# Parallel implementation of Conjugate Gradient Linear System Solver was updated.

## Parallel implementation of Conjugate Gradient Linear System Solver was updated.

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

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
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.

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