A Computational Algorithm for the Greens Function

8/3/2019 A Computational Algorithm for the Greens Function

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A Computational Algorithm for theGreens Function Method of SensitivityAnalysis in Chemical Kinetics

EUGENE P. DOUGHERTY* and HERSCHEL RABITZIChemis t ry Depar tment , Pr ince ton Uniuers i t y ,Pr inceton, New Jersey 08540

AbstractThe recent interest in numerical modeling of chemical kinetics has generated the need forproper analysis of the system sensitivities in such models. This paper describes the logic fora program developed by the authors to implement the Greens function method of sensitivityanalysis in complex kinetic schemes. The relevant equations and numerical details of thealgorithm are outlined, two flow charts are provided, and some special programming con-siderations are discussed in some detail. Computer storage and computational time con-siderations are also treated. Finally, applications of sensitivity information to understandingcomplex kinetic system behavior and analyzing experimental results are suggested.

IntroductionThe recent interest in the numerical solution of the differential equations

of chemical kinetics [1,2] has brought about considerable research in thearea of sensitivity analysis of reaction mechanisms [3-lo]. Sensitivityanalysis of the coupled differential equations of reaction kinetics involvesdetermining the sensitivity of output concentrations at various times withrespect to the input parameters of the model (e.g., the rate constant pa-rameters and initial species concentrations). The Greens function method(GFM) has already been shown [9,10]to be a very promising approach tothe sensitivity analysis of large kinetic models. The present paper outlinesthe methods and logic for an algorithm which implements the GFM to studycomplex kinetic schemes. It is expected that extensions of this technologyshould be applicable in areas other than chemical kinetics such as quantumdynamics [Ill,chemical engineering [12,13],and world models for resourcesand population [14,15]. This paper is organized as follows: The secondsection outlines the theoretical and numerical considerations relevant tothe GFM algorithm. Two flow charts are provided to allow the potential

*Present address: Research Laboratories, Rohm and Haas Company, Spring House, PAt Alfred P. Sloan Fellow and Camille and Henry Dreyfus Teacher-Scholar.19477.

International Journal of Chemical Kinetics, Vol. X I, 1237-1248 (1979)8 1979 John Wiley & Sons, Inc. 0538-8066/79/0011 1237$01.00


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1238 DOUGHERTY AND RABITZuser to construct his own sensitivity code. Th e third section treats somespecial programming considerations and procedures necessary for a properanalysis of sensitivities. The fourth section considers potential applicationsof sensitivity analysis for understanding and predicting system behaviorand for treating experimental results. The concluding section of the paperpresents a few suggestions for possible improvements to the algorithmproposed in this paper.

Theory and Numerical ConsiderationsHere we summarize the basic equations and discuss some relevant nu-

merical considerations for implementation of the GFM to complex kineticschemes. Although some of these details have already been treated in twoearlier papers [9,10],a brief discussion is given here so that the programlogic may be more easily understood by the reader.

The coupled first-order ordinary differential equations (ODES) de-scribing chemical kinetic systems are of the following general form:(1) Y = f(Y,cy,t) , Y(0)= YO)Here y is an n vector of species concentrations, cy is an m vector of inputparameters, and the dot indicates differentiation with respect to time. Foreach species, i is determined by the law of mass action in chemical kinetics[16], which defines the rate of reaction in terms of the concentrations of itsreactants; f i is a polynomial up to third order in the species concentrationsy. Equation (1) s generally numerically stiff [17,18],and should be dealtwith by special software designed to treat stiff ODES [19].

The first-order sensitivity coefficient is defined as the partial derivativeof the ith species concentration with respect to the j t h input parameterbyilbaj. By differentiating eq. (1)with respect to aj, we obtain

where the matrix elements of the n X n Jacobian matrix J are given by(3 )and the notation(4) b

dXZ,(t) =- ( t )

is used. The form of the Jacobian is known analytically once the reactionmodel is specified. The initial condition y$) is the zero vector, unless ajis the initial concentration of the i th species concentration, in which casey$) = 6i. The symbol 6i is a vector whose components are all zero except


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S ENS I TI VI TY AND ANALYS I S 1239the ith component, which has the value 1. An n X n Greens functionmatrix K(t ,7) can be constructed for eq. (2) which satisfies

d- ( t , 7 )- J ( t ) K ( t , 7 )= 0, t > Td t5a)(5b) K ( t , 7 ) = 1It can be shown [9] that y,(t) is given by

