An Acceleration Method for Dogleg Methods in Simple SingularRegions

Angelo Lucia* and Delong Liu

Department of Chemical Engineering, University of Rhode Island, Kingston, Rhode Island 02881-0805

The behavior of dogleg methods in singular regions that have a one-dimensional null space isstudied. A two-tier approach of identifying singular regions and accelerating convergence to asingular point is proposed. It is shown that singular regions are easily identified using a ratioof the two-norm of the Newton step to the two-norm of the Cauchy step since Newton stepstend to infinity and Cauchy steps tend to zero as a singular point is approached. Convergenceacceleration is accomplished by bracketing the singular point using a projection of the gradientof the two-norm of the process model functions onto the normalized Newton direction inconjunction with bisection, thus preserving the global convergence properties of the doglegmethod. Numerical examples for a continuous-stirred tank reactor and vapor-liquid equilibriumflash are used to illustrate the reliability and effectiveness of the proposed approach. Severalgeometric illustrations are presented.

1. Introduction

In a sequence of recent papers, Lucia and co-workers(Lucia and Xu, 1992; Lucia et al., 1993; Sridhar andLucia, 1995) have studied the behavior of trust regionmethods in the real domain and presented an extensionof the dogleg method to the complex domain. Inparticular, Lucia and Xu (1992) studied the behavior ofthe dogleg strategy and extended it to the complexdomain. Lucia et al. (1993) showed the advantages innumerical performance of the complex domain doglegstrategy over Newton’s method, which can behaveperiodically or aperiodically, and Powell’s original doglegstrategy, which can terminate at a singular point (orlocal minimum or saddle point in the two-norm of thefunction in the real domain). They also proposed asingular-point perturbation for moving iterates from asingular point to a solution. Sridhar and Lucia (1995)gave an analysis for the extended dogleg method show-ing that all nondegenerate singular points are saddlepoints of the two-norm of the model functions in thecomplex domain. This analysis also showed that aneigenvalue-eigenvector decomposition of the Hessianmatrix of the two-norm of the function can be used toconstruct a path from a singular point to a solution. Asummary of this equation-solving methodology can befound in the recent paper by Gow et al. (p 2843, 1997).In this paper, we concentrate on methods for the rapidand reliable convergence to singular points.There is considerable literature in the area of applied

mathematics on the computation of singular points andrelated convergence acceleration methods. See, forexample, Keller (1977), Rheinboldt (1978), Abbott (1978),Moore and Spence (1980), Decker and Kelley (1980),Griewank (1980), Griewank and Osborne (1981), Doedel(1981), Georg (1981), Decker and Kelley (1981), Kearfott(1983), Griewank and Reddien (1984), and others.However, most of these papers are concerned with thecalculation of singular solutions (or turning and bifur-cation points) and not necessarily singular points that

do not represent solutions to the given set of nonlinearalgebraic equations. Moreover, most of these algorithmsare based on Newton’s method and do not consider trustregion methods, and none address the issue of complex-valued points of singularity. In fact, there has been nosystematic study addressing the performance of doglegstrategies in singular regions. Therefore, the primaryobjective of this research is to address the difficultiesassociated with slow convergence and termination insingular regions by developing a convergence accelera-tion method for the dogleg method. Accordingly, thispaper is organized in the following way.In section 2 we present new methodologies for im-

proving the numerical performance of the dogleg algo-rithm. These improvements consist of new computertools for (1) the identification of singular regions and(2) convergence acceleration in the Newton directionwhile locating simple singularities. The ideas developedin section 2 are tested on a two-dimensional continuous-stirred tank reactor (CSTR) problem and a multidimen-sional vapor-liquid equilibrium (VLE) problem in sec-tion 3. Conclusions based on the numerical results arepresented in section 4.

2. Numerical Behavior of the Dogleg Method inSingular Regions

The dogleg algorithm of Powell (1970) chooses be-tween Newton, steepest descent (or Cauchy), and doglegsteps. At any iteration, the desired step is selectedusing a linear combination of the steepest descent step,µ, in the least-squares function and the Newton step,γ, constrained to lie within a trust region, to produce amonotonically decreasing sequence of least-squaresfunction values that converge to a solution. Becausethe size of the trust region radius is also adjustediteratively and because the rules of adjustment requirethat the trust region radius decrease when norm reduc-tion does not occur, steepest descent steps are guaran-teed to be selected in regions of difficulty and hereinlies the global convergence characteristics of the doglegstrategy, which guarantees convergence to either asolution or a singular point of f(Z,p). We refer the

* To whom correspondence should be addressed. Tel.: (401)874-2814. Fax: (401) 874-4689. E-mail: lucia@egr.uri.edu.

