rcc02007 / CompMech04-LinearAlgebra

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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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