🚀 go-pugleaf

In article <1992Oct23.162602.21514@cronkite.ocis.temple.edu>, neilfp@astro.ocis.temple.
edu (Neil F. Palumbo) writes:
> [ Article crossposted from sci.math.num-analysis ]
> [ Author was Neil F. Palumbo (neilfp@astro.ocis.temple.edu) ]
> [ Posted on Fri, 23 Oct 1992 16:20:16 GMT ]
>
>  Suppose I have set of linear matrix equations of the form:
>
>               Q1 * t = a1 * z1
>               Q2 * t = a2 * z1
>               Q3 * t = a3 * z1
>               ...............
>               ...............
>              Q11 * t =  1 * z1
>
> where the known matrices (Qi's) are (11x101) and full rank. The ai's
> are unknown scalars, z1 is an unknown (11x1)vector and t is an unknown
> (101x1)vector. A lot of unknowns BUT notice that z1 appears on the right
> side of every equation (scaled by a constant) and that the 11'th equation
> has z1 alone on the right hand side. I want to find ai's, z1 and t to
> make the system true. Anyone with any ideas, comments, suggestions on
> a procedural method of solution?

Why not try the following:

The System could be written as 121 nonlinear (but nearly linear) equations
in variables t1...t101, a1..a10, z1...z11 (rename z1 -> z) ,that means 122
variables. -> solution could be not unique.

One could start with 101 Gaussian elimination steps on both sides
(just treat r.h.s as constants) and one gets:

Line nr.
1               (1 ........................)           ...................
2               (0 1 ......................)           ...................
...             (                          )
...             (                          )  * t  =
...             (                          )
101             (0 0 0 .................. 1)           ...................
--------------------------------------------------------------------------
102                                             0   =  ...................
..
...
...
121                                             0   =   ..................

The last 20 lines are equations in 21 variables (a1...a10,z1...z11) and
nonlinear. The right hand side of this part reads as:

        a1 * M1 * z + a2 * M2 * z + ... + a10 * M10 * z + M11 * z,

where a1..a10 are the input-unknown scalars, z is the input-unknown (11x1)
vector and M1...M11 are (20x11) KNOWN matrices.

This homogenous nonlinear system of 20 equations in 21 variables could be
solved for instance with CA-Systems (maple :kernel. Mathematica: NullSpace,
which has a bug and will most probably not do it. If there is a need: I
have written my own version, which is quit fast!) or with other methods.

The rest is not too hard: substitute these results on the left hand side
of equations 1 ... 101 und solve a linear right upper triangular system.

Gruetzi, Kurt.

P.S.: If one is only interested in ONE solution, one could try t101=0
and then solve 21 eqns with 21 unknowns ...