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.solveCreating 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 decompositionHow 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, \(r^2~ and~ r\)
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