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Linear Stage

Iterative Method: Obtaining Accurate Solutions in Solving Linear Equation

Linear Equation is a mathematical or algebraic equation that has one or more variables, equal sign and linear expressions. The Iterative Method is one good approach in solving linear equations. A wide array of techniques is being Used in obtaining more accurate solutions for the linear system.

Linear equation consists of simple variables like x and y or any letter in the alphabet, along with equal signs and expressions. Each variable can either be a constant or product of a constant.

Considerations on using variables:

• Should not consist of exponents; x2
• Should not be multiplied or divided with each other; 3xy + 4.
• Should not be found under a square root sign.

Thus, linear expression is a statement used in performing certain functions of adding, subtracting, multiplying and dividing of numbers. These mathematical components can generate an equation such as X + 3; 2x + 5; 3x + 5y.

Learning the basics is useful in solving equations. One common form is the equation; X + 2 = 5

To find the value of x, let x be equal to 1. Both sides must be equal to 5 so as to remain to be true. It must have both one correct answer. To balance the equation, both sides should use an equal sign. Terms being added to one side should be also added to the other side. This is similar in multiplying and dividing both sides of the equation.

The iterative method is being used to solve a problem by finding the exact solution, basing from an initial guess. The basic idea repeats a set of steps which will generate an approximate final answer. It contrasts direct methods which aim to solve problems via a limited sequence of operations.

The iterative method is useful in solving linear equations which involve a large number of variables. The iterative method depends on the pre-conditioners in order to improve its performance. Pre-conditioners are the transformation matrix which ensures a fast convergence in overcoming extra cost for its construction. Without it, the method may fail to converge.

The two main classes of iterative methods are:

• Stationary Iterative Method
• And the Non-stationary Method.

The Stationary Iterative Method can perform the same operation of iteration on current vectors. It solves a linear system with the use of an operator (a function which operates on another function).

It then forms a correction equation based on the error of measurement, repeating the process entirely. The Stationary Method is simple to implement and analyze but its convergence can be limited to a class of matrices (mathematical tables). It works well with sparse matrices (a matrix populated primarily with zeros) which are easy to parallelize.

The Stationary Iterative Method is one of the oldest methods. It is simple to understand although it is not as effective. Two examples of this method would include the

• Jacobi Method
• and Gauss-Seidel Method

The so-called Jacobi Method is regarded as an algorithm (sequence of finite instructions) that determines the solution in each row and column, having the largest absolute value. It solves each diagonal element and plugs in an approximate value. The process is iterated but the convergence is still slow. It is termed after Carl Gustav Jakob Jacobi, a German mathematician.

On the other hand, the Gauss-Seidel method was named after Carl Friedrich Gauss and Philipp Ludwig von Seidel. It is an improved version of Jacobi. If Jacobi converges, Gauss-Seidel converges faster. The method can be defined diagonally on matrices with non-zero values. Thus, Convergence still guarantees that the matrix can be diagonally dominant and definitely positive.

Non-stationary pertains to the recent development in our modern mathematics. It is harder to understand but it is highly effective. Non-stationary is based on sequential orthogonal vectors that mainly depend on the iteration co-efficient. Thus, it also goes with the computations involving data changes at each stage of iteration. Here are some of the method types being used:

• Conjugate Gradient Method
• MINRES and SYMMLQ
• CG on the Normal Equations
• Generalized Minimal Residual
• BiConjugate Gradient
• Quasi Minimal Residual
• Conjugate Gradient Square Method
• BiConjugate Gradient Stabilized
• Chebyshev Iteration

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