BMTE-144 Solved Assignment January 2026 | NUMERICAL ANALYSIS | IGNOU BSCG

Q: State whether the following statements are true or false. Give a Short proof or a counter-example in support of your answer.

As an expert IGNOU tutor for BMTE-144 Numerical Analysis, I will analyze each given statement regarding fundamental numerical methods. For each statement, I will determine its truth value (True/False) and provide a concise proof or a relevant counter-example, drawing upon the core principles and formulas typically covered in the BMTE-144 course material. The solutions will adhere to the specified word limits and formatting guidelines, including step-by-step derivations for calculations.

Q: Find the missing values in the following table: X 0 1 2 3 4 5 y 0 2 - 18 - 90

To find the missing values in the given table, we employ the finite difference method, a fundamental technique in Numerical Analysis for interpolating or extrapolating data points, particularly when the underlying function is assumed to be a polynomial. The principle states that if a function f(x) is a polynomial of degree 'n', then its nth forward differences will be constant, and all subsequent differences will be zero. Given the data: X 0 1 2 3 4 5 and y 0 2 ? 18 ? 90. Let the missing values

Q: Using Classical Runge-Kutta fourth order method, find an approximate value of y(1.2) for the IVP dy/dx = xy, y(1) = 2 with h = 0.2.

The Classical Runge-Kutta fourth-order method (RK4) is a widely used and highly accurate numerical technique for approximating solutions to initial value problems (IVPs) involving ordinary differential equations (ODEs). As detailed in the BMTE-144 course material, it is a single-step method that achieves a high degree of precision by evaluating the derivative function at multiple points within each interval `h`. This method is particularly favored in numerical analysis due to its excellent bala

Q: Find the approximate root of the equation 2x³ = 3x + 6 using Newton-Raphson method. Perform only 3 iterations with x0 = 2.

The Newton-Raphson method is a powerful iterative technique used in Numerical Analysis, as discussed in Block 2 of BMTE-144, for finding successively better approximations to the roots (or zeroes) of a real-valued function f(x) = 0. It is based on the idea of approximating the function by its tangent line at a given point. The core formula for the Newton-Raphson iteration is given by xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ), where xₙ is the current approximation, xₙ₊₁ is the next approximation, f(xₙ) is the

Q: The roots of the quadratic equation x² + ax + b = 0 are given by α and β. Show that the iteration X_k+1 = -(ax_k + b)/X_k will converge near x = α when |α| > |β|.

To show that the iteration X_k+1 = -(ax_k + b)/X_k converges near x = α when |α| > |β|, we utilize the concept of the functional iteration method (or fixed-point iteration) and its convergence criterion as discussed in Unit 2 of BMTE-144. The given quadratic equation is x² + ax + b = 0, with roots α and β. From Vieta's formulas for quadratic equations, we know that the sum of the roots, α + β, is equal to -a, and the product of the roots, αβ, is equal to b. Therefore, we can express the coeffic

Q: The Gauss-Seidel method is used to solve the system of equations: 4 0 2 [X₁] = [4]; 0 5 2 [X₂] = [-3]; 5 4 10 [X₃] = [2]. Determine the rate of convergence of the method.

The Gauss-Seidel method is an iterative technique used for solving systems of linear equations of the form AX = B. To determine its rate of convergence for a given system, we first need to formulate the coefficient matrix, check its diagonal dominance, and then calculate the spectral radius of its associated iteration matrix. From the given system of equations: 4X₁ + 0X₂ + 2X₃ = 4 0X₁ + 5X₂ + 2X₃ = -3 5X₁ + 4X₂ + 10X₃ = 2 The coefficient matrix A is: A = [[4, 0, 2], [0, 5, 2], [5, 4,

Q: Find the interpolating polynomial by Newton's divided difference formula for the following data: X 0 1 2 4 y 1 1 2 5

Newton's Divided Difference Interpolation Formula is a method used in Numerical Analysis (BMTE-144) to construct an interpolating polynomial for a given set of data points. This method is particularly useful because it does not require equally spaced data points and allows for easy addition of new data points without recomputing all previous coefficients. The general form of Newton's divided difference polynomial of degree n passing through (x₀, f(x₀)), (x₁, f(x₁)), ..., (x_n, f(x_n)) is given

Q1. State whether the following statements are true or false. Give a Short proof or a counter-example in support of your answer.

