Recent Posts

Spline interpolation

Just like before with polynomial interpolation, we have a list of $n+1$ given point $(x_i, y_i)$ with $x_0 < x_1 < \dots < x_n$. We want to find a function that goes through those points and approximates the underlying function that produced that points as good as possible. Polynomial interpolation The problem … Read More »

Polynomial interpolation

Suppose you have a list of $n+1$ given point $(x_i, y_i)$ with $i \in {0, \dots, n}$ and $\forall i,j \in {0, \dots, n}: i \neq j \Rightarrow x_i \neq x_j$. Now you want to find a polynomial $\displaystyle p(x) = \sum_{i=0}^n a_i \cdot x … Read More »

Solving equations of upper triangular matrices

Suppose you have an equation like $R \cdot x = b$ with $R \in \mathbb{R}^{n \times n}$ and $x,b \in \mathbb{R}^n$. $b$ and $R$ are given and you want to solve for $x$. Example With $n=5$, the problem could look like this: $$\begin{pmatrix} 2 … Read More »

Solving equations of lower unitriangular matrices

Suppose you have an equation like $L \cdot x = b$ with $L \in \mathbb{R}^{n \times n}$ and $x,b \in \mathbb{R}^n$. $b$ and $L$ are given and you want to solve for $x$. Example With $n=5$, the problem could look like this: $$\begin{pmatrix} 1 … Read More »