COR RESPONDlENCE D E T E R M I N A T I O N OF BEST-FIT RATE CONSTANTS IN C H E M I C A L K I N E T I C S SIR: I n a recent paper, Ball and Groenweghe (7) presented a method for kinetic rate constant evaluation based on Marquardt's algorithm ( 2 ) . Their Equation l is:
(PTI?
+ XD)d = P T e
where P = the matrix of partial derivatives, (3fi/db,) e . = Yi - fi Y i = actual value of variable f . = value of variable calculated by model This equation is different from Marquardt's original formulation. H e develops the: Taylor series correction as:
Since Ball and Groenweghe use D (the diagonal of the PTP matrix) instead of a unit matrix, their formulation is different from Marquardt, although not necessarily incorrect. I would like to ask if this procedure has been compared to the original formulation and found to be superior. Marquardt transforms the PTP matrix and the PTe matrix by a scaling procedure designed to make the gradient correction scale invariant. I n this process the diagonal elements become unity. If Ball and Groenweghe use this scaling procedure the methods a r e effectively identical. Could the authors offer clarification on this point?
(PTP)tjT = PTe
John D . Wright
and the gradient correction as:
Pace Co. Houston, Tex.
(3, = P T e
Therefore his method of interpolation between these two corrections is: (PI') AI)d = PTe with X as the weighting factor for the interpolation.
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SIR: Wright raises a n interesting point concerning the Marquardt technique (;?). Letting A = P T P , as in Marquardt's original paper, ithen the Marquardt algorithm for the rth iteration is based on: (A*(')
+ X(')I)8*(') = g*(')
literature Cited
( 1 ) Ball, W. E., Groenweghe, L. C. D., IND.ENG.CHEM.FUNDAMENTALS 5 , 181 (1966). (2) Marquardt, D. W., J . SOC. Ind. Appl. Math. 11, 431 (1963).
After simplification these equations become : (all
+ Auld61 +
a12&
+ ... +
= gl
(Marquardt Equation 31)
Hence we obtain the equation presented in our paper ( 7 )
(dG
8* = (6,*) = Si) Written out in component form, this scaled equation becomes : (all*
+ X)61* +
akl*61*
a12:'62*
+
,
..
+
alk*6k* =
+ ak?*a2* + . . . + (akk* + A)&*
gl*
=
gk*
Substituting for the scaled variables in terms of the original variables yields :
This unscaled form of the iteration equation has thus been shown to be equivalent to the original formulation using scaled variables. We tried both the scaled and unscaled forms of the algorithm in solving a number of sample problems. Identical numerical results were always obtained. The kinetics problem described in our paper was solved using the unscaled form to save the small amount of time necessary to scale the correction equations. As a n additional item of interest, E. M . Rosen, Monsanto Co., investigated the effect of forward difference derivative approximations on the convergence rates of the Marquardt algorithm. T h e problem solved was: Determine the best estimates? in the least squares sense, of the function: y i = crl exp(cusxi) a3 exp(aaxi). Nine data points were used in the experiment. In addition, the cy1 variables were restricted to not changing sign and the numerical derivatives were based on a step size of 0.170 of the value of the independent variable.
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al,a2,as,ad,in
VOL. 6
NO. 3
AUGUST
1967
475
