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README.md

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.

01_Linear-Algebra

  • 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

02_Gauss_elimination

  • 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.lu to create the PLU decomposition
  • How to use the PLU efficient structure to solve our linear algebra problem (solveLU)

03_Linear-regression-algebra

  • 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
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