It seems that a somewhat subtle point in all these comparisons using Hilbert matrices is being overlooked. What I'm going to say only applies to calculators *not* in exact mode. Consider what happens when you enter the 2x2 Hilbert matrix manually. You probably enter the matrix editor on a TI or HP machine and type in 1, 1/2, 1/2, 1/3; but the calculator cannot accept exact rational numbers (fractions), so it converts these numbers to floating point numbers. The result is that you have the numbers 1, .5, .5, .33333333333333 on a TI-85 and similar recent TI machines that use 14 digits to store numbers, or you have 1, .5, .5, .333333333333 on Saturn based HP calculators.
The same thing happens for the higher order Hilbert matrices; the matrix stored in the calculator is not the exact Hilbert matrix, but an approximation to it. So when you then calculate the determinant of the approximate matrix stored in the calculator, you shouldn't expect to get the exact determinant for that particular Hilbert matrix.
What should you reasonably expect? Use a calculator that can do exact arithmetic such as the TI89 or HP49 and use these values for the Hilbert matrix (2x2 for this example):
To determine what a TI calculator which uses 14 digits to store its numbers *should* get, use
1, 1/2, 1/2, 33333333333333/100000000000000 and calculate the determinant of *this* matrix on the TI89 or HP49 (or use a PC program such as Derive, Maple, or Mathematica which can do exact rational arithmetic).
To determine what an HP Saturn based calculator *should* get, use 1, 1/2, 1/2, 333333333333/1000000000000 and calculate the determinant of *this* matrix with exact rational arithmetic.
Extend this technique for the higher order Hilbert matrices. For example, here are the last three numbers in the last row of the matrix which should be used with exact rational arithmetic to determine what a Saturn based calculator *should* get for the determinant of a 7x7 Hilbert matrix:
......909090909091/1000000000000, 833333333333/1000000000000, 769230769231/1000000000000
It isn't reasonable to expect *any* calculator which uses floating point arithmetic (and not exact rational arithmetic) to get the *exact* determinant of a Hilbert matrix because the input to the determinant calculation *isn't* the Hilbert matrix, but an approximation to it. The best it could possibly do is to get the *exact* determinant of the approximation to the Hilbert matrix which your calculator has stored. So don't compare the result a floating point calculator gets for a Hilbert determinant to the *exact* value for that particular Hilbert Matrix. You can't expect your calculator to get that result. It should in fact get the result from the procedure I described above, and you should use that result to compute the relative error for various calculator's calculated Hilbert determinants. For example, adding to Palmer's table:
7x7 8x8 9x9 10x10
HP-41 Math Pac 4.820822E-25 2.437673E-33 -1.605584E-42 -1.046127E-51
HP-41 Advantage 4.836648E-25 2.704536E-33 6.435130E-43 -1.747106E-52
HP-49 Nonexact 4.835583E-25 2.736296E-33 9.802414E-43 3.014075E-53
HP-28S 4.835592E-25 2.736365E-33 9.819514E-43 3.281917E-53
TI-59 ML-02 4.835807E-25 2.737082E-33 9.687516E-43 Note 4
TI-95 Math Module 4.835770E-25 2.736821E-33 9.728025E-43 2.338510E-53
CC-40/TI-74 4.835789E-25 2.736781E-33 9.689026E-43 1.898080E-53
TI-83+ 4.835795E-25 2.737004E-33 9.721266E-43 Note 5
TI-85 4.835795E-25 2.737004E-33 9.721266E-43 2.207089E-53
HP-49 Exact 4.835803E-25 2.737050E-33 9.720234E-43 2.164179E-53
HP-41 should get 4.835822505E-25 2.709988486E-33 8.587780880E-43
HP Saturn should get 4.83558199919 2.73630981825 9.80342803933
TI-85 should get 4.8357998501314 2.7370410057623 9.7218653313429
I left out the exponents of the last two lines to save space.
HP Saturn includes the HP28, HP48S, HP48G, etc., but the HP48G and HP49 (approximate mode) will give better results than the earlier Saturn machines because the matrix arithmetic was reworked to use 15 digits for all internal matrix calculations.
According to these numbers, the relative error the TI-85 gets for the 7x7 Hilbert matrix is -1.0029e-6. The relative error for the HP-49 nonexact is 2.068E-7. This value is not quite right because Palmer reported an insufficient number of digits to accurately calculate the relative error. (There may be a similar problem with the reported results for the TI-85.) More digits would allow an improved result for relative error. The HP48G gets 4.83558259986E-25 for the determinant for a relative error of 1.24E-7 which is also what the HP49 (nonexact) gets.
Another way around the problem is to multiply a Hilbert matrix by the least common multiple (LCM) of all the denominators in the original Hilbert matrix, and find the determinant of *this* matrix. This replaces the Hilbert matrix with another which contains only integers, which can be represented *exactly* on a floating point calculator. For a Hilbert matrix of order n, divide the determinant of this "integerized" matrix by the LCM of the denominators of the original Hilbert matrix, raised to the power n, and that will be the determinant of the original Hilbert matrix. Or, just calculate the determinant of the "integerized" matrix exactly on a TI89 or HP49 (or Derive, etc.) and compare that to the result gotten from the floating point calculator.
As an example, the LCM of the denominators of the 7th order Hilbert matrix is 360360. The exact determinant of the 7x7 Hilbert matrix which has been multiplied by 360360, is 381614277072600. The HP48G gets 3.81614292044E14, for a relative error of 3.92E-8. Readers could try this on their own calculators and report the results. And beware, you can't just take the 7th order Hilbert matrix already stored in your floating point calculator and multiply it by 360360 to get the "integerized" version. In the process where you created the Hilbert matrix by inputting 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, etc., instead input 360360/1, 360360/2, 360360/3, 360360/4, 360360/5, 360360/6, 360360/7, etc. This guarantees that you get integers.