H=: % @: >: @: (+/~) @: i.
det=: -/ .*
0j_6 ": det@H"0 i.3 5x
1.000000e0 1.000000e0 8.333333e_2 4.629630e_4 1.653439e_7
3.749295e_12 5.367300e_18 4.835803e_25 2.737050e_33 9.720234e_43
2.164179e_53 3.019095e_65 2.637781e_78 1.442897e_92 4.940315e_108
The preceding computes the determinant of the Hilbert matrix of order
0-14. The later numbers indicate that the matrices are very nearly
singular. In such cases it is better to not to produce a result on matrix
inverse but instead more truthfully signal domain error.
As well, it's misleading to judge the quality of ⌹H n by comparing it to ⌹H
n+0x. Better, I think, to look at how close h+.×⌹h←H n is to the identity
matrix. You said Nars does well through 12. I fired up Nars, and:
⎕fx 'z←H n' 'z←÷1+∘.+⍨⍳n'
H
0 ¯4 ⍕ h+.×⌹h←H 12
1.000E0 ¯1.516E¯6 9.162E¯5 ¯5.689E¯4 8.397E¯3 ¯4.418E¯2 8.586E¯2
¯1.787E¯1 1.740E¯1 ¯2.069E¯1 7.577E¯2 ¯1.666E¯2
¯4.262E¯10 1.000E0 9.401E¯6 3.702E¯4 6.376E¯4 ¯1.662E¯2 ¯3.609E¯3
4.427E¯2 ¯8.380E¯3 ¯5.246E¯2 ¯8.840E¯4 ¯7.011E¯3
¯2.106E¯9 1.310E¯7 1.000E0 4.874E¯4 1.928E¯3 ¯3.517E¯3 1.780E¯2
¯6.267E¯3 ¯5.421E¯2 ¯8.853E¯2 4.773E¯3 ¯3.261E¯3
1.439E¯10 ¯2.240E¯7 1.361E¯5 1.000E0 1.561E¯3 ¯1.930E¯2 ¯9.595E¯3
¯8.351E¯3 1.277E¯2 ¯8.644E¯2 ¯1.020E¯4 ¯2.995E¯3
¯3.341E¯9 ¯8.130E¯8 4.757E¯6 2.152E¯4 1.003E0 ¯1.248E¯2 ¯9.875E¯3
¯2.338E¯2 ¯2.646E¯2 ¯2.180E¯2 3.163E¯3 ¯3.569E¯3
2.867E¯9 ¯3.154E¯7 3.221E¯5 ¯1.622E¯4 1.797E¯3 9.907E¯1 2.314E¯2
¯5.096E¯2 2.835E¯3 ¯7.728E¯2 8.604E¯3 ¯3.089E¯3
2.387E¯9 ¯4.019E¯7 3.047E¯5 2.491E¯4 2.982E¯3 ¯1.269E¯2 1.009E0
¯4.721E¯2 4.133E¯2 ¯5.898E¯2 ¯5.427E¯5 ¯3.163E¯3
¯1.555E¯9 ¯2.262E¯7 3.253E¯5 3.957E¯5 1.979E¯3 ¯1.016E¯2 ¯5.039E¯3
9.740E¯1 1.559E¯2 ¯3.719E¯2 1.313E¯2 ¯3.457E¯3
¯9.249E¯10 1.446E¯7 2.011E¯5 1.630E¯4 1.268E¯3 ¯9.304E¯3 ¯6.022E¯3
¯1.894E¯2 1.009E0 ¯3.097E¯2 9.581E¯3 ¯1.927E¯3
1.453E¯9 1.888E¯7 1.126E¯5 1.977E¯4 5.719E¯4 ¯9.099E¯3 ¯3.797E¯3
¯2.931E¯2 ¯1.234E¯2 9.723E¯1 ¯9.762E¯4 ¯2.198E¯3
¯1.307E¯9 3.232E¯9 7.623E¯6 1.050E¯4 3.457E¯4 ¯6.768E¯3 1.071E¯2
3.447E¯2 ¯1.285E¯2 ¯2.097E¯2 1.001E0 ¯1.842E¯3
¯1.039E¯9 1.911E¯7 6.941E¯6 3.105E¯4 8.445E¯4 ¯6.431E¯3 ¯1.150E¯2
1.330E¯2 ¯1.604E¯3 ¯3.245E¯2 2.224E¯3 9.974E¯1
On Tue, Jun 19, 2012 at 9:43 PM, Paul Jackson <[email protected]> wrote:
> I don't expect floating answers to be exact. However, most of my
> use of computers begins with inexact numbers, so getting the floating
> routines as good as possible seems like a worthy effort to me.
>
> Unfortunately, I stopped my reseach when J's answers went very
> wrong at H 9. Nars also supports extended percision and rational
> numbers, so I've continued the work. While it does well though 12, it
> goes horribly wrong at 13.
>
> http://home.comcast.net/~paul.l.jackson/PLJsAPL/JandNars.txt
>
> Paul
>
> On Mon, Jun 18, 2012 at 7:01 PM, Roger Hui <[email protected]>
> wrote:
> > Don't you become even a bit suspicious when the answers don't agree?
> > Consider the number on the lower right hand corner:
> >
> > PLJsAPL: 2815820646.25489
> > Dyalog: 2815825985
> > NARS2000: 2815820182
> > J Rational: 2815827300 (true answer).
> >
> > If you are proud of implementation X, I can bring it down to earth real
> > quick by asking it to invert H 2*n, or by working in rational numbers.
> >
> > H=: %@>:@(+/~)@i.
> > (+/ .* %.) H 20x
> ----------------------------------------------------------------------
> For information about J forums see http://www.jsoftware.com/forums.htm
>
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
