Matrices sparse

An alternative to split operator methods is to use iterative approaches. In these metiiods, one notes that the wavefiinction is fomially "tt(0) = exp(-i/7oi " ), and the action of the exponential operator is obtained by repetitive application of //on a function (i.e. on the computer, by repetitive applications of the sparse matrix... 

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. 

A block Lanczos algorithm (where one starts with more than one vector) has been used to calculate the first 120 normal modes of citrate synthase [4]. In this calculation no apparent use was made of symmetry, but it appears that to save memory a short cutoff of 7.5 A was used to create a sparse matrix. The results suggested some overlap between the low frequency normal modes and functional modes detennined from the two X-ray conformers. 

Different processes like eddy turbulence, bottom current, stagnation of flows, and storm-water events can be simulated, using either laminar or turbulent flow model for simulation. All processes are displayed in real-time graphical mode (history, contour graph, surface, etc.) you can also record them to data files. Thanks to innovative sparse matrix technology, calculation process is fast and stable a large number of layers in vertical and horizontal directions can be used, as well as a small time step. You can hunt for these on the Web. 

The highest level of integration would be to establish one large set of equations and to apply one solution process to both thermal and airflow-related variables. Nevertheless, a very sparse matrix must be solved, and one cannot use the reliable and well-proven solvers of the present codes anymore. Therefore, a separate solution process for thermal and airflow parameters respectively remains the most promising approach. This seems to be appropriate also for the coupling of computational fluid dynamics (CFD) with a thermal model. ... 

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. 

Fig. 9. Sparse matrix storage schemes (a) a square matrix, (b) scheme I, (c) scheme II (linked lists).

Sparse matrices are ones in which the majority of the elements are zero. If the structure of the matrix is exploited, the solution time on a computer is greatly reduced. See Duff, I. S., J. K. Reid, and A. M. Erisman (eds.), Direct Methods for Sparse Matrices, Clarendon Press, Oxford (1986) Saad, Y., Iterative Methods for Sparse Linear Systems, 2d ed., Society for Industrial and Applied Mathematics, Philadelphia (2003). The conjugate gradient method is one method for solving sparse matrix problems, since it only involves multiplication of a matrix times a vector. Thus the sparseness of the matrix is easy to exploit. The conjugate gradient method is an iterative method that converges for sure in n iterations where the matrix is an n x n matrix. 

SQP. This is a sister code to GRG2 and available from the same source. The interfaces to SQP are very similar to those of GRG2. SQP is useful for small problems as well as large sparse ones, employing sparse matrix structures throughout. The implementation and performance of SQP are documented in Fan, et al. (1988). 

More variables are retained in this type of NLP problem formulation, but you can take advantage of sparse matrix routines that factor the linear (and linearized) equations efficiently. Figure 15.5 illustrates the sparsity of the Hessian matrix used in the QP subproblem that is part of the execution of an optimization of a plant involving five unit operations. 

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. 

Spin System Dynamics via Sparse Matrix Methodology. 

Both methods require the use of a broad range of crystallization solutions for the initial screening of crystals. These screens usually come in two types, grid screens and sparse matrix screens ... 

Sparse matrix screens These screens are based on the factorial or incomplete factorial statistical approach for designing the screening experiment (Jancarik and Kim, 1991). An extended version of this approach, called sparse matrix sampling has been developed to cover a very large number of conditions for initial screening. 

Jancarik J, Kim SH. 1991. Sparse matrix sampling a screening method for crystallization of proteins. J Appl Cryst 24 409-411. 

It has been proposed that a sparse matrix provides superior coverage to the selection of apparently optimal diversity reagents. However, the final collection of molecules that... 

One of the most popular refinement programs is the state-of-the-art package Refmac (Murshudov et ah, 1997). Refmac uses atomic parameters (xyz, B, occ) but also offers optimization of TLS and anisotropic displacement parameters. The objective function is a maximum likelihood derived residual that is available for structure factor amplitudes but can also include experimental phase information. Refmac boasts a sparse-matrix approximation to the normal matrix and also full matrix calculation. The program is extremely fast, very robust, and is capable of delivering excellent results over a wide range of resolutions. 

Cate, J. H. and Doudna, J. A. (1997). A sparse matrix approach to crystallizing ribozymes and RNA motifs. Method Mol. Biol. 74, 379-386. 

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