In practice K(t ,7) is computed by first solving eq. (1) o obtain the con-centration at an adequate number of grid points so that simple interpolationformulas can be employed to calculate a sufficiently accurate J(t)for usein eq. (5). Suppose th at time t = t l is the first instance where sensitivityinformation is sought. It is more convenient to calculate the adjointGreens function matrix Kt(7,t) [9], and the differential equations for eachrow of this matrix are solved by proceeding backward in time from t 1 to zero.The relevant initial value problem is

d- f 7 , t l )+ k{ 7 , t l )J(T)= 0, 7 < tld7(7b) kf (tlJ1) = 6ilwhere kf, row vector, refers to the lth row of the adjoint Greens functionmatrix. This row corresponds to the lth species concentration. Equation(7), like eq. ( l ) ,s generally stiff; a special ODE solver is then required forefficient solution. Linear interpolation of logy is used to construct J(7).Since 191(8) K f ( 7 , t ) = K ( t , 7 )we may construct the integrand of eq. (6) for the Zth species for all m inputparameters by a simple row-column vector multiplication. For the j t hinput parameter the integrand is(9 ) ILj(7, t l ) = kf (7,tl) * f a j ( 7 )The integrands may be constructed a t the same time as kf s being solvedon [0, t l ]by evaluating faj(7). To obtain the sensitivity integrals, we usethe step sizes (or some portion of them) taken automatically by the ODEsolver in treating eq. (7). Since the behavior of the integrands is mainlydictated by the behavior of kf 7 , l ) , hese step sizes taken by the solverare used as a guide in constructing the integration grid. Every IJUMPthstep is taken to ensure accuracy. Generally, IJUMP = 5 or 6 was foundto be sufficient. By using the solvers own mesh points to construct thisgrid, we are able to deal with even rapidly varying integrands which result


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1240 DOUGHERTY AND RABI TZfrom the stiff nature of eq. (7). To carry out the actual numerical inte-gration, either one of two possible quadrature schemes was used. Forrapidly varying integrals, locally exponential behavior is assumed to holdbetween two adjacent grid points 7 1 and 72 ,

where the constants Alj and Rlj are determined by Ilj(7-1)and Ilj(72). Forslowly varying integrals and for the case in which the integrand changessign or goes to zero, a simple trapezoid rule was used. The value of theentire sensitivity integral is obtained by summing up the individual integralcontributions from each pair of adjacent grid points. By the above pro-cedure the sensitivity of the Ith concentration to all m rate constant pa-rameters can be obtained simultaneously with the solution of kf n [ t i ,O ] .The same procedure is repeated for the other n-1 rows.To obtain sensitivity information a t times longer than t l , he recursionformula [9] is used

Thus it is necessary to compute Kt(7,t) only on short 7 intervals [ t k , O ] ,[ t 2 ,t I], etc., rather than repeatedly on the longer intervals [tk,O]. Altogethern + 1 sets of n-vector stiff ODESare solved and mN integrals must beevaluated. Here N is the number of times sensitivies are to be computedand m is the number of rate constant parameters. For the m initialcondition parameters, no integrals need to be evaluated in eq. (6) since inthis case fa, = 0.As one can see from the preceding section, evaluation of sensitivitiesinvolves reasonably straightforward programming. Simple interpolation,numerical integration, indexing, matrix arithmetic, and numerical solutionof stiff ordinary differential equations are required. All are standardprocedures in numerical analysis. To give the contents of this section ina more concise form, we have included two somewhat detailed flow charts(Figs.1and 2). The first gives a general outline of the sensitivity code. Thesecond shows how the sensitivity integrals are actually evaluated in thecourse of solving eq. (7). The notation corresponds to tha t used in thissection.

Some Special Programming Considerations for SensitivityAnalysis of Kinetic ModelsIn developing the equations and considering the relevant numerical as-

pects in the previous section, we gave a brief overview of what is involvedin determining sensitivities for reaction mechanisms. In this section we


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S E N S I T I V I T Y A N D A N A L Y S I S 1 2 4 1

Read data: n(No. o f s p ec i e s ) , m ' ( N o . of r a t e c o n s t a n t s p a r a m e t e rs ) ,m"(No. of i n i t i a l c o n c e n t r a t i o n p a r am e t er s ), m=m '+m" , ra te c o n s t a n t s ,i n i t i a l c o n c en t r at i o ns , e r r o r t o le r a n c e, i n t e g r a t i o n o pt i on , IJUMP

S ol ve d i f f e r e n t i a l e q u a t i on s f o r c o n c e n t r a t i o n s by c a l l i n g ODE s o l v e r .S o l ve r e v a l u a t e s a nd 2 a s needed. Save y a t a g r i d of t i m e s f o r f u t u ri n t e r p o l a t i o n . C h e c k mass c o n s e r v a t i o n p e r i o d i c a l l y .

