I forgot to mention that I'm running Maxima 5.20.1. This problem is
present both on Windows version, compiled with GCL, and Linux version,
compiled with CLISP.
2010/3/10 Stefano Ferri <ferriste at gmail.com>:
> I was trying to solve a linear system (4 equations in 4 unknowns) by
> reducing the complete matrix, and I came to this issue with
> triangularize. When reducing the matrix in symbolic ?form, one gets a
> (wrong?) result, a different one (correct) if the triangularization is
> made over numerical values.
>
> This is the example: a system which, for h=-1, has 1 free unknown,
> with infinite^1 solutions, with the following complete matrix:
>
>
> (%i1) display2d:false;
> (%o1) false
>
> (%i2) M:matrix([h-1,h,h+2,h+1,1],[1-h,2,-h-2,1,1],[h-1,h,1,0,1],[0,-h-2,0,-2,1]);
> (%o2) matrix([h-1,h,h+2,h+1,1],[1-h,2,-h-2,1,1],[h-1,h,1,0,1],[0,-h-2,0,-2,1])
>
> (%i3) Mt:triangularize(M);
> (%o3) matrix([h-1,h,1,0,1],[0,-h^2-h+2,0,2-2*h,h-1],
> ? ? ? ? ? ? [0,0,-h^3-2*h^2+h+2,-h^3-2*h^2+h+2,0],
> ? ? ? ? ? ? [0,0,0,-h^4-2*h^3+h^2+2*h,-3*h^3-6*h^2+3*h+6])
>
> (%i4) ev(Mt,h=-1);
> (%o4) matrix([-2,-1,1,0,1],[0,2,0,4,-2],[0,0,0,0,0],[0,0,0,0,0])
>
> (%i5) Mr:triangularize(ev(M,h=-1));
> (%o5) matrix([-2,-1,1,0,1],[0,-2,0,-2,-4],[0,0,0,2,-6],[0,0,0,0,0])
>
> (%i9) rank(ev(Mt,h=-1));
> (%o9) 2
>
> (%i10) ?rank(Mr);
> (%o10) 3
>
> (%i11) rank(ev(M,h=-1));
> (%o11) 3
>
>
> I have checked by hand, and Mt seems wrong (see the rank in the last
> outputs). What's happening here? Is this a bug or I am missing
> something?
> The triangularization is correct if made over the only coefficient matrix.
>
> Moreover, triangularize yelds to complicated expressions. See, in
> example, the triangularization of the coefficient matrix above:
>
>
> (%i20) Mc:submatrix(M,5);
> (%o20) matrix([h-1,h,h+2,h+1],[1-h,2,-h-2,1],[h-1,h,1,0],[0,-h-2,0,-2])
> (%i21) Mtc:triangularize(Mc);
> (%o21) matrix([h-1,h,1,0],[0,-h^2-h+2,0,2-2*h],[0,0,h^3+2*h^2-h-2,h^2+h-2],
> ? ? ? ? ? ? ?[0,0,0,h^4+2*h^3-h^2-2*h])
>
>
> while with some simple rows transformations, we get:
>
>
> (%i12) Mc[2]:Mc[2]+Mc[1];
> (%o12) [0,h+2,0,h+2]
>
> (%i13) Mc[3]:Mc[3]-Mc[1];
> (%o13) [0,0,-h-1,-h-1]
>
> (%i14) Mc[4]:Mc[4]+Mc[2];
> (%o14) [0,0,0,h]
>
> %i15) Mc;
> (%o15) matrix([h-1,h,h+2,h+1],[0,h+2,0,h+2],[0,0,-h-1,-h-1],[0,0,0,h])
>
> The latter expression looks better to me (it has ony first order terms).
>
>
> Can someone please check these results?
>
> Thanks.
> Stefano
>