(a)) The equation x³ - 4x - 16 = 0 has no root in the interval [3,5]. (200 words)
(b)) ΔΕ = ΕΔ, where E is the shift operator and Δ is the forward difference operator. (200 words)
(c)) Every 3x3 system of linear equations can be solved using the LU decomposition method. (200 words)
(d)) For the data (2,4), (1,5), (3,6) the Newton's divided difference f[x0, x1, x2] is 3/2 (200 words)
(e)) The Newton-Raphson method cannot be used to find a cube root of a positive real number. (200 words)

Key Points:

Intermediate Value Theorem: If f(x) is continuous and f(a)f(b) < 0, a root exists in (a,b).
Shift and Forward Difference Operators: E f(x) = f(x+h) and Δ f(x) = f(x+h) - f(x).
Operator Commutativity: ΔE f(x) = EΔ f(x) = f(x+2h) - f(x+h).
LU Decomposition Limitations: Fails for singular matrices or without pivoting for some non-singular matrices.

Answer: As an expert IGNOU tutor for BMTE-144 Numerical Analysis, I will analyze each given statement regarding fundamental numerical methods. For each statement, I will determine its truth value (True/False) and provide a concise proof or a relevant counter-example, drawing upon the core principles and formulas typically covered in the BMTE-144 course material. The solutions will adhere to the specified word limits and formatting guidelines, including step-by-step derivations for calculations.

Q2.a. Find the missing values in the following table: X 0 1 2 3 4 5 y 0 2 - 18 - 90

Key Points:

Finite difference method is used to find missing values by assuming a polynomial function.
For a polynomial of degree 'n', the (n+1)th order differences are constant.
The given table has 6 points and 2 missing values (y₂, y₄).
Assuming a cubic polynomial (degree 3) allows the 3rd differences to be constant.

Answer: To find the missing values in the given table, we employ the finite difference method, a fundamental technique in Numerical Analysis for interpolating or extrapolating data points, particularly when the underlying function is assumed to be a polynomial. The principle states that if a function f(x) is a polynomial of degree 'n', then its nth forward differences will be constant, and all subsequent differences will be zero. Given the data: X 0 1 2 3 4 5 and y 0 2 ? 18 ? 90. Let the missing values...

Q2.b. Using Classical Runge-Kutta fourth order method, find an approximate value of y(1.2) for the IVP dy/dx = xy, y(1) = 2 with h = 0.2.

Key Points:

Classical Runge-Kutta fourth-order (RK4) is a single-step, highly accurate ODE solver.
RK4 calculates next y value using a weighted average of four slope estimates.
The four slopes (k₁, k₂, k₃, k₄) are determined by evaluating f(x,y) at different points.
k₁ = h*f(xᵢ, yᵢ), k₂ = h*f(xᵢ+h/2, yᵢ+k₁/2), k₃ = h*f(xᵢ+h/2, yᵢ+k₂/2), k₄ = h*f(xᵢ+h, yᵢ+k₃).

Answer: The Classical Runge-Kutta fourth-order method (RK4) is a widely used and highly accurate numerical technique for approximating solutions to initial value problems (IVPs) involving ordinary differential equations (ODEs). As detailed in the BMTE-144 course material, it is a single-step method that achieves a high degree of precision by evaluating the derivative function at multiple points within each interval `h`. This method is particularly favored in numerical analysis due to its excellent bala...

Q3.a. Find the approximate root of the equation 2x³ = 3x + 6 using Newton-Raphson method. Perform only 3 iterations with x0 = 2.

Key Points:

Newton-Raphson is an iterative method for finding roots of f(x)=0.
The formula is xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ).
Requires deriving f(x) from the given equation and finding f'(x).
An initial guess, x₀, is essential to start the iterative process.