Loop over No. of t i m e s s e n s i t i v i t i e s are t o be computed , i . et l , ..., r , ..., N . ( t o = 0 )

9.=1,. . n~ ~ ~~~~ ~~~

I n t e r p o l a t e t o f i n d y ( t r ) . Compute a j / y t ( t r ) . j = l ,...,n or ma li ze d s e n s i t i v i t i e s l a t e r . E va lu at e f ( t r ) , j=l,I -cL:

l c he ck s e n s i t i v i t y mass c o n se r va t io n r e l a t i o n s 1

Figure 1. Flow chart for the Green's function method of sensitivity analysis ofchemical reaction mechanisms.

amplify some of these latter points to show how to carry out a thorough andefficient determination of sensitivities for problems of chemical interest.We consider a few preliminaries necessary to set up the study, certainprogramming hints that should be useful, and estimates of programmingeffort in terms of execution time and storage requirements. The first stepin kinetic modeling is to build a mechanism suitable to the system of in-terest. Naturally a thorough literature search is required to become fa-miliar with the expected system behavior and to obtain rate constant pa-rameters for the elementary steps involved. Information from thermo-dynamics as well as kinetics is useful in building the reaction set and de-termining the best rate constants. Estimates are often necessary [20],andappropriate caution should be used here. A conservative approach is de-sirable to ensure that no important step is omitted, since sensitivity analysiscan be useful in refining the model, as discussed below.

The second step involves translating the reaction mechanism into an


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1242 D O U G H E R T Y A N D R A B I T ZOD E s o l v er t a ke s f i r s t s t e p . I n i t i a l i z e i n t e g r a l s t o 0 .

S e t T 2 = t r ( ti me t h a t s e n s i t i v i t i e s a r e d e s ir e d ) .

I n i t i a l i z e i nt e gr a nd I (T2)= - , ( t T ) , j = l , ..., '= c u r r e n t ti me . I n t e r p o l a t e ( i f n e c es s a ry ) t o g e t Y( T )Compute J a 2 ( ~ l )=1,. .. m' +---

.LLoop over r a t e cons t an t

i n t e g r a t i o n

Add r e s u l t of t h i s p i e c e t o o l d v al u eof i n t e g r a l . S t o r e new i n t e g r a l .

4.~

ODE s o l v e r t a k e s ne xt s t e pS t e p i s c h o s e n s o t h a tand normalized 'tr-l ) ime . r eached does no t ge ts m a l l e r t h a n t r - l . In -c r emen t a coun te r a t eachs t e p t a k e n .

s e n s i t i v i t y c o e f f i c i e n t sa t t r f o r m ' r a t e con-s t a n t s a n d m" i n i t i a ls p e c i e s c o n c e n t r a t i o n s .Use r ecu r s ion f o r mula

Figure 2. Flow chart to illustrate computation of sensitivity coefficients in thecourse of solving k t ( T , t r ) , T c(t,-l, h).

appropriate code to compute concentrations and sensitivities. For smallproblems the algebra involved is trivial and can be performed by the mo-deler. Otherwise an automated precompiler [21] should be utilized. Anefficient method must be used to construct the algebraic formulas involvingy necessary for evaluating the species derivatives on the right-hand sideof eq. (I). For treating both eqs. (1)and (7), an analytical expression forthe Jacobian matrix [see eq. (3)] is desirable. Finally, it is desirable toconstruct the fa, array analytically as well. This involves simple differ-


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SENSITIVITY AND ANALYSIS 1243entiation, yet can become tedious for larger systems, so an automatedprocedure is clearly useful.

The third step is to carry out test calculations of y(t) and a few sensi-tivities y , ( t ) o become familiar with the system of interest. Test calcu-lations are used to assign proper stepsizes, tolerances, the interpolation gridof y, etc. The times where concentrations are changing rapidly, the timeswhere sensitivity information is most desirable, and the behavior of a fewsensitivity integrands should particularly be checked. From the resultsof these preliminary calculations, the quadrature scheme, the interpolationprocedure, and tolerances can be chosen for an efficient and accurate studyof the kinetics.A few additional programming hints are also useful. It was found to bemost convenient to divide the program into several modular units. Firstthere is a solver package for treating both stiff and nonstiff ODES. Forsensitivity analysis the authors recommend a package written by Hind-marsch [ 22 ]which is a revision of an earlier code by Gear [23]. Besides thesolver package, it is convenient to have additional program modules, eachof which carries out a special task for sensitivity analysis. We have foundit useful to have separate modules to read in the rate constants and otherparameters, set up the mechanism and rate constant arrays, perform thenumerical integration, carry out interpolation, and evaluate the arrays f ,J, and fB,.

