Multiplication vectors

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

This construction requires one matrix-vector multiplication with S and two inner products in each recursive step. Therefore, it is not necessary to store S explicitly as a matrix. The Lanczos process yields the approximation [21, 7, 12]... 

The rules of matrix-vector multiplication show that the matrix form is the same as the algebraic form, Eq. (5-25)... 

Following fee general rules for the development of determinants (see Section 7.4), it is apparent that vector multiplication is not commutative, as A x B = -B x A. However, the normal distributive law still applies, as, for example,... 

As indicated by the Kronecker deltas in the above equation, the resulting Hamiltonian matrix is extremely sparse and its action onto a vector can be readily computed one term at a time.12,13 This property becomes very important for recursive diagonalization methods, which rely on matrix-vector multiplication ... 

PIST distinguishes itself from other spectral transform Lanczos methods by using two important innovations. First, the linear equation Eq. [38] is solved by QMR but not to a high degree of accuracy. In practice, the QMR recursion is terminated once a prespecified (and relatively large) tolerance is reached. Consequently, the resulting Lanczos vectors are only approximately filtered. This inexact spectral transform is efficient because many less matrix-vector multiplications are needed, and its deficiencies can subsequently... 

Like the time propagation, the major computational task in Chebyshev propagation is repetitive matrix-vector multiplication, a task that is amenable to sparse matrix techniques with favorable scaling laws. The memory request is minimal because the Hamiltonian matrix need not be stored and its action on the recurring vector can be generated on the fly. Finally, the Chebyshev propagation can be performed in real space as long as a real initial wave packet and real-symmetric Hamiltonian are used. 

Because of round-off errors, symmetry contamination is often present even when the initial vector is properly symmetrized. To circumvent this problem, an effective scheme to reinforce the symmetry at every Lanczos recursion step has been proposed independently by Chen and Guo100 and by Wang and Carrington.195 Specifically, the Lanczos recursion is executed with symmetry-adapted vectors, but the matrix-vector multiplication is performed at every Lanczos step with the unsymmetrized vector. In other words, the symmetrized vectors are combined just before the operation Hq, and the resultant vector is symmetrized using the projection operators ... 

Effectively, vector r has 3 x n x 1 components since each r, in (47) is itself a three-dimensional vector. Technically speaking, in place of Ak in (46), one should write the Kronecker product A with being the 3 x 3 identity matrix. However, to simplify notations and avoid writing routinely this obvious Kronecker product, below in this section we will be using the following convention for matrix-vector multiplications involving such vectors ... 

Prediction is a simple vector multiplication of the regression vector by the pre-processed spectrum of the unloiown to yield a concentration estimate (see Equation 5-23). Using this procedure, the predicted values for component A are obtained for 20 unknown samples. 

Poynting s theorem for the energy flow of plane waves in vacuo thus applies to the EM and EMS modes, but not to the S mode. Vector multiplication of Eqs. (52) and (53) by k, and combination with Eq. (49) and the result E C = 0, is easily shown [16,20] to result in a Poynting vector that is parallel with the group velocity C of Eq. (56). Later in Section VII.C.3 we shall return to Poynting s theorem in the case of axisymmetric photon wavepackets. 

After the required sums have been obtained and normalized they become the elements of a matrix, which must be inverted. The resultant inverse matrix is the basis for the derivation of the final regression equation and testing of its significance. These last steps are accomplished in part through matrix-by-vector multiplications. Anyone who has attempted the inversion of a high-order matrix will appreciate the difficulty of performing this operation through hand calculation. 

In either case, carrying out the matrix-vector multiplication reveals the meaning of the stress vector as... 

Notice that the construction of the updated Hessian involves simple matrix and vector multiplications of gradient and step vectors. 

Matrix and vector multiplication using the dot product is denoted by the symbol between matrices. It is only possible to multiply two matrices together if the number of columns of the first matrix equal the number of rows of the second matrix. The number of rows of the product will equal the number of rows of the first matrix, and the number of columns equal the... 

Here V.ft " stands for the 3 x 3 matrix representing the vector multiplication by vRm VRmx = vRm x x and the gRmLm subblock is defined by eq. (3.101). These subblocks couple small pseudo- and quasirotations of the hybridization tetrahedra corresponding to the right- and left-end atoms of the bond (in the specified order) in a bilinear fashion. Their form particularly simplifies for the sp3 carbon atom in a symmetric environment for which we have ... 

Finally, most doubly or triply subscripted array operations can execute as a single vector instruction on the ASC. To demonstrate the hardware capabilities of the ASC,the vector dot product matrix multiplication instruction, which utilizes one of the most powerful pieces of hardware on the ASC, is compared to similar code on an IBM 360/91 and the CDC 7600 and Cyber 174. Table IV lists the Fortran pattern, which is recognized by the ASC compiler and collapsed into a single vector dot product instruction, the basic instructions required and the hardware speeds obtained when executing the same matrix operations on all four machines. Since many vector instructions in a CP pipe produce one result every clock cycle (80 nanoseconds), ordinary vector multiplications and additions (together) execute at the rate of 24 million floating point operations per second (MFLOPS). For the vector dot product instruction however, each output value produced represents a multiplication and an addition. Thus, vector dot product on the ASC attains a speed of 48 million floating point operations per second. 

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