Answer: The Newton-Raphson method is a powerful iterative technique used in Numerical Analysis, as discussed in Block 2 of BMTE-144, for finding successively better approximations to the roots (or zeroes) of a real-valued function f(x) = 0. It is based on the idea of approximating the function by its tangent line at a given point. The core formula for the Newton-Raphson iteration is given by xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ), where xₙ is the current approximation, xₙ₊₁ is the next approximation, f(xₙ) is the ...

Q3.b. The roots of the quadratic equation x² + ax + b = 0 are given by α and β. Show that the iteration X_k+1 = -(ax_k + b)/X_k will converge near x = α when |α| > |β|.

Key Points:

Vieta's formulas relate roots (α, β) to quadratic coefficients: a = -(α+β), b = αβ.
The iterative formula X_k+1 = -(ax_k + b)/X_k transforms into g(x) = α + β - αβ/x.
Convergence of fixed-point iteration X_k+1 = g(X_k) requires |g'(p)| < 1 at the fixed point 'p'.
The derivative of the iteration function is g'(x) = αβ/x².

Answer: To show that the iteration X_k+1 = -(ax_k + b)/X_k converges near x = α when |α| > |β|, we utilize the concept of the functional iteration method (or fixed-point iteration) and its convergence criterion as discussed in Unit 2 of BMTE-144. The given quadratic equation is x² + ax + b = 0, with roots α and β. From Vieta's formulas for quadratic equations, we know that the sum of the roots, α + β, is equal to -a, and the product of the roots, αβ, is equal to b. Therefore, we can express the coeffic...

Q3.c. If δ²f(x) = C₁h²f''(x) + C2h⁴f⁽⁴⁾(x) + …, find the values of C₁ and C2.

Key Points:

Central difference operator δ²f(x) is defined as f(x+h) - 2f(x) + f(x-h).
Taylor series expansion is used to express f(x+h) and f(x-h) in terms of derivatives.
Substituting Taylor series into δ²f(x) and combining terms results in an expansion.
Odd derivative terms cancel out due to the symmetric nature of central differences.

Answer: Numerical analysis heavily relies on finite difference operators to approximate derivatives. The second central difference operator, δ²f(x), is a common approximation for the second derivative of a function, defined as δ²f(x) = f(x + h) - 2f(x) + f(x - h). Here, 'h' represents the step size, and our objective is to express this operator in terms of derivatives of f(x) using Taylor series expansion to determine the coefficients C₁ and C₂. To achieve this, we expand f(x + h) and f(x - h) around t...

Q4.a. The Gauss-Seidel method is used to solve the system of equations: 4 0 2 [X₁] = [4]; 0 5 2 [X₂] = [-3]; 5 4 10 [X₃] = [2]. Determine the rate of convergence of the method.

Key Points:

Gauss-Seidel convergence requires spectral radius ρ(T_GS) < 1.
The matrix A is strictly diagonally dominant, guaranteeing convergence.
The Gauss-Seidel iteration matrix is T_GS = -(D+L)⁻¹U.
For the given system, T_GS = [[0, 0, -1/2], [0, 0, -2/5], [0, 0, 41/100]].

Answer: The Gauss-Seidel method is an iterative technique used for solving systems of linear equations of the form AX = B. To determine its rate of convergence for a given system, we first need to formulate the coefficient matrix, check its diagonal dominance, and then calculate the spectral radius of its associated iteration matrix. From the given system of equations: 4X₁ + 0X₂ + 2X₃ = 4 0X₁ + 5X₂ + 2X₃ = -3 5X₁ + 4X₂ + 10X₃ = 2 The coefficient matrix A is: A = [[4, 0, 2], [0, 5, 2], [5, 4,...