Computational times for both linear and nonlinear kinetic systems havebeen determined in previous studies [9,10] and comparisons with othermethods of sensitivity analysis [3,8] indicate the GFM to be a practicalapproach. In ref. 10 concentrations and sensitivities a t a single time weredetermined both by the GFM and the direct method [3-51 for a system with13 species and 24 parameters. The direct method, which solves eqs. (1)and (2) together for each parameter, required 2.8 min of execution time.The GFM required only 24 sec. While the direct method and the GFMcalculate exactly the same time of sensitivity information (i.e., the sensi-tivity coefficientsdy,ldcr,), the Fourier amplitude sensitivity test (FAST)[6-81 computes a slightly different measure of sensitivity. Nevertheless,this earlier study showed that all three approaches predict essentially thesame set of important rate constants. Th e computational time requiredto obtain the same sensitivities by the FAST approach was estimated tobe 1hr. These estimates are for double-precision calculations on an IB M360/91 computer.

In general the computation time required for the GFM should be com-petitive with that of the DM provided that m , the number of input pa-rameters, is equal to or greater than n , he number of species. (For mostkinetic models this is indeed the case.) The reason for the difference incomputational effort is that the direct method requires solution of a set of2tz [see eqs. (1)and (2)] coupled equations for each of the m parameters;


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1244 DOUGHERTY AND RABIrZwhereas the GFM requires only solutions of n + 1sets of n coupled equa-tions plus some extra time for numerical integration. Actually, practicallyall the effort for the GFM is due to solving the relevant ODES. Numericalquadrature is not very time-consuming, as evidenced by the fact that therequired execution time is generally independent of the number of pa-rameters present. A useful rule of thumb is that the Greens-functionapproach for determining all relevant sensitivities a t a single time requiresabout 1.1n times the labor for a single kinetic calculation. Each additionaltime at which sensitivities are computed requires a small fractional increasein computational effort mainly because of additional operations by the ODEsolver to start-up the solution of a new initial-value problem. Althoughthe amount of start-up time is problem-dependent, a typical increase isabout 0.1 n times additional effort for each new time considered. Gener-ally, computation of sensitivities at three or four times is sufficient. TheGFM can be very inexpensive in cases where sensitivities for only a singlespecies at a single time are of interest. Then only tw o sets of n-coupledequations-one from the concentrations and one for the proper row of theadjoint Greens function matrix-must be solved; the effort may even beless than that required for two kinetic calculations!

The following empirical formula is useful for estimating the amount ofstorage required to carry out a sensitivity calculation for a particularsystem:(12) STORAGE(ki1obytes) = 95 + 2.4n + 0.016n2+ 0.024mn + 0.08mThe formula above refers to the FORTRAN program we have developed,which includes the ODE solver package. Special overlaying proceduresor out-of-core matrix handling techniques could be used to reduce thisstorage.

Applications of Sensitivity InformationSensitivity information can be utilized in a variety of different ways tounderstand system behavior. The sensitivity results should be useful not

only in kinetic modeling but in analyzing results of kinetic experiments.Sensitivity information can be applied to study and improve the kineticmodel, to elucidate mechanistic details, to estimate system behavior atdifferent values of the parameters, to optimize species profiles, and to de-termine relevant sensitivities for the design of kinetics experiments. Wenow discuss each of these applications briefly. The reader is referred toanother paper [ lo ] for more extensive discussions.

First, by computing the normalized sensitivity coefficients (i.e.,b 1ny;lbIn aj) the relative importance of the parameters in the model can be as-sessed. Rate constants are often not accurately known and identificationof the rate constant exhibiting the largest sensitivities aids in determining


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SENSITIVITY AND A N A L Y S I S 1245which ones require additional experimental study. If there is a large sen-sitivity to a rate constant whose value is in error, the results of the modelitself may be suspect. On the other hand, if very small normalized sensi-tivities are observed for a particular rate constant parameter, its uncertaintydoes not really matter. Thus sensitivity analysis may be useful in deter-mining the important reactions in the model, and the unimportant onesmay be eliminated.

