84
H. J. BREMERMANN
enough for the modified NEWTON'S method to converge. The latter turned
out to be very effective in a case of linear ~i-S, derived from a system of
linear equations in 64 variables with condition number 2. A solution
accurate to 13 decimal places was obtained in 10 sec of CDC 6400 time.
The program was written in FORTRAN IV. In some experiments the balance equations (2) were not included in the ~i but were solved to reduce the
number of independent variables. This approach, however, was not very
successful. Convergence difficulties occurred and the expressions for the
derivatives off became cumbersome.
Experimentation has been in progress only for a short time. It is hoped
that it will lead to efficient programs that can solve equations (4) as well
as (1) up to M = 100 and greater. A different approach based on minimizing a free energy function rather than jhas been pursued by SHAPIRO [17 a].
Our optimization method could be applied to the free energy function as
well.
References
1. ANTHONY, R. G., and D. M. HIMMELBLAU: J. Phys. Chern. 67, 1080 (1963).
2. BARD, A. J.: Chemical Equilibrium. New York: Harper and Row 1966.
3. BREMERMANN, H. J., and H. DE GRASSE: Logistics Review 3,15 (1967).
4. -, M. ROGSON and S. SALAFF: In: E. A. EDELSACK, L. FEIN, A. B. PATTEE
and A. B. CALLAHAN (Eds.), Fundamental Biological Models. Washington: Spartan 1965.
5. - In: H. L. OESTREICHER and D. R. MOORE (Eds.), Cybernetic Problems in
Bionics. New York: Gordon and Breach 1968.
6. CLASEN, R. J.: The numerical solution of the chemical equilibrium problem,
RAND Corporation (Santa Monica, Calif.) Report RM-4345-RP Jan. 1965.
7. DANTZIG, G. B, J. FOLKMAN and N. SHAPIRO: J. Math. Analysis and Appl. 17,
519 (1967).
7a. -, J. C. DE HAVEN, I. COOPER, S. M. JOHNSON, E. C. DE LAND, H. E. KANTER
and C. F. SAMS: Persp. BioI. Med. 4, 324 (1961).
8. DE TAR, D. F.: J. Chern. Educ. 44, 191 (1967).
9. - J. Chern. Educ. 44, 193 (1967).
10. - and D. E. DE TAR: J. Phys. Chern. 70,3842 (1966).
11. FORSYTHE, G., and C. B. MOLER: Computer Solutions of Linear Algebraic
Systems. New Jersey: Prentice-Hall, Englewood Cliffs 1967.
12. HEINMETS, F.: Analysis of Normal and Abnormal Cell Growth. New York:
Plenum Press 1966.
13. HENRICI, P.: Elements of Numerical Analysis. New York: Wiley 1964.
14. HOOKE, R., and T. A. JEEVES: J. Assn. Compo Mach. 8,212 (1961).
15. GARFINKEL, D.: J. BioI. Chern. 241, 3918 (1966).
16. - J. Theoret. BioI. 14, 46 (1967).
16a. LICKO, V.: Bull. Math. Biophys. 25, 141 (1963).
16b. - Bull. Math. Biophys. 28, 379 (1966).
17. ROSENLICHT, M.: Pac. J. Math. early 24,153 (1968).
17 a. SHAPIRO, N. Z.: A generalized technique for eliminating species in complex
chemical equilibrium calculations. RAND Corporation (Santa Monica,
Calif.) Report RM-4205-PR, Sept. 1964.
18. WATSON, J.: Molecular Biology of the Gene. New York: Benjamin 1966.
H. J. BREMERMANN
enough for the modified NEWTON'S method to converge. The latter turned
out to be very effective in a case of linear ~i-S, derived from a system of
linear equations in 64 variables with condition number 2. A solution
accurate to 13 decimal places was obtained in 10 sec of CDC 6400 time.
The program was written in FORTRAN IV. In some experiments the balance equations (2) were not included in the ~i but were solved to reduce the
number of independent variables. This approach, however, was not very
successful. Convergence difficulties occurred and the expressions for the
derivatives off became cumbersome.
Experimentation has been in progress only for a short time. It is hoped
that it will lead to efficient programs that can solve equations (4) as well
as (1) up to M = 100 and greater. A different approach based on minimizing a free energy function rather than jhas been pursued by SHAPIRO [17 a].
Our optimization method could be applied to the free energy function as
well.
References
1. ANTHONY, R. G., and D. M. HIMMELBLAU: J. Phys. Chern. 67, 1080 (1963).
2. BARD, A. J.: Chemical Equilibrium. New York: Harper and Row 1966.
3. BREMERMANN, H. J., and H. DE GRASSE: Logistics Review 3,15 (1967).
4. -, M. ROGSON and S. SALAFF: In: E. A. EDELSACK, L. FEIN, A. B. PATTEE
and A. B. CALLAHAN (Eds.), Fundamental Biological Models. Washington: Spartan 1965.
5. - In: H. L. OESTREICHER and D. R. MOORE (Eds.), Cybernetic Problems in
Bionics. New York: Gordon and Breach 1968.
6. CLASEN, R. J.: The numerical solution of the chemical equilibrium problem,
RAND Corporation (Santa Monica, Calif.) Report RM-4345-RP Jan. 1965.
7. DANTZIG, G. B, J. FOLKMAN and N. SHAPIRO: J. Math. Analysis and Appl. 17,
519 (1967).
7a. -, J. C. DE HAVEN, I. COOPER, S. M. JOHNSON, E. C. DE LAND, H. E. KANTER
and C. F. SAMS: Persp. BioI. Med. 4, 324 (1961).
8. DE TAR, D. F.: J. Chern. Educ. 44, 191 (1967).
9. - J. Chern. Educ. 44, 193 (1967).
10. - and D. E. DE TAR: J. Phys. Chern. 70,3842 (1966).
11. FORSYTHE, G., and C. B. MOLER: Computer Solutions of Linear Algebraic
Systems. New Jersey: Prentice-Hall, Englewood Cliffs 1967.
12. HEINMETS, F.: Analysis of Normal and Abnormal Cell Growth. New York:
Plenum Press 1966.
13. HENRICI, P.: Elements of Numerical Analysis. New York: Wiley 1964.
14. HOOKE, R., and T. A. JEEVES: J. Assn. Compo Mach. 8,212 (1961).
15. GARFINKEL, D.: J. BioI. Chern. 241, 3918 (1966).
16. - J. Theoret. BioI. 14, 46 (1967).
16a. LICKO, V.: Bull. Math. Biophys. 25, 141 (1963).
16b. - Bull. Math. Biophys. 28, 379 (1966).
17. ROSENLICHT, M.: Pac. J. Math. early 24,153 (1968).
17 a. SHAPIRO, N. Z.: A generalized technique for eliminating species in complex
chemical equilibrium calculations. RAND Corporation (Santa Monica,
Calif.) Report RM-4205-PR, Sept. 1964.
18. WATSON, J.: Molecular Biology of the Gene. New York: Benjamin 1966.