1358 Ind. Eng. Chem. Res. 1998, 37, 1358-1363

S0888-5885(97)00681-7 CCC: $15.00 © 1998 American Chemical SocietyPublished on Web 02/25/1998

reader to the original paper of Powell (1970) and to thoseof Lucia and co-workers for the mathematical detailsof the dogleg strategy and its extension to the complexdomain.In the material that follows, we present a strategy

for improving the numerical performance of the doglegmethod in simple singular regions. By a simple singularregion, we mean a region which contains a singularpoint at which the null space of the Jacobian matrix ofthe model functions has dimension one. The proposedstrategy is comprised of two parts: (1) the identificationof a singular region by a ratio test; (2) convergenceacceleration of iterates in the Newton direction.2.1. Identification of Singular Regions. The

normal convergence condition that ||f|| be less than aprescribed tolerance will not usually be satisfied in aregion of singularity, especially if ||f|| is bounded wellaway from zero in this region. Therefore, the identifica-tion of singular regions is the first step in improvingthe performance of the dogleg method. When a localminimum of ||f|| (or equivalently a singular point) isbeing approached, any trust region algorithm willusually take Cauchy (or steepest descent) steps andconverge linearly to this stationary point in the two-norm of a function. Furthermore, this convergence willusually be slow. The original condition for terminatingthe calculations proposed by Powell (1970) is

where M is an arbitrarily chosen positive numbergoverning the conditions for finishing the iterativeprocess, usually set to an overestimate of the distancefrom Zk to the solution. Here, Z represents a vector ofunknown variables, p is a vector of parameters, F(Z,p)) ||f(Z,p)||2 ) fTf, where f(Z,p) is a vector of processmodel functions, ||g(Z,p)|| is the two-norm of the gradi-ent of ||f||2 at (Z,p), where g(Z,p) ) JTf(Z,p), and J isthe Jacobian matrix of f.The disadvantages of this original convergence condi-

tion are that it does not clearly characterize a region ofsingularity so that the number M is difficult to selectand an excessive number of Cauchy steps (or trustregion reductions) is usually required to meet condition1. This also means that it requires many functionevaluations. Moreover, sometimes the calculationsterminate prior to reaching a stationary point or failbecause the size of the trust region becomes extremelysmall (i.e., <10-10). Some of these disadvantages areillustrated in the numerical examples section.To overcome these deficiencies, we prefer to use a test

that directly detects a singular region rather than afinishing condition. If our test is satisfied, furtherverification of the existence of a singular point (orconvergence acceleration to the singular point) willproceed. This procedure avoids excessive Cauchy stepsand prevents the trust region algorithm from terminat-ing prematurely at an unexpected point when ap-proaching a singular point.The proposed test for the identification of singular

regions is given by the pair of conditions

and

where again M is some positive number and γk and µk

are the Newton step and Cauchy step, respectively, onthe kth iteration. We call condition 3 a ratio test sinceit is based on the ratio of the two-norm of the Newtonand Cauchy steps. When conditions 2 and 3 aresatisfied, the singular region can contain either anondegenerate or degenerate singular point. The rea-sons for preferring conditions 2 and 3 are due to thefacts that in a region of singularity the ratio of the two-norm of the Newton step to the two-norm of the Cauchystep tends to be a large number, and at a singular point,this ratio is infinite. In contrast, in nonsingular regions,the ratio is bounded.Compared with other methods, such as using a

condition number for detecting singularity, conditions2 and 3 capture the main characteristics of any singu-larity and are easily implemented within the doglegalgorithm.2.2. Convergence Acceleration in Singular Re-