Q4.b. Find the interpolating polynomial by Newton's divided difference formula for the following data: X 0 1 2 4 y 1 1 2 5

Key Points:

Newton's divided difference formula constructs an interpolating polynomial.
It uses a systematic table to compute divided differences, serving as polynomial coefficients.
First-order divided differences are calculated as (f(xⱼ) - f(xᵢ)) / (xⱼ - xᵢ).
Higher-order differences are derived from differences of the next lower order.

Answer: Newton's Divided Difference Interpolation Formula is a method used in Numerical Analysis (BMTE-144) to construct an interpolating polynomial for a given set of data points. This method is particularly useful because it does not require equally spaced data points and allows for easy addition of new data points without recomputing all previous coefficients. The general form of Newton's divided difference polynomial of degree n passing through (x₀, f(x₀)), (x₁, f(x₁)), ..., (x_n, f(x_n)) is given ...

Q4.c. Using synthetic division method, show that 2 is a simple root of the equation p(x) = x⁴ - 2x³ + x² - x - 2 = 0.

Key Points:

Synthetic division is used to divide polynomials by linear factors (x-c).
If p(c) divided by (x-c) has a zero remainder, then c is a root of p(x).
A simple root 'c' implies p(c)=0 and p'(c)≠0, meaning it has multiplicity one.
Using synthetic division, 'c' is a simple root if the first remainder is 0 and the second remainder (on the quotient) is non-zero.

Answer: To show that 2 is a simple root of the equation p(x) = x⁴ - 2x³ + x² - x - 2 = 0, we employ the synthetic division method. As per the course material (Block 1, Unit 1/2), a number 'c' is a simple root of p(x) if p(c) = 0 and p'(c) ≠ 0. Using synthetic division, this means the remainder of dividing p(x) by (x-c) is zero, and the remainder of dividing the resulting quotient by (x-c) is non-zero. First, we perform synthetic division of p(x) = x⁴ - 2x³ + x² - x - 2 by (x - 2). We use the coefficien...

Q5.a. Obtain the interpolating polynomial in simplest form which fits the following data: X -1 0 1 2 f(x) 3 -4 5 -6

Key Points:

Newton's Divided Difference method is suitable for interpolating data with unequal intervals.
The interpolating polynomial uses a systematic table of divided differences for its coefficients.
First-order divided differences are ratios of function value differences to x-value differences.
Higher-order differences are computed recursively from lower-order divided differences.

Answer: The problem requires obtaining an interpolating polynomial that fits the given data points. Newton's Divided Difference Interpolation formula is an effective method for constructing such a polynomial, especially when the data points are not equally spaced, as discussed in Unit 3 of the BMTE-144 course. This method systematically builds the polynomial by calculating successive divided differences. First, we list the given data points and proceed to construct the divided difference table: x₀ = -1...

Q5.b. Prove that μ² = 1 + δ²/4

Key Points:

The average operator μ is defined as (E^(1/2) + E^(-1/2))/2.
The central difference operator δ is defined as E^(1/2) - E^(-1/2).
Expanding μ² yields (E + 2 + E^(-1))/4 using algebraic properties.
Expanding δ² results in E - 2 + E^(-1) in terms of shift operators.

Answer: The proof for μ² = 1 + δ²/4 involves understanding the definitions of the average operator (μ) and the central difference operator (δ) in terms of the shift operator (E). The average operator, μ, is defined as μ f(x) = [f(x + h/2) + f(x - h/2)] / 2. In terms of the shift operator E, where E f(x) = f(x + h), this translates to μ = (E^(1/2) + E^(-1/2)) / 2. To find μ², we square this expression: μ² = [(E^(1/2) + E^(-1/2)) / 2]² μ² = [E^(1/2))² + 2 * E^(1/2) * E^(-1/2) + (E^(-1/2))²] / 4 μ² = ...

Q5.c. Determine the order of convergence of the iterative method X_n+1 = (X_n-1 f(X_n) - X_n f(X_n-1))/(f(X_n) - f(X_n-1)) for finding a simple root of the equation f(x) = 0.

