So I have the following equation:

$$ \sum_{n=1}^{N} S_n f_n(x,y,z) = g(x,y,z) $$

And then for every particular set $\xi$ of $N$ random $(x,y,z)$ points, $\forall x,y,z \in {\mathbb R} $, I can construct an ${\bf F}_{N \times N}(\xi)$ matrix and a ${\bf g}(\xi)$ column vector and then have a system of equations represented by the matrix equation of the form $ {\bf F}(\xi) \cdot {\bf s} = {\bf g}(\xi)$

So my question is this: Is there a way to find the solution column vector ${\bf s}$ which solves $ {\bf F}(\xi) \cdot {\bf s} = {\bf g}(\xi)$ for multiple $\xi$ sets simultaneously?

  • $\begingroup$ This means the same $s$ solves all those systems, or? $\endgroup$ – mvw Nov 7 '15 at 19:41
  • $\begingroup$ @mvw Yes, a single ${\bf s}$ that solves all systems at once. Notice that ${\bf s}$ isn't a function of $\xi$. $\endgroup$ – urquiza Nov 7 '15 at 20:37
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    $\begingroup$ if your coefficients are exact elimination should work, if not you might want to solve $A^tA s = A^tb$. $\endgroup$ – mvw Nov 7 '15 at 20:40
  • $\begingroup$ And by that you mean the $A$ and $b$ are created the way you outlined below? $\endgroup$ – urquiza Nov 7 '15 at 20:42
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    $\begingroup$ Yes, that large system of $kn$ equations in $n$ unknowns. See en.wikipedia.org/wiki/… $\endgroup$ – mvw Nov 7 '15 at 20:44

Just stack the matrices and vectors top to bottom: $$ \left( \begin{array}{c} F_{(1)}\\ \vdots\\ F_{(k)} \end{array} \right) s= \left( \begin{array}{c} g_{(1)}\\ \vdots\\ g_{(k)} \end{array} \right) \iff As = b $$ A solution $s$ must fulfill all the equations from the $k$ subsystems $F_{(i)}s= g_{(i)}$, thus simultaneously.

  • $\begingroup$ The $t$ stands for transpose? $\endgroup$ – urquiza Nov 7 '15 at 19:26
  • $\begingroup$ Okay so to construct the "total" ${\bf F}$ I take each ${\bf F}(\xi)$, transpose it, and pile one on top of the others? $\endgroup$ – urquiza Nov 7 '15 at 19:28
  • $\begingroup$ I usually refer to a vector as a column vector. But you can state your terminology in the answer to make it clear, feel free to explain in any way you feel okay with, I'm listening. :) $\endgroup$ – urquiza Nov 7 '15 at 19:45
  • $\begingroup$ Yeah, $F s = g$. Sorry, I missed that mistake =P. It's the equation of a linear system of equations, so yeah. $Ax=b$. $\endgroup$ – urquiza Nov 7 '15 at 19:53

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