gions. Once a region of singularity has been identifiedby conditions 2 and 3, a method of acceleration to moveiterates quickly toward the singular point is needed. Thegoal of the acceleration step is to reduce the two-normof a gradient of ||f(Z,p)||2 to an acceptable tolerance ina given direction by an efficient and reliable meanswithout sacrificing the global convergence characteris-tics of the dogleg method. To achieve this goal in asingular region, higher order derivatives and/or geo-metric information associated with the function f or ||f||2must be utilized. However, for a set of nonlinearequations in many variables, it can be difficult to obtainthe second-order information of f and ||f||2 analytically.Here, we present a geometric approach for locating asimple singular point more efficiently on the basis ofthe reduction of ||g|| in the normalized Newton directionin a region of singularity. Figure 1 shows the underly-ing geometry for a two-dimensional problem.The basic idea of this method is to reduce the

magnitude of the directional derivative of ||f||2 given by

in the normalized Newton direction to a desired toler-ance, where gi is the ith component of the gradient ofthe function ||f||2 with respect to Zi and where γi is theith component of the normalized Newton step. This isbecause at the projected minimum (i.e., singular point),the Newton step aligns with the null space of theJacobian matrix of f(Z,p) and the gradient of ||f||2 iszero. This is a fact that we have known for some time(see Venkataraman and Lucia (1986)). One way oflocating a projected local minimum is by maintaining achange in the sign of the directional derivative of ||f||2in a bracketed region. To do this, the singular point isbracketed by placing an iterate on the trust regionboundary in the Newton direction until a change in signof the directional derivative occurs and the bracketedregion is reduced in size by discarding the current endpoint whose directional derivative is the same as thatat the bisection midpoint. This procedure is repeateduntil ||DD|| is reduced to a small number, say 10-7.The advantages of this approach are that it is reliable,

efficient, and geometrically straightforward in acceler-ating convergence to a singular point. In particular,

F(Z,p)k > M||g(Z,p)k|| (1)

||f(Z,p)k|| > ||g(Z,p)k|| (2)

||γk||||µk||

g M (3)

DD(Z,p) ) gTγ ) [g1 g2]T[γ1 γ2] (4)

Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1359

bisection requires only one function evaluation periteration once the singular region has been bracketed.Note, however, we have limited our attention to prob-lems with a one-dimensional null space. For problemswith higher-order singularities (e.g., critical points ofmixtures, azeotropes, cusps, etc.) a similar approach canbe used but additional research is needed to identifyuseful ways of accelerating the calculations since thenull space has a dimension greater than one.

3. Numerical Examples

In this section, we demonstrate the usefulness of thetechniques developed in section 2 using an adiabaticcontinuous-stirred tank reactor (CSTR) with first-order,irreversible kinetics and the vapor-liquid equilibriumflash of a binary mixture in the retrograde region.3.1. A CSTR Example. One mathematical model

for an adiabatic CSTR derived from the conservation ofmass and energy is given by

and

where T denotes the reaction temperature in degreesKelvin, θ denotes the residence time in seconds, X isthe conversion of the chemical reactant, F is the densityof the fluid, Cp is the specific heat, C0 and T0 are theinlet concentration and temperature, respectively, ∆Hdenotes the heat of reaction, E is activation energy, Adenotes the pre-exponential factor, and R is the uni-versal gas constant. In the illustrations in this section,all other quantities were held fixed and their particularnumerical values and units are given in Table 1. Figure2 shows the complex bifurcation diagram for the physi-

cally meaningful solutions to eqs 5 and 6 parametrizedin residence time from 0 to 800 s. Note that there areregions to the left and right of the two turning pointsin which there are real- and complex-valued solutionsand a region between the two turning points in whichthere are three real-valued solutions. However, only thereal-valued high-temperature (and corresponding high-conversion) solution represents the desired operation ofthe reactor. The low-temperature solution correspondsto essentially no reaction and the intermediate solutionis an unstable steady state. Moreover, the left-handturning point, which occurs at 62.19 s, represents alower limit of feasibility for the high-temperature/high-conversion solution and consequently places limitationson the feed flow and/or volume that result in properoperation of the reactor.Now we study the behavior of the multivariable

acceleration method on this two-variable CSTR problem,for which eqs 5 and 6 are solved simultaneously for theconversion, X, and temperature, T. However, first wegeometrically demonstrate the numerical behavior ofthe trust region method in a singular region withoutacceleration. For initial values (X, T) ) (0.7, 450 K) witha specified residence time of 60 s and an initial trustregion radius of 5, the values of X and T at eachiteration are shown in Table 2. The associated levelcurves of ||f|| are shown in Figure 3.At a residence time of 60 s, only one real-valued

solution at (X, T) ) (0.00277, 298.415 K) exists sincewe are to the left of the turning point at 62.19 s. SeeFigure 2. Note also that iterates 1-6 lie outside theregion shown in Figure 3; only iterates 7-12 are shown.Observe that all iterates are located in a narrow strip,

Figure 1. Projection of the gradient of ||f(Z,p)||2 onto the Newtondirection.

