Matrix polynomial product

We also define the matrix polynomial product, using the symbol o as the operator ... 

We will use the matrix polynomial product in the context of DWT factorisation (see Chapter 7). 

The Chebyshev filtering in Step 3 costs 0(s N/p) flops. The discretized Hamiltonian is sparse and each matrix-vector product on one processor costs 0 N/p) flops. Step 3 requires m s matrix-vector products, at a total cost of 0(s m N/ p) where the degree w of the polynomial is small (typically between 8 and 20). 

If, in particular, we convert a matrix L —xl into its SCF, where now the (0,1)-entries of L are elements of -Fig], and factor the similarity invariants into products of powers of monic irreducible polynomials Pj x), so that fi x) =... 

The energy Ea is a quantum term associated with the proton reaction coordinate coupling to the Q vibration, Ea = h1 /2m. and Co is the tunneling matrix element for the transfer from the 0th vibrational level in the reactant state to the 0th vibrational level in the product state. The term AQe is the shift in the oscillator equilibrium position and F L(Eq, Ea, Laguerre polynomial. For a thorough discussion of Eq. (8), see [13],... 

It is possible to interconnect the matrices J and Qk through a compact matrix equation [48]. This follows from the tridiagonal structure of the matrix in Eq. (60), which allows one to deduce Eq. (78) directly from the product JQt at the set of the eigenvalues uk. By the same reasoning, this conclusion can also be extended to encompass Eq. (84) at values of u other than uk. Thus, the polynomials Q ( ) satisfy the matrix equation ... 

The basic fact we need is that on an algebraic matrix group S SL + t(/c) the functions xi bx, xh- x 1, and xi- x ibx for fixed b are continuous. This is clear, since they are given by polynomials, and polynomial maps are always continuous in the Zariski topology. It is worth mentioning only because multiplication is not jointly continuous (it is a continuous map S x S- S, but the topology on S x S is not the product topology). 

Reaction systems with mass action polynomials containing terms of order greater than unity are obtained when interaction is allowed between free and adsorbed molecular species— that is, when condition (4) is relaxed. Such interactions always cause the amount of the free species involved to appear in columns of the matrix A other than the one corresponding to the free sites 3lo. Consequently, this introduces terms containing products of the amounts of various species into the mass action polynomial. 

The series in (171) terminates and the expansion is exact. The coefficients -ifl form a large but very sparse matrix that can be precalculated and stored. What we have done here is to expand a product of two Coulomb Sturmians in terms of a single Coulomb Sturmian with double the k value. When this is done, the exponential part is automatically correct, and only the polynomial parts need to be taken care of. Hence, the sparseness of Cpt p p. Then... 

If one or more isotopic substitutions are performed on the moleciile, new frequencies will be obtained, as well as new G matrix elements. Within the Born-Oppenheimer approximation, however, the F matrix elements will transfer intact to the new molecule. The amotint of new information which can be obtained about the elements in the F matrix in this way is, however, limited by several isotope rules >rtiich the sets of harmonic frequencies of each symmetry type must obey. One of these, the form of the Teller-Redlich product rule which applies to two isotopic variants having the same molecular symmetry, may be deduced immediately from the secular equation Itself. When the nxn secular determinant is expanded in polynomial form, the constant term, which must be equal to the product of the roots, n. ... 

The choice of Laguerre polynomials and the inner product in (6.2.11) using the weight function (6.2.14) makes the set Aj tj rj") orthogonal. Thus, some analytical computation is facilitated for the matrix coefficients Xij. It is thus possible to show that... 

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