Conjugate Gradient Method

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Introduction

In civil engineering, conjugate gradient method can be defined as an algorithm for

solving numerical problems mainly involving linear equations. The coefficients of those

matrices are asymmetric and non-singular. The conjugate gradient method is mostly applied in

implementing iterative algorithms. This is again used in the analysis of sparse systems that are

too large to be broken down by a direct method or techniques like Cholesky decomposition and

LU decomposition.

Large sparse systems have equations that come up when trying to solve numerical

ordinary differential equations or optimization of problems. The conjugate gradient method is

used by civil engineers to solve optimization problems like the engineering problems involving

energy optimization. This approach was developed by mathematicians and engineers who

specialized in applied mathematics and advanced calculus. E. Stiefel and Hestenes discovered

this method of conjugate gradients. Currently, the Eigenvalue problem can be solved using the

conjugate gradient method.

Importance of the method

The conjugate gradient method gives a generalization to coefficient matrices. Most of

the nonlinear conjugate gradient method is used to provide the minima of any non-linear

equations.

For example with the following system of equations

Ax=B

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Vector x is taken to be known m× m matrix. A is a symmetric matrix, i.e. (A

=A). B is known.

We can denote the solution of this system by using vector x

∗

. We can prove that two vectors e

and f are conjugates if and only if e

A f=0. If A is symmetric non–zero, the left-hand side

gives the inner product. Vectors are conjugate if they are orthogonal in relation to the inner

product. The conjugate is a symmetric relation: if e is conjugate to f, then also f is conjugate to

Taking the conjugate vectors g and k, we can use them to approximate the value for x.

We can also use the conjugate gradient method as an iterative method. With the iterative

method, we can approximate the solution of systems though it takes a long time to carry out

the iterations.

Let’s assume the initial values for x

∗

by x

. The values can be assumed without any

loss, x = 0. If we start with x, then the solution from the successive iterations can be used to

get the approximate value of the system of equation. The solution comes from the fact that any

answer to the system of the equations is unique. For example in the following equation, the

conjugate constraint has an algorithm which bears a resemblance to orthonormalization. After

a long derivation, the formula gives the following expression.

From the above expression M, the algorithm provides a direct explanation of the

conjugate gradient method. The algorithm above states has to have storage of all previous

directions and vectors; in addition to many matrix-vector manipulations, thus, they are complex

to compute manually. However, one matrix-vector manipulation is required in each iteration.

The algorithm is used for solving AX =b where A is a real and positive-definite matrix-vector.

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The input vector is an estimate of the initial solution or zero. It has a different derivation and

formulation of the exact procedure.

The code used in Matlab

Function[a] =conjgrad(e,f,g)

q=f-e*a;

p=q;

qsold=q'*q;

form=1:length(f)

ap=a*p;

alpha=qsold/(p'*ap);

a=a+alpha*p;

q=q-alpha*Ap;

rsnew=q'*q;

ifsqrt(rsnew)<1e-10

break;

end

p=q+(rsnew/rsold)*p;

rsold=rsnew;

end

Example

Consider an example with the linear system Ax = b

The following are the two steps of the performance of the conjugate gradient method starting

with the initial guess to get the approximate solution to the system.

For reference, the exact solution is as follows:

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The first step is to determine vector r

associated with x

. This vector is calculated from the

expression r

= b - Ax

Being the first iteration, we will employ the residual vector r

as our preliminary search

direction p

; the method of choosing p will alter in auxiliary iterations. We now compute the

scalar α

using the relationship. We can determine the x using the formula

The above formula performs the first iteration. The result is an approved solution for the

approximated value of the system x

In the next step, we can determine the next residual vector by use of the formula provided

below. The equation simplifies the lengthy processes of calculating the approximated value

The next step includes the computation of the step scalar β

that is used to determine

the incoming search direction P1.

By use of the value of β0, determined from the above formulas the next search direction, p1

can be computed using the following formulae.

The values of the quantity α

1 can

be calculated using the value from the previous calculation.

Use of the following mathematical equations can calculate the new value.

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In the final step now the value of x

can be computed using similar steps like the ones used

to determine the value of x

The solution that is got after the calculations for x

is a more accurate method for estimation

of the solution than the solution for x

and x

.Thus, we can conclude that with more

iterations, the solution for a given system becomes more accurate.

The conjugate gradient method can be termed as a direct method that gives the exact value of

the solution after carrying out a finite number of iterations. The technique has a limit of being

unstable with regard to small perturbations.

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Works Cited

Adams, Loyce M., and J. L. Nazareth. Linear and nonlinear conjugate gradient-

related methods. Philadelphia: Society for Industrial and Applied

Mathematics, 1996. Print.

Hanke, Martin. Conjugate gradient type methods for ill-posed problems. Harlow,

Essex, England New York: Longman Scientific & Technical Wiley, 1995.

Mlek, Josef, and Zdenk Strako. Preconditioning and the conjugate gradient

method in the context of solving PDEs. Philadelphia, Pennsylvania: Society for

Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6,

Philadelphia, PA 19104, 2015. Print.

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