§1 - National Tsing Hua University



§3.3 Gaussian elimination

Consider the following example

Example 3.1:

[pic]

([pic](, [pic]

([pic](, [pic]

[pic]

([pic](, [pic], yieds

[pic]

Here [pic], [pic], [pic] are called multipliers.

Then we solve [pic], [pic], [pic] by backward substitution [pic], [pic], [pic]. Use the matrix notation, we have

[pic]

From Example 3.1 we see that we reduce the problem of [pic] by eliminations into [pic] where [pic] is a upper triangular matrix.

[pic]

where we assume [pic], [pic], [pic]are called pivots. [pic] becomes

[pic]

We solve it by backward substitution,

[pic]

[pic], [pic].

In fact Gaussian elimination is equivalent to [pic] decomposition of matrix [pic]

[pic]

where

[pic] [pic]

where [pic] are multipliers. From Example 3.1

we have

[pic]

Remark 3.1: For the linear system of [pic] equations in [pic] unknowns i.e. [pic], [pic], [pic], [pic], we can apply elimination process to obtain a kind of “staircase pattern” or echelon form

[pic][pic] [pic]

and

[pic], [pic] is a [pic] lower triangular matrix of the form [pic]

Remark 3.2: Finding [pic] by Gaussian elimination. Let [pic] and the identity matrix [pic] can be written as [pic] where [pic].

Then [pic] can be written as

[pic]

or

[pic], [pic] [pic]

We say solve [pic] by Gaussian eliminations

[pic] [pic]

If we do more elimination process to reduce [pic] to [pic] (This is called Gauss-Jordan elimination) Then

[pic].

Example 3.2: Find [pic] where [pic] is given in Example 3.1

[pic]

[pic]

[pic]

[pic]

Remark 3.3: There is a analytic formula for [pic], [pic] where [pic]

[pic] [pic]

where [pic] is the [pic] submatrix of [pic] obtained by deleting [pic]-th row and [pic]-th column.

Remark 3.4: If we solve [pic] by Gaussian elimination, it costs [pic] multiplications. It cost to [pic] compute [pic] by Gauss-Jordan method. Hence in numerical computation, we don’t solve [pic] by [pic]. But in MATLAB, they solve it this way.

For the analytic formula [pic] of [pic], we never use it in numerical computation. If we compute [pic] directly by definition of [pic], it will cost [pic] multiplication. In fact if you want to compute [pic], you may use Gaussian elimination. For [pic], [pic].

-----------------------

((( (

((( (

((( (

yieds

( [pic]th

0

((( (

((( (

((( (

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download