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Iterative Method – Obtaining Accurate Solutions in Solving Linear Equations
The linear equation consists of simple variables such as x and y or any letter of the alphabet, as well as signs and equal expressions. Each variable can be either a constant or the product of a constant.
Considerations for using variables:
o Must not be made up of exhibitors; x2
o Must not be multiplied or divided with each other; 3xy + 4.
o Must not be found under a square root sign.
Thus, linear expression is an instruction used to perform certain functions of adding, subtracting, multiplying and dividing numbers. These math components can generate an equation such as X + 3; 2x + 5; 3x + 5 years.
Learning the basics is useful for solving equations. A common form is the equation;
X + 2 = 5
To find the value of x, let x equal 1. Both sides must be equal to 5 to hold true. It must have both a correct answer. To balance the equation, both sides must use an equal sign. Terms added on one side must also be added on the other side. This is similar in multiplying and dividing both sides of the equation.
The iterative method is used to solve a problem by finding the exact solution, based on an initial guess. The basic idea repeats a set of steps that will generate an approximate final answer. It is opposed to direct methods which aim to solve problems via a limited sequence of operations.
The iterative method is useful for solving linear equations that involve a large number of variables. The iterative method depends on pre-conditioners in order to improve its performance. Preconditioners are the transformation matrix that ensures rapid convergence by overcoming the overhead of its construction. Without this, the method may fail to converge.
The two main classes of iterative methods are:
o Stationary iterative method
o And the non stationary method.
The stationary iterative method can perform the same iteration operation on the current vectors. It solves a linear system using an operator (a function that operates on another function).
It then forms a correction equation based on the measurement error, 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 mostly by zeros) which are easy to parallelize.
The stationary iterative method is one of the oldest methods. It is simple to understand although not as effective. Here are two examples of this method:
o Jacobi method
o and Gauss-Seidel method
The so-called Jacobi method is considered an algorithm (sequence of finite instructions) which determines the solution in each row and column, having the greatest absolute value. It solves each diagonal element and inserts an approximate value. The process is iterated but the convergence is still slow. It is named 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 over matrices with non-zero values. Thus, Convergence always guarantees that the matrix can be diagonally dominant and definitely positive.
Non-stationary relates to the recent development of our modern mathematics. It is more difficult to understand but it is very effective. The non-stationary is based on sequential orthogonal vectors which mainly depend on the iteration coefficient. Thus, this also accompanies calculations involving data changes at each iteration step.
Here are some of the types of methods used:
o Conjugate gradient method
o MINRES and SYMMLQ
o CG on normal equations
o Generalized minimal residual
o Biconjugate gradient
o Almost minimal residue
o Conjugate gradient square method
o Stabilized biconjugate gradient
o Chebyshev iteration
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