Solving a matrix equation

A special property of solving a matrix equation in this way is that the LU decomposition does not involve the right-hand side vector b, in contrast both to the Gauss-Jordan method and to the Gaussian elimination. This is... 

Once the equilibrium geometry has been reached, the right-hand term in Eq. [36] vanishes and the vibrational frequencies can be found by solving a matrix equation... 

The maximum-likelihood parameter estimates for an MA process can be obtained by solving a matrix equation without any numerical iterations. 

The linear integral equation (5) is solved by a standard technique, including expansion of the unknown An z) by some basis functions and transformation of (5) into a matrix equation to... 

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... 

The absorbance A is proportional to 1 through Beer s law (see Computer Projeet 2-1). The analytical problem is to solve the matrix equation... 

The respiratory quotient (RQ) is often used to estimate metabolic stoichiometry. Using quasi-steady-state and by definition of RQ, develop a system of two linear equations with two unknowns by solving a matrix under the following conditions the coefficient of the matrix with yeast growth (y = 4.14), ammonia (yN = 0) and glucose (ys = 4.0), where the evolution of C02 and biosynthesis are very small (o- = 0.095). Calculate the stoichiometric coefficient for RQ =1.0 for the above biological processes ... 

Similar to the parameter sensitivity matrix, the initial state sensitivity matrix, P(t), cannot be obtained by a simple differentiation. P(t) is determined by solving a matrix differential equation that is obtained by differentiating both sides of Equation 6.1 (state equation) with respect to p( ... 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

The numerical values were obtained from a simple Hartree-Fock variational treatment, as described in Section 1.3 below. Note that, in this simple case, we could obtain the exact eigenvalues of the 2x2 matrix H by solving a quadratic equation. The present use of perturbation theory to approximate these eigenvalues is for illustrative purposes only. 

There is no matrix version of simple division, as with scalar quantities. Rather, the inverse of a matrix (A-1), which exists only for square matrices, is the closest analog to a divisor. An inverse matrix is defined such that AA"1 = A-1 A = I (all three matrices are n X n). In scalar algebra, the equation a-b = c can be solved for b by simply multiplying both sides of the equation by la. For a matrix equation, the analog of solving... 

The Euler-Lagrange equations can he formed for the dynamical variables q—Rji, Pji, Zph, Zph and collected into a matrix equation which, when solved, yields the wave function for the compound system at each time step. 

The second step in this equation involves a property called Green s identity. Using either method brings one to the point where the solutions of both require the same basic approaches solving a matrix problem. As in the case of collocation, the L sample points are used to generate the rows of the A matrix and b vector whose elements are written m,k = y) (x, y) dx dy and = b x, y)[Pg.257]

To solve a diffusion equation, one needs to diagonalize the D matrix. This is best done with a computer program. For a ternary system, one can find the two eigenvalues by solving the quadratic Equation 3-lOOe. The two vectors of matrix T can then be found by solving... 

As you learned in the previous sections, LU decomposition with built-in partial pivoting, followed by backsubstitution is a good method to solve the matrix equation Ax = b. You can use, however, considerable simpler technics if the matrix A has some special structure. In this section we assume that A is symmetric (i.e., AT = A), and positive definite (i.e., x Ax > 0 for all x 0 you will encounter the expression x Ax many times in this book, and hence we note that it is called quadratic form.) The problem considered here is special, but very important. In particular, estimating parameters in Chapter 3 you will have to invert matrices of the form A = X X many times, where X is an nxm matrix. The matrix X X is clearly symmetric, and it is positive definite if the columns of X are linearly independent. Indeed, x (x" X)x = (Xx) (Xx) > 0 for every x since it is a sum of squares. Thus (Xx) (Xx) = 0 implies Xx = 0 and also x = 0 if the columns of X are linearly independent. 

The method (ref. 2) is based an solving the matrix equation = Ax, where is not a fixed right-hand side, but a vector of variables 1 2 " > with completely "free" values. To solve the equation for x in terms of notice that an 0 due to positive definiteness of A, since an = (e ) Ae. We can therefore solve the first equation for x, and replace x by the resulting expression in the other equations ... 

Solving the matrix equation Ax = b by LU decomposition or by Gaussian elimination you perform a number of operations on the coefficient matrix (and also on the right-hand side vector in the latter case). The precisian in each step is constrained by the precision of your computer s floating-point word that can deal with numbers within certain range. Thus each operation will introduce some round-off error into your results, and you end up with same... 

In this case, in order to solve Eq. (1.25), it is necessary to take into account the boundary conditions. Thus, by taking into account conditions (I.22)-(I.24), it is possible to make an arrangement of the set of linear equations for the concentrations into a matrix equation ... 

Density operator equations were converted into coupled integrodifferential equations suitable for numerical processing, and an extended Runge-Kutta algorithm has been implemented for solving the matrix equations in diadic form. A similar procedure can be followed for the original density matrix. 

Hess et al.119 utilized a Hamiltonian matrix approach to determine the spin-orbit coupling between a spin-free correlated wave function and the configuration state functions (CSFs) of the perturbing symmetries. Havriliak and Yarkony120 proposed to solve the matrix equation... 

In Eq. (1-6), E) , vcnt refers to the total solvent electric field and it contains a sum of contributions from the point charges and the induced dipole moments in the MM part of the system. Such a field (and hence the induced dipole) depends on all other induced dipole moments in the solvent. This means that Eq. (1-6) must be solved iteratively within each SCF iteration. As an alternative, Eq. (1-6) may be reformulated into a matrix equation... 

A typical problem of interest at Los Alamos is the solution of the infrared multiple photon excitation dynamics of sulfur hexafluoride. This very problem has been quite popular in the literature in the past few years. (7) The solution of this problem is modeled by a molecular Hamiltonian which explicitly treats the asymmetric stretch ladder of the molecule coupled implicitly to the other molecular degrees of freedom. (See Fig. 12.) We consider the the first seven vibrational states of the mode of SF (6v ) the octahedral symmetry of the SF molecule makes these vibrational levels degenerate, and coupling between vibrational and rotational motion splits these degeneracies slightly. Furthermore, there is a rotational manifold of states associated with each vibrational level. Even to describe the zeroth-order level states of this molecule is itself a fairly complicated problem. Now if we were to include collisions in our model of multiple photon excitation of SF, e wou d have to solve a matrix Bloch equation with a minimum of 84 x 84 elements. Clearly such a problem is beyond our current abilities, so in fact we neglect collisional effects in order to stay with a Schrodinger picture of the excitation dynamics. 

HF theory with expansion of the MOs in a finite basis set. The equation to be solved is now a matrix equation for the expansion coefficients... 

In early instruments, the detectors consisted of a series of half rings [143,144] (Figure 10.8) so that a matrix equation developed. Sliepcevich and co-workers [145,146] inverted this equation to obtain the particle size distribution. The equation was solved by assuming the distribution fitted a standard equation and carrying out an iteration to obtain the best fit. A matrix inversion was not possible due to the large dynamic range of the coefficients and experimental noise that could give rise to non-physical results. An inversion procedure that overcame these problems was developed by Philips [147] and Twomey [148] that eliminated the need to assume a shape for the distribution curve. 

These equations are a matrix equation of the form AX=Y, with complex elements where A is a known matrix, Y is a known vector. The unknown vector X is solved through the conjugate gradient method [38. ... 

---