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More explanation so that you understand me more..
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amin...@gmail.com
Dec 2, 2019, 5:37:14 PM12/2/19
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Hello,
More explanation so that you understand me more..
You have to understand me more, as you have noticed my opensource software projects are not just opensource projects, because i am taking care of explaining how to use them inside the README file or inside
an HTML file when they are too complex, and you can for example notice it with my following software project of my Parallel C++ Conjugate Gradient Linear System Solver Library that scales very well that is very interesting, because i am simplifying the complex mathematics behind it by presenting it to you with my easy C++ interface, the Conjugate Gradient Method is the most prominent iterative method for solving sparse systems of linear equations. Unfortunately, many textbook treatments of the topic are written with neither illustrations nor intuition, and their victims can be found to this day babbling senselessly in the corners of dusty libraries. For this reason, a deep, geometric understanding of the method has been reserved for the elite brilliant few who have painstakingly decoded the mumblings of their forebears. Conjugate gradient is the most popular iterative method for solving large systems of linear equations. CG is effective for systems of the form A.x = b where x is an unknown vector, b is a known vector, A is a known square, symmetric, positive-definite (or positive-indefinite) matrix. These systems arise in many important settings, such as finite difference and finite element methods for solving partial differential equations, structural analysis, circuit analysis, and math homework
The Conjugate gradient method can also be applied to non-linear problems, but with much less success since the non-linear functions have multiple minimums. The Conjugate gradient method will indeed find a minimum of such a nonlinear function, but it is in no way guaranteed to be a global minimum, or the minimum that is desired.
But the conjugate gradient method is great iterative method for solving large, sparse linear systems with a symmetric, positive, definite matrix.
In the method of conjugate gradients the residuals are not used as search directions, as in the steepest decent method, cause searching can require a large number of iterations as the residuals zig zag towards the minimum value for ill-conditioned matrices. But instead conjugate gradient method uses the residuals as a basis to form conjugate search directions . In this manner, the conjugated gradients (residuals) form a
basis of search directions to minimize the quadratic function f(x)=1/2*Transpose(x)*A*x + Transpose(b)*x and to achieve faster speed and result of dim(N) convergence.
And here is My Parallel C++ Conjugate Gradient Linear System Solver Library that scales very well version 1.76:
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)
Thank you,
Amine Moulay Ramdane.
More explanation so that you understand me more..
You have to understand me more, as you have noticed my opensource software projects are not just opensource projects, because i am taking care of explaining how to use them inside the README file or inside
an HTML file when they are too complex, and you can for example notice it with my following software project of my Parallel C++ Conjugate Gradient Linear System Solver Library that scales very well that is very interesting, because i am simplifying the complex mathematics behind it by presenting it to you with my easy C++ interface, the Conjugate Gradient Method is the most prominent iterative method for solving sparse systems of linear equations. Unfortunately, many textbook treatments of the topic are written with neither illustrations nor intuition, and their victims can be found to this day babbling senselessly in the corners of dusty libraries. For this reason, a deep, geometric understanding of the method has been reserved for the elite brilliant few who have painstakingly decoded the mumblings of their forebears. Conjugate gradient is the most popular iterative method for solving large systems of linear equations. CG is effective for systems of the form A.x = b where x is an unknown vector, b is a known vector, A is a known square, symmetric, positive-definite (or positive-indefinite) matrix. These systems arise in many important settings, such as finite difference and finite element methods for solving partial differential equations, structural analysis, circuit analysis, and math homework
The Conjugate gradient method can also be applied to non-linear problems, but with much less success since the non-linear functions have multiple minimums. The Conjugate gradient method will indeed find a minimum of such a nonlinear function, but it is in no way guaranteed to be a global minimum, or the minimum that is desired.
But the conjugate gradient method is great iterative method for solving large, sparse linear systems with a symmetric, positive, definite matrix.
In the method of conjugate gradients the residuals are not used as search directions, as in the steepest decent method, cause searching can require a large number of iterations as the residuals zig zag towards the minimum value for ill-conditioned matrices. But instead conjugate gradient method uses the residuals as a basis to form conjugate search directions . In this manner, the conjugated gradients (residuals) form a
basis of search directions to minimize the quadratic function f(x)=1/2*Transpose(x)*A*x + Transpose(b)*x and to achieve faster speed and result of dim(N) convergence.
And here is My Parallel C++ Conjugate Gradient Linear System Solver Library that scales very well version 1.76:
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)
Thank you,
Amine Moulay Ramdane.
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