Computation of Kinetics of Chemical Systems with Many Species
83
and the solutions can be approximated by the solutions of a linear system
which can be written as a sum of exponentials (a fact much used and abused
in metabolism studies). Perturbation methods can also be used.
Computation of the stead\ state means finding the solution of equations
(1) and (2) with );";
O. Alternatively one may use the equilibrium equations of the s\'stem which are of the form
i= 1, .. . N;
(4)
where the pij are integers between - 2 and 2, K; the equilibrium constants.
(In some systems an accurate description is more complicated.) In both
cases we have a system of simultaneous non-linc:ar equations of many
variables. Linear equations pose a challenge to computers when the number
of variables is large (> 100) and the upper limit of today's technology
seems to be of the order of 1000. Non-linear equations are more difficult.
Methods found in the literature are sometimes impractical except for small
n. H. ZASSENHAUS pointed out in his lecture at the Annual Meeting of the
German Mathematical Society at Karlsruhe, Sept. 14, 67, that the most
efficient computational methods (for many tasks) must be determined by
experimentation and through competitive comparison. At present little
such comparison of methods developed by different groups takes place due
to lack of communication.
In cooperation with J. and N. GOGUEN this author has recently begun
experimentation with a variety of computational procedures for equilibrium
computation, Let the left hand terms of the equations (4) and (2) be denoted
M
by (/J;, let ,11- t!J
LV and let f = 2.: (/J~ (Xl' ... X n)· The point at which
; ~,
I = 0 is a minimum of/and is a solution of (4) with balance equations (2).
The solution of equations (4) and (2) is thus reduced to optimizing the
function f(X 1 , ••• "\'n). ;\NTHOKY and HIMylELBLAU [1] have followed a
similar approach except that the\, used a different method (by HOOKE and
]EEVES [14]) to minimize f To optimize I we tried to solve Ix; = 0,
i = 1, ... M, where /x; is the partial derivative off with respect to X;.
NEWTON'S method for several variables may be used for this purpose
(Comp. [11] Sect. 25, and [13]). The method is computationally rather cumbersome for large M since it involves at each iteration the inversion of an
AI by M matrix of the second partials of f. Instead we have tried to use
NEWTON'S method in each variable separately. In the proximity of the
solution good convergence was obtained while with a "bad initial guess"
divergence occurred in some cases. Previous experimentation with optimization procedures by the author (BREMERMANN, ROGSON, and SALAFF [4],
[5 J, [3]) has led to a method of "optimization along random rays" (unpublished) which can be used alone, or in order to provide an "initial guess" close
6*
83
and the solutions can be approximated by the solutions of a linear system
which can be written as a sum of exponentials (a fact much used and abused
in metabolism studies). Perturbation methods can also be used.
Computation of the stead\ state means finding the solution of equations
(1) and (2) with );";
O. Alternatively one may use the equilibrium equations of the s\'stem which are of the form
i= 1, .. . N;
(4)
where the pij are integers between - 2 and 2, K; the equilibrium constants.
(In some systems an accurate description is more complicated.) In both
cases we have a system of simultaneous non-linc:ar equations of many
variables. Linear equations pose a challenge to computers when the number
of variables is large (> 100) and the upper limit of today's technology
seems to be of the order of 1000. Non-linear equations are more difficult.
Methods found in the literature are sometimes impractical except for small
n. H. ZASSENHAUS pointed out in his lecture at the Annual Meeting of the
German Mathematical Society at Karlsruhe, Sept. 14, 67, that the most
efficient computational methods (for many tasks) must be determined by
experimentation and through competitive comparison. At present little
such comparison of methods developed by different groups takes place due
to lack of communication.
In cooperation with J. and N. GOGUEN this author has recently begun
experimentation with a variety of computational procedures for equilibrium
computation, Let the left hand terms of the equations (4) and (2) be denoted
M
by (/J;, let ,11- t!J
LV and let f = 2.: (/J~ (Xl' ... X n)· The point at which
; ~,
I = 0 is a minimum of/and is a solution of (4) with balance equations (2).
The solution of equations (4) and (2) is thus reduced to optimizing the
function f(X 1 , ••• "\'n). ;\NTHOKY and HIMylELBLAU [1] have followed a
similar approach except that the\, used a different method (by HOOKE and
]EEVES [14]) to minimize f To optimize I we tried to solve Ix; = 0,
i = 1, ... M, where /x; is the partial derivative off with respect to X;.
NEWTON'S method for several variables may be used for this purpose
(Comp. [11] Sect. 25, and [13]). The method is computationally rather cumbersome for large M since it involves at each iteration the inversion of an
AI by M matrix of the second partials of f. Instead we have tried to use
NEWTON'S method in each variable separately. In the proximity of the
solution good convergence was obtained while with a "bad initial guess"
divergence occurred in some cases. Previous experimentation with optimization procedures by the author (BREMERMANN, ROGSON, and SALAFF [4],
[5 J, [3]) has led to a method of "optimization along random rays" (unpublished) which can be used alone, or in order to provide an "initial guess" close
6*