Note
-Here we have been a bit sloppy with our notation. Technically, the functions \(u(x)\) and \(u(\hat{x})\) are two different functions, so we should have used e.g. notation like \(u_x(x)\) and \(u_{\hat{x}}(\hat{x})\). The key thing is that the functions are related as \(u_x(x) = u_{\hat{x}}(\hat{x})\).
+Here we have been a bit sloppy with our notation. Technically, due to our change of variable, the functions \(u(x)\) and \(u(\hat{x})\) are two different functions, so we should have used e.g. notation like \(u_x(x)\) for the original function and \(u_{\hat{x}}(\hat{x})\) for the function after the variable change. The key thing is that the functions are related as \(u_{\hat{x}}(\hat{x}(x)) = u_x(x)\).
Discretization: We discretize this by dividing our \(\hat{x}\) range into \(n\) parts, i.e. we will have \(n+1\) points \(\hat{x}_0\), \(\hat{x}_1\), \(\ldots\), \(\hat{x}_{n-1}\), \(\hat{x}_{n}\). Thus we have a stepsize
diff --git a/lecture_notes/2024/README.html b/lecture_notes/2024/README.html index 846fbbd..3e99b44 100644 --- a/lecture_notes/2024/README.html +++ b/lecture_notes/2024/README.html @@ -595,6 +595,11 @@LU decomposition
What it is
What it’s good for
What it’s good for:
+Solving matrix equations
Continue discussion LU decomposition
+What it’s good for:
+Finding the determinant
Finding the inverse
How: algorithm for determining the L and U matrices
PLU decomposition
Overview of topics for Project 2
Scaling equations