Tridiagonal

Dubai H-R and Quack M 1984 Tridiagonal Fermi resonance structure in the IR spectrum of the excited CH chromophore in CFgH J. Chem. Phys. 81 3779-91... 

Segall J, Zare R N, Dubai H R, Lewerenz M and Quack M 1987 Tridiagonal Fermi resonance structure in the vibrational spectrum of the CM chromophore in CHFg. II. Visible spectra J. Chem. Phys. 86 634-46... 

A tridiagonal matr ix has nonzero elements only on the pr incipal diagonal and on the diagonals on either side of the pr incipal diagonal. If the diagonals on either side of the principal diagonal are the same, the matrix is a symmetr ic tr idiagonal matr ix. 

The full bond order matrix is a symmetric tridiagonal matrix (Chapter 2). It is symmetric because the bond order Cji Y Na,ack is the same as tiie bond order Na,i, a,j. Elements off the tridiagonal (-.4472 in the butadiene example) are artifacts of the minimization atid should be disi egarded. The full bond order matrix for butadietie is... 

Run benzene using HMO. Write out the full bond order matr ix, enter ing zero for any element off the tridiagonal. What is the bond order of benzene Is there any Kekule-type alternation in this model ... 

A tridiagonal matrix is one in which the only nonzero entries he on the main diagonal and the diagonal just above andjust below the main diagonal. The set of equations can be written as... 

The sets of equations can he solved using the Newton-Raphson method. The first form of the derivative gives a tridiagonal system of equations, and the standard routines for solving tridiagonal equations suffice. For the other two options, some manipulation is necessary to put them into a tridiagonal form (see Ref. 105). 

To avoid such small time steps, which become smaller as Ax decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an /i X n grid to a series of 2n one-dimensional problems on an n grid. 

Since the continuity conditions apply only for i = 2,. . . , NT — 1, we have only NT — 2 conditions for the NT values of y. Two additional conditions are needed, and these are usually taken as the value of the second derivative at each end of the domain, y, y f. If these values are zero, we get the natural cubic splines they can also be set to achieve some other purpose, such as making the first derivative match some desired condition at the two ends. With these values taken as zero in the natural cubic spline, we have a NT — 2 system of tridiagonal equations, which is easily solved. Once the second derivatives are known at each of the knots, the first derivatives are given by... 

Equation-Tearing Procedures Using the Tridiagonal-Matrix... 

EQUATION-TEARING PROCEDURES USING THE TRIDIAGONAL-MATRIX ALGORITHM... 

Tridiagonal-Matrix Algorithm Both the BP and the SR equation-tearing methods compute hqnid-phase mole fractions in the same way by first developing linear matrix equations in a manner shown by Amundson andPontinen [Jnd. ng. Ch m., 50, 730 (1958)]. Equations (13-69) and (13-68) are combinedto eliminate yjj and yij + i (however, the vector yj stiU remains imphcitly in K j) ... 

FIG. 13 50 Tridiagonal -matrix equation for a column with five theoretical stages, (a) Original equation, (h) After forward elimination. 

Compute values of Xi j by solving Eqs. (13-75) through (13-79) by the tridiagonal-matrix algorithm once for each component. Unless all mesh equations are converged, X, Xi j 1 for each stage J. 

Compute a new set of values of the T) tear variables by solving simultaneously the set of N energy-balance equations (13-72), which are nonlinear in the temperatures that determine the enthalpy values. When linearized by a Newton iterative procedure, a tridiagonal-matrix equation that is solved by the Thomas gorithm is obtained. If we set gj equal to Eq. (13-72), i.e., its residual, the hnearized equations to be solved simultaneously are... 

In the inner-loop calculation sequence, component flow rates are computed from the MESH equations by the tridiagonal matrix method. The resulting bottoms-product flow rate deviates somewhat from the specified value of 50 lb mol/h. However, by modifying the component stripping factors with a base stripping factor, S, in (13-109) of 1,1863, the error in the bottoms flow rate is reduced to 0,73 percent. 

This prescription transforms the effective Hamiltonian to a tridiagonal form and thus leads directly to a continued fraction representation for the configuration averaged Green function matrix element = [G i]at,. This algorithm is usually continued... 

For general matrices the reduction by triangular matrices requires less computation and is probably to be preferred. But if A is hermi-tian, observe that the use of the unitary reduction produces a matrix H that is again hermitian, hence, that is tridiagonal in form, having zeros everywhere except along, just above, and just below the main diagonal. 

Once the tridiagonal form has been obtained, the method recommended for finding the roots depends, as remarked, upon a special property of such a matrix. Let the matrix be written... 

This method has very little other than its simplicity to recommend it in the form just described. But when a binary base is used, the corresponding procedure is to bisect the interval successively. Each bisection determines one additional binary digit to the approximation, it requires only the evaluation of the function, and the method is often efficient and accurate. The principle is used by Givens (Section 2.3) in finding the roots of a tridiagonal symmetric matrix. 

Transformations in Hilbert space, 433 Transition probabilities of negatons in, external fields, 626 Transport theory, 1 Transportation problems, 261,296 Transversal amplitude, 552 Transversal vector, 554 Transverse gauge, 643 Triangular factorization, 65 Tridiagonal form, 73 Triple product ensemble, 218 Truncation error, 52 Truncation of differential equations/ 388... 

The starting point in more a detailed exploration is the simplest systems of linear algebraic equations, namely, difference equations with special matrices in simplified form, for example, with tridiagonal matrices. 

Moreover, a lot of rather complicated problems in numerical analysis gives rise to the canonical problem, where a square (A -f-1) x (N-f l)-matrix of the corresponding system acquires a tridiagonal form... 

Solution estimation for difference bormdary-value problems by the elimination method. In tackling the first boundary-value problem difference equation (21) has the tridiagonal matrix of order TV — 1... 

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