Direct Methods for Solving Linear Equations Systems

Matrix Notation

A matrix is a rectangular array of elements represented by a single simbols. A set of horizontal elements called line ( or row ) and vertical , column.
The First subscript i always designates the number of the row in which is the element. The Second subscript j designates the column. ViewPDF

Matrix Operating Rules

Addition of two matrices, say, [A] and [B], is accomplished by adding corresponding terms in each matrix. The elements of the resulting matrix [C] are :

cij= aij + bij

Similarly, the Substraction of two matrices, say, [E] minus [F] is obtain by subtracting corresponding terms:

dij= eij-fij

The addition and Subtraction can be performed only between matrices having the same dimensions.

Both Addition and subtraction are Conmutative:

[A] + [B] = [B] + [A]

Addition and Subtraction are also associative, that is,

([A] + [B]) + [C] = [A] + ([B] +[C])

Matrix Multiplication and Division

NAIVE GAUSS ELIMINATION

Elimination of unknows

The Procedure:

1. The Ecuations were manipulated to eliminate one of the unknows from the ecuations.The result of this elimination step was that we had one equation with one unknown.

2. Consequently, this equation could be solved directly and the result back- substituted into one of the original equations to solve for the reaming unknown.

This basic approuch can be extended to large sets of equation by developing a systematic scheme or algorithm to eliminate unknows and to back- sustitute. Gauss Elimination is the most basic of these schemes ( The first phase is designed to reduce the set of equations to an upper triangular system).

PITFALLS OF ELIMINATION METHODS

Some pitfalls:

* Division by Zero. The primary reason that the foregoing technique is called " naive" is that

during both the elimination and the back-substitution phases, it is possible that a division by zero can occur.

* ill-Conditioned Systems. Well conditional systems are those where a small change in one or more of the coefficients results in a similar small change in the solution. Ill- Conditional systems are those where small changes in coefficients result in large changes in the solution.

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GAUSS JORDAN