### Table 7: Short Vector Speedups (Scalar time / vector time) Machine length

1992

"... In PAGE 13: ... It is also important to assess the penalty for using the vectorized codes when vector lengths are short. Table7 lists sample speedups for each function on the three machines for vectors of length 2 and 20. Arguments for these tests are distributed over two argument subranges so that internal vector lengths are actually half that listed in the table.... ..."

Cited by 2

### Table I. Test Matrices and Their Disciplines Discipline Matrices

2003

Cited by 38

### Table 5. Performance comparison on two-dimensional FFT codes. .0

"... In PAGE 9: ... Then the FORTRAN subroutine in SGEFA performing matrix-vector multiplications was replaced with the corresonding CAL routine, MXV A. A consider- able increase in performance resulted, as shown in Table5 . However, the times reported here for the CRA Y -2 should not be taken as optimal for solving linear equations on the CRA Y -2.... In PAGE 9: ...Table5... ..."

### Table 1: 7-dimensional vector product algebra

"... In PAGE 30: ... Hence, classifying the exible dissident maps in 7-dimensional Euclidean space, and thereby the exible quadratic division algebras of dimension 8, com- pletely and irredundantly, is equivalent to solving the normal form problem of the right group action Pds(R7) G2 ! Pds(R7); ( ; ) 7! = 1 (2) We denote by e1 = (1; 0) and e2 = (0; 1) the standard basis vectors in R2. The algebra AE(a; b) we de ne as the vector space R2 with multiplication in the standard basis given by Table1 , and AF (a; b) as R2 with multiplication given by Table 2. 5By we denote the adjoint operator.... In PAGE 31: ...e1 e2 e1 e1 ae1 + be2 e2 ae1 + be2 e1 Table1 : AE(a; b) e1 e2 e1 e1 ae1 + be2 e2 ae1 + be2 e2 Table 2: AF (a; b) Theorem 1.4 Let N 2 Pds(R7) be a cross-section for Pds(R7)=G2, the orbit set of the above group action (2).... In PAGE 60: ... By induction, we deduce that any two vector product algebras having multiplicative bases which contain the same number of elements (equivalently, having the same dimension) must be isomorphic. On the other hand, anyone may convince himself, by a straightforward cal- culation, that V = R7 with multiplication given by Table1 satis es the axioms for a vector product algebra. Then it is also clear that span(;), spanfe1g and spanfe1; e2; e3g are vector product subalgebras of V of the types listed in the proposition.... ..."

### Table 5: Asymptotic Speedups (Scalar time / vector time) Machine

1992

"... In PAGE 12: ... The test includes cases where all arguments are in a single range as well as cases where only one per cent of the arguments are in a given range. Table5 lists minimum, average, and maximum speedups observed with the single precision codes on each computer. The speedups for the Convex are in the range of 4.... ..."

Cited by 2

### Table 1- Methods for solving differential equations Method Message Description

"... In PAGE 3: ... In the more optimistic synchronization strategy we are currently developing, the objects encapsulating analog behavior may use a sufficient small time step as needed by the numerical integration, while synchronization with other objects will be made only when objects have to communicate events on interface signals. The analog objects have default methods, where the equations and parameters are included, as shown in Table1 . The Start method is responsible for the initialization of parameters, such as the integration step.... ..."

### Table 3: Number of Plane Equations Solved (Uniform)

1993

"... In PAGE 52: ...The following two tables and graphs show the average execution time of LSSOL in sec- onds as a function of dimensionality and number of data points. Table3 0: Number of LSSOL Iterations (Gaussian) Dim 128 256 512 1024 2048 4096 8192 2 1.9 1.... In PAGE 52: ...3 95.8 Table3 1: Execution Time in Seconds (Uniform) Dim 128 256 512 1024 2048 4096 8192 2 0.01 0.... In PAGE 53: ... Table3 2: Execution Time in Seconds (Gaussian) Dim 128 256 512 1024 2048 4096 8192 2 0.01 0.... ..."

