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Introduction
Linear Eqn. Solvers
Systems of linear algebraic equations occur abundantly in most fields of science and engineering.
Examples are:
structural analysis (civil engineering),
heat conduction (mechanical and chemical engineering),
analysis of power grids (electrical engineering),
production planning (economics),
regression analysis (statistics),

One of the most common sources of linear algebraic from the discretisation of ordinary or partial differential equations or integral equations.

A linear equation is an equation that can be expressed as,
a0x0 + a1x1 + … = b

where a0, a1, … and b are constants.

A finite set of linear equations is called a system of linear equations or a linear system. A set of numbers s0, s1, … is a solution to a linear system if and only if the substitutions a0 = s0, a1 = s1, … satisfies every equation in the system.

A linear system of n linear equations in n variables
a0,0x0+ a0,1x1+…a0,n-1xn-1=b0
a1,0x0+ a1,1x1+…a1,n-1xn-1=b1
an-1,0x0+ an-1,1x1+…an-1,n-1xn-1=bn-1
is usually expressed as Ax = b, where A is an n × n matrix containing the ai,js and x and b are n-element vectors storing xis and bis.

There are two methods to solve a system of linear equations, direct or iterative.
 
Direct Methods
Direct Methods are suitable for solving dense systems of linear equations. Upper triangular systems can be solved using the Back substitution algorithm. The Gaussian Elimination algorithm transforms a dense system into an upper triangular system.
 
Iterative Methods
The discretisation of partial differential equations often results in the creation of spare systems of linear equations. Iterative methods are more appropriate for these systems than Gaussian elimination. Iterative methods solve the system of linear equations by generating a series of increasingly better approximations to the solution vector. Algorithms include the Jacobi method, the Gauss-Seidel method, (which converge slowly to the solution), and the conjugate gradient method which has a higher rate of convergence.

Last modified 14 Jan 06