KINETICS AND MECHANISM OF ISOMERIZATION OF 5-CHOLESTEN-3-ONE
41
model of the reaction (multivariate modeling kinetic).
We performed the treatment by kinetic methodologies
and later numerical computational analysis using a
multivariate nonlinear regression technique, based in
all cases on the use of the AGDC (controlled descent
general algorithm) mathematical unconstrained opti-
mization algorithm. It is a robust algorithm that can
be used for such purposes, both in a regression tech-
nique and in any other different ones as long as the
aim is to search for the minimum of a function in
the hyperspace defined by the parameters to be opti-
mized. This package of calculation programs has been
assigned the generic name of ANALKIN (AGDC) [4]
because it is applied to chemical systems for ANALytic
purposes, via the use of KINetic techniques, using the
AGDC mathematical unconstrained optimization algo-
rithm. The generic expression of a DMM according to
the IUPAC recommendations [19] will be
the rate constants, occasionally leading to “stiff” prob-
lems. The proposed treatment uses the Gear algorithm
to solve the sets of differential equations and affords
excellent results even in the case of complex systems
with notable “stiff” characteristics. Next, the actual op-
timization process is to begin; this is carried out by
application of the AGDC algorithm. It consists of the
minimization of SQD through the development of an
iterative process in which the vector of movement is de-
termined and at all times is subject to strict control. A
rigorous analysis is made of its elements, being suitably
corrected in the event of detecting any errors, thereby
ensuring that the minimum will be reached.
ANALKIN (AGDC) can be represented schemati-
cally, step by step, thus:
1. Generate the model
1.1 Input data (matrix of νj,r number of reactions,
species, experimental data, vector of εj , etc.
1.2 Input of the initial estimates of the concentra-
ꢁ
0 =
νj,r Bj r = [1, Nr ]
(15)
tions of the species to optimize ([Bj,i ]), X(m)
Let m = 1 (m = number of the iteration).
.
j
where the stoichiometric coefficients νj,r are less than 0
for the chemical species acting as reagents and νj,r > 0
for those acting as products in the rth reaction con-
sidered. The differential rate equation for the reactant
Bj,i is
2. Establishtheratedifferentialequationsystemandits
solution (the Gear algorithm) [28] obtaining [Bj,i
3. Calculate the absorbance (Ai )cal
]
cal
4. Determinate of the SQD(m) function (Eq. (17)).
5. AGDC algorithm
5.1. Computepartialnumericalderivativesof(Ai )cal
with respect to the parameters to be determined
X(m), by the method of central-differences
5.2. Compute the g(m) and H(m) (gradient vector and
Hessian matrix)
−d[Bj,i ]/dt = kr |νBj,r |ꢀj [Bj,i ]|νBj,r
|
(16)
that represents a set of Nr differential equations com-
posed of as many equations as species whose con-
centrations we might wish to know. The mathemati-
cal optimization consisting of the minimization of the
numerical function of the sum of quadratic deviations
(SQD), extended for a Nd pair of data and Nc species,
is given by the developed expression
5.3. Compute [H(m) −1
by the Gauss elimination
]
method and improvement by successive ap-
proximations method.
5.4. Calculate the components of the vector of
movement ( p(m) = −(H(m) −1
)
g(m)
)
5.5. Control and correction of the vector of move-
ꢂ
ꢃ
2
Nd
Nc
ꢁ
ꢁ
ment p(m)
SQD =
εj [Bj,i ] − (Aj,i
)
(17)
exp
5.5.1 Direction of p(m)
i=1
j=1
5.5.1.1 If H(m) is singular, set p(m)
−g(m), and go to 5.5.2
=
where
5.5.1.2 If p(m) g(m) < ε(ε = scalar close
to zero), set p(m) = −g(m) and go
to 5.5.2.
Nc
ꢁ
(Ai )cal
=
εj [Bj,i ]
(18)
j=1
5.5.1.3 If p(m) g(m) > 0 set p(m) = −p(m)
(Aj,i
)
exp
represents the experimental values of the
5.5.2. Length of p(m)
monitored total absorbance, Aj,i,ꢂ represents the ab-
sorbance of each of the j species present in the mix-
ture at time I, [Bj,i ] represents the concentrations of the
Nc species, and εj is the molar absorption coefficient
of each species. The solution of this type of differen-
tial equation is sometimes difficult, owing both to the
characteristics of the systems and to the values taken by
5.5.2.1 Compute the scalar (α(m)) by the
method of Hartley
5.5.2.2 X(m + l) = X(m) + α(m) p(m) ([Bj,i
]
optimized concentrations)
5.5.2.3 If the Goldstein–Armijo criterion
is satisfied go to 5.6.
5.5.2.4 α(m) = α(m)/2 and go to 5.5.2.2