Notes

1. Interpolation & Newton's Forward Interpolation

Foundations of interpolation versus extrapolation, forward difference operator $\Delta$, derivation of forward difference tables, Newton-Gregory forward interpolation formula for equispaced nodes near the beginning of a table, and polynomial reconstruction.

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2. Newton's Backward Interpolation

Backward difference operator $\nabla$, construction of backward difference tables, Newton's backward interpolation formula for points near the end of a dataset, advantages/disadvantages, and practical applications in digital image resizing and game graphics linear interpolation (lerp).

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3. Lagrange Interpolation

Lagrange polynomial formulation for unequally or equally spaced points, algebraic derivation via divided difference zeroing conditions, construction of Lagrange cardinal basis polynomials $L_i(x)$, and high-order numerical evaluation.

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4. Gauss Central Difference Interpolation

Central difference operators, shifting operator identities ($E = 1 + \Delta$), derivation of Gauss's Forward and Backward difference interpolation formulas for points located near the center of a tabular dataset, Stirling's mean formula, and Bessel's formula with practical thermal and demographic applications.

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5. Trapezoidal Rule

Fundamentals of closed Newton-Cotes numerical quadrature, geometric interpretation of trapezoidal strip approximation, uniform step size computation ($h = \frac{b - a}{n}$), composite trapezoidal formula, and numerical evaluation of $\int_1^2 \frac{1}{x}\,dx$ with $n = 10$.

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6. Numerical Quadrature: Trapezoidal, Simpson's 1/3 & Simpson's 3/8 Rules

Comparative study of Newton-Cotes numerical quadrature formulas: Trapezoidal rule, Simpson's 1/3 rule (parabolic interpolation, requiring even number of intervals $n$), and Simpson's 3/8 rule (cubic polynomial interpolation, requiring $n$ to be a multiple of 3), applied to the rational integral $\int_0^1 \frac{2}{1 + x^3}\,dx$ with $n = 6$.

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7. Systems of Linear Equations: Gauss-Jacobi & Gauss-Seidel Methods

Iterative solution techniques for linear systems $A\mathbf{x} = \mathbf{b}$, strict diagonal dominance criteria, iterative formulation rearrangement, simultaneous updating in Gauss-Jacobi versus immediate in-place updating in Gauss-Seidel, convergence rate acceleration, and termination error criteria ($E_A \le \epsilon$).