Band Jacobian

Hindmarsh, A. C. GEARB Solution of Ordinary Differential Equations Having Banded Jacobian, UCID-.30059, Rev. 1 Computer Documentation, Lawrence Livermore Laboratory, University of California (1975). 

Hindmarsh A. C. (1976) Preliminary Documentation of GEARIB. Solution of Implicit Systems of Ordinary Differential Equations with Banded Jacobians, Rep. UCID - 30130, Lawrence Livermore Laboratory, Livermore. 

To maintain a banded Jacobian structure, the eigenvalue is represented through a trivial differential equation,... 

The pressure p(z) is a function of z alone. Thus it could be carried as a single scalar dependent variable, rather than defined as a variable at each mesh point. However, analogous to the reasoning used in Section 16.6.2 for one-dimensional flames, carrying the extra variables has the important benefit of maintaining a banded Jacobian structure in the differential-equation solution. 

The main reason for slow computations is the dimensions of the sparse 160 x 160 Jacobian. Although other routines are available through the work of Hindmarsh (1, 2) to handle banded Jacobians or Jacobians with certain structures there are no commercial routines that could be used for the particular Jacobian that arises in the modeling of the low pressure pyrolysis of polydispersed coal particles. 

Hindmarsh, A.C., nGEARB Solution of Ordinary Differential Equations Having Banded Jacobians," Lawrence Livermore Laboratory Report UCID-30059, June 1977, Rev. 2. 

When the independent variables and functions are ordered as indicated by Eqs. (10-20) and (10-21), a banded jacobian matrix is obtained in which most of the elements lie along the principal diagonal as shown in Fig. 10-1. 

Selected entries from Methods in Enzymology [vol, page(s)] Computer programs, 240, 312 infrared S-H stretch bands for hemoglobin A, 232, 159-160 determination of enzyme kinetic parameter, 240, 314-319 kinetics program, in finite element analysis of hemoglobin-CO reaction, 232, 523-524, 538-558 nonlinear least-squares method, 240, 3-5, 10 to oxygen equilibrium curve, 232, 559, 563 parameter estimation with Jacobians, 240, 187-191. 

The effect is to keep the iteration matrix (Jacobian) banded, which considerably improves the efficiency of the Newton iteration that is used to solve the discrete problem. This procedure is equivalent to solving a simple first-order differential equation,... 

The Jacobian of the system is a square matrix, but importantly, because the residuals at any mesh point depend only on variables at the next-nearest-neighbor mesh point, the Jacobian is banded in a block-tridiagonal form. Figure 16.10 illustrates the structure of the Jacobian in the form used by the linear-equation solution at a step of the Newton iteration,... 

FIGURE 10 Change of the torus and the angular function with W o. (a) A succession of computed sections of invariant tori for various values of >/ >o (Brusselator, a = 0.0072). The centre point is indicated by (+). (b, c) The occupancy of the converged Jacobian for (u/gjo = 1.186667 and 1.3, respectively. The bumps on some of the circles are artifacts of the mesh and are associated with the almost vertical parts of the nonzero band. They can be eliminated by mesh adaptation. 

The computational domain is the unit square in u and v, and this was divided into a 15 x 15 mesh i.e., 225 elements, and 16 x 16 = 256 nodes, so 256 basis functions and 256 residual equations. The Jacobian matrix was banded with a total bandwidth of 35. The first solution computed was the minimal surface, for which the initial estimate was an hyperbolic paraboloid. The nonlinear system of residual equations was solved by Newton iteration on a Cyber 124, each iteration using about 1 second cpu time. For nearly all the surfaces calculated, the mesh was an even mesh over the entire unit square. However, for the surfaces just near the close-packed spheres (CPS) limit, the nodes were evenly spaced in the u-direction but placed as follows in the i -direction i = 0,1/60,1/30,0.05,0.075,0.1,0.15,0.2,0.3,0.4,0.5,0.6,0.7,... 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

The Jacobian for this problem is a sparse constant matrix (banded with bandwidth 5). The property... 

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