Table 1. Data for a CSTR Example

F (mol/mL) 1.0Cp (cal/(mol‚K)) 1.0C0 (mol/mL) 0.003T0 (K) 298.0θ (s) 300A (1/s) 4.48 × 106E (cal/mol) 1.50 × 104∆H (cal/mol) -5.0 × 104

X )θA exp(- E

RT)1 + θA exp(- E

RT)(5)

X )FCp(T - T0)

C0(-∆H)(6)

Figure 2. Complex bifurcation diagram for a CSTR.

Table 2. Numerical Results for a CSTR Example withoutAcceleration

ni (nr)a X T (K) ||f|| ||g||1 (0) 0.700 00 450.000 47.1285 710.2072 (0) 1.045 98 454.988 1.817 91 18.63113 (0) 1.012 40 449.988 1.191 91 1.477 624 (0) 0.979 48 444.988 0.745 86 0.561 185 (0) 0.946 38 439.988 0.437 73 0.089 466 (0) 0.913 17 434.988 0.235 77 0.110 247 (0) 0.879 89 429.988 0.113 81 0.155 718 (0) 0.846 58 424.989 0.050 56 0.133 319 (3) 0.813 26 419.989 0.028 98 0.084 8110 (5) 0.804 92 418.739 0.028 45 0.001 8411 (6) 0.807 00 419.051 0.028 42 0.001 7812 (7) 0.805 96 418.895 0.028 42 0.000 7013 (10) 0.806 48 418.973 0.028 42 0.000 4014 (16) 0.806 42 418.963 0.028 42 0.000 0515 (18) 0.806 42 418.964 0.028 42 7.49152 × 10-7

16 (20) 0.806 42 418.964 0.028 42 1.87266 × 10-7

17 (21) 0.806 42 418.964 0.028 42 9.36373 × 10-8

a ni ) number of iterations, nr ) number of trust regionreductions. M ) 10 for condition 1 and 50 000 for condition 3.

1360 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998

and iterates 9-11 are alternately distributed on bothsides of the 12th iterate as ||f|| decreases on each step.On iteration 12, ||f|| ) 2.84218 × 10-2, ||g|| ) 7.04209× 10-4, and ||γ||/||µ|| ) 5.46 × 104. However, neitherthe two-norm of the gradient of ||f||2 nor the two-normof f is close to vanishing and the calculations requireincreasingly excessive trust region reductions (andtherefore function evaluation) in the Cauchy directionon this and all subsequent iterations. In fact, fromiteration 9-17, 106 additional function calls are re-quired by trust region reduction to reach a value of ||g||< 10-7, as shown in Table 2. We can still, however,recognize that there is a singular point (a local mini-mum in the real domain but saddle point in the complexdomain) near (X, T) ) (0.80596, 418.895 K). Note alsothat with M ) 10 in condition 1, a value suggested byPowell, the calculations terminate on iteration 14, veryclose to the singular point. However, 47 additionalfunction calls are needed to reach this point.Now we apply acceleration to this example to see if

the singular point shown in Table 2 (i.e., iterate 17) canbe located more efficiently. On the 10th iteration, thetwo-norm of the gradient is 1.8419× 10-3, the two-normof f(Z,p) is 2.8445 × 10-2, and the ratio of the two-normof the Newton step to the two-norm of the Cauchy stepis 6.7722× 104. Accordingly, conditions 2 and 3 indicatea region of singularity is nearby and acceleration isautomatically invoked on iteration 10sfour iterationssooner than Powell’s condition. Following the identifi-cation of this singular region, the singular point isbracketed and reduction of the directional derivative of||f||2 in the normalized Newton direction (i.e., (∆X, ∆T)) (0.00667, 0.99998 K)) is accomplished by bisection.The bisection results are shown in Tables 3 and 4.The value of the directional derivative on the ninth

