### Table 4.4 Relative error Err and Ave Rate for k = 10. The predicted rate of convergence is from Theorem 3.5.

1999

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### Table 2.1: For a symmetric bilinear form a choose the smallest such that lt; . Now, we can prove the convergence of the general cycle. Theorem 4 (Convergence of the general cycle) Let

### Table 2: This table gives the theoretical convergence rate for the example prob- lem. The columns correspond to the convergence terms derived in Theorem 2.1. The upper block of results lt; r and in the lower gt; r. In this example, max j (K(x))j = 2:8538, at the solution point. Observe that this result is the theoretical upper bound. In practice the direction often lies in quot;nicer quot; subspaces that may give faster convergence.

### TABLE 2: This table gives the theoretical convergence rate for the example prob- lem. The columns correspond to the convergence terms derived in Theorem 2.1. For r 10 then lt; r otherwise gt; r at ^ x. In this example, max j (K(x))j = 2:8538, at the solution point. Observe that this result is the theoretical upper bound. In practice the search directions often lay in quot;nicer quot; subspaces that may give faster convergence.

in Regularization Tools for Training Feed-Forward Neural Networks Part I: Theory and basic algorithms

### Table 2. Convergences for Example 7.2

"... In PAGE 23: ...1 are listed in Table 1 and those for 7.2 in Table2 . One can see that almost perfect O(h2) convergence rates are achieved as Theorem 6.... ..."

### Table 1: Observed errors and respective orders of convergence for both steps of the two-step QSC method applied to Problem 1. Here, si xi ? hx and ti yi ? hy. The constant is defined in Theorem 1.

2000

"... In PAGE 17: ... The first component of the solution, u, is x9=2(x ? 1)2y9=2(y ? 1)2, and the second, v, is x9=2y9=2, that is, both u and v have finite continuity in the domain of the problem definition. Table1 shows that the errors and respective orders of convergence of the solution obtained by the first and second steps of the QSC method conform with those predicted by Theo- rem 1. It is worth noting that, according to the theory in [3] and [4], C6 continuity of the solution suffices to get the optimal order of convergence for the QSC approximation.... ..."

Cited by 3

### Table 2: Proof of Theorem 3.2.

"... In PAGE 5: ... Let c S;#0F and c X;#0F refer to the closest gridpoint functions of c S and c X , respectively. We now explain the chain of inequalities as shown in Table2 needed for the proof. Note that by the sample size given in the statement of the theorem wehave uniform convergence for each f 2 F #0F 6 .... In PAGE 5: ... Thus the sample and true costs for each gridpointor#0F-net clustering are close. This implies that the values #282#29 and #283#29 as well as the values #285#29 and #286#29 in Table2 are close. Further, the #28sample or true#29 cost of any clustering and its nearest gridpoint clustering is no more than #0F 6 , hence the values #281#29 and #282#29 as well as the values #283#29 and #284#29, as well as the values #286#29 and #287#29 are close.... ..."

### Table 1: Convergence rates for the Dirichlet problem on the square. Due to Theorem 3.1 we expect for the hp-version with geometric mesh numerically an exponentially fast convergence. In Figure 3 we compare the errors for the di erent ver- sions on the square plate: The error curves are linear for the pure h- and p-versions and

1996

"... In PAGE 10: ...6 in [1] for the h-version and [16] for the p- version).In Table1 we present the experimental convergence rates j for the error jC?CNj, which underline the theoretical estimates (4.4).... ..."

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### Table 1: Iteration counts for iterative solutions of FIT2P. Algo- rithm switched phase at step 17.

"... In PAGE 22: ... The results for FIT2P are tabulated in Table 1. Table1 : Iteration counts for iterative solutions of FIT2P. Algo- rithm switched phase at step 17.... In PAGE 23: ...Indeed, as shown in Table1 , the number of PCG iterations taken to solve the normal equations generally increases as the IPM converges to a solution. On the other hand, when the two-phase algorithm switches to the RAE system (which occurs at the 17th IPM step), the number of SQMR iterations taken to solve the preconditioned RAE system generally decreases as the IPM solution converges.... ..."

### Table 3. Rates of convergence for various test functions

2007

"... In PAGE 16: ... Our nu- merical results show that the convergence rates are almost the same as that in Example 5.1 (See Table3 .) This confirms Theorem 4.... ..."

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