

Equation Solvers 
 Numerical MethodsEquation SolversLinear Algebraic Equation 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,
a_{0}x_{0} + a_{1}x_{1} + … = b
where a_{0}, a_{1}, … 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 s_{0}, s_{1}, … is a solution to a linear system if and only if the substitutions a_{0} = s_{0}, a_{1} = s_{1}, … satisfies every equation in the system.
A linear system of n linear equations in n variables
 a_{0,0}x_{0}  + a_{0,1}x_{1}  +…  a_{0,n1}x_{n1}  =  b_{0} 
 a_{1,0}x_{0}  + a_{1,1}x_{1}  +…  a_{1,n1}x_{n1}  =  b_{1} 
 …      
 a_{n1,0}x_{0}  + a_{n1,1}x_{1}  +…  a_{n1,n1}x_{n1}  =  b_{n1} 
is usually expressed as Ax = b, where A is an n × n matrix containing the a_{i,j}s and x and b are nelement vectors storing x_{i}s and b_{i}s.
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 GaussSeidel method, (which converge slowly to the solution), and the conjugate gradient method which has a higher rate of convergence.
Last modified 14 Jan 06

