Introduction to Scientific Computing

Introduction to Scientific Computing#

AMATH 301 covers numerical methods and programming skills needed to solve physical, biological, and engineering problems. Content is transcribed from MATLAB (originally taught in the course) to Python for open source reference. Taken in Sp20 with Dr. Craig Gin.

Note: Starred topics (*) are not covered in Sp20 but covered in previous years.

Root-Finding and Optimization#

Topic	Applications	Numerical Methods Python Skills	Jupyter Notebook	Online
Root finding algorithms	Solving function values	Bisection method Newton’s method `scipy.optimize.root_scalar()`	ipynb	html
Unconstrained optimization	Finding max and min of convex functions	Golden section search Gradient descent `scipy.optimize.fminbound()` `scipy.optimize.fmin()`	ipynb	html
Constrained optimization*	Finding max and min of functions with constraints	Linear programming Genetic algorithm `scipy.optimize.linprog()` `scipy.optimize.differential_evolution()`	ipynb	html
Curve fitting and interpolation	Fitting data Interpolating between data	Sum of squared error Sum of absolute error Maximum absolute error `scipy.optimize.fmin()` `scipy.optimize.curve_fit()` `numpy.polyfit()` `numpy.polyval()` `scipy.interpolate.interp1d()`	ipynb	html

Linear Algebra#

Topic	Applications	Numerical Methods Python Skills	Jupyter Notebook	Online
Matrix operations in python	Dot product Norm Matrix multiplication	`@` `numpy.linalg.norm()`	ipynb	html
Solving linear systems with direct methods	Solving linear systems repeatedly	Matrix inversion LU decomposition `np.linalg.lu()` `scipy.linalg.solve()`	ipynb	html
Solving linear systems with iterative methods	Solving sparse linear systems	Jacobi method Gauss-Seidel method `numpy.linalg.eig()` `numpy.linalg.eigvals()`	ipynb	html
Singular value decomposition (SVD) (Principle component analysis, PCA)	Dimensionality reduction Image compression	SVD, PCA algorithm `scipy.linalg.svd()`	ipynb	html

Numerical Calculus#

Topic	Applications	Numerical Methods Python Skills	Jupyter Notebook	Online
Numerical differentiation	Differentiate functions Differentiate data	Forward difference Backward difference Central difference Other second order methods `numpy.gradient()`	ipynb	html
Numerical integration	Single integrals Double integrals Triple integrals (functions and data)	Left end point rule Right endpoint rule Midpoint rule Trapezoidal rule Simpson’s rule `scipy.integrate.quad()` `scipy.integrate.dblquad()` `scipy.integrate.tplquad()`	ipynb	html

Ordinary Differential Equations (ODEs)#

Topic	Applications	Numerical Methods Python Skills	Jupyter Notebook	Online
Solving first-order ODEs	Solving first-order ODEs	Forward Euler Backward Euler Midpoint method Fourth-order Runge-Kutta (RK4) `scipy.integrate.solve_ivp()`	ipynb	html
Solving higher-order ODEs	Solving higher-order ODEs Systems of first-order ODEs	Forward Euler Backward Euler `scipy.integrate.solve_ivp()`	ipynb	html
Stability and stiffness of ODEs	Choosing integration methods Choosing time steps	Forward Euler Backward Euler `scipy.integrate.solve_ivp()`	ipynb	html
Chaotic systems part 1*	Lorenz system	`scipy.integrate.solve_ivp()`	ipynb	html
Chaotic systems part 2*	Lorenz system	`scipy.integrate.solve_ivp()`	ipynb	html

Phase Portrait Applications#

Topic	Applications	Numerical Methods Python Skills	Jupyter Notebook	Online
Spring-mass-damper system	Physics	`scipy.integrate.solve_ivp()`	ipynb	html
Linear, nonlinear pendulum	Physics	`scipy.integrate.solve_ivp()`	ipynb	html
Two-eyed monster	Applied math	`scipy.integrate.solve_ivp()`	ipynb	html
Population dynamics Predator-pray model Competition model	Ecology	`scipy.integrate.solve_ivp()`	ipynb	html
FitzHugh-Nagumo neuron excitation model part 1*	Neuroscience	`scipy.integrate.solve_ivp()`	ipynb	html
FitzHugh-Nagumo neuron excitation model part 2*	Neuroscience	`scipy.integrate.solve_ivp()`	ipynb	html
FitzHugh-Nagumo neuron excitation model part 3*	Neuroscience	`scipy.integrate.solve_ivp()`	ipynb	html
Coupled harmonic oscillator*	Physics	`scipy.integrate.solve_ivp()`	ipynb	html

Fourier Transform#

Topic	Applications	Numerical Methods Python Skills	Jupyter Notebook	Online
Fourier transform*	Power spectrum density Noise filtering Image compression	Discrete Fourier transform Fast Fourier transform `numpy.fft.fft()` `numpy.fft.ifft()` `numpy.fft.fftfreq()` `numpy.fft.fft2()`	ipynb	html