Re: Valentin's AM7 determinant on the WP 34S Message #55 Posted by Rodger Rosenbaum on 19 Oct 2011, 5:55 a.m., in response to message #38 by Werner
AM#8 has a condition number of about 5E19 according to Mathematica, but my HP50 says the condition number is about 5E15. If the goal is to determine the true condition number, then arithmetic with many more digits than the 15 digits internally in the HP50 would be necessary.
However, if one's goal is to solve some system using the given matrix, knowing that the condition number is at least 5E15 is enough to know that any solution derived from that matrix is likely to have no correct digits (on an HP50)we don't need to know that the true condition number is 5E19.
Testing the COND function on the HP50, I have been unable to find a matrix with a true condition number greater than E12 which was inaccurately calculated to have a smaller condition number. The calculator doesn't seem to ever seriously underestimate the condition number.
Using a column matrix of:
b=[1289 202 209 1905 1182 210 1689 1728]T along with A=AM#8, we have a linear system Ax=b. The HP50 solution of this system, using the / key, is:
x=[85.97057 1328.61068 452.27432 61.401457 1570.67801 1770.08111 2566.08833 471.67052]T
but, the exact solution is:
[1 1 1 1 1 1 1 1]T
We can see that the high condition number makes for no correct digits in the solution.
Using the LSQ function to solve the system rather than the / key method gives much better results on the HP50showing the first 4 digits of the results:
[.7876 1.181 .7576 1.015 .7726 .8039 1.211 1.212]T
This shows the advantage of orthogonal methods of solution rather than the Gaussian method.
