lec16.md

Lecture #16: Introduction to SciPy and Linear Algebra Review

Lecture Objectives

1. Become aware of the problem solving utilities in the SciPy library
2. Identify the functions that correspond to previously seen utilites from other language libraries, e.g. MATLAB's fsolve, ode45, \-operator, etc.
3. Be able to translate mathematical summation over an index to loops
4. Be able to write a second-order difference equation as a matrix/banded-matrix
5. Be able to handle Dirichlet boundary conditions for a second-order difference equation

Textbook Reference

Activities

1. Tour of SciPy library
   • Discrete Fourier transforms (scipy.fftpack)
   • Integration and ODEs (scipy.integrate)
   • Linear algebra (scipy.linalg)
   • Optimization and root finding (scipy.optimize)
   • Special functions (scipy.special)
2. Review of basic linear algebra
   • matrix multiplication
   • Gaussian elimination
3. Turning the second order difference equation into a matrix
   • banded matrix
     • identify the structure and define the translation from a full matrix
     • scipy.linalg.solve_banded
   • Dirichlet boundary conditions
     1. write out difference equations involving boundary nodes
     2. move known values to the right hand side