Jacobi matrix diagonalization

Check it out. Linear algebra texts describe an analytical procedure using determinants, but computational chemistry employs a numerical iterative procedure called Jacobi matrix diagonalization, or some related method, in which the off-diagonal elements are made to approach zero. 

The Jacobi state vectors without diagonalization of the Jacobi matrix... 

This recurrence relation and the available triple set an, fin uk] are sufficient to completely determine the state vector Q without any diagonalization of the associated Jacobi matrix U = c0J, which is given in Eq. (60). Of course, diagonalization of J might be used to obtain the eigenvalues uk, but this is not the only approach at our disposal. Alternatively, the same set uk is also obtainable by rooting the characteristic polynomial or eigenpolynomial from... 

Equation (3.12) clearly illustrates that the roots of Pn(0, namely the nodes of the quadrature approximation a, are the eigenvalues of the tridiagonal matrix appearing in the equation. This matrix can be made symmetric (preserving the eigenvalues) by a diagonal similarity transformation to give a Jacobi matrix ... 

THE JACOBI TRANSFORMATION, DIAGONALIZATION OF A SYMMETRIC MATRIX AND CANONICAL ORTHOGONALIZATION... 

The Jacobi matrix remains a triangular matrix. The only difference from eq. (2.88) is given by a change in the rate constant which amounts to k22 =-( 2 + 3)- concentration-time curve is given in Fig. 2.8. The eigenvalues are the elements of the diagonal. By use of the final values calculated in Example 2.20 in Section 2.2.1.3 one finds the following time functions ... 

The model of measurement will specify in addition which variables mj are measured, and (assuming uncorrelatedness and absence of systematic errors) the standard deviations measurement errors in /Wj and of errors in Vj will be given. The covariance matrix F is diagonal of elements and. Given the measured values, one can start the nonlinear reconciliation. Here, any 7 is a fixed parameter for each measurement. TTie nonlinear reconciliation makes use of the Jacobi matrix of the system of equations. In the given case, the matrix can be easily generated a priori. For example if V3 is measured, in the column we ll have... 

RHS of transport equation for weights in DQMOM Coefficient in generic algebraic equation in TDMA outline Components of the diagonal of the Jacobi matrix in PD algorithm... 

It is convenient to start from (8.2.31), with defined in (8.2.10) or (8.2.14), according to the choice of basis. An obvious way to satisfy this equation would be to make a series of 2 x 2 rotations , as in the Jacobi method for matrix diagonalization, in order to reduce the elements of - to zero. This approach has been used by Hinze and coworkers (see e.g. Hinze, 1973 Hinze and Yurtsever, 1979), who choose the 0-basis and consider the reduction of... 

The Jacobi method is probably the simplest diagonalization method that is well adapted to computers. It is limited to real symmetric matrices, but that is the only kind we will get by the formula for generating simple Huckel molecular orbital method (HMO) matrices just described. A rotation matrix is defined, for example. 

Thus the Jacobi procedure, by making many rotations of the elements of the operand matrix, ultimately arrives at the operator matrix that diagonalizes it. Mathematically, we can imagine one operator matr ix that would have diagonalized the operand matr ix R, all in one step... 

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

A comparison of the performance of the three algorithms for eigenvalue decomposition has been made on a PC (IBM AT) equipped with a mathematical coprocessor [38]. The results which are displayed in Fig. 31.14 show that the Householder-QR algorithm outperforms Jacobi s by a factor of about 4 and is superior to the power method by a factor of about 20. The time for diagonalization of a square symmetric value required by Householder-QR increases with the power 2.6 of the dimension of the matrix. 

There are several ways to diagonalize a real-symmetric matrix. The common strategy is to use orthogonal matrices to gradually reduce the magnitude of off-diagonal elements.18,19 The most rudimental and well-known approach is the Jacobi rotation, which zeros one such element at a time. The alternative rotation scheme devised by Givens can be used to make the process more efficient. Perhaps the most efficient and most commonly used approach is that... 

The eigenvalues of A can be find by solving the characteristic equation of (1.61). It is much more efficient to look for similarity transformations that will translate A into the diagonal form with the eigenvalues in the diagonal. The Jacobi method involves a sequence of orthonormal similarity transformations, 12,... such that A(<+1 = TTkAkTk. The matrix Tk differs from the identity... 

The Jacobi method is generally slower than these other methods unless the matrix is nearly diagonal. In SCF calculations one is faced with the non-orthogonal eigenvalue equation... 

Usually on the first iteration of an SCF calculation W is computed by the Schmidt orthogonalization method but thereafter W is chosen to be the C matrix from the previous iteration. This produces an F matrix which is nearly diagonal so the Jacobi method becomes quite efficient after the first iteration. Further, in the Jacobi method, F is diagonalized by an iterative sequence of simple plane-rotation transformations... 

The N x N matrix AmT1 AT will in general not be diagonal, so there will be cross terms of the kind PiPj in the expression for the kinetic energy. This may sometimes be inconvenient, and we shall see in the following how one may choose the matrix A in such a way that the kinetic energy is still diagonal in the new momenta. This leads to the so-called Jacobi coordinates that are often used in reaction dynamics calculations. 

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