Computational Mechanics 4 - Linear Algebra
Welcome to Computational Mechanics Module #4! In this module we will explore applied linear algebra for engineering problems and revisit the topic of linear regression with a new toolbox of linear algebra. Our main goal, is to transform large systems of equations into manageable engineering solutions.
- How to solve a linear algebra problem with
np.linalg.solve - Creating a linear system of equations
- Identify constants in a linear system
$\mathbf{A}$ and$\mathbf{b}$ - Identify unknown variables in a linear system
$\mathbf{x}$ - Identify a singular or ill-conditioned matrix
- Calculate the condition of a matrix
- Estimate the error in the solution based upon the condition of a matrix
- Graph 2D and 3D linear algebra problems to identify a solution (intersections
- of lines and planes)
- How to solve a linear algebra problem using Gaussian elimination (
GaussNaive) - Store a matrix with an efficient structure LU decomposition where
$\mathbf{A=LU}$ - Solve for
$\mathbf{x}$ using forward and backward substitution (solveLU) - Create the LU Decomposition using the Naive Gaussian elimination process (
LUNaive) - Why partial pivoting is necessary in solving linear algebra problems
- How to use the existing
scipy.linalg.luto create the PLU decomposition - How to use the PLU efficient structure to solve our linear algebra problem (
solveLU)
- How to use the general least squares regression method for almost any function
- How to calculate the coefficient of determination and correlation coefficient for a general least squares regression,
$r^2~ and~ r$ - How to plot and read a training-testing plot
- How to divide data into training and testing data for analysis
- Why we need to avoid overfitting
- How to construct general least squares regression using the dependent and independent data to form
$\mathbf{y}=\mathbf{Za}$ . - How to construct a piecewise linear regression