bisection iteration clearly indicates that a singular pointis found at (X, T) ) (0.80642, 418.963 K). Note thatthe calculated singular point is, indeed, reasonably closeto the 10th-12th iterates shown in Table 2, and thatthe line connecting the 10th through the 12th iteratescan be approximately considered the null space of theJacobian matrix of f(X,T). As a consequence, we can

use any of these normalized Newton steps to ap-proximate the null space of the Jacobian matrix off(X,T). In addition, we note that each bisection iterationrequires only one function evaluation, and thus we havereduced the number of function evaluations required tofind this singular point from 44 required by Powell’scondition to 9 needed by our ratio test.Finally, for a residence time of 750 s, with initial

values of (X, T) ) (0.20, 350 K), a real-valued localminimum (or singular point) is found at (X, T) )(0.09372, 312.059 K) with five acceleration steps reach-ing a final directional derivative value of 5.13290× 10-9.Note again that this singular point is a saddle point inthe complex domain.3.2. A Vapor-Liquid Equilibrium (VLE) Flash

Example. Figure 4 shows the experimental (Olds etal., 1949) and calculated VLE for carbon dioxide andn-butane at 344.26 K using the Soave-Redlich-Kwong(SRK) equation of state. The calculated results in thisfigure were generated by solving the component massbalances,

the phase equilibrium equations,

the vapor specification equations,

Figure 3. Level curve of ||f|| for CSTR model equations.

Table 3. Numerical Results for a CSTR Example withAcceleration

bisection step X T (K) directional derivatives

1 0.821 59 421.239 8.04000 × 10-5

2 0.813 25 419.989 3.18320 × 10-5

3 0.809 09 419.364 1.18966 × 10-5

4 0.807 00 419.051 2.56577 × 10-6

5 0.805 96 418.895 -1.98890 × 10-6

6 0.806 48 418.973 2.80545 × 10-7

7 0.806 22 418.934 8.56060 × 10-7

8 0.806 35 418.954 -2.88237 × 10-7

9 0.806 42 418.963 -3.96758 × 10-9

fi - li - vi ) 0, i ) 1, 2, ..., nc (7)

Ki(li/∑lj) - (vi/∑vj) ) 0, i, j ) 1, 2, ..., nc (8)

Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1361

and

at fixed temperature and feed conditions, for li, vi, V/F,and P, where fi, li, and vi are the ith component molarflow rates of the feed, liquid and vapor respectively, Kiis the ith component equilibrium ratio, F is the totalfeed flow, V/F is the vapor-to-feed ratio, which isspecified at a value of O. This mixture exhibits retro-grade behavior and has a turning point at yCO2 ) 0.7855and P ) 69.40 bar. Moreover, because of continuity offirst partial derivatives, there are singular points whichare not solutions for feed specifications either just tothe left or just to the right of the turning point.Consider then 1 mol of feed with a composition of z

) (z1, z2) ) (0.79, 0.21), which is just to the right of theturning point and for which the two-phase solution iscomplex-valued, indicating that the only real-valuedsolution is a single-phase solution. See Figure 4. Forthese feed specifications, the extended dogleg methodwithout acceleration finds a point at Z ) (l1, l2, v1, v2,V/F, P) ) (0.5921, 0.4075, 0.790, 0.2102, -1.318 × 10-4,71.67) in 100 iterations, at which F ) fTf is 1.1 × 10-6

and ||g|| is 8.5 × 10-6, where the subscripts 1 and 2denote carbon dioxide and n-butane, respectively. How-ever, Powell’s condition 1 with M ) 10 is still notsatisfied even after 100 iterations. The eigenvalues ofthe Hessian matrix of the two-norm of the modelfunctions at the singular point are (0.02122, 0.0119,0.0008, 0.00009, 0.00002, 0.0) which clearly show thatthe two-norm is convex in this region and that thesingular point is truly a local minimum in the two-normwhen the calculations are restricted to or remain in thereal domain. This means that Cauchy steps willconverge to the singular point. However, it is important

