Matrices tridiagonal

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... 

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... 

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. 

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... 

Thus emerged the system of algebraic equations with a tridiagonal matrix. Because of this form, the elimination method may be useful (see Chapter 1, Section 1). 

Proper evaluation of the necessary actions in solving problem (5) by the matrix elimination method is stipulated, as usual, by the special structures of the matrices involved. Because all the matrices are complete in spite of the fact that C is a tridiagonal matrix, O(iVf) arithmetic operations are required for determination of one matrix on the basis of all of which are known to us in advance. Thus, it is necessary to perform 0 Ni N2) operations in practical implementations with all the matrices j = 1,2,N-2- Further, 0 N ) arithmetic operations are required for determination of one vector with knowledge of and 0 Nf N2) operations for determination of all vectors Pj. 

The subsequent diagonalization of the symmetric tridiagonal matrix T is relatively straightforward and numerically inexpensive. It can be carried out by root-searching methods such as bisection if a small number of eigenvalues is of interest.18,19 When the entire spectrum is needed, on the other hand, one... 

Finally, the conversion of the tridiagonal matrix to a diagonal form yields the approximate eigenvalues of H.20 In particular... 

The Cullum-Willoughby test,27,37 on the other hand, was designed to identify the so-called spurious eigenvalues, rather than the converged eigenvalues. In particular, the tridiagonal matrix T and its submatrix, obtained... 

Equation (7.30) is solved with a tridiagonal matrix algorithm as described in Patankar (1980). First reform equation (7.29) into... 

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