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Numerical Analysis - Coggle Diagram
Numerical Analysis
Interpolation
Newton Forward: Uses forward differences to construct a polynomial, ideal for estimating values near the beginning of a data set.
Newton Backward: Uses backward differences, making it best suited for approximating values near the end of a data set.
Gauss Forward: A central difference interpolation method that utilizes points alternatingly above and below the starting origin.
Gauss Backward: A central difference interpolation method that begins its calculations from a central point and proceeds backward.
Stirling: The average of Gauss forward and backward formulas, offering high accuracy for interpolating values near the center of a data set.
Bessel: A central difference formula optimized for interpolating values precisely midway between two given data points.
Lagrange: A direct, versatile approach that fits a single polynomial through a set of points without requiring them to be equally spaced.
Integration
Simpson 1/3: Approximates the integrand by fitting a parabolic arc through sets of three equally spaced data points.
Composite 1/3: Applies Simpson's 1/3 rule across multiple subintervals to improve accuracy over a large domain.
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Gaussian Quadrature: Chooses optimal, non-equally spaced points and weights to achieve maximum accuracy.
Errors
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Relative Error: The absolute error scaled by the true value, showing the error's significance relative to the size of the number.
Percentage Error: The relative error expressed as a percentage to provide an intuitive measure of accuracy.
Root Finding Methods
Bisection: A reliable, bracketed method that repeatedly halves an interval containing a root until it converges.
Newton-Raphson: A fast, calculus-based method that uses tangents to rapidly home in on a root, requiring a good initial guess.
Secant: An iterative method similar to Newton-Raphson that approximates the derivative using a line through two points instead of calculus.
Linear Systems
Jacobi: An iterative algorithm that calculates the next approximation of each variable independently using values from the previous step.
Gauss-Seidel: An enhanced iterative method that immediately uses newly computed values within the same iteration for faster convergence.
QR Decomposition: A direct method that factors a matrix into orthogonal (Q) and upper triangular (R) matrices to solve systems stably.
ODEs
Euler Method: The simplest first-order numerical technique that projects the next value along the tangent line of the current slope.
Runge-Kutta: A powerful family of iterative methods that sample multiple intermediate slopes to achieve highly accurate solutions.
Numerical ODE Methods: The broader category of predictor-corrector and multi-step algorithms for solving differential equations.