to point out that this singular point is a saddle point inthe complex domain, that 74 of the 100 iterationswithout acceleration are Cauchy steps, that many ofthese Cauchy steps require trust region reductionsresulting in more than 322 function evaluations to reacha value of ||g|| < 10-5 and that condition 1 still cannotidentify the singular point. In our opinion, this iscomputationally wasteful.If, on the other hand, the parameter M in eq 3 is set

to 107 and the acceleration procedure in section 2 isused, the ratio test invokes acceleration on the 27thiteration, for which the normalized Newton direction is(∆l1, ∆l2, ∆v1, ∆v2, ∆V/F, ∆P) ) (1.2187 × 10-2, -1.2187× 10-2, 6.769 × 10-7, -6.645 × 10-7, -2.2601 × 10-7,0.99985) and represents a very good approximation tothe null space of the Jacobian matrix. After this, 12bisection iterations are needed to find the approximatesingular point at which the value of ||g|| is 1.9244 ×10-4. Thus, a total of 39 function evaluations (insteadof more than 322) from start to finish using ouracceleration procedure are required and this, in ouropinion, represents a considerable computational sav-ings.

4. Conclusions

A strategy comprised of the identification of singularregions by a ratio test and an acceleration method usingthe reduction of the gradient of the square of the two-norm of the model functions in the normalized Newtondirection in a singular region was proposed and testedon two process engineering examples, a CSTR example,and a VLE problem. Numerical results showed that theproposed strategy avoids excessive Cauchy steps or trustregion radius reduction failures in singular regions. Itwas also shown that the normalized Newton directionin a singular region can be considered a reasonableapproximation of the null space of the Jacobian matrixand that reducing the projection of the gradient of||f(Z,p)||2 in that direction is a reliable means fordetermining a singular point. The concepts that un-derlie this strategy are simple and its implementationwithin a dogleg method is straightforward. Also, thisstrategy is suitable for use in other types of trust regionalgorithms. Furthermore, the enhanced dogleg algo-rithm can be used for locating simple local minima of||f(Z,p)||2. With some modifications, it can also be usedto determine turning points in a given domain.

Acknowledgment

This work was supported by the National ScienceFoundation under Grant Nos. CTS-9409158 and CTS-9696121.

Nomenclature

A ) frequency factor (s-1)C0 ) inlet concentration (mol mL-1)

Table 4. Numerical Results for Bisection for a CSTR Example

bisection step mid point end points

1 (0.821 59, 421.239) [(0.804 92, 418.739), (0.838 25, 423.748)]2 (0.813 25, 419.989) [(0.804 92, 418.739), (0.821 59, 421.239)]3 (0.809 09, 419.364) [(0.804 92, 418.739), (0.813 25, 419.989)]4 (0.807 00, 419.051) [(0.804 92, 418.739), (0.809 09, 419.364)]5 (0.805 96, 418.895) [(0.804 92, 418.739), (0.807 00, 419.051)]6 (0.806 48, 418.973) [(0.805 96, 418.895), (0.807 00, 419.051)]7 (0.806 22, 418.934) [(0.805 96, 418.895), (0.806 48, 418.973)]8 (0.806 35, 418.954) [(0.806 22, 418.934), (0.806 48, 418.973)]9 (0.806 42, 418.963) [(0.806 35, 418.953), (0.806 48, 418.973)]

Figure 4. Isothermal VLE for CO2(1)-n-C4H10(2) at 344.26 Kusing the SRK EOS.

(∑vi - V)/F ) 0 (9)

V/F - O ) 0 (10)

1362 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998

Cp ) specific heat (cal/mol‚K)DD ) directional derivative of gradient of ||f(Z,p)||2 in anormalized Newton direction

E ) activation energy (cal mol-1)fi ) component i feed molar flow rate to flash (mol s-1)f(Z,p) ) model functionsF ) total feed molar flow rate (mol s-1)F(Z,p) ) ||f||2, square of two-norm of model functionsF(Z,p)k ) two-norm of model functions at kth iterategi ) ith component of the gradient of F(Z,p)g(Z,p)k ) gradient of F(Z,p) at kth iterate evaluated at(Z,p)

