisection Method:** What is the formula to calculate the midpoint of the interval? 2. **Newton-Raphson Method:** Provide the formula used to compute the next approximation. 3. **Regula-Falsi Method:** What is the equation used to update the root approximation? 4. **Secant Method:** Give the formula for the secant line slope. 5. **Bisection Method Interval Update:** Express the formula for updating the interval after each iteration. 6. **Newton-Raphson Iteration Formula:** What is it? 7. **Regula-Falsi Method Interval Update:** Write down the formula using linear interpolation. 8. **Secant Method Next Guess:** What is the equation used to calculate it. 9. **Bisection Method Interval Determination:** Provide the formula given the initial interval and the midpoint. 10. **Secant Method Guess Update:** Write the formula based on the previous two guesses. ## Interpolation 1. **Forward Difference:** What is the formula for calculating the forward difference of a function f(x) at a given point using the finite difference method? 2. **Backward Difference:** Provide the formula for computing the backward difference of a function f(x) at a specific point using the finite difference method. 3. **Newton's Forward Difference Interpolation Formula:** What is the expression used to compute the divided difference table? 4. **Backward Difference Coefficients:** What is the formula to calculate them in Newton's backward difference interpolation? 5. **Lagrange's Interpolation Polynomial:** Write down the formula using the given data points (x0,y0), (x1,y1),..., (xn,yn). 6. **Newton's Forward Difference Interpolation Polynomial Coefficients:** What is the formula to find them? ## Linear Algebra 1. **Gauss-Jordan Method:** What is the formula for transforming a given system of linear equations into row-echelon form using Gauss-Jordan elimination? 2. **Gauss-Seidel Method:** Provide the iterative formula used in the Gauss-Seidel method to solve a system of linear equations. ## Numerical Differentiation and Integration 1. **Numerical Differentiation:** Write down the formula for approximating the derivative of a function f(x) at a given point using forward difference numerical differentiation. 2. **Trapezoidal Rule:** What is the formula for calculating the approximate integral of a function f(x) over the interval [a,b] using the Trapezoidal Rule? 3. **Simpson's 1/3 Rule:** Give the formula for approximating the integral of a function f(x) over the interval [a,b] using Simpson's 1/3 Rule. 4. **Simpson's 3/8 Rule:** Write down the formula for estimating the integral of a function f(x) over the interval [a,b] using Simpson's 3/8 rule. ## Numerical Solution of Ordinary Differential Equations (ODEs) 1. **Taylor Series Method:** Provide the formula for the Taylor series expansion of a function y(x) around a point x0 up to the n-th order. 2. **Euler's Method:** Write down the formula used in Euler's method to approximate the solution of a first-order ordinary differential equation $dy/dx = f(x,y)$ with initial condition $y(x_0) = y_0$. 3. **Modified Euler's Method:** What is the formula for computing the next approximation in Modified Euler's method, an improvement over Euler's method? 4. **Runge-Kutta Method (RK2):** Provide the formula for the second-order Runge-Kutta method (RK2) used to solve a first-order ordinary differential equation. 5. **Runge-Kutta Method (RK4):** Write down the formula for the fourth-order Runge-Kutta method (RK4) used to solve a first-order ordinary differential equation. ## Regression 1. **Linear Regression:** What is the equation of the straight line being fitted to the data in linear regression, and what are the normal equations used to determine its parameters? 2. **Parabolic Curve Fitting:** What is the equation of the parabolic curve being fitted to the data in parabolic regression, and what are the normal equations used to determine its coefficients? 3. **Multiple Linear Regression:** In multiple linear regression, what is the equation representing the linear combination of predictors, and what are the normal equations used to estimate the regression coefficients? ## Linear Programming 1. **Linear Optimization Problem:** What defines a linear optimization problem, and how is it typically formulated in terms of objective function and constraints? 2. **Formulation and Graphical Solution:** How do you formulate a linear programming problem mathematically, and how can its solution be represented graphically on a two-dimensional plane? 3. **Convex Polygon Theorem:** Explain the Convex Polygon Theorem in the context of linear programming, and how does it relate to the feasible region of a linear programming problem? ## Partial Differential Equations (PDEs) 1. **Partial Differential Equations (PDEs):** What is a partial differential equation (PDE), and how does it differ from ordinary differential equations (ODEs)? 2. **Classification of PDEs:** Describe the classification of second-order partial differential equations (PDEs) based on their coefficients and principal parts, and provide examples for each classification.