### Table 3. Dynamic simulation of reactor600 (45 600 equations) Simulation with Processors Blocks CPU time Wall clock time

"... In PAGE 8: ...72 5.02 In Table3 the performance of BOP using di erent implemented block structured Newton-type methods is compared to that of SPEEDUP [1] at a Cray J90. The example is a reactor model built up modularly by a multi phase cell model which might be associated to a simpli ed reactive separation... ..."

### Table 4. Performance of Successive Substitution and Inexact Successive Substitution. Influence of the maximum tolerance choice. GMRES(45), Re = 100, 500 and 1000.

"... In PAGE 7: ...igure 5. Flow over a backward facing step. (a) Problem domain and (b) finite element mesh (1,800 elements and 1,021 nodes). Table4 presents the performance results obtained for the Newton-type methods. The definitions adopted here are the same employed in the previous example.... ..."

### Table 3. Dynamic simulation of reactor600 (45 600 equations)

946

"... In PAGE 8: ...72 5.02 In Table3 the performance of BOP using di erent implemented block structured Newton-type methods is compared to that of SPEEDUP [1] at aCray J90. The example is a reactor model built up modularly byamulti phase cell model which might be associated to a simpli ed reactive separation... ..."

### Table 5.4: Steepest Descent and Newton apos;s method The approach in a Damped Newton method is to combine the two methods by adding a multiple of the identity matrix to f 00(x). Hence, the framework for this type of method is Damped Newton step Solve (f 00(x) + I) hdN = ?f 0(x) ( 0) Adjust If x + hdN is acceptable, then x := x + hdN

1999

### Table 2. Comparison of computational performance Type Newton Secant BNS

"... In PAGE 11: ...ander (1977). For additional criteria see, e.g., Sect. 2.1.2 in von Matt (1993). In Table2 are listed, for each method (improved Newton, improved secant and BNS) and problem type, the average number of iterations needed to determine a single root. From the number of iterations, averaged over all roots, we see that the improved Newton method is about 10% faster than the BNS method.... ..."

### Table 17: Problem 3 Description Areal Extent of Reservoir (acres) 160.

"... In PAGE 9: ... There are four time periods, so the model must deal with a total of eight decision variables. All other features of the problem are enumerated in Table17 . As in Problem 2, the Newton type technique was not used to solve this problem.... ..."

### Table 1: Number of articles on each algorithm class through the decades. Note that the three papers [Vid84, Vid86, Vid87] on Lagrange multiplier methods are only counted once.

2006

"... In PAGE 30: ...Analysis, comments and future research We summarize the above bibliographies of the two main algorithm approaches for the problem (1), by listing the|in our opinion|main contributions, sorted in chronological order: [Bec52] The rst algorithm [ChC58b] The rst practical and convergent algorithm [ChC58b] The rst explicit use of breakpoints in a Lagrange multiplier algorithm [Sri63] The rst bisection algorithm [Sri63] The rst algorithm for a general form of j [Bod69] The rst algorithm for the parametric problem (over the values of the RHS b) [DaS69] The rst numerical experiment with n gt; 10 [San71] The rst (recursive) pegging algorithm [San71] The rst article to discuss both pegging and Lagrange multiplier algorithms [LuG75] The rst article to discuss the value of having an explicit formula for a0 ( ) [BiH77] The rst true pegging algorithm, together with convergence theory [HKL80] The rst complexity analysis of a Lagrange multiplier algorithm [Zip80b] The rst survey on algorithms [Zip80b] The rst discussion on the reoptimization of the problem for small changes in the data; utilizes the previous value of [Ein81] The rst numerical solution of the Lagrangian minimization problem [Zie82] The rst Newton-type algorithm for the problem [FeZ83] The rst mention of reoptimization of the problem through the re-ordering of the list of breakpoints [CDZ86] The rst serious computational study [CaM87] The rst discussion on the value of solving the Lagrangian dual problem even when the original problem is inconsistent [IbK88] The rst comprehensive survey [IbK88] The rst collected treatise on extensions of the problem (to integer variables, maximin problems, non-di erentiable functions j, etcetera) [RoW88] The rst algorithm for a general form of gj [Ven89] The rst numerical comparison between pegging and Lagrange multiplier algorithms [Ven89] The rst hybrid pegging/Lagrange multiplier algorithm [NiZ92] The rst theoretical analysis of a Newton algorithm for the Lagrangian dual problem [NiZ92] The rst (massively) parallel implementation [KoL98] The most complete computational study [KoL98] The rst pegging algorithm for the general problem [BrS02a] The rst pegging algorithm analyzed for the general problem In Table1 we summarize the appearance of articles on the two main algorithmic approaches, among... In PAGE 30: ... Note that the three papers [Vid84, Vid86, Vid87] on Lagrange multiplier methods are only counted once. It is apparent from the above list and Table1 that most of the development of Lagrange multiplier algorithms occurred in the 1980s and the early 1990s while the development of pegging algorithms has con- tinued to increase over the decades, albeit at a smaller scale. Notice that noone actually has yet proposed a Lagrange multiplier algorithm for the general problem (1), although it is of course straightforward.... ..."

Cited by 1

### Table 7: Observed mental methods of the PETAL group learners

"... In PAGE 36: ...In fact, Table7 indicates that T5 engaged the loop method in solving the first problem, the Exponent problem. Afterwards T5 abstracted the syntactic method or apos;the structure apos; of recursion.... ..."

### Table 1. Semiclassical resonances and multiplicities for the three-disc scattering problem (A1 subspace) with R D 1, d D 6.

1999

### Table 2. Nontrivial zeros wk and multiplicities dk for the Riemann zeta function. k

1999