Key Points:

The given iterative method is the Secant Method for root finding.
Order of convergence measures how fast an iterative method approaches the true root.
Secant Method error analysis leads to the characteristic equation λ² - λ - 1 = 0.
The positive root of the characteristic equation, φ = (1 + √5)/2, defines the order.

Answer: The iterative method provided, X_n+1 = (X_n-1 f(X_n) - X_n f(X_n-1))/(f(X_n) - f(X_n-1)), is recognized as the Secant Method for finding a simple root of the equation f(x) = 0. This method is typically discussed in Unit 3, 'Iterative Methods for Solving Non-linear Equations,' of the BMTE-144 Numerical Analysis course. To determine the order of convergence, we first understand what order of convergence signifies. It is a measure of the speed at which an iterative sequence converges to its limit...

Q6.a. Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix [[1 6 1], [1 2 0], [0 0 3]] using the power method. Take (1, 0, 0)ᵀ as the initial approximation and perform 4 iterations.

Key Points:

Power Method approximates the dominant eigenvalue (largest in magnitude) and its eigenvector.
It is an iterative method involving repeated matrix-vector multiplication and normalization.
The initial vector is multiplied by the matrix in each step: y_k+1 = A * x_k.
The approximate eigenvalue at iteration k+1 is the largest component of y_k+1.

Answer: The Power Method is an iterative numerical technique used to approximate the largest eigenvalue in magnitude (dominant eigenvalue) and its corresponding eigenvector of a matrix. This method is particularly useful for large matrices where finding eigenvalues by solving the characteristic equation directly can be computationally intensive, as discussed in Unit 3 of BMTE-144 Numerical Analysis. To apply the Power Method, we start with an initial non-zero approximation vector. In each iteration, th...

Q6.b. The method X_n+1 = (1/9)[5x_n + 5N/x_n² - N²/x_n⁵], n = 0, 1, 2, ... where N is a positive constant, converges to N¹/³. Find the rate of convergence of the method.

Key Points:

Rate of convergence (order p) defines how fast an iterative method approaches the root.
For x_n+1 = φ(x_n), order p is found from derivatives of φ(x) at the fixed point α.
If φ'(α) = 0, φ''(α) = 0, and φ'''(α) ≠ 0, the method exhibits cubic convergence (p=3).
The given method's fixed point is α = N^(1/3), as confirmed by φ(α) = α.

Answer: The rate of convergence of an iterative method, also known as the order of convergence, quantifies how quickly the sequence of approximations generated by the method approaches the true root. For a fixed-point iteration of the form X_n+1 = φ(x_n), the order of convergence 'p' is determined by the derivatives of the function φ(x) evaluated at the fixed point (the true root). If the first 'p-1' derivatives of φ(x) are zero at the fixed point, and the p-th derivative is non-zero, then the method ha...

Q7.a. Evaluate ∫ from 0 to 1 of 1/(1+x²) dx by using trapezoidal rule with h = 0.5 and h = 0.25. Use Romberg's method to find the best value of π.

Key Points:

The integral ∫ from 0 to 1 of 1/(1+x²) dx analytically equals π/4.
Trapezoidal Rule approximates integral I ≈ (h/2) * [f(x₀) + 2Σ f(xᵢ) + f(xₙ)].
T(0.5) for f(x)=1/(1+x²) on [0,1] is calculated as 0.775.
T(0.25) for f(x)=1/(1+x²) on [0,1] is calculated as approximately 0.782794.

Answer: To evaluate the definite integral ∫ from 0 to 1 of 1/(1+x²) dx and use Romberg's method to find an approximation for π, we first recognize that the analytical solution to this integral is [arctan(x)] from 0 to 1, which equals arctan(1) - arctan(0) = π/4 - 0 = π/4. Therefore, four times the numerical approximation of this integral will yield an approximation for π. We begin by applying the Composite Trapezoidal Rule, as described in Unit 10, Section 10.3 of BMTE-144 (Block 3, page 19). The formu...

Q7.b. Estimate the eigenvalues of the matrix [[1 -1 2], [-1 1 2], [2 2 -2]] using the Gerschgorin bounds.

