### Table 9 Hybrid Lanczos Outer Iteration. corresponding to i are given by

1992

"... In PAGE 26: ... This hybrid Lanczos approach which incorporates inner iterations of single-vector Lanczos bidiagonalization within the outer iterations of a block Lanczos SVD recursion is outlined in Tables 9 and 10. As an alternative to the outer recursion in Table9 , which is derived from the equivalent eigenvalue in the 2-cyclic matrix B, Table 11 depicts the simpli ed outer block Lanczos recursion for approximating the eigensystem of ATA. Combining the equations in (34), we obtain ATA^ Vk = ^ VkHk ; where Hk = JT k Jk is the k k symmetric block tridiagonal matrix Hk 0 B B B B B B B B @ S1 RT 1 R1 S2 RT 2 R2 RT k?1 Rk?1 Sk 1 C C C C C C C C A ; (40) having block size b.... In PAGE 26: ... Analogous to the diagonalization of Bk in (36), the computation of eigenpairs of the resulting tridiagonal matrix in this case can be performed via a Jacobi or QR-based symmetric eigensolver. The conservation of computer memory for our iterative block Lanczos iterative SVD method is insured by enforcing an upper bound, c, for the order (bk) of any Jk constructed (see Table9 ). This technique was suggested by Golub, Luk, and Overton in [20].... In PAGE 27: ...iteration in Table9 we have determined that p0 singular triplets are acceptable to a user-supplied tolerance for the residual error de ned in (6). Then, we update the values of the block size (b), the maximum allowable order for Jk (c), the number of diagonal blocks for Jk (d), and the number of triplets yet to be found (p) as follows: bnew = bold ? p0 ; if b pold, (42) = min fbold ; pold ? p0g otherwise, cnew = cold ? p0 ; pnew = pold ? p0 ; dnew = bcnew=bnewc : All converged left and right singular vector approximations are respectively stored in matrices U0 and... In PAGE 28: ... V0 so that U0 (U0j u1; u2; : : :; up0) ; V0 (V0j v1; v2; : : :; vp0) ; where U0 = V0 = 0 initially (prior to any restart). For restarting the block Lanczos outer iteration in Table9 , we simply rede ne V1 to be the unconverged right singular vector approximations from the previous iteration, i.e.... In PAGE 29: ... Table 11 Hybrid Lanczos Outer Iteration for the Equivalent Symmetric Eigensystem of ATA. estimate the residual for some k (see 6) by kykk2 of Step (1a) in Table9 for iteration l + 1, where yk is the k-th column of the n b matrix Yi. Hence, at the start of iteration l + 1 we can determine the accuracy of our approximations from iteration l.... In PAGE 29: ... As with the previous iterative SVD methods, we access the sparse matrices A and AT for this hybrid Lanczos method only through sparse matrix-vector multiplications. Some e ciency, however, is gained in the outer (block) Lanczos iterations by the multiplication of b vectors (Steps (1a), (1b) in Table9 ) rather than by a single vector. These dense vectors may be stored in a fast local memory (cache) of a hierarchical memory-based archtitecture (Alliant FX/80, Cray-2S) and thus yield more e ective data reuse.... In PAGE 29: ... These dense vectors may be stored in a fast local memory (cache) of a hierarchical memory-based archtitecture (Alliant FX/80, Cray-2S) and thus yield more e ective data reuse. The total reorthogonalization strategy and de ation of converged singular vector approximations is accomplished in Steps (1b), (1e) in Table9 and Steps (2b), (2d) in Table 10. A stable variant of Gram-Schmidt orthogonalization ([37]), which requires e cient dense matrix-vector muliplication (level-2 BLAS) routines ([14]), is used to produce the orthogonal projections of Yi (i.... In PAGE 33: ... Table 13 also indicates that a signi cant proportion of time (24% of total CPU time) is spent in the level-2 (matrix-vector) and level-3 (matrix-matrix) BLAS kernels. The outer block Lanczos recursion for ATA (see Table 11), as with the outer recursion in Table9 , primarily consists of these higher- level BLAS kernels (also supplied by the Alliant FX/Series Scienti c Library) which are designed for execution on all 8 processors of the Alliant FX/80. The modi ed Gram-Schmidt procedure we employ for re-orthogonalization is also driven by the higher-level BLAS kernels.... In PAGE 37: ...2 from the eigensystem of AT A. The parameters for BLSVD (see Table9 in Section 3:5) include the initial block size, b, an upper bound on the dimension of the Krylov subspace, c. For LASVD, we also include a similar upper bound, q, for the order of the symmetric tridiagonal matrix Tj in (31).... In PAGE 38: ... The consequence of doubling p in terms of memory is discussed in Section 4:3. As mentioned in the preceding section, BLSVD requires an initial block size b, where b p, and the bound c on the Krylov subspace generated within the outer block Lanczos recursion given in Table9 . The choice of b can be di cult, and as mentioned in Section 3:5 we have made some gains... ..."

Cited by 4

### Table 12. Solution steps for Example 5 Outer

1914

"... In PAGE 22: ...terations. The first selected structure required 4 inner iterations each to check globality. The gridpoint sets were updated in each inner optimization using the middle point of the active subinterval. Details of the solution in each iteration can be seen in Table12 , as well as the global and local lower and upper bounds. Figure 11 shows the progress of the bounds.... In PAGE 37: ... Solution steps for Example 4 Table 8. Model sizes and solution time for Example 4 Table 9: Distribution of the pollutant Hej and concentration of pollutant in organic phase Coj Table 10: Inlet Streams data for Example 5 Table 11: Cost and removal ratio data for the equipments in Example 5 Table12 . Solution steps for example 5 Table 13: Global lower and upper bounds and CPU time using BARON in Example Table 1: Results using GAMS/DICOPT with different initial points in Example 1 CPU time (sec) Initialization for variables x Optimal Solution Stopping Criterion MIP NLP Major iterations Units x=xup 82.... ..."

