### Table 1 Examples of rational reductions

2002

"... In PAGE 5: ...4. Rationalreductions It is easy to see that all the rational linear reductions are based on a modular relation au + bu = 0 (mod k) and the associated reduction are then deFFned by R(u; v)=|au+bv|=k (see Table1 ).Let c denotes the modular quotient of u by v modulo k, i.... ..."

### Table 4. Example of Rational Buyer Procedure Existing Procedure Rational Buyer

"... In PAGE 45: ... As an example, consider the following AS requirements: 1,500 MW of regulation, and 1,000 MW each of spinning, non-spinning, and replacement reserves. Under the existing procedure, market-clearing prices (MCP) presented in Table4 are obtained. If the rational buyer procedure is applied, then the MW purchased and the resulting prices change according to the right side of Table 4.... In PAGE 45: ... That is, 2,500 MW of regulation would be paid at $20/MW, and so on. The total payments to AS providers would be $95,000 (see Table4 ). AS buyers settlement would be based on the preliminary AS requirements (before the Rational Buyer procedure) and the final MCPs.... ..."

### Table 3. The coe cients of the six lters (two low-pass ones and four high-pass ones) of the example in x6. The three reconstruction lters are listed rst, followed by the three decomposition lters. Note that all the coe cients are actually rational.

2001

Cited by 50

### Table 1: Error in rational interpolant.

2006

"... In PAGE 14: ... Figure 1 shows plots of the rational interpolant with d = 3 for respectively n = 10, 20, 40, 80. The second column of Table1 shows the numerically computed errors in this example, for n up to 640, and they confirm the fourth order approximation predicted by Theorem 2. Figure 2 shows plots of the rational interpolant of the function f(x) = sin(x) at the same equally spaced points as in the previous example, but this time with d = 4.... In PAGE 14: ... Figure 2 shows plots of the rational interpolant of the function f(x) = sin(x) at the same equally spaced points as in the previous example, but this time with d = 4. The third column of Table1 shows the computed errors, which confirm the fifth order approximation predicted by Theorem 2. One advantage of the rational interpolants is the ease with which we can change the degree d of the blended polynomials.... ..."

Cited by 2

### Table 1: How online escrow benefits traders

"... In PAGE 6: ...In addition to the benefits of online escrow listed in Table1 , we assume that online escrow can effectively protect trades from fraud and facilitate transactions. This implies that cheaters never initiate the OES, or, conversely, if a trader adopts online escrow, she must be an honest trader.... ..."

Cited by 2

### Table 6.1: Quality of the reconstructions of 3D mirror-symmetric con gu- rations from noisy 2D projections. The di erences between the reconstructions and the original 3D mirror- symmetric con gurations are measured by the mean squared-distance re- quired to move points of the reconstructed con guration in order to obtain the original con guration. As seen in the examples, reconstruction of 3D mirror-symmetric con gurations from noisy 2D projected data, can be greatly improved by correcting for symmetry either prior and/or following reconstruction. Although, correcting for symmetry prior to recon- struction improves the result, correcting for symmetry following reconstruction generally gives a greater improvement. Not surprisingly, the greatest improvement in reconstruc- tion is obtained when correction for symmetry is performed both prior and following reconstruction.

1993

Cited by 9

### Table 3: Rational Lanczos Parameters in Example 4.4

"... In PAGE 20: ...ashion. In this case, pm equals m ? K for m gt; K. Thus, ^ ET has a lower bandwidth of 1 and an upper bandwidth of K. Example 5 Consider executing the RL algorithm again, as in Example 4, except with the interpolation points alternating in the order indicated by Table3 . Because seven iterations are still performed at (1) and four at (2), this new reduced-order model is equivalent (up to a similarity transform) with the results of Example 4.... ..."

Cited by 1

### Table 2: Rational Lanczos Parameters in Example 4.3

"... In PAGE 20: ... Example 4 Consider executing the RL algorithm for M = 11 iterations that use (1) in the rst four iterations, (2) in iterations ve through eight, and (1) again in the last three iterations. The values of the parameters m, pm, j1 and j2 as m varies are presented in Table2 . The structure of the matrix ^ ET is shown in Figure 1.... In PAGE 25: ...The data in both columns are only accurate to the order of the dominant terms. Table2 utilizes the special notation A, E, F and X to represent the cost of speci c matrix operations utilized in the RK method. The cost to acquire the triangular factors of (A ? (k)E) is denoted by F, X is the cost to solve a system of equations given these factors, E is the cost of multiplying a dense vector by E, and A is the cost of multiplying a dense vector by A.... ..."

Cited by 1

### Table 1: Orthogonalization Choices in the Rational Krylov Algorithm

"... In PAGE 8: ... It is stressed that the placement of orthogonality or biorthogonality type constraints on V and Z is purely an implementational decision. Various biorthogonality/orthogonality possibilities are explored in Table1 of Section 4. Yet these orthogonalization choices are in no way fundamental to model reduction via projection.... In PAGE 12: ... Furthermore, it is these vectors that primarily distinguish the speci c implementations. Several important options (but certainly not all) are summarized in Table1 . The rst, second and fourth cases in Table 1 are implemented in detail in Sections 4.... In PAGE 13: ... q and v. The last column of Table1 , titled , restriction, lists the conditions that must be met in each case by the choice of the four q, v, w and z parameters. In the second row, for example, orthogonal V and Z are required, V T V = ZT Z = I.... In PAGE 15: ...ection 4.4. Insights into alternatives to pm are provided by Example 3. Example 3 Consider the construction of an orthogonal V3 (see second row of Table1 ) with the interpolation point-ordering 1 = (1), 2 = (2) and 3 = (1). The rst two columns of V are therefore V2 = 1(A ? (1)E)?1b 2(A ? (2)E)?1b + 2;1v1 ; where the parameters 1, 2 and 2 are chosen so that V2 is orthogonal.... In PAGE 16: ... By taking this step, one need only store two rather than four sequences of vectors in memory. In general, one or more of the qm, vm, wm, or zm equals the corresponding vectors ~ qm, ~ vm, ~ wm, and ~ zm (see Table1 ). When these equalities occur, it is possible that the corresponding vector sequence need not be stored in memory.... ..."

Cited by 1

### Table 1: Reconstruction properties of our examples.

"... In PAGE 10: ... 4 Experiments and Discussion We applied the presented algorithms to a number of point clouds with varying size and complexity. Table1 summarizes the properties of our examples used in this section. All figures printed in red were reconstructed using the Neural Gas algorithm, whereas the green figures originate from the Growing Neural Gas algorithm.... ..."