linear alg equ2

8/13/2019 Linear Alg Equ2

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Numerical Methods for Eng [ENGR 391] [Lyes KAEM !""#]

)ooooooooouffffffffff ***.

+or a general nn matri" !ou have to appl! the following e"pressions to find the LU

decomposition of a matri" [A]#

=

=

1

1

j

kkjikijij ulal , i-, i$1/n

=

=

ii

i

k

kjikij

ijl

ula

u

1

1, i0-, -$!(/n

and 1=iiu , i$1/n

$$$%!% -olution of e,uations

2ow to solve our s!stem of linear equations we can e"press our initial s!stem#

[ ]{ } { }bxA =Under the following form

[ ]{ } [ ][ ]{ } { }bxULxA ==

&o find the solution { }x we define first a vector{ }z #

{ } [ ]{ }xUz =

)ur initial s!stem becomes then# [ ]{ } { }bzL =

Linear Alge+raic E,uations 53

345)6&A2& 2)&7

As for the (( matri" 8see above9 it is better to follow a certain order when computing the

terms of the [L] and [U] matrices. &his order is# l i1u1-, liu-, /,lin:1un:1-, lnn.

7"ample

+ind the LU decomposition of the following matri" using 'routs method#

[A]$

333231

232221

131211

aaa

aaa

aaa

$

223

134

112


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As [L] is a lower triangular matri" the { }z can be computed starting b! ;1until ;n. &hen the valuesof { }x can be found using the equation#

{ } [ ]{ }xUz =

as [U] is an upper triangular matri" it is possible to compute { }x using a bac< substitutionprocess starting "nuntil "1. [=ou will better understand with an e"ample /]

&he general form to solve a s!stem of linear equations using LU decomposition is#

nil

zlb

z

l

bz

ii

i

k

kiki

i ,...,3,2;

1

1

11

11

=

=

=

=

And

1,2,...,2,1;1

==

=

+=

nnixuzx

zx

n

ik

kikii

nn

$$$%3% (holes.i)s method for symmetric matrices


7"ample

>olve the following equations using the LU decomposition#

==+

=+

634

42

321

321

xxx

xxx

2ote on storage of [A], [L], [U]

1: 3n practice the matrices [L] and [U] do not need to be stored separatel!. ?! omittingthe ;eroes in [L] and [U] and the ones in the diagonal of [U] it is possible to store theelements of [L] and [U] in the same matri".

: 2ote also that in the general formula for LU decomposition once an element of thematri" [A] is used it is not needed in the subsequent computations. Hence the

elements of the matri" generated in point 819 above can be stored in [A]


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3n man! engineering applications the matrices involved will be s!mmetric and defined positive. 3tis then better to use the 'holes


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$$$%2%1% ector and matri norms

A norm is a real:valued function that provides a measure of the si;e or length of multicomponentmathematical entities#

+or a vectorx #

=

nx

x

x

x

.

.

.

2

1

&he 7uclidean norm of this vector is defined as#

( )21

22

2

2

1 ... nxxxx +++=

3n general Lp@norm of a vector x is defined as#

Pn

i

P

iP xL

1

1

= =

+or a matri", the first and infinit! norms are defined as#

[ ] =

=

n

i

ijnj

aA1

11 max $ ma"imum column sum

[ ] =

=

n

j

ijni

aA1

1max $ ma"imum of row sum.

$$$%2%!% Matri condition num+er

&he matri" condition number is defined as#


2ote

3f the value of 8p9 is increased to infinit! in the above e"pression the value of the Lnorm will

tend to the value of the largest component ofx #

ixL max=


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[ ] 1= AAACond

+or a matri" [A] we have that# cond [A] 1

and

[ ] AA

ACondx

x

&herefore the error on the solution can be as large as the relative error of the norm of [A]multiplied b! the condition number.3f the precision on [A] is t:digits 81:t9 and 'ond[A]$1' the solution on ["] ma! be valid to onl! t:cdigits 81c:t9.

$$$%4% 5aco+i iteration Method

&he Bacobi method is an iterative method to solve s!stems of linear algebraic equations.'onsider the following s!stem#

=+++++

=++++

nnnnnnnn

nn

bxaxaxaxa

bxaxaxaxa

...

.

.

...

332211

11313212111

&his s!stem can be written under the following form#

( )

( )

=

=

1,1,22111

13132121

11

1

...1

.

.

...1

nnnnnnn

nn

nn

xaxaxaba

x

xaxaxaba

x

&he general formulation is#

=

=

n

ijj

jiji

ii

i xab

a

x,1

1, i$1/n

Here we start b! an initial guess for "1, ", /, "n and we compute the new values for the ne"titeration. 3f no good initial guess is available we can assume each component to be ;ero.%e generate the solution at the ne"t iteration using the following e"pression#

=

=

+n

ijj

k

jiji

ii

k

i xaba

x,1

1 1, i$1/n, and for


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&he calculation must be stopped if#

+

k

i

k

i

k

i

x

xx 1

, is the desired precision.

3t is possible to show that a sufficient condition for the convergence of the Bacobi method is#

=

>n

ijj

ijii aa,1

$$$%#% Gauss6-eidel iteration Method

3t can be seen that in Bacobi iteration method all the new values are computed using the valuesat the previous iteration. &his implies that both the present and the previous set of values have tobe stored. Gauss:>eidel method will improve the storage requirement as well as theconvergence.

3n Gauss:>eidel method the values11

2

1

1 ,...,,

+++ ki

kk xxx computed in the current iteration as

well as knkiki xxx ,...,, 32 ++ are used in finding the value 11++kix . &his implies that alwa!s the

most recent a''roimationsare used during the computation. &he general e"pression is#

=

= +=

++1

1 1

11 1 i

j

n

ijOLD

k

iij

NEW

k

jiji

ii

k

i xaxaba

x

, i$1...n,


8/8


OLD

i

NEW

i

NEW

i xxx )1( +=

And usuall! 00

3f 001 we are using under:rela"ation used to ma

linear alg equ2

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