Block-tridiagonal method

A better alternative approach is what will be called the Rudolph method [476], after the person who introduced it into electrochemical simulation. It was known before 1991 under various names, notably block-tridiagonal [280,412,470,471,528,570]. This comes from the fact that if one lumps the large matrix into a matrix of smaller matrices and vectors, the result is a tridiagonal system that is amenable to more efficient methods of solution. In the present context, we define some vectors... 

Standard specifications for the Naphtali-Sandholm method are Q-(including zero values) at each stage at which heat transfer occurs and sidestream flow ratio Sj or Sj (including zero values) at each stage at which a sidestream is withdrawn. However, the desirable block tridiagonal structure of the Jacobian matrix can still be preserved when substitute specifications are made if they are associated with the same stage or an adjacent stage. For example, suppose that for a reboiled absorber, as in Fig. 13- it is desired to specify a boil-up ratio rather than reboiler duty. Equation (13-95) for function is removed from the N(2C + 1) set of equations and is replaced by the equation... 

The residues (Nb-kNp at each node) are reduced to zero (a small positive number fixed by specifying an error tolerance at input) iteratively by computing corrections to current values of the unknowns using the Newton-Raphson method (14). Elements of the Jacobian matrix required by this method are computed from analytical expressions. The system of equations to be solved for the corrections has block tridiagonal form and is solved by use of a published software routine (1.5b... 

Equations (15-58), (15-59), and (15-60) are solved simultaneously by the Newton-Raphson iterative method in which successive sets of the output variables are produced until the values of the M, E, and H functions are driven to within some tolerance of zero. During the iterations, nonzero values of the functions are called discrepancies or errors. Let the functions and output variables be grouped by stage in order from top to bottom. As will be shown, this is done to produce a block tridiagonal structure for the Jacobian matrix of partial derivatives so that the Thomas algorithm can be applied. Let... 

Revise Equations (15-58) to (15-60) to allow two interlinked columns of the type shown below to be solved simultaneously by the SC method. Does the matrix equation that results from the Newton-Raphson procedure still have a block tridiagonal structure ... 

Apart from triangular factorization and Gaussian elimination, there exist other effective direct methods for block-banded systems. For instance, the odd-even reduction (the cyclic reduction) is effective for solving symmetric block tridiagonal systems Ax=b with (2 — 1) x (2 — 1), matrices of the form... 

Because of the stiffness which is inherent to most Chemical Engineering problems, implicit integrators are commonly used in simulations. These integration methods require multiple solutions of the underlying linear equation system. The SFD discretization (1.1) leads to block-tridiagonal matrices that can be solved very efficiently. But, as the order of approximation is only two,... 

The unknowns are taken in a large vector in the order [Rq, Pq, Co,ifo,Ri,Pi,Ci, ifi,..., RN,PN,CN,ifNV but are lumped into the vector of four-point vectors U = [Ri, Pi, Ci, ifiY, i = 0... N, to prepare for the block-tridiagonal procedure for solving the system. The system of equations (13.37) is nonlinear, and the Newton method is used to solve it. At each index i we have three 4x4 blocks in the Jacobian matrix L the left-hand block for the elements at index / — 1 M,, the middle block for index /, and Q the right-hand block for index t -I-1, that symbol chosen here in order to avoid clashes with the concentration symbol R. They produce a tridiagonal block system. For this example, three-point BDF was used, started with one BI step. 

The basic algorithm is Newton s method, which exploits the structured form of the system (tridiagonal block). 

The comparison of the results obtained by direct CO and by NFC calculation of the DOS-s show even in the case of the more complicated periodic poly(gly-gly-ser) chain excellent agreement (the band limits and the peaks in the DOS-s curves agree within 0.002 a.u. 0.05 eV which was the step length applied in the NFC procedure). Therefore one can expect that also in the case of disordered chains the NFC method in its matrix block form provides highly reliable results. For quasi-lD-systems there seems to be no other method which can compete with this technique. For 2- and 3D disordered systems unfortunately there is no way to write down a tridiagonal (or triblockdiagonal) matrix and therefore the method is not applicable (for further details see the paper of Ladik at ref. /2/). 

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