Sparse matrix techniques

The LIN method (described below) was constructed on the premise of filtering out the high-frequency motion by NM analysis and using a large-timestep implicit method to resolve the remaining motion components. This technique turned out to work when properly implemented for up to moderate timesteps (e.g., 15 Is) [73] (each timestep interval is associated with a new linearization model). However, the CPU gain for biomolecules is modest even when substantial work is expanded on sparse matrix techniques, adaptive timestep selection, and fast minimization [73]. Still, LIN can be considered a true long-timestep method. 

Gunn, D. J. (1977) Inst. Chem. Eng., 4th Annual Research Meeting, Swansea, April. A sparse matrix technique for the calculation of Unear reactor-separator simulations of chemical plant. 

Like the time propagation, the major computational task in Chebyshev propagation is repetitive matrix-vector multiplication, a task that is amenable to sparse matrix techniques with favorable scaling laws. The memory request is minimal because the Hamiltonian matrix need not be stored and its action on the recurring vector can be generated on the fly. Finally, the Chebyshev propagation can be performed in real space as long as a real initial wave packet and real-symmetric Hamiltonian are used. 

Warner [176] has given a comprehensive discussion of the principal approaches to the solution of stiff differential equations, including a hundred references among the most pertinent books, papers and application packages directed at simulating kinetic models. Emphasis has been put not only on numerical and software problems such as robustness, improving the linear equation solvers, using sparse matrix techniques, etc., but also on the availability of a chemical compiler, i.e. a powerful interface between kineticist and computer. 

Barker, V. A. (ed.). Sparse Matrix Techniques—Copenhagen 1976, Lecture Notes in Mathematics 572, Springer-Verlag, New York (1977). 

Two extremes are encountered in flowsheeting software. At one extreme, the entire set of equations (and inequalities) representing the process is employed. This representation is known as the equation-oriented method of flowsheeting. The equations can be solved in a sequential fashion analogous to the modular representation described below or simultaneously by Newton s method, Broyden s method, or by employing sparse matrix techniques to reduce the extent of matrix manipulations. Refer to the review by Evans and Chapter 5. ... 

The recent application of sparse-matrix techniques combined with computer optimization (vectorization) techniques has, however, improved the speed of Gear s code substantially, so that this advanced algorithm can now be used to study complex problems in multi-dimensional models (see e.g., the SMVGEAR package developed by Jacobson (1995 1998) and Jacobson and Turco, 1994). 

Vol. 572 Sparse Matrix Techniques, Copenhagen 1976. Edited by V A. Barker. V, 184 pages. 1977. 

Jacobian is not usually calculated at each iteration, and not even at every timestep. Further time is saved by using sparse matrix techniques to take advantage of the fact that the Jacobian usually possesses many zero elements (cf. equation (2.52) for example). Sparse matrix techniques are similarly used in solving equation (2.78) once the Jacobian has been found. Finally, the integration routine will seek to lengthen the timestep to the maximum extent consistent with a defined accuracy criterion, to take advantage of the strong stability properties of the implicit method. 

The use of sparse matrix techniques makes computation times unpredictable but far better than the would suggest. 

Given this sinprising behavior, the computation time depends on how efficiently the individual steps can be executed. This is a very active area of research and one that is exploring new parallel and vector computer architectures. (As with the simplex method, if the A matrix were completely dense, the computation time would increase with the cube of the number of constraints, but sparse matrix techniques make this rather meaningless.)... 

George J.A. (1977) Solutions of Linear Systems of Equations Direct Methods for Finite Element Probelms. In Barker V.A. (ed) Sparse Matrix Techniques. Lecture notes in mathematic, Vol. 572, Springer Berlin, Heidelberg, New York, pp 52-101. 

As the constrained system is described redundantly, the resulting ODEs are not minimal. A considerable reduction of the computational cost of the right hand side of the ODEs results from describing the free system with a constant and diagonal mass-matrix. In addition, the linear algebra computations have to be well organised and the sparsity of the system-matrices exploited. Since the sparsity structure is independent of time, the reduction obtained by sparse matrix techniques is considerable even for relatively small systems. This is discussed in section 3. 

The Lua language is ideally suited to implement sparse matrix techniques. A table in Lua is an associative array that is efficiently implemented in flie native language in terms of both storage allocation and access speed. A table array can be defined in the language and only the non-zero elements simply be defined. The... 

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