Re: Obtaining the coefficient matrix of multivariable homogeneous linear equation system.

[Hongyi Zhao <hongyi.zhao@gmail.com>]

On Sun, Nov 20, 2022 at 1:52 AM Thomas D. Dean <tomdean@wavecable.com> wrote:
>
> On 11/19/22 04:58, Charles Greathouse wrote:
>
> >
> >     ? A     = [a,b;c,d];
> >     ? basis = [matrix(2,2,k,l,k==i&&l==j)|i<-[1..2];j<-[1..2]]
> >     %2 = [[1,0;0,0],[0,1;0,0],[0,0;1,0],[0,0;0,1]]
> >     ? C     = [A * B - B * A  | B<-basis]
> >     %3 = [[0,-b;c,0],[-c,a-d;0,c],[b,0;-a+d,-b],[0,b;-c,0]]
> >     ? D     = Mat([concat(Vec(c))  | c<-C]~)
> >     %4 = [0,-c,b,0;-b,a-d,0,b;c,0,-a+d,-c;0,c,-b,0]
> >     ? matker(D)
> >     %5 = [1/c*a-d/c,1;1/c*b,0;1,0;0,1]
> >
> >     Cheers,
> >     Bill.
> >
>
> Ubuntu 20.04 latest gp.
>
> I get an error with this code. The value of D is not the same order and
> is enclosed with brackets.
>
>
>
>
>  > /usr/local/bin/gp
>
> Reading GPRC: /home/tomdean/.gprc
>
> GPRC Done.
>
>
>
>          GP/PARI CALCULATOR Version 2.16.0 (development 28175-12b43bac1)
>
>            amd64 running linux (x86-64/GMP-6.2.1 kernel) 64-bit version
>
>      compiled: Nov 19 2022, gcc version 9.4.0 (Ubuntu
> 9.4.0-1ubuntu1~20.04.1)
>
>                             threading engine: pthread
>
>                   (readline v8.0 enabled, extended help enabled)
>
>
>
>                       Copyright (C) 2000-2022 The PARI Group
>
>
>
> PARI/GP is free software, covered by the GNU General Public License, and
> comes
>
> WITHOUT ANY WARRANTY WHATSOEVER.
>
>
>
> Type ? for help, \q to quit.
>
> Type ?18 for how to get moral (and possibly technical) support.
>
>
>
> parisizemax = 4000002048, primelimit = 1000000, nbthreads = 4
>
> ? A     = [a,b;c,d];
>
> ? basis = [matrix(2,2,k,l,k==i&&l==j)|i<-[1..2];j<-[1..2]]
>
> %2 = [[1, 0; 0, 0], [0, 1; 0, 0], [0, 0; 1, 0], [0, 0; 0, 1]]
>
> ? C     = [A * B - B * A  | B<-basis]
>
> %3 = [[0, -b; c, 0], [-c, a - d; 0, c], [b, 0; -a + d, -b], [0, b; -c, 0]]
>
> ? D     = Mat([concat(Vec(c))  | c<-C]~)
>
> %4 =
>
> [     [0, c, -b, 0]~]
>
>
>
> [ [-c, 0, a - d, c]~]
>
>
>
> [[b, -a + d, 0, -b]~]
>
>
>
> [     [0, -c, b, 0]~]
>
>
>
> ? matker(D)
>
>    ***   at top-level: matker(D)
>
>    ***                 ^---------
>
>    *** matker: forbidden division t_INT / t_COL (4 elts).
>
>    ***   Break loop: type 'break' to go back to GP prompt
>
> break> break

I encountered the same error as shown below:

werner@X10DAi-00:~$ gp
Reading GPRC: /etc/gprc ...Done.

                  GP/PARI CALCULATOR Version 2.11.2 (released)
          amd64 running linux (x86-64/GMP-6.1.2 kernel) 64-bit version
        compiled: Jul 12 2019, gcc version 9.1.0 (Ubuntu 9.1.0-8ubuntu1)
                           threading engine: pthread
                 (readline v8.0 enabled, extended help enabled)

                     Copyright (C) 2000-2018 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes
WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?17 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000, nbthreads = 88
? A     = [a,b;c,d];
? basis = [matrix(2,2,k,l,k==i&&l==j)|i<-[1..2];j<-[1..2]]
%2 = [[1, 0; 0, 0], [0, 1; 0, 0], [0, 0; 1, 0], [0, 0; 0, 1]]
? C     = [A * B - B * A  | B<-basis]
%3 = [[0, -b; c, 0], [-c, a - d; 0, c], [b, 0; -a + d, -b], [0, b; -c, 0]]
? D     = Mat([concat(Vec(c))  | c<-C]~)
%4 =
[     [0, c, -b, 0]~]

[ [-c, 0, a - d, c]~]

[[b, -a + d, 0, -b]~]

[     [0, -c, b, 0]~]

? matker(D)
  ***   at top-level: matker(D)
  ***                 ^---------
  *** matker: forbidden division t_INT / t_COL (4 elts).
  ***   Break loop: type 'break' to go back to GP prompt
break>

>
>
> ? E = [0,-c,b,0;-b,a-d,0,b;c,0,-a+d,-c;0,c,-b,0]
>
> %5 =
>
> [ 0    -c      b  0]
>
>
>
> [-b a - d      0  b]
>
>
>
> [ c     0 -a + d -c]
>
>
>
> [ 0     c     -b  0]
>
>
>
> ? matker(E)
>
> %6 =
>
> [1/c*a - d/c 1]
>
>
>
> [      1/c*b 0]
>
>
>
> [          1 0]
>
>
>
> [          0 1]
>
>
> Tom Dean
>

Best,
Zhao