• About the convergence properties of the conjugate gradient method

    From Wisdom91@21:1/5 to All on Fri Jul 3 17:40:48 2020
    Hello..

    Read this:


    About the convergence properties of the conjugate gradient method

    The conjugate gradient method can theoretically be viewed as a direct
    method, as it produces the exact solution after a finite number of
    iterations, which is not larger than the size of the matrix, in the
    absence of round-off error. However, the conjugate gradient method is
    unstable with respect to even small perturbations, e.g., most directions
    are not in practice conjugate, and the exact solution is never obtained. Fortunately, the conjugate gradient method can be used as an iterative
    method as it provides monotonically improving approximations to the
    exact solution, which may reach the required tolerance after a
    relatively small (compared to the problem size) number of iterations.
    The improvement is typically linear and its speed is determined by the condition number κ(A) of the system matrix A: the
    larger is κ(A), the slower the improvement.

    Read more here:

    http://pages.stat.wisc.edu/~wahba/stat860public/pdf1/cj.pdf


    So i think my Conjugate Gradient Linear System Solver Library
    that scales very well is still very useful, read about it
    in my writing below:

    Read the following interesting news:

    The finite element method finds its place in games

    Read more here:

    https://translate.google.com/translate?hl=en&sl=auto&tl=en&u=https%3A%2F%2Fhpc.developpez.com%2Factu%2F288260%2FLa-methode-des-elements-finis-trouve-sa-place-dans-les-jeux-AMD-propose-la-bibliotheque-FEMFX-pour-une-simulation-en-temps-reel-des-
    deformations%2F

    But you have to be aware that finite element method uses Conjugate
    Gradient Method for Solution of Finite Element Problems, read here to
    notice it:

    Conjugate Gradient Method for Solution of Large Finite Element Problems
    on CPU and GPU

    https://pdfs.semanticscholar.org/1f4c/f080ee622aa02623b35eda947fbc169b199d.pdf


    This is why i have also designed and implemented my Parallel Conjugate
    Gradient Linear System Solver library that scales very well,
    here it is:

    My Parallel C++ Conjugate Gradient Linear System Solver Library
    that scales very well version 1.76 is here..

    Author: Amine Moulay Ramdane

    Description:

    This library contains a Parallel implementation of Conjugate Gradient
    Dense Linear System Solver library that is NUMA-aware and cache-aware
    that scales very well, and it contains also a Parallel implementation of Conjugate Gradient Sparse Linear System Solver library that is
    cache-aware that scales very well.

    Sparse linear system solvers are ubiquitous in high performance
    computing (HPC) and often are the most computational intensive parts in scientific computing codes. A few of the many applications relying on
    sparse linear solvers include fusion energy simulation, space weather simulation, climate modeling, and environmental modeling, and finite
    element method, and large-scale reservoir simulations to enhance oil
    recovery by the oil and gas industry.

    Conjugate Gradient is known to converge to the exact solution in n steps
    for a matrix of size n, and was historically first seen as a direct
    method because of this. However, after a while people figured out that
    it works really well if you just stop the iteration much earlier - often
    you will get a very good approximation after much fewer than n steps. In
    fact, we can analyze how fast Conjugate gradient converges. The end
    result is that Conjugate gradient is used as an iterative method for
    large linear systems today.

    Please download the zip file and read the readme file inside the zip to
    know how to use it.

    You can download it from:

    https://sites.google.com/site/scalable68/scalable-parallel-c-conjugate-gradient-linear-system-solver-library

    Language: GNU C++ and Visual C++ and C++Builder

    Operating Systems: Windows, Linux, Unix and Mac OS X on (x86)

    --

    As you have noticed i have just wrote above my Parallel C++ Conjugate
    Gradient Linear System Solver Library that scales very well, but here is
    my Parallel Delphi and Freepascal Conjugate Gradient Linear System
    Solvers Libraries that scale very well:

    Parallel implementation of Conjugate Gradient Dense Linear System solver library that is NUMA-aware and cache-aware that scales very well

    https://sites.google.com/site/scalable68/scalable-parallel-implementation-of-conjugate-gradient-dense-linear-system-solver-library-that-is-numa-aware-and-cache-aware

    PARALLEL IMPLEMENTATION OF CONJUGATE GRADIENT SPARSE LINEAR SYSTEM
    SOLVER LIBRARY THAT SCALES VERY WELL

    https://sites.google.com/site/scalable68/scalable-parallel-implementation-of-conjugate-gradient-sparse-linear-system-solver


    Thank you,
    Amine Moulay Ramdane.

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