In identifying the important parameters, sensitivity analysis is also usefulin elucidating mechanistic behavior. Often, the mechanism for a particularprocess is difficult to obtain. Tedious experimentation must be carriedout to confirm or deny certain mechanistic details. Rate constants ex-hibiting the largest normalized sensitivities are inevitably the rate constantsfor the most important elementary reaction steps in the mechanism.Sensitivity analysis is a useful tool for studying complicated reactionmechanisms. In a recent study of a 57-reaction model for the H2-02 system[ 2 4 ] , he thirteen elementary reactions exhibiting the largest normalizedsensitivities for most species of a particularly well-studied pressure tem-perature regime of the system are virtually ident ical to the elementaryreactions of a mechanism [25] deduced by much more tedious analysis.Knowledge of sensitivities to initial species concentrations is useful inidentifying which reactants are most critical in determining species con-centrations at later times. In the design of a particular chemical processinvolving complicated reactions, such knowledge is useful for optimizationand control of important species profiles.

Sensitivity information can be used to estimate system behavior whena parameter value is changed. In a previous paper [ lo ]we have developedan exponential formula which estimates such behavior quite well. For onespeciesy and one parameter a , he formula for the estimated concentrationat the new value a + A is

The success of eq. (13) (and its generalization to m parameters) in pre-dicting the concentrations at new parameter values likely reflects theremnant of exponential behavior which is ubiquitous in chemical kinetics.Equation (13)is useful for obtaining a quick estimate of the effect on a givenspecies concentration when a new value of a particular parameter has beenobtained.The e l e m e n t a r y sensitivity coefficients described up to this point,d y J d a j , can also be employed to obtain so-called derived sensitivitycoefficients, which are useful in various experimentallmodeling situations.For example, derived sensitivities are applicable when it is more appropriateto consider a rate constant hi as an output quantity and a species concen-tration y~ as an input parameter in the model. This is the case when a rate


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1246 DOUGHERTY AND RABITZconstant is determined from an experiment and a particular concentrationprofile is monitored. This situation can be generalized to several of therate constants and concentrations [lo],although the total number of inputsand outputs (i.e., independent and dependent variables) must remain thesame. The relevant derived sensitivity coefficients are of the form (bk,/bk J ) t l , +or (dhl/dYl)k,,k, in contrast to the elementary sensitivity (by [ /dhJ )k , , k , Techniques from multivariable calculus may be used to evaluatethese derived sensitivities from the usual set of elementary sensitivitycoefficients. No additional differential equations need to be treated; onlya set of simple linear algebraic equations has to be solved [lo]. Derivedsensitivities of the type (dk, ldk, ) j L , k rare useful in investigating the corre-lation of h, with k, These correlations can aid in assigning realistic errorbounds on the experimental result for k , . If such correlations are very high,the results are suspect. Finally, derived sensitivities of the type ( b k Jdy[)k,,+ may be useful in planning experimental studies. Their magnitudesserve as a guide in deciding which species profiles to monitor in order todetermine an unknown rate constant. The reader is referred elsewhere[10,24,26] for concrete applications of these sensitivities to particular ex-perimentallmodeling situations.

Concluding RemarksIn this paper we have described an algorithm designed to implement the

Greens function method for sensitivity analysis of reaction mechanisms.The theory of the Greens function approach, relevant computationalprocedures, and various applications of sensitivity information were dis-cussed. Although the computational algorithm described here is efficienteven for complicated kinetic analyses, it is likely that improvements canbe made [ 2 7 ] . In this section four possible future areas for improvementsare suggested. One suggestion concerns the methods used for numericalintegration. In the current version of the code, numerical integration isdone by means of the trapezoid rule or the exponential formula [eq. (lo)].By improved quadrature procedures or fuller utilization of the potentiallyvaluable information easily available from the differential equation solver,it may be possible to speed up the integration. A second possible area ofimprovement concerns optional programming to reduce the storage re-quired. Special overlaying procedures, out-of-core matrix handlingtechniques. and/or proper use of special buffers or disks may be imple-mented to reduce storage. This is clearly important for large and morerealistic kinetic models. A third suggestion for possible improvementconcerns reduction in star t-up time for the differential equation solver.The choice of a particular ODE solver is an important question for the ki-netics alone. For the Greens function approach, special considerationshould be given to the amount of start-up time required by a particular


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S E N S I T I V I T Y A N D A N A L Y S I S 1247solver, especially if sensitivities are desired a t a large number of times. Afinal suggestion for efficient computation of sensitivity information dealswith the manner of performing the operations described in the secondsection. For example, since the same Jacobian occurs for each row of theadjoint Greens function matrix, it may be more efficient to save its matrixelements a t various times instead of saving y for interpolation purposes.Careful timing studies will probably be useful to determine the optimalarrangement of computational procedures. These suggestions have notbeen implemented at this time, but they arose as natural possible areas ofimprovement from evaluating the present code.Although coding or certain numerical improvements can likely still bemade, the present procedure outlined in this paper is now practical andreasonably efficient. We hope this paper will stimulate further applicationsas well as basic development of the GFM in chemical kinetics.