∆H ) heat of reaction (cal mol-1)J ) Jacobian matrix JKi ) component i equilibrium ratioli ) component i liquid molar flow rate from flash (mol s-1)M ) a positive numbernc ) number of componentsR ) universal gas constant (cal mol-1‚K-1)p ) a real-valued parameter vectorP ) pressure (bar)X ) conversion of reactantT ) temperature (K)T0 ) inlet temperature (K)vi ) component i vapor molar flow rate from flash (mol s-1)V ) total overhead vapor molar flow rate from flash (mols-1)

z ) feed compositionZ ) vector of unknown variables

Greek Letters

γ ) normalized Newton stepφ ) specified vapor-to-feed ratioF ) density of reactant (mol ml-1)θ ) residence time (sec)µ ) Cauchy step

Subscripts

i ) ith component

Superscripts

k ) iteration countern ) dimension of vector of unknowns

Literature Cited

Abbott, J. P. An Efficient Algorithm for the Determination ofCertain Bifurcation Points. J. Comput. Appl. Math. 1978, 4, 19.

Decker, D. W.; Kelley, C. T. Newton’s Method at Singular Points.SIAM J. Numer. Anal. 1980, 17, 66.

Decker, D. W.; Kelley, C. T. Convergence Acceleration for Newton’sMethod at Singular Points. SIAM J. Numer. Anal. 1981, 19,219.

Doedel, E. J. Auto: A Program for the Automatic BifurcationAnalysis of Autonomous Systems. Congr. Numer. 1981, 30, 265.

Georg, K. On Tracing an Implicitly Defined Curve by Quasi-Newton Steps and Calculating Bifurcation by Local Perturba-tions. SIAM J. Sci. Stat. Comput. 1981, 2, 35.

Gow, A. S.; Guo, X.; Liu, D.; Lucia, A. Simulation of RefrigerantPhase Equilibria. Ind. Eng. Chem. Res. 1997, 36, 2841.

Griewank, A. O. Starlike Domains of Convergence for Newton’sMethod at Singular Points. Numer. Math. 1980, 35, 95.

Griewank, A. O.; Osborne, M. R. Newton’s Method for SingularProblems When the Dimension of the Null Space is > 1. SIAMJ. Numer. Anal. 1981, 18, 145.

Griewank, A. O.; Reddien, G. W. Characterization and Computa-tion of Generalized Turning Points. SIAM J. Numer. Anal. 1984,21, 176.

Kearfott, R. B. Some General Derivative-Free Bifurcation Tech-niques. SIAM J. Sci. Stat. Comput. 1983, 4, 52.

Keller, H. B. Numerical Solution of Bifurcation and NonlinearEigenvalue Problems. In Applications of Bifurcation Theory;Rabinowitz, P. H., Ed.; Academic Press: New York, 1977.

Lucia, A.; Xu, J. Global Minima in Root Finding. In RecentAdvances in Global Optimization; Floudas, C. A., Pardalos, P.M., Eds.; Princeton University Press: Princeton, NJ, 1992.

Lucia, A.; Guo, X.; Richey, P. J.; Derebail, R. Simple ProcessEquations, Fixed Point Methods, and Chaos. AIChE J. 1990,36, 641.

Lucia, A.; Guo, X.; Wang, X. Process Simulation in the ComplexDomain. AIChE J. 1993, 39, 461.

Moore, G.; Spence, A. The Calculation of Turning Points ofNonlinear Equations. SIAM J. Numer. Anal. 1980, 17, 567.

Olds, R. H.; Reamer, H. H.; Sage, B. H.; Lacey, W. N. PhaseEquilibrium in Hydrocarbon System. The n-Butane-CarbonDioxide System. Ind. Eng. Chem. Res. 1949, 41, 475.

Powell, M. J. D. A Hybrid Method for Nonlinear Equation. InNumerical Methods for Nonlinear Algebraic Equations; Rabinow-itz, P., Ed.; Gordon and Breach: London: 1970.

Rheinboldt, W. C. Numerical Methods for a Class of FiniteDimensional Bifurcation Problems. SIAM J. Numer. Anal. 1978,15, 1.

Sridhar, L. N.; Lucia, A. Process Analysis in the Complex Domain.AIChE J. 1995, 41, 585.

Venkataraman, S.; Lucia, A. Exploiting the Gibbs-Duhem Equa-tions in Separation Calculations. AIChE J. 1986, 32, 1057.

Received for review September 24, 1997Revised manuscript received January 14, 1998

Accepted January 15, 1998

IE970681F

Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1363