HP Forum Archive 14

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Strange matrix results for HP-71B Math ROM Message #1 Posted by J-F Garnier on 5 July 2004, 7:49 a.m. Following a recent long discussion, about matrix calculation accuracy on various calculators, I was quite surprised by the HP-71B Math ROM matrix performance, and investigated a little bit. First, using the Kahan exemple, I noticed that determinant calculation (DET function) gives a not so accurate answer: Kahan matrix: DET=-.999999999998 transposed matrix: DET=-.999999999989 First, why is the determinant of the transposed matrix different? Second, why is the result non-integer, as all elements are integer and no overflow occurs? Then I built a simpler exemple, in order to check that the DET function uses internal 15-digit accuracy (this was stated many times, but no clear evidence was given). I used this matrix: 1+1E-7 1+2E-7 1+3E-7 1+4E-7 If DET uses 15-digits accuracy, result should be -2E-7, otherwise loss of accuracy will occur and result is expected to be 0. HP-71B Math ROM gives 1.0000003E-14 so it seems that 15-digit accuracy is used but not only is the value half than expected, but the sign is wrong. Transposing the matrix gives DET= 2.0000004E-14. Sign is still wrong. Can we call this a bug? I don't know, but I checked that all next models starting from the HP-28S give correct DET value, so algorihms were changed at some point. J-F

Re: Strange matrix results for HP-71B Math ROM Message #2 Posted by J-F Garnier on 5 July 2004, 7:57 a.m., in response to message #1 by J-F Garnier Sorry, my test matrix was not correctly reproduced in this post (a bug?): | 1.0000001 1.0000002 | 1.0000003 1.0000004 I hope it's OK now. J-F

Re: Strange matrix results for HP-71B Math ROM Message #3 Posted by Werner Huysegoms on 5 July 2004, 8:29 a.m., in response to message #1 by J-F Garnier As to why the results are non-integer and the transposed matrix has a 'different' result: The numerical operations in the algorithm used (some variant of LU-decomposition followed by the multiplication of the elements on the diagonal) are exposed to roundoff errors, of course. The operations are different in the transposed matrix - that's why the roundoff errors are different, too. As regards the 'internal 15-digit accuracy'. That only holds for single operations and dot products (that can be accumulated without extra storage). The numbers themselves are still stored in 12-digit format. The reason the 71B gets the determinant wrong, and the 28S gets it right is probably (guessing): the determinant of 2-by-2 matrices is calculated using the definition, in full 15-digit accuracy (28S). From the 48G onwards, a check for integer elements has been added - if all elements are integer, then so is the determinant, and the computed value is rounded off to the nearest integer. Also, the matrix is stored in full 15-digit form, reducing roundoff errors. BTW the determinant for the sample matrix you show is of course -2e-14. Cheers, Werner

Re: Strange matrix results for HP-71B Math ROM Message #4 Posted by Werner Huysegoms on 5 July 2004, 9:02 a.m., in response to message #3 by Werner Huysegoms My guess does seem to be correct: If I try your original matrix on a 42S and a 48SX, they return the correct answer, but if I try the equivalent 3-by-3 matrix 1.0000000 0.0000000 0.0000000 0.0000000 1.0000001 1.0000002 0.0000000 1.0000003 1.0000004 Then the result is (in both cases) 1.0000003e-14, the result the 71B gave. And yes, the transpose's determinant is 2.0000004e-14. I'm pretty sure I can find out the exact order of 15-digit arithmetic operations and 12-digit rounded stores to come up with these answers. Valentin will probably post it in the next 2 minutes ;-) Cheers, Werner

Re: Strange matrix results for HP-71B Math ROM Message #5 Posted by J-F Garnier on 5 July 2004, 12:10 p.m., in response to message #4 by Werner Huysegoms Thanks Werner, I fill better: the HP-71B Math ROM is OK, just the HP28/HP48 added some filtering against the most visible rounding effects. J-F

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