Key Points:

Gerschgorin's Theorem estimates eigenvalue location using matrix elements.
Each row `i` defines a Gerschgorin disk Dᵢ for eigenvalues.
Disk center is the diagonal element `aᵢᵢ` of the row.
Disk radius `Rᵢ` is the sum of absolute off-diagonal elements in the row.

Answer: The Gerschgorin Circle Theorem is a fundamental tool in numerical analysis for estimating the bounds of eigenvalues of a matrix. It provides a simple yet effective way to localize eigenvalues in the complex plane without complex computations. This theorem is crucial for understanding matrix properties, especially in iterative methods where eigenvalue magnitudes determine convergence rates. According to Gerschgorin's Circle Theorem, every eigenvalue of an n×n matrix A lies within at least one of...

Q8.a. Solve the initial value problem using Euler method y' = 1/(x²-3y), y(3) = 2. Find y(3.1) taking h = 0.1.

Key Points:

Euler's method numerically approximates solutions to initial value problems.
The formula is `y_{i+1} = y_i + h * f(x_i, y_i)`.
`h` is the step size, determining `x` increments and approximation accuracy.
First, evaluate `f(x₀, y₀)` from the given differential equation.

Answer: The Euler method is a fundamental first-order numerical technique for approximating solutions to initial value problems (IVPs) of the form `y' = f(x, y)` with an initial condition `y(x₀) = y₀`. As discussed in Unit 1 of BMTE-144, this method provides a sequence of approximations `y₁, y₂, ..., y_n` for `y(x₁, y₂), ..., y(x_n)` using a given step size `h`. The core formula for Euler's method is `y_{i+1} = y_i + h * f(x_i, y_i)`. Here, `y_i` represents the approximation of `y` at `x_i`, and `f(x_i...

Q8.b. Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations [[1 1 1], [4 3 -1], [3 5 3]] * [[x], [y], [z]] = [[1], [6], [4]]. Show that the method is convergent and hence find its rate of convergence.

Key Points:

Gauss-Seidel matrix form: (D+L)x^(k+1) = -Ux^(k) + b.
Convergence requires spectral radius of iteration matrix ρ(T_GS) < 1.
Iteration matrix T_GS = -(D+L)^(-1)U.
Calculated eigenvalues of T_GS are 0, 0.3964, -0.8409.

Answer: The Gauss-Seidel iteration scheme is an iterative method used to solve a system of linear equations, Ax = b. This method is particularly useful for large systems where direct methods can be computationally expensive. It builds upon the Jacobi method by using the most recently computed values of the variables during the current iteration, leading to faster convergence in many cases, as discussed in Unit 3 of BMTE-144, 'Iterative Methods for Solving Linear Systems'. Given the system of equations:...

Q9.a. For the following data, use Gauss backward difference method to obtain the interpolating polynomial f(x): X 0.1 0.2 0.3 0.4 0.5 f(x) 1.40 1.56 1.76 2.00 2.28. Hence, find the value of f(0.45).

Key Points:

Gauss Backward method interpolates values near the end of a data table.
A difference table is essential to identify the required differences.
Choose reference point x₀ close to the interpolation value x.
Parameter p is calculated as p = (x - x₀) / h for interpolation.

Answer: As an expert IGNOU tutor, and drawing upon the standard curriculum for BMTE-144 Numerical Analysis (specifically concepts related to interpolation methods), I will address this question. Since the course PDF was not provided, the following explanation and solution are based on the general principles and formulas typically taught in this course. Gauss's Backward Difference formula is a method of interpolation primarily used when interpolating values of f(x) that lie towards the end of a given se...

Q9.b. The velocity of a vehicle beginning from rest is given in the following table for part of the first four. Using Simpson's 1/3 - rule, find the distance travelled by the vehicle in this hour: t = time in min. 10 20 30 40 50 60 v = velocity in km/hr. 80 60 70 75 70 80

Key Points:

Simpson's 1/3 Rule requires an even number of subintervals (n) for direct application.
Distance travelled is calculated by numerically integrating the velocity function over time.
When 'n' is odd, a common method is to use Simpson's 1/3 Rule for 'n-1' intervals and Trapezoidal Rule for the last interval.
Ensure consistent units: convert time step 'h' to hours if velocity is in km/hr.

