Numerical Methods
Semester IV · BCA 253 · 3 credits
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chapters done
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The syllabus asks for
- Types of equations and locating roots
- Errors in computing
- The bisection method
- The method of false position (regula falsi)
- The Newton–Raphson method
- The secant method
- Fixed-point iteration and convergence
- Systems of nonlinear equations
- Choosing a method
- Seeing the ideas: Newton's tangent, bracketing and how fast each method converges
Key points
- Numerical root-finding replaces an impossible formula by a convergent repetition; errors (round-off, truncation, cancellation) set the achievable accuracy.
- Bisection and false position are safe but linear; Newton is quadratic but needs f' and a good start; the secant method trades derivative for a 1.618 order; fixed-point iteration depends on |g'| < 1.
- Newton's method extends to systems through the Jacobian.
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The syllabus asks for
- Introduction and uniqueness
- Errors in polynomial interpolation
- Lagrange interpolation
- Newton's interpolation with divided differences
- Newton's forward and backward difference formulas (equal spacing)
- Cubic spline interpolation
- Least-squares approximation
- Seeing the ideas: one polynomial, divided differences, Runge's warning and the best-fit line
Key points
- Through n + 1 points there is one polynomial of degree ≤ n; Lagrange, Newton divided-difference and Newton difference formulas all give it.
- The error depends on the (n + 1)th derivative and the product of distances to the nodes: interpolate near the centre and avoid very high degrees.
- Cubic splines give a smooth piecewise fit by solving a tridiagonal system.
- Least squares fits noisy data by minimising squared residuals; many non-linear models reduce to straight lines after a transformation.
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The syllabus asks for
- Numerical differentiation
- Numerical integration: Newton–Cotes rules
- Romberg integration
- Numerical double integration
- Seeing the ideas: areas under curves, Romberg's table and the error of a derivative
Key points
- Derivatives come from differentiating interpolation polynomials; central differences are second-order and best; too small an h lets round-off dominate.
- Newton–Cotes rules integrate interpolating polynomials; Simpson's rules beat the trapezoidal rule at the same cost.
- Romberg extrapolation turns cheap trapezoidal values into a high-order estimate.
- Double integrals reuse the one-dimensional rules with product weights.
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The syllabus asks for
- Gauss elimination
- Gauss–Jordan method and matrix inversion
- LU factorisation
- Iterative methods: Jacobi and Gauss–Seidel
- Ill-conditioning
- Eigenvalues: the power method
- Seeing the ideas: elimination, iteration speed and the geometry of ill-conditioning
Key points
- Direct methods (Gauss, Gauss–Jordan, LU) give the exact answer in a fixed number of steps; use partial pivoting.
- Iterative methods suit large sparse systems; Gauss–Seidel usually converges faster than Jacobi, and diagonal dominance guarantees it.
- Ill-conditioned systems amplify data errors regardless of method; the condition number quantifies this.
- The power method estimates the dominant eigenpair using repeated multiplication.
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The syllabus asks for
- Initial value problems
- Taylor series method
- Picard's method
- Euler's method
- Heun's (improved Euler) method
- Runge–Kutta methods
- Higher-order equations and systems
- Boundary value problems and the shooting method
- Seeing the ideas: stepping along slopes, the four slopes of RK4 and how the methods compare
Key points
- An IVP is solved by stepping from the initial condition: Taylor and Picard give series; Euler, Heun and RK4 give numbers.
- Euler's method is first order, Heun second, RK4 fourth; halving h reduces the error by about 2, 4 and 16 respectively.
- Higher-order equations become first-order systems.
- BVPs are solved by shooting (guess the slope, integrate, correct with interpolation or secant steps) or by finite differences.
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The syllabus asks for
- Introduction and classification
- Deriving difference equations
- Laplace's equation on a square plate
- Poisson's equation
- Accuracy and practical notes
- Seeing the ideas: the five-point stencil, the solved plate and Gauss-Seidel on the grid
Key points
- Elliptic PDEs such as Laplace's and Poisson's equations describe steady states and are determined by their boundary values.
- Replacing derivatives by central differences turns the PDE into one algebraic equation per grid point: the value is the average of its neighbours (Laplace) plus a source term (Poisson).
- The resulting diagonally dominant system is solved directly or by Liebmann's iteration.
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The paper
TU assessment: 60% final examination set by the university, 40% internal (assignments, lab work, attendance). Confirm the split in your college's notice.
Where the hours go
Teaching hours are the board's guide to how much of the paper each part carries.
- Unit 1 · Solution of Nonlinear Equations10 h
- Unit 4 · Solution of Linear Algebraic Equations10 h
- Unit 2 · Interpolation and Approximation8 h
- Unit 5 · Solution of Ordinary Differential Equations7 h
- Unit 3 · Numerical Differentiation and Integration5 h
- Unit 6 · Solution of Partial Differential Equations5 h