Summary

Summary#

Things you should know about or how to do#

Floating point representation and roundoff error: what sets the scale of roundoff errors and the maximum and minimum floats that can be stored.
Finite differencing of derivatives, including the idea of the order of the approximation and the optimal step size
Interpolation: how to do simple linear interpolation; the difference between cubic interpolation and a cubic spline; how to do interpolation in >1D
Integration: Newton-Cotes methods; Gaussian quadrature and how to implement it; how to estimate the error in a numerical integration; Monte-Carlo integration and its error
Random numbers: how to generate samples from a distribution, including built-in NumPy/SciPy functions, rejection/transformation/ratio of uniforms methods, or using Metropolis-Hastings.
Matrix operations and decompositions in Numpy, especially SVD. The idea of condition number and how to calculate it.
Newton’s method for minimizing a function or finding roots
Integrating differential equations: the difference between explicit and implicit methods. Advantages and disadvantages of both methods. What is meant by the term stiff equation.
Different types of errors: numerical instability, amplitude errors, phase errors, von Neumann stability analysis
How to implement an adaptive step-size. The difference between absolute and relative errors.
What makes an integrator symplectic. The idea of a pseudo-Hamiltonian. The leapfrog method.
Differences between initial value and boundary value problems. Basic idea of relaxation methods.
Methods for solving the linear system \(\mathbf{Ax} = \mathbf{b}\): inversion including with SVD, conjugate gradient for sparse matrices.
How to implement boundary conditions in finite difference schemes
The discrete Fourier transform: frequency spacing, maximum spacing, aliasing, leakage
The difference between convolution and cross-correlation and how to implement them in Fourier space. Zero-padding and why/when it is needed.