10.3 The physics of springs  (Page 6/11)

It would seem that all networks that we had to modify are significantly worse than networks we did not have to modify. Looking more closely at the singular-value decomposition, we find that there is one very large singular value corresponding to the row we added because the entries in this row are much larger than all the other entries in B . There is no particular reason why we need a 1 and a $-1$ (or any other pair of opposite numbers), so we scale this row down. We would like to improve the condition number as much as possible, so we look at the condition number of B as a function of the scaling factor of the last row. We find that, as the scaling factor increases, the condition number decreases.

[link] shows the relationship between the scaling factor on the additional row and the condition number of B for Network g, using the first 4 experiments of Data Set A.

We repeated the simulations, scaling the additional row by a factor of $1/200$ . The scaled results are presented in [link] ; for reference, the unscaled results are also included. This method, then, greatly improves the condition number of the matrix.

Ultimately, we decided to work with Network g because it was the tautest, so it eliminated some experimental error. It was unfortunate that this network had the worst average condition number, but the benefit of accurate data was greater than the benefit of an accurate matrix. For future experimentation, however, building a new structure and using Network a might be the best.

We were also interested in how stacking impacts the condition number of a network. To study this, we add two experiments from Data Set A at a time and compute the condition number after each step. [link] shows the singular values of B as we add more experiments; the lighter the color, the more experiments have been added. [link] shows how the condition number of B changes as we add more experiments; the red line is the minimum. From this, we conclude that adding more experiments improves the condition number to a certain point, but it is close to constant from there.

When we stack all of the results from Data Set A together and scale them appropriately, we get 145.6% error.