AcknowledgmentsThe authors are grateful to the Department of Energy and the ExxonCorporation for support of this work, and to Dr. Dave Edelson and Dr.

Charles Westbrook for helpful discussions. One of us (E.P.D.) would liketo thank Doren Indritz for help with the computer codes and Jenn-TaiHwang for several helpful suggestions.

Bibliography[ l]D. Edelson, J . Chem. Ed., 52,642 (1975).[2] J . Phys. Chem., 81(25) (1977) (this issue contains many articles on numerical modeling[3] Y. Bard, Nonlinear Parameter Estimation, Academic Press, New York, 1974.141 R. W. Atherton, R. B. Schainker, and E. R. Ducot, AZChE J . , 21,441 (1975).[5] R. P. Dickinson and R. J. Gelinas, J . Cornp. Phys., 21,123 (1976).[S] R. I. Cukier, C. M. Fortuin, K. E. Shuler, A. G .Petschek, and J. H. Schaibly, J . Chrrn.[7] M. Koda, G. J. McCrae, and J. H. Seinfeld, Znt. J . Chem Kine t . , 9,427 (1979) .[8] A. A. Boni and R. C. Penner, Combust. Sci. Technol., 15,9 9 (1976).[9] J.-T. Hwang, E. P. Dougherty, S. Rabitz, and H. Rabitz, J . Chem. Phys., 69, 5180

in chemical kinetics).

Phys., 59,3873 (1973).

(1978).[ lo ] E. P. Dougherty, J.-T. Hwang, and H. Rabitz, J . Chem. Phys., 7, 1794 (1979).[ l l] J.-T. Hwang and H. Rabitz, J . Chem. Phys., 70, 4609, (1979).[In] R. Tomovic and M. Vukobratovic, General Stability Theory, Academic Press, New[13] P. M. Frank, Introduction to System Sensitivity Theory, Academic Press, New York,[14] D. H. Meadows et al., The Limits to Growth, Potomac AssociatesKJniverse Books,1151 P. J. Vermeulen and D. C. H. de Jongh, Appl. M a t h . Modeling, 1,2 9 (1976).1161 A. A. Frost and R. G. Pearson, Kinetics and Mechanism, Wiley, New York, 1961.[17] C. F. Curtiss and J. 0. Hirschfelder, Proc. Nat l . Acad. Sci., U.S.A., 38,235 (1952).

York, 1978.1978.New York, 1972.


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1248 DOUGHERTY AND RABI TZ[18] C. W. Gear, Num erical Initial Value Problems in Ordinary Differential Equations,[19] D. D. Wagn er, J . Phys. Chem., 81,232 9 (1977); see also references therein.[20] S. W. Benson, Proceedings of the Sym posium on Chemical Kinetics Da ta for the Upper

and Lower Atmo sphere,Znt. J . Chem. Kinet., S u p p l . 1 , 7 , 3 5 9 (1975).[21] D. Edelson, Cornput. Chem., 1,2 9 (1976).[22] A. C. Hindm arsch, Gear: Ordinary Differential Equa tion System Solver, UCID-30001,[23] C. W. Gear, ACM Commun., 14, 176 (1971).[24] E. P. Dougher ty and H. R abitz, J . Chem. Phys., to appear .[25] R . R . Baldwin, M. E. Fuller, J. S.Hillman, D. Jackson , and R . W. Walker, J . Chem.SOC.

Faraday Trans. 1 , 4,63 5 (1974).[26] M. W . Slack, Combust. Flame, 28,241 (1977).[27] D . Ede lson, Priv ate comm unication.

Pre ntic e-H all, Englewood C liffs, N.J., 1971, Ch ap. 11.

Rev. 3, Lawrence Livermore Lab oratory , Livermore, CA, December 1974.

Received Ju ne 27 ,1979Accepted August 6,19 79

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A Computational Algorithm for the Greens Function