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When using Gauss-Jordan elimination to convert a matrix to an upper triangular matrix, truncation errors can drastically change the answer.

For example, when performing row operations with excel, with the following matrix:

mat1

Is reduced to:

mat2

The value in the red cell is a truncation error and should be zero. The next step in the algorithm would be to divide the 2nd last row by a constant to make the red cell equal to 1. If no truncation errors had occurred then this cell would have a value of zero. This drastically changes the final matrix. How could I correct for this?

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    $\begingroup$ This is known as loss of significance. $\endgroup$ – David Richerby Apr 10 '17 at 12:52
  • $\begingroup$ Whole books are written on numerical instability in linear algebra, as written your question is too broad and a reference request at best. $\endgroup$ – orlp Apr 11 '17 at 0:10
  • $\begingroup$ @orlp I'm just looking for some solution to this particular problem. Not trying to learn about the whole field. Like looking for a solution to a particular differential equation without learning how to solve differential equations in general. Since this is such a well know algorithm, I expect that someone has dealt with this particular problem before. $\endgroup$ – Nathan Apr 13 '17 at 16:09

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