When we have circular references in our Excel spreadsheet, we can have Excel do a (large) number of iterations in the hope this converges to a solution. Mathematically speaking we could say this is like:
| \[x_{i,j} := f_{i,j}(X)\] |
where \(x_{i,j}\) is the cell in row \(i\) and column \(j\). This will converge to a fixed point:
| \[X = F(X)\] |
if the stars are aligned. Of course we can look at this as if we are solving a system of non-linear equations:
| \[G(X)=F(X)-X=0\] |
For a project, I am looking at some spreadsheets that have a few hundred thousand of such formulas.
Convergence can be a problem for a scheme like this. Below is a nice example of solving the equation \(x^2-x-1=0\) using two different fixed point iteration schemes:
- \(x^{(k+1)} := 1 +\displaystyle\frac{1}{x^{(k)}}\), this one converges
- \(x^{(k+1)} := \displaystyle\frac{1}{x^{(k)}-1}\), this one diverges
Details in the youtube video below:
See (2) and (3) for more information how Excel does these recalculations.
References
- Oscar Veliz, Fixed point Iteration, https://www.youtube.com/watch?v=OLqdJMjzib8
- Recalculation in Excel 2002, https://msdn.microsoft.com/en-us/library/aa140058
- Multithreaded recalculation in Excel, https://msdn.microsoft.com/en-us/library/office/bb687899.aspx