I am reading through my numerical mathematics script and I am currently in the chapter 4 (see listing)
- computer arithmetic
- direct solution of linear systems of equations
- polynomial interpolation
- numerical integration
the script continues but i wanted you to know how much we covered so far.
My Problem:
they claim that the condition of any quadrate formula is given by: $\sum_{j=0}^{n} \frac{\vert \alpha_j \vert}{\vert a - b \vert}$
My thoughts:
The $\alpha_j$ are the weights (of the quadrate formula) which are defined as: $$ \alpha_j := \int_a^b l_j(x) \ \mathrm{d}x$$ where $l_j$ are the $j^{th}$ Lagrangian polynom and: $$l_j(x) := \prod_{i=0 \atop i\neq j}^n \frac{x-x_i}{x_j-x_i}$$
so therefore: $$\sum_{j=0}^{n} \frac{\int_a^b \prod_{i=0 \atop i\neq j}^n \frac{x-x_i}{x_j-x_i}}{\vert a-b \vert} $$
but furthermore I don't see how this claim can be proven. Thanks for any advice and help.