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Iterative Methods
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
The method of Example 3 is referred to as Gauss-Seidel iteration. It simply uses newer information as soon as it becomes available (a simple idea that is widely applicable in scientific computing). Gauss-Seidel iteration corresponds to the matrix splitting Ax=b(U+L+D)x=b(L+D)x=−Ux+bx=−(L+D)−1Ux+(L+D)−1b
Cardiac Propagation Simulation
Published in Theo C. Pilkington, Bruce Loftis, Joe F. Thompson, Savio L-Y. Woo, Thomas C. Palmer, Thomas F. Budinger, High-Performance Computing in Biomedical Research, 2020
Andrew E. Pollard, Nigel Hooke, Craig S. Henriquez
The differential system in Equation 22 was integrated using the TRBDF method,33 which stepped from the known state ut at time t to the new state ut+ Δt via an intermediate state uγ. The unknown states uγ and ut+ Δt were computed as the solutions of two nonlinear algebraic systems fTR(uγ) and fBDF(ut+ Δt). The nonlinear systems were solved using a Newton-iterative method 34–36 This method searches for the solution by solving a sequence of linear systems whose coefficient matrices are Jacobians (∂f/∂u). The portion of each linear system corresponding to the membrane state equations was eliminated by a block factorization,37 and the remainder was solved by an iterative method. We have used a number of iterative methods and have found preconditioned conjugate-gradient type methods such as GMRES38 to be very effective for monodomain problems, but ineffective for bidomain problems. The latter are solved relatively efficiently by a matrix splitting induced by the Jacobian of a nearby state û.35,36
Cryogenic Solutions as a Tool to Characterize Red-and Blue-Shifting C–H···X Hydrogen Bonding
Published in Leonid Khriachtchev, Physics and Chemistry at Low Temperatures, 2019
Wouter A. Herrebout, Benjamin J. van der Veken
During the last decades, matrix isolation spectroscopy has been one of the most active fields in low-temperature vibrational spectroscopy.1 Although this technique remains unsurpassed for a variety of problems including, e.g., the study of free radicals and other unstable species, it has several disadvantages. Because it is difficult to produce sufficiently thick samples and because of strong scattering of the infrared (IR) beam by thicker matrixes, this type of spectroscopy is rarely used to explore weak bands. Moreover, the spectra of matrix isolated species are often complicated by various effects: bands may show a multiplet structure which can arise from rotation of the solute molecule in its trapping site, from the presence of different trapping sites in the matrix, from aggregation of solute molecules, etc. This matrix splitting is often difficult to distinguish from other spectral effects that are of interest for our studies and that arise from the formation of molecular complexes or from the occurrence of multiple conformations. Also, since species are studied in a rigid, solid medium, no direct information on the thermodynamic properties of the species studied is accessible.
On SOR-like iteration methods for solving weakly nonlinear systems
Published in Optimization Methods and Software, 2022
If A = M−N is a splitting of the matrix A and where then we can obtain the following fixed point equation where the parameter . That is Based on the fixed point equation (5), we can establish the following matrix splitting iteration methods for solving the weakly nonlinear equation (1), called the SOR-like iteration method.