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### Table 1: Some constructions with random outer codes.

2005

"... In PAGE 9: ... (16) Interestingly, both the bounds on epsilon1I and epsilon1II are independent of r, and hence we are going to choose r = 1 to minimise the length. In Table1 , we show some constructions of RS-RC-Soft. The parameters have been found by trial and error, and cannot be expected to be optimal.... ..."

### Table 7. Solution steps for Example 4 Outer

1914

"... In PAGE 21: ... The global optimum solution is shown in Figure 10. Six outer iterations were necessary to prove globality of the solution as seen in Table7 . In the third outer iteration, (MILP-1) selected the optimal equipment, and obtained a lower bound within a tolerance of 0.... In PAGE 37: ...37 List of tables Table 1: Results using GAMS/DICOPT with different initial points in Example 1 Table 2: Solution steps and problem sizes for Example 1 Table 3: Steps and problem sizes for Example 2 Table 4: Solution steps for Example 3 Table 5: CPU time and model size for Example 3 Table 6: CPU time using BARON for solving the NLP subproblems in Example 3. Table7 . Solution steps for Example 4 Table 8.... ..."

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### Table 6: The outer normals to the de cient faces of system D1.

1998

"... In PAGE 14: ... d1 For this system there are six de cient faces. In Table6 we list their outer normals, as derived from the computed path directions.... ..."

Cited by 19

### Table 4: The achieved and bound performance for LFK 2, 4, 6, 9, and 10 using the derived k, h and c values of formula 2.1.

1993

"... In PAGE 20: ... The constant term, c, is thus considered to be the actual measured CPF during steady-state execution of the inner loop iterations. Table4 summarizes the k, h, and c values obtained from curve tting for LFK2, 4, 6, 9 and 10.... In PAGE 20: ...nd 10. The relative errors of the curve tting are small (less than 1.5%). Note that h is close to 1 and outer loop overhead is approximately inversely proportional to n. Figure 6 displays the nal achieved performance as in gure 5, except that the measured CPF for LFK2, 4, 6, 9 and 10 are taken to be the values of c in Table4 . With this adjustment, all loops are seen to achieve at least 94% of the performance bound in steady state, and the average is 97.... ..."

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### Table 11 Hybrid Lanczos Outer Iteration for the Equivalent Symmetric Eigensystem of ATA.

1992

"... In PAGE 26: ... This hybrid Lanczos approach which incorporates inner iterations of single-vector Lanczos bidiagonalization within the outer iterations of a block Lanczos SVD recursion is outlined in Tables 9 and 10. As an alternative to the outer recursion in Table 9, which is derived from the equivalent eigenvalue in the 2-cyclic matrix B, Table11 depicts the simpli ed outer block Lanczos recursion for approximating the eigensystem of ATA. Combining the equations in (34), we obtain ATA^ Vk = ^ VkHk ; where Hk = JT k Jk is the k k symmetric block tridiagonal matrix Hk 0 B B B B B B B B @ S1 RT 1 R1 S2 RT 2 R2 RT k?1 Rk?1 Sk 1 C C C C C C C C A ; (40) having block size b.... In PAGE 26: ... Combining the equations in (34), we obtain ATA^ Vk = ^ VkHk ; where Hk = JT k Jk is the k k symmetric block tridiagonal matrix Hk 0 B B B B B B B B @ S1 RT 1 R1 S2 RT 2 R2 RT k?1 Rk?1 Sk 1 C C C C C C C C A ; (40) having block size b. We then apply the block Lanczos recursion ([23]) in Table11 for computing the eigenpairs of the n n symmetric positive de nite matrix ATA. The tridiagonalization of Hk via an inner Lanczos recursion follows from simple modi cations to Table 10.... In PAGE 33: ... Table 13 also indicates that a signi cant proportion of time (24% of total CPU time) is spent in the level-2 (matrix-vector) and level-3 (matrix-matrix) BLAS kernels. The outer block Lanczos recursion for ATA (see Table11 ), as with the outer recursion in Table 9, primarily consists of these higher- level BLAS kernels (also supplied by the Alliant FX/Series Scienti c Library) which are designed for execution on all 8 processors of the Alliant FX/80. The modi ed Gram-Schmidt procedure we employ for re-orthogonalization is also driven by the higher-level BLAS kernels.... In PAGE 45: ... To assess the speci c gains in performance (time reduction) associated with the eigenvalue problem of order n, we list the speed improvements for LASVD and BLSVD (and the other candidate methods) when eigensystems of ATA are approximated in Table 21. The limited improvement for BLSVD, in this case, stems from the fact that although the less time is spent in re-orthogonalization (see Table 12) the number of outer iteration steps for the ATA-based recursion (see Table11 ) can be as much as 1:5 times greater than the number of outer iterations for the cyclic-based hybrid recursion (Tables 9 and 10). Hence, the de ation associated with larger gaps among the p = 100 eigenvalues of AT A is not quite as e cient.... ..."

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### Table 2. Derived Operators

1995

"... In PAGE 9: ... The logic L s is su#0Eciently expressive that we may specify a number of safety and bounded liveness properties. In particular, restricting #20 to c and p in Table2... ..."

Cited by 85