Answer: To determine the distance travelled by the vehicle, we need to numerically integrate the velocity function over the given time interval. The distance is represented by the definite integral of velocity with respect to time, i.e., Distance = ∫v(t) dt. The problem specifies using Simpson's 1/3 rule, a method for approximating definite integrals from discrete data points, as covered in Unit 5, Section 5.3 of BMTE-144. First, let's organize the given data. The time interval spans from 10 minutes to...

Q10.a. Find the inverse of the matrix [[2 1 1], [3 2 1], [2 1 2]] using Gauss-Jordan method.

Key Points:

Gauss-Jordan method transforms augmented matrix [A|I] into [I|A⁻¹] using row operations.
Elementary row operations include scaling rows, swapping rows, and adding row multiples.
The primary goal is to convert the original matrix (A) into an identity matrix.
Each row operation must be applied consistently across the entire augmented matrix.

Answer: The Gauss-Jordan method is a systematic approach to find the inverse of a matrix by performing a sequence of elementary row operations. As covered in Unit 3, Block 1 of BMTE-144, this method transforms an augmented matrix [A | I] into [I | A⁻¹], where A is the given matrix, I is the identity matrix, and A⁻¹ is its inverse. First, we construct the augmented matrix by placing the given matrix A on the left and the identity matrix I of the same order on the right. For the given matrix A = [[2 1 1]...

Q10.b. Divide the polynomial x⁵ - 6x⁴ + 8x³ + 8x² + 4x - 40 by (x-3) by the synthetic division method and find the remainder.

Key Points:

Synthetic division is a shortcut for dividing polynomials by linear divisors (x-k).
Coefficients of the dividend are used, with '0' for missing terms.
The value 'k' is taken from the divisor (x-k).
Multiply 'k' by the bottom row number, add to the next coefficient.

Answer: Synthetic division is an efficient method for dividing a polynomial P(x) by a linear divisor of the form (x - k). It simplifies the long division process by operating only on the coefficients of the polynomial. This method is particularly useful in Numerical Analysis for tasks like evaluating polynomials at a specific point or finding roots. To divide x⁵ - 6x⁴ + 8x³ + 8x² + 4x - 40 by (x-3), we identify the coefficients of the dividend as 1, -6, 8, 8, 4, -40, and from the divisor (x-3), we take...

Q10.c. Determine a unique polynomial f(x) of degree ≤3 such that f(x_0) = 1, f'(x_0) = 2, f(x_1) = 2, f'(x_1) = 3, where x_1 - x_0 = h.

Key Points:

Hermite interpolation constructs polynomials using function and derivative values.
A unique polynomial of degree ≤ 2n-1 exists for n distinct points with derivative data.
Extended Newton's divided difference table is used by repeating nodes (e.g., z₀=x₀, z₁=x₀).
For repeated nodes (zᵢ=zᵢ₊₁), the first divided difference f[zᵢ,zᵢ₊₁] is equal to f'(xᵢ).

Answer: To determine a unique polynomial f(x) of degree ≤3 that satisfies the given conditions, we employ Hermite Interpolation. This method is specifically designed for cases where both function values and derivative values are known at certain points, as discussed in Unit 3, Section 3.4 of the BMTE-144 course material. Since we have four conditions (two function values and two derivative values), a polynomial of degree at most 2n-1, where n is the number of distinct points, is unique. Here, n=2 (x₀ an...

BMTE-144 Solved Assignment — January 2026 / July 2026

NUMERICAL ANALYSIS | IGNOU Bachelor of Science (General) (CBCS) (BSCG)

BMTE-144—January 2026 / July 2026

NUMERICAL ANALYSIS

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BMTE-144 Assignment Questions — January 2026

Q1. State whether the following statements are true or false. Give a Short proof or a counter-example in support of your answer.

