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##

Function involving determinants

A determinant of a matrix `A` may appear in an equation using
one of the syntax:

Fast_Determinant(A) Medium_Determinant(A) Slow_Determinant(A)

the differences between the syntax being only in the computation
time (from the fastest to the slowest) and in the width of the
interval evaluation (from the largest to the narrowest).
If you are interested in determining if a determinant may cancel
you may also use `Slow_NonZero_Determinant` which is faster than
`Slow_Determinant`.
Note that if determinants are present
in the equations there are different ways to compute the
determinant:

- by using Gaussian elimination. This method will be used only if
the flag
``ALIAS/use_gaussian_elim_det`` is set to 1
- either by computing only the intervals for each component
of the matrix and using a row or column expansion to determine the
interval evaluation of the determinant
- same as above but the gradient of the coefficients are used to
improve their interval evaluation
- by computing symbolically every minors of dimension of the
matrix and using the interval evaluation of these expressions to
compute the interval evaluation of the determinant. The value of
is given by
``ALIAS/minor22` (default value: 2)
- a mix of method 2 and 3

or pre-computing an interval expression for all the
dimension 2 minors (thereby using, for example, either the
`Fast_Determinant` or the `Fast_Determinant22` procedures. The
second procedure will, in general,
lead to better interval evaluation but Maple may take some time to
produce the corresponding source code.
You may choose one of these
ways by setting the flag
``ALIAS/det22``
to 0, 1, 2 or 3 (default value:
0). Clearly the values 2 and 3 should be avoided for large matrix as
the Maple computation time and the size of the generated code may be
large. If the determinant of minors is used, then the
expansion will be done according either to the row or to the column
according to the value of
``ALIAS/row22``. A value of 0 (the default value means that the
expansion will done along the rows and for any other value according
to the column).
Note that you may speed up the interval evaluation of functions including
determinant of matrices (which are often computer intensive) by
defining intermediate interval variables and substituting these
variables in the expression. For example assume that the coefficients
of a matrix `A` involve a large number of sine and cosine depending upon
the unknown , the first unknown in our list of unknowns:
you first define a Maple procedure `intro_A`
that return an
array of strings that contain all the definition of the intermediate
variables. In our case the procedure is:

intro_A=proc()
local h;
h:=["INTERVAL SX;SX=Sin(v_IS(1));","INTERVAL SY;SY=Cos(v_IS(1));"]
RETURN(h);
end:

Note that the name of the unknowns in the code generated by `ALIAS` is `v_IS`.
The two strings of the array `h` will be automatically inserted in
the C++ procedure generated by `ALIAS`. Hence the intermediate
variable `SX`, `SY` will contain the value of the interval
evaluation of
. You may then write the simplification
procedure that will substitute every occurrence of `Sin(v_IS(1))`
and `Cos(v_IS(1))` by `SX` and `CX`. The name of this
procedure must be `simplify_A`
and is written as:
simplify_A:=proc(eq)
local eq1;
eq1:=subs(Sin(v_IS(1))=SX,Cos(v_IS(1)=CX,eq):
RETURN(eq1);
end:

** Next:** Dealing with undefined expressions
** Up:** Equations, Gradient and Hessian
** Previous:** Improving the efficiency of
** Contents**
Jean-Pierre Merlet
2012-12-20