Transforming differential equations
Using a matrix to transform the differential equations
It frequently happens that a differential system can be written
in terms of y(i) (i = 1, 2, ..., n) but the user is only able
to measure, or is only interested in, some sub-set of the y(i).
This is easily achieved interactively by selecting the i-values
of special interest.
However the user might wish to simulate or
fit linear combinations of the y(i).
For instance, in a chemical
scheme such as A->B->C->D it might only be possible to measure
B + C.
More complications are possible, for instance if B and C
had different molar absorbtivities, or fluorescence parameters,
then something like 0.75*B + 1.12*C might be the combination of
interest.
This program allows you to input a real square matrix A which
then overwrites a sub-matrix of the identity matrix to redefine
variables using: y(new) = A*y(old).
For instance:
A = 0 1
1 0
would just interchange y(1) and y(2).
Consult the readme files and files deqmat.tf? to see how to do
this.
Note that whatever the value of A, the initial y0(i) will
always be the actual y0(i), so be careful when curve-fitting.
Some important technical details
Irrespective of what sub-set of y(i) are selected or whatever matrix A
is used for transformation, the program always integrates all of
the y(i) from y0(i) on.
If A is not the identity, y(i) selected
will be transformed y(i). Using selected sub-sets restricts the
output to components of interest when the number of equations is large.
Once an A has been used or a sub-set selected they remain until
changed interactively.
The range and no. of points are as set by
the user by menu or input file unless a simulation is requested
as an overlay over curve-fitting data, when the range is set by
the data (temporarily), but the number of points used for best-fit
curves will still be as selected for simulation.
If the problem you are trying to solve is ill-conditioned, then
an error message (with the IFAIL value etc.) will be output, so
you can change tolerances etc. and try again with more suitable
settings.
This procedure is turned off during curve-fitting, to
avoid screen after screen of error messages.
So, before fitting,
you should integrate and overlay the simulation on the data, to
be sure curve-fitting is sensible.
As a check after fitting, in some versions of DEQSOL, the
error detection is turned on for a final integration.
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