Q2.a. Find the missing values in the following table: X 0 1 2 3 4 5 y 0 2 - 18 - 90

Q2.b. Using Classical Runge-Kutta fourth order method, find an approximate value of y(1.2) for the IVP dy/dx = xy, y(1) = 2 with h = 0.2.

Q3.a. Find the approximate root of the equation 2x³ = 3x + 6 using Newton-Raphson method. Perform only 3 iterations with x0 = 2.

Q3.b. The roots of the quadratic equation x² + ax + b = 0 are given by α and β. Show that the iteration X_k+1 = -(ax_k + b)/X_k will converge near x = α when |α| > |β|.

Q3.c. If δ²f(x) = C₁h²f''(x) + C2h⁴f⁽⁴⁾(x) + …, find the values of C₁ and C2.

Q4.a. The Gauss-Seidel method is used to solve the system of equations: 4 0 2 [X₁] = [4]; 0 5 2 [X₂] = [-3]; 5 4 10 [X₃] = [2]. Determine the rate of convergence of the method.

Q4.b. Find the interpolating polynomial by Newton's divided difference formula for the following data: X 0 1 2 4 y 1 1 2 5

Q4.c. Using synthetic division method, show that 2 is a simple root of the equation p(x) = x⁴ - 2x³ + x² - x - 2 = 0.

Q5.a. Obtain the interpolating polynomial in simplest form which fits the following data: X -1 0 1 2 f(x) 3 -4 5 -6

Q5.b. Prove that μ² = 1 + δ²/4

Q5.c. Determine the order of convergence of the iterative method X_n+1 = (X_n-1 f(X_n) - X_n f(X_n-1))/(f(X_n) - f(X_n-1)) for finding a simple root of the equation f(x) = 0.

Q6.a. Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix [[1 6 1], [1 2 0], [0 0 3]] using the power method. Take (1, 0, 0)ᵀ as the initial approximation and perform 4 iterations.

Q6.b. The method X_n+1 = (1/9)[5x_n + 5N/x_n² - N²/x_n⁵], n = 0, 1, 2, ... where N is a positive constant, converges to N¹/³. Find the rate of convergence of the method.

Q7.a. Evaluate ∫ from 0 to 1 of 1/(1+x²) dx by using trapezoidal rule with h = 0.5 and h = 0.25. Use Romberg's method to find the best value of π.

Q7.b. Estimate the eigenvalues of the matrix [[1 -1 2], [-1 1 2], [2 2 -2]] using the Gerschgorin bounds.

Q8.a. Solve the initial value problem using Euler method y' = 1/(x²-3y), y(3) = 2. Find y(3.1) taking h = 0.1.

Q8.b. Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations [[1 1 1], [4 3 -1], [3 5 3]] * [[x], [y], [z]] = [[1], [6], [4]]. Show that the method is convergent and hence find its rate of convergence.

Q9.a. For the following data, use Gauss backward difference method to obtain the interpolating polynomial f(x): X 0.1 0.2 0.3 0.4 0.5 f(x) 1.40 1.56 1.76 2.00 2.28. Hence, find the value of f(0.45).

Q9.b. The velocity of a vehicle beginning from rest is given in the following table for part of the first four. Using Simpson's 1/3 - rule, find the distance travelled by the vehicle in this hour: t = time in min. 10 20 30 40 50 60 v = velocity in km/hr. 80 60 70 75 70 80

Q10.a. Find the inverse of the matrix [[2 1 1], [3 2 1], [2 1 2]] using Gauss-Jordan method.

Q10.b. Divide the polynomial x⁵ - 6x⁴ + 8x³ + 8x² + 4x - 40 by (x-3) by the synthetic division method and find the remainder.

Q10.c. Determine a unique polynomial f(x) of degree ≤3 such that f(x_0) = 1, f'(x_0) = 2, f(x_1) = 2, f'(x_1) = 3, where x_1 - x_0 = h.

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