### Table 2: Numerical results for the domain transformation technique for solving the 2D Laplace equation with prescribed surface shapes 1, 2 and 3. 3.2.2 Numerical results for the PCG method Now we want to study the number of CG-iterations needed to solve the linear system associated with the two-dimensional test problem described above. A vector 0 with

"... In PAGE 11: ... Gaussian elimination is used as the equation solver, and linear elements (triangles with three nodes) are used for the discretization. The numerical results are shown in Table2 , where nite element solutions are denoted by b apos;. Clearly, second order convergence is obtained for all test problems.... ..."

### Table 8: Short Vector Break-even Point (Vector time = scalar time) Machine

1992

"... In PAGE 13: ... For vector length 2 the vector code runs twice as long on the Sun and the Cray, and ve times as long on the Convex. Table8 lists the vector length for which the speedup is 1, i.... ..."

Cited by 2

### Table 1: List of all symmetric spaces obtained by dimensional reduction from four to three dimensions of theories with scalars and vectors (reproduced from Table 2 in [1]).

1998

"... In PAGE 7: ... is faithful, i.e., that all scalars of the 4-dimensional G= H -model couple to the vector elds. The group G will then be simple, and Table1 (a reproduction of Table 2 in [1]) lists all possible cases. 3... In PAGE 21: ... The conditions that all these scalars form one G=H -model have been discussed in [1]. In Table1 , reproduced from that paper, we list 15 di erent possibilities, all with a simple Lie group G. In the following we discuss some of these cases in detail and nally indicate a general procedure applicable to all cases.... In PAGE 28: ... A.6 The General Procedure For the remaining cases of Table1 the group G is one of the exceptional groups G2, F4, E6, E7, or E8. Some of these cases describe the bosonic sector of supergravity theories, e.... In PAGE 30: ... In these new bases Bi = 0 except when i gt; 0 and Ci = 0 except when i lt; 0 as desired. In the following we demonstrate the existence of such an X for each of the Cases 7{9 and 11{15 of Table1 . For Case 10, where no such X can be found, we will directly demonstrate how to express the matrices M in terms of symmetric matrices ~ and ~ .... In PAGE 30: ...6.1 The 14-Dimensional Representation of Sp(6; R) In Case 7 of Table1 there are 7 vector elds and the electromagnetic po- tentials transform under one of the two inequivalent 14-dimensional repre- sentations of Sp(6; R). Decomposing this representation with respect to the subgroup SL(2) SL(2) SL(2) (using the isomorphism Sp(2; R) = SL(2)) yields 2 2 2 2 1 1 1 2 1 1 1 2.... In PAGE 30: ...6.2 The 20-Dimensional Representation of A5 In Cases 8{10 of Table1 there are 10 vector elds and the electromagnetic potentials transform under the 20-dimensional representation of one of the noncompact forms sl(6), su(3; 3), or su(5; 1) of the Lie algebra A5. As repre- sentation space we may take the totally antisymmetric 3-index tensors apos;ijk, real for sl(6) and subject to the reality condition apos;ijk il jm kn( apos;lmn) = 1 6 ijklmn apos;lmn for su(3; 3) and su(5; 1), where ij is the su(p; q) metric (cho- sen as = 1p 0 0 ?1q ! for simplicity).... In PAGE 32: ...6.3 The 32-Dimensional Representation of D6 In Cases 11{13 of Table1 there are 16 vector elds and the electromagnetic potentials transform under a 32-dimensional real `spinor apos; representation of one of the noncompact forms so(6; 6), so (12), or so(10; 2) of the Lie alge- bra D6. The Lie algebra Dn has two inequivalent 2n?1-dimensional `chiral apos; spinor representations Sn .... In PAGE 32: ...6.4 The 56-Dimensional Representation of E7 In Cases 14 and 15 of Table1 there are 28 vector elds and the electromag- netic potentials transform under the 56-dimensional representation of one of the noncompact forms E7(+7) or E7(?25) of the Lie algebra E7. We rst decompose the representation with respect to the subalgebra sl(2) + D6, with D6 = so(6; 6) for E7(+7) or D6 = so(10; 2) for E7(?25), and obtain 2 V 6 1 S6 +, where V 6 denotes the vector representation of D6.... ..."