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Numerical Methods (Metric)

Learn, or revise, numerical integration, iterative solution of equations, fitting curves to data and numerical solution of differential equations.

Numerical Integration

Trapezium rule, Simpson's rule (A-level topics) and the Richardson extrapolation (not on lost A-level courses).

Estimating definite integrals using the ordinates \(y_0,\,y_1,\,y_2,\,\dots,y_n\).

Getting an improved estimate by extrapolating.

Iterative Solution of Equations

Fixed point iteration and Newton's method (on some A-level courses).

Solving the equation \(x=f(x)\) using the iteration \(x_{n+1}=f(x_n)\).

Solving the equation \(g(x)=0\) using the iteration \(x_{n+1}=x_n-f(x_n)/f'(x_n)\).

Data Fitting

Interpolation using polynomials; least-squares regression (mostly First Year university topics).

Fitting a polynomial exactly to a data set.

The least-squares curve of best fit.

Differential Equations

Euler's method for the approximate solution of differential equations.

Solving the differential equation \(dy/dx=f(x,\,y)\) approximately using the iteration \(y_{n+1}=y_n+h\,f(x_n,\,y_n)\).