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Hello all,

Parallel implementation of Conjugate Gradient Linear System Solver was updated.

I have corrected a bug so that it works correctly when you useonlya
asingle thread.

Description:

The Parallel implementation of Conjugate Gradient Linear System Solver that

i programmed here is designed to be used to solve large sparse systems of

linear equations where the direct methods can exceed available machine memory

and/or be extremely time-consuming. for example the direct method of the Gauss

algorithm takes O(n^2) in the back substitution process and is dominated by

Description:

The Parallel implementation of Conjugate Gradient Linear System Solver that

i programmed here is designed to be used to solve large sparse systems of

linear equations where the direct methods can exceed available machine memory

and/or be extremely time-consuming. for example the direct method of the Gauss

algorithm takes O(n^2) in the back substitution process and is dominated by

the O(n^3)forward elimination process, that means, if for example an
operation

takes 10^-9 second and we have 1000 equations , the elimination process in

the Gauss algorithm will takes 0.7 second, but if we have 10000 equations
in the

system , the elimination process in the Gauss algorithm will take 11
minutes !.

This is why i have develloped for you the Parallel implementation of
Conjugate

Gradient Linear System Solver in Object Pascal, that is very
fast.

You have only one method to use that is Solve()

function TParallelConjugateGradient.Solve(var A: arrarrext;var B,X:VECT;var

RSQ:DOUBLE;nbr_iter:integer;show_iter:boolean):boolean;

The system: A*x = b

The important parameters in the Solve() method are:

A is the matrix , B is the b vector, X the initial vector x,

nbr_iter is the number of iterations that you want

and show_iter to show the number of iteration on the screen.

RSQ is the sum of the squares of the components of the residual vector

A.x - b.

I have got over 3X scalability on a quad core.

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,

You have only one method to use that is Solve()

function TParallelConjugateGradient.Solve(var A: arrarrext;var B,X:VECT;var

RSQ:DOUBLE;nbr_iter:integer;show_iter:boolean):boolean;

The system: A*x = b

The important parameters in the Solve() method are:

A is the matrix , B is the b vector, X the initial vector x,

nbr_iter is the number of iterations that you want

and show_iter to show the number of iteration on the screen.

RSQ is the sum of the squares of the components of the residual vector

A.x - b.

I have got over 3X scalability on a quad core.

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,

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

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

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

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.

Jacobi serial complexity is O(N^2) and Conjugate gradient serial complexity

is O(N^3/2).

Jacobi serial complexity is O(N^2) and Conjugate gradient serial complexity

is O(N^3/2).

Please look at the test.pas example inside the zip file, compile and
execute

it...

You can download Parallel implementation of Conjugate Gradient Linear

System Solver from:

http://pages.videotron.com/aminer

Thank you,

Amine Moulay Ramdane.

it...

You can download Parallel implementation of Conjugate Gradient Linear

System Solver from:

http://pages.videotron.com/aminer

Thank you,

Amine Moulay Ramdane.

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