Vectors vector multiplication

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,... 

Note that 0 is an element of the field F, whereas 0 is the null element of L. Henceforth the two types of zero will no longer be distinguished. The elements of a vector space are called vectors. The multiplication of two elements of a vector space is not necessarily defined. 

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 ... 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

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 ... 

Working with Markov chains, confusion is bound to arise if the indices of the Markov matrix are handled without care. As stated lucidly in an excellent elementary textbook devoted to finite mathematics,24 transition probability matrices must obey the constraints of a stochastic matrix. Namely that they have to be square, each element has to be non-negative, and the sum of each column must be unity. In this respect, and in order to conform with standard rules vector-matrix multiplication, it is preferable to interpret the probability / , as the probability of transition from state. v, to state s (this interpretation stipulates the standard Pp format instead of the pTP format, the latter convenient for the alternative 5 —> Sjinterpretation in defining p ), 5,6... 

The contamination and the solutions in remediation have three common elements. The first is the source. If the source is contained, the contamination will stop spreading. The second element is the vector. The vector can be gravity, groundwater, air, surface water, wind, fire, or whatever moves the contamination off or away from the source. The third is the receptor. The receptor is the most critical element because it is where the harm comes in or where the problem is destroyed. Sometimes, however, a single source will have multiple vectors and multiple receptors. The Figure 1 indicates this. 

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 ... 

This notation is more convenient, since in (j,.IMIsy) the states of the pair i and j are specified while the subscripts in specify the location of the element in the /th row and Mi column. The symbol (s,IMIS , ) is the factor contributed to the GPF by the nn pair i and / + 1, being in states s,. and s- j, respectively. This matrix element is obtained by multiplying the matrix M by a row vector on the left and by a column vector on the right. Since (s,l and b j) are unit vectors, this multiplication produces the element of the matrix that corresponds to the state of the nn pair i and I + 1. 

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. 

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