Eigenvalue problems eigenvalues

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

Another step that is common to most, if not all, approaches that compute orbitals of one fomi or anotiier is tlie solution of matrix eigenvalue problems of the fomi... 

The solution of any such eigenvalue problem requires a number of computer operations that scales as the dimension of the F matrix to the third power. Since the indices on the F matrix label AOs, this means... 

This is because no four-indexed two-electron integral like expressions enter into the integrals needed to compute the energy. All such integrals involve p(r) or the product p(/)p(r) because p is itself expanded in a basis (say of M functions), even the term p(r)p(r) scales no worse than tvF. The solution of the KS equations for the KS orbitals ([). involves solving a matrix eigenvalue problem this... 

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

After the assembling of the stochastic matrix Pd we have to solve the associated non-selfadjoint eigenvalue problem. Our present numerical results have been computed using the code speig by Radke AND S0RENSEN in Matlab,... 

The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

B. N. Parlett. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, N.J., 1980. 

The two sets of coeflicien ts, one for spin-up alpha electrons and the other for spin-down beta electrons, are solutions of iw O coupled matrix eigenvalue problems ... 

The eigenvalue problem defined by equations (12.56) and (12.37) has been studied by Lee and Luss l79j and, more recently, in considerable detail by Villadsen and Michelsen When - I it is easy to show... 

If the functions Oj are orthonormal, then the overlap matrix S reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form ... 

This matrix eigenvalue problem then beeomes ... 

As a result, the 9x9 mass-weighted Hessian eigenvalue problem ean be sub divided into two 3x3 matrix problems ( of ai and b2 symmetry), one 2x2 matrix of bi symmetry... 

As in all matrix eigenvalue problems, we are able to express (n-1) elements of the eigenveetor v(k) in terms of one remaining element. However, we ean never solve for this one last element. So, for eonvenienee, we impose one more eonstraint (equation to be... 

Remember that ai is the representation of g(x) in the fi basis. So the operator eigenvalue equation is equivalent to the matrix eigenvalue problem if the functions fi form a complete set. 

Sehrodinger equation, multiplying on the left by and integrating over the eoordinates of the eleetron generates the following orbital-level eigenvalue problem ... 

J. L. Ging, Eigenvalue Problem—Eimit of Tow Angular Speedy Rept. EP-3912-64U, Research Laboratories for the Engineering Sciences, University of Virginia, ChadottesviUe, 1964. 

Wilkinson, J. H. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1988). 

Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundaiy conditions are prescribed. 

An eigenvalue problem is a homogeneous equation of the second land, and solutions exist only for certain A. 

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