Surely you need to recognize the difficulties here. If not, then you need to consider what is involved.
Let me give a trivial example. Consider this easy to create matrix:
M8 = magic(8),rank(M8)
M8 =
64 2 3 61 60 6 7 57
9 55 54 12 13 51 50 16
17 47 46 20 21 43 42 24
40 26 27 37 36 30 31 33
32 34 35 29 28 38 39 25
41 23 22 44 45 19 18 48
49 15 14 52 53 11 10 56
8 58 59 5 4 62 63 1
ans =
3
Now, if I try to compute null(M8), you will see it has 5 dimensions, thus is spanned by 5 vectors.
But now, lets create an associated matrix, of size 13x8, with symbolic entries. Idon't need to be any more creative that to write it as:
syms a b c d e
M = [M8;a*rand(1,8);b*rand(1,8);c*rand(1,8);d*rand(1,8);e*rand(1,8)];
What does computation of the null space involve now? Depending on the values of a,b,c,d,e, the null space can have dimension anywhere from 3 to 8.
This is a more complex problem than merely computing the null space for a purely numerical matrix, since there are many possibilities. And that is for a trivial problem. Had I been more creative in the above matrix, things can get highly nonlinear, with the nullspace vectors themselves changing shape as well as dimension.
So seriously, can you be remotely surprised that null is inefficient for a large dimensional symbolic matrix?
Can you force the use of multiple threads? Not really easy here, since multiple threads work well when you can pass things to linear algebra routines like the BLAS. MATLAB does that automatically. But symbolic computations are not something the BLAS can handle. So the symbolic toolbox logically tends to stay single threaded, and I doubt that will change soon.
Finally, just consider tat the null space of a 10000x20 array will be of size 10000x9980.
