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Lecture 13 - LU Decomposition

CVEN 302

September 24, 2001

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Lectures Goals

LU Decomposition Crouts technique Doolittles technique Choleskys technique

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LU DecompositionA modification of the elimination method,called the LU decomposition. Thetechnique will rewrite the matrix as the

product of two matrices.

A = LU

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LU Decomposition

The technique breaks the matrix into a product of two matrices, L and U , L is alower triangular matrix and U is an upper triangular matrix.

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LU Decomposition

There are variation of the technique usingdifferent methods. Crouts reduction ( U has ones on the diagonal) Doolittles method( L has ones on the diagonal) Choleskys method ( The diagonal terms are the

same value for the L and U matrices)

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Decomposition

-

-

!

-

-

!

-

-

!

-

300

120

112

103

020

001

300

240

112

103

010

001100

5.010

5.05.01

306

040

002

036

240

112

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LU Decomposition SolvingUsing the LU decomposition

[A]{x} = [ L][U]{x} = [ L]{[U]{x}} = {b}Solve

[L]{y} = {b}

and then solve[U]{x} = {y}

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LU DecompositionThe matrices are represented by

-

-

-

44434241

34333231

24232221

14131211

44

3433

242322

14131211

44434241

333231

2221

11

000

00

0*

0

00

000

aaaa

aaaa

aaaa

aaaa

F

F F

F F F

F F F F

EEEE

EEE

EE

E

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Equation SolvingWhat is the advantage of breaking up one

linear set into two successive ones?

The advantage is that the solution of triangular set of equations is trivial to solve.

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Equation SolvingFirst step - forward substitution

,2,i ,y b1y

by

1i

1 j jiji

iii

11

11

-!

-!

!

!

EE

E

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Equation SolvingSecond step - back substitution

,11, Ni ,xy1x

yx

N

1iii

iii

NN

N N

-!

-!

!

!

F F

F

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LU Decomposition (Doolittles method)Matrix decomposition

-

!

-

-

44434241

34333231

24232221

14131211

44

3433

242322

14131211

434241

3231

21

000

00

0*

1

01

001

0001

aaaa

aaaa

aaaa

aaaa

u

uu

uuu

uuuu

l l l

l l

l

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Choleskys methodMatrix is decomposed into:

where, l ii = u ii

-

!

-

-

44434241

34333231

24232221

14131211

44

3433

242322

14131211

44434241

333231

2221

11

000

00

0*

0

00000

aaaa

aaaa

aaaaaaaa

u

uu

uuuuuuu

l l l l

l l l

l l l

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LU Decomposition (Crouts reduction)Matrix decomposition

-

!

-

-

44434241

34333231

24232221

14131211

34

2423

141312

44434241

333231

2221

11

1000

100

10

1

*0

00

000

aaaa

aaaa

aaaa

aaaa

u

uu

uuu

l l l l

l l l

l l

l

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Crouts ReductionThe method alternates from solving from thelower triangular to the upper triangular

4141

3131

2121

1111

al

al

al

al

!!!!

11

1414141411

11

1313131311

11

1212121211

l a

uaul

l auaul

l a

uaul

!p!

!p!

!p! Step 1 : sweep down Step 2 : sweep a cross

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Crouts ReductionSecond set of steps through the reduction

1421424242421241

1321323232321231

1221222222221221

ul al al ul

ul al al ul ul al al ul

!p!!p! !p!

Step 3 : sweep down

22

142124242424221421

22

132123232323221321

l

ul auaul ul

l ul a

uaul ul

!p!

!p! Step 4 : sweep a cross

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General formulation of Crouts

These are the general equations for thecomponent of the two matrices

n,2, ,i l

ulau

N,1,i i, ulal

ii

1i

1k k ik i

i

1

1k k ik ii

-

-

!e!

!e!

!

!

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ExampleThe matrix is broken into a lower and upper triangular matrices.

-

-

!

-

100

110

3/23/11

13/42

03/71

003

122

321

213

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ExampleSolve problem using LU decomposition

!

!

-

!

!

-

!

-

-

7143.0

0000.0

1429.1

7143.0

7143.0

6667.0

100

110

3/23/117143.0

7143.0

6667.0

3

1

2

13/42

03/71

003

31

2

100110

3/23/11

13/4203/71

003

3

2

1

3

2

1

3

2

1

3

2

1

3

2

1

x

x

x

x

x

x

y

y

y

y

y

y

x x

x

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LU ProgramsThere are two programs LU_ crout _ factor - the program does a Crout

decomposition of a matrix and returns the L and U matrices

LU_ solve uses an L and U matrix combinationto solve the system of equations.

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SummaryDefined a LU decomposition.Setup of the LU Crouts decomposition technique.Showed an example

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Homework

Check the H omework webpage