Special Methods to solve linear equations

8/9/2019 Special Methods to solve linear equations

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LINEAR EQUATIONS

NON-CONVENTIONAL METHODS TO SOLVE L.E


2/16

HOW DO WE SOLVE AN EQUATION

SYSTEM?

Algebraic methods like matrixes are used tosolve linear equation systems, furthermore, this

method is used to solve some another non-linear system in which we need to give a

solution.

As a consequence of using matrixes the methodsto find solutions are the result of algebraic

solution for matrixes.


3/16

SolutionMethods

Direct Methods

EliminacinGaussiana

Gauss conPivoteo

Gauss-Jordan

SistemasEspeciales

IterativeMethods

Jacobi

Gauss-Seidel

Gauss-Seidelwith relaxation


4/16

THESE ARE EXAMPLE OF MATRICES


5/16

ANOTHER EXAMPLE

A way to write matrices that is commonly to find at any place:

-

!

-

y

-

m

i

n

i

mnmm

inii

n

n

c

c

c

c

y

y

y

y

fff

fff

fff

fff

2

1

2

1

21

21

22221

11211

.

/

.

//

.

.

F y c

Fy = c


6/16

THOMAS METHOD THOMAS

ALGORITHM

This method emerge as a simplification of LU factorization but only if wehave a tri-diagonal matrix .

-

!

-

y

-

n

n

n

n

nn

nnn

r

r

rr

r

x

x

xx

x

ba

cba

cbacba

cb

1

3

2

1

1

3

2

1

111

313

222

11

//111

A x r

Note that a simply

form to identify when

to use this method iswhen your matrix is

banded.

We are going to solve

the system as usual as

LU for other matrices.

WE ALSO CAN SOLVE THIS METHOD AS A SIMPLIFICATION OF GAUSSIAN ELIMINATION


7/16

-

!

-

y

-

nn

nnn

nn

nnnn

nn

nn

ba

cba

cba

cba

cb

U

UU

UU

UU

UU

L

L

L

L

111

333

222

11

,

,11,1

3433

2322

1211

1,

2,1

32

21

1

1

1

1

1

111111

As what is usual on LU we are going to say that A = LU and using Doolitle whereLii=1 for i=1 till n, we finally have:

L U A

ote that the Lower atrix and the U er were si lify as LU ethod re uire utote that the Lower atrix and the U er were si lify as LU ethod re uire ut

what we o tain for oth of the are two diagonal of nu ers. ence the way towhat we o tain for oth of the are two diagonal of nu ers. ence the way to

solve had een si lified in order to find a solution; s ecially L.solve had een si lified in order to find a solution; s ecially L.


8/16

00

,

1

,11,,

1,1

1,1

1,

111

!!

!

!

!

!

n

nnnnnnn

nnn

nn

nnn

cy

Donde

ULbU

cU

U

aL

bU

Based on the matrix product showed before we

obtain these expressions

kkkkkkk

kkk

kk

k

kk

ULbU

cU

U

a

L

,11,,

1,1

1,11,

!

!

!

Now scanning from k=2 till n we finally have


9/16

-

!

-

y

-

n

n

n

n

nn

nn

r

r

r

r

r

d

d

d

d

d

L

L

L

L

1

3

2

1

1

3

2

1

1,

2,1

32

21

1

1

1

1

1

//11

11,

11

2

!

!

!

kkkkkdLrd

tillkFrom

rd

If LUx=r and Ux=d then Ld=r, hence:

L

d r

Base on a regressive

substitution


10/16

-

!

-

-

n

n

n

n

nn

nnnn

d

d

d

dd

x

x

x

xx

U

UU

UU

UUUU

1

3

2

1

1

3

2

1

,11,1

3433

2322

1211

//11

U x d

Finally we solveFinally we solve UxUx=d based on the regressive substitution=d based on the regressive substitution

nn

n

n

U

dx

Where

,

,

!

kk

n

kj

jkjk

kU

xUd

x

tillnkTo

,

1

,11


11/16

CHOLESKY DECOMPOSITION

Is a decomposition of a symmetric, positive-definite matrix into the product of a

lower triangle matrix and its conjugate transpose. When is applicable this

method is twice as efficient as LU decomposition for solving systems

TLU !

HENCE

bxLL

bAx

T !

!


12/16

-

-

y

-

nnnnnnnn

nnnnnnnn

nnnnnnnn

nnn

nnn

nn

nnnn

nnnnnn

nnn

nnn

nnnnnnnn

nnnnnn

nnnn

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

L

LL

LLL

LLLL

LLLLL

LLLLL

LLLL

LLL

LL

L

,1,2,2,1,

,11,12,12,11,1

,21,22,22,21,2

,21,22,22221

,11,12,11211

,

1,1,1

2,2,12,2

2,2,12,222

1,1,11,22111

,1,2,2,1,

1,12,12,11,1

2,22,21,2

2221

11

.

.

.

.

.

.

.

.

.

.

L LT

A

A =LLT

What was mention before shows that:


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From the product of the nth row of L and the nth columnFrom the product of the nth row of L and the nth column LLTT of we obtainof we obtain

that:that:

!

!

!

!

!

!

1

1

2

,

1

1

2

,2

2

1,2

2,2

2,2

1,2

22

1,

2

2,

2

2,

2

1,

n

j

jnnnnn

n

j

jnnnnn

nnnnnnnnnn

nnnnnnnnnn

a

a

aa

.

.

Once again

scanning fromk=1 till n we

obtain

!

!1

1

2

,

k

j

jkkkkk LaL


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!

!

!

!

2

1

,1,1,1,

1,1

2,12,2,12,1,11,1,

1,

1,1,11,2,12,2,12,1,11,

n

j

jnjnnnnn

nn

nnnnnnnnnnnn

nnnnnnnnnnnnnn

LLaL

L

LLLLLLaL

aLLLLLLLL

.

.

In the other way if we multiply the nth row of L with the (nIn the other way if we multiply the nth row of L with the (n--1) column of1) column ofLLTT we willwe will

have:have:

11

1

1

,,,,

ee

!

!

kidonde

LLaLi

j

jijkikik

sc i g fr till t i


15/16

APPLICATIONS

LinearLinear leastleast squaressquares:: Systems of the form Ax = b with A symmetric and

positive definite arise quite often in applications. For instance, the normal

equations in linear least squares problems are of this form.

MonteMonte CarloCarlo SimulationSimulation:: The Cholesky decomposition is commonly used inthe Monte Carlo method for simulating systems with multiple correlated

variables: The matrix of inter-variable correlations is decomposed, to give

the lower-triangular L.

NonNon--linearlinear optimizationoptimization:: Non-linear multi-variate functions may beminimized over their parameters using variants of Newton's method called

quasi-Newton methods.


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BIBLIOGRAPHY

CHAPRA, Steven C. y CANALE, Raymond P.:

Mtodos Numricos ara Ingenieros.

McGraw Hill 2002.

http://en.wikipedia.org/wiki/Cholesky_decomp

osition#Applications

http://math.fullerton.edu/mathews/n2003/Ch

oleskyMod.html

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Special Methods to solve linear equations

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