Matrix system solver

Parallelism-in-Matrix-ComputationsThis repository contains 5 sections of programs to solve various matrix computation problems. The following is based on the MPI/OpenMP programming paradigm, and was tested on four nodes of the MC cluster.1. Implement BBTSBlock banded triangular solver to solve Lx = b, where L is a banded lower triangular matrix with m subdiagonals.2. Implement a SPIKE-like algorithmSolve Lˆy = f, where Lˆ is a banded lower triangular matrix of order N = 32, 768 = 215, with t = 128 subdiagonals on a cluster of multicore nodes.The SPIKE-like algorithm should be implemented using MPI. In this algorithm, use BBTS (from Part (a)) to solve the system involvings Lˆi.3. Block LU Factorization with boostingSolve the dense linear system Ax = b using the approximate block LU Factorization algorithm with the procedure of ”diagonal boosting”Let α be a multiple of the unit roundoff, e.g. 10−6, and aj be the jth column of the updated matrix A after step j − 1. In step j, if the diagonal pivot does not satisfy |pivot| > α||aj ||1, its value is ”boosted” as, pivot = pivot + β||aj ||1, if pivot > 0, pivot = pivot − β||aj ||1, if pivot 4. The Spike algorithmImplement the Spike algorithm in Chapter 5.2.1 to solve Ax = b, where A is a banded matrix, on a cluster of multicore nodes.5. Sparse matrix solverData Source: a parallel C++ solver with libraries MPI, MKL and LAPACK based on Kaczmarz scheme to solve a sparse linear system Ax = f of order 1 millionImplemented the RCK scheme to do reordering in sparse matrix and utilized Block Gauss-Seidel and permutations to transform the model into independent least square problems within a parallel Conjugate Gradient framework, resulting in a more robust and scalable method compared with GMRES using ILU preconditionerTextbook: Parallelism in Matrix ComputationsJul.3 2016

Developer’s DescriptionWill guide you how to solve your math homework and textbook problems, anytime, anywhere.FX Math Solver is a comprehensive math software, based on...Will guide you how to solve your math homework and textbook problems, anytime, anywhere.FX Math Solver is a comprehensive math software, based on an automatic mathematical problem solving engine, and ideal for students preparing term math exams, ACT, SAT, and GRE:- Over 1,500 sample math problems and fully animated solution steps - Scientific calculator supported - Graphing calculator supported - Automatic problem solving and generation of fully animated step-by-step procedures for problems typed in by users - User friendly math problem expression editor (WYSIWYG mode)Math Solver covers problems at the level of Pre-Algebra, Algebra1, Algebra2, and Calculus courses:- Number operations: add, subtract, multiply, divide- Prime factorization- Mixed number- Complex number- Basic expression simplification: - polynomial - factorization, long-division - rational - radical - exponential and log/ln - sin, cos, tan, ..- Matrix- Equations, system of equations- Inequalities- Function- Graphing- Limit- Differentiation- Integral

1 to the fundamental polymer mode at port 2 which is given by |S21|^2 with port 1 at the input side, and port 2 at the output. However, since the device behaves symmetrically, we can get the same result by looking at |S12|^2. For more information about the S-matrix index mapping see EME solver analysis.Analysis and ResultsPressing the run button will calculate the modes at each cell. You can visualize the calculated modes by expanding the EME solver and cell group in the Objects Tree, then right-clicking the individual cell and selecting the result to visualize.To see the final field profile of the device as well as the S-matrix results, press the EME PROPAGATE button in the EME analysis window. Once the propagation is complete, profile monitor results and S-matrix results will be available, and can be visualized by right-clicking on the objects in the Objects Tree. The results for different propagation lengths can also be changed without having to re-calculate any modes. The field profile for a tapered region of length 10 um and 100 um are shown below.10 um taper (xz plot)100 um taper (xz plot)Scattering parameters relate the transmission and reflection coefficients for each port and input/output modes of the device. This is automatically calculated by the EME solver, and returned as the result of an EME solver region. The internal s-matrix includes all of the s-parameters for all the modes of all the ports, whereas the user s-matrix will contain only the s-parameters for the modes selected in the ports. Since we have 2 ports, and we are only interested in the fundamental mode at each port, the user s-matrix will be a 2 by 2 matrix, with elements S11, S12, S21 and S22.Length scanningThe propagation sweep widget allows you to scan the length of any cell group and calculate s-matrix results automatically. The S-matrix index mapping table allows you to quickly identify which s-matrix components correspond to which port and mode.Below, the transmission through the taper is plotted over taper lengths from 10 um to 200 um.The length scanning can also be done by running the script spot_size_converter.lsf.EME vs 3D FDTDWe also compare the EME results with 3D FDTD. The results between two solvers agree reasonably well, however they are done with completely different time scale. The EME simulation takes 3 minutes to simulate 101 different taper lengths (blue squares), whereas 3D FDTD takes 6