School of Mathematics · Review MatrixEquations ConvergenceProperties Summary CONVERGENCE...

ReviewMatrix Equations

Convergence PropertiesSummary

CONVERGENCE PROPERTIES OF ITERATION

SCHEMES

Dr. Johnson

School of Mathematics

Semester 1 2008

Dr. Johnson MATH65241



OUTLINE

1 REVIEW

2 MATRIX EQUATIONS

Rearranging the MatrixIteration Matrix Equations

3 CONVERGENCE PROPERTIES

The Iteration MatrixProperties of ConvergenceFurther Convergence Properties

4 SUMMARY




OUTLINE

1 REVIEW

2 MATRIX EQUATIONS




4 SUMMARY




OUTLINE

1 REVIEW

2 MATRIX EQUATIONS




4 SUMMARY




OUTLINE

1 REVIEW

2 MATRIX EQUATIONS




4 SUMMARY




RELAXATION METHODS

The model equations can be rewritten

wi+1,j − 2wi,j +wi−1,j

∆x2+

wi,j+1 − 2wi,j +wi,j−1

∆y2= fi,j,

How up-to-date the values are define different schemes

We can over-relax to improve convergence

wq+1i,j = (1− ω)w

qi,j + ωw∗

i,j

and also solve a line (or block) using a direct method.




RELAXATION METHODS

The model equations can be rewritten

wi+1,j − 2wi,j +wi−1,j

∆x2+

wi,j+1 − 2wi,j +wi,j−1

∆y2= fi,j,

How up-to-date the values are define different schemes

We can over-relax to improve convergence

wq+1i,j = (1− ω)w

qi,j + ωw∗

i,j

and also solve a line (or block) using a direct method.





OUTLINE

1 REVIEW

2 MATRIX EQUATIONS




4 SUMMARY





MATRIX EQUATION

Given some PDE problem, it can always be reduced to thesolution of a linear system

Ax = b

where A = (ai,j) is an (n× n) matrix, and x, b are (n× 1)column vectors.





MATRIX EQUATION

Let write the matrix A in the form A = D+ L+U where D,L, U are the diagonal matrix, lower and upper triangularparts of A, ie

D =

a1,1 00 a2,2 0

0 a3,3 0. . .

. . .

0 an−1,n−1 00 an,n





MATRIX EQUATION

U =

0 a1,2 . . .0 0 a2,3

0 0 a3,4

0. . .

0. . .

...0 0 an−1,n

0 0





MATRIX EQUATION

L =

0 0a2,1 0 0... a3,2 0 0

. . .. . .

an−1,n−2 0 0. . . an,n−1 0





MATRIX EQUATION

Then we may write

Ax = b

as

Lx+Dx+Ux = b.

In this way we can express our iteration schemes in termsof matrices.





OUTLINE

1 REVIEW

2 MATRIX EQUATIONS




4 SUMMARY





ITERATIVE METHODS

Jacobi iteration may be written as:

x(k+1) = D−1(b− (L+U)x(k))

Gauss-Seidel iteration may be written as:

x(k+1) = (D+ L)−1(b−Ux

(k))





ITERATIVE METHODS

Jacobi iteration may be written as:

x(k+1) = D−1(b− (L+U)x(k))

Gauss-Seidel iteration may be written as:

x(k+1) = (D+ L)−1(b−Ux

(k))





OUTLINE

1 REVIEW

2 MATRIX EQUATIONS




4 SUMMARY





GENERAL ITERATION SCHEME

In general an iteration scheme may be written as

x(k+1) = Px(k) +Q,

where P is the iteration matrix

For the Jacobi scheme scheme we have

P = PJ = D−1(−L−U)

For the Gauss-Seidel scheme

P = PG = (D+ L)−1(−U).





GENERAL ITERATION SCHEME

In general an iteration scheme may be written as

x(k+1) = Px(k) +Q,

where P is the iteration matrix

For the Jacobi scheme scheme we have

P = PJ = D−1(−L−U)

For the Gauss-Seidel scheme

P = PG = (D+ L)−1(−U).





OUTLINE

1 REVIEW

2 MATRIX EQUATIONS




4 SUMMARY





ERROR VECTORS

The exact solution to the linear system satisfies

x = Px+Q.

The approximation at the kth iteration has some errordefined by

x(k) = x+ e

(k)

then the error satisfies the equation

e(k+1) = Pe(k) = P2

e(k−1) = Pk+1

e(0).





DIMINISHING ERRORS

In order that the error diminishes as k → ∞ we must have

||Pk|| → 0 as k → ∞,

Since

||Pk|| = ||P||k

we see that we require

||P|| < 1.





OUTLINE

1 REVIEW

2 MATRIX EQUATIONS




4 SUMMARY





SPECTRAL RADIUS

From linear algebra

||P|| < 1

is equivalent to the requirement that

ρ(P) = maxi

|λj| < 1

where λi are the eigenvalues of the matrix P, and ρ(P) iscalled the spectral radius of P.





CONVERGENCE RATE

Note also that for large k

||e(k+1)|| = ρ||e(k)||

How many iterations does it takes to reduce the initialerror by a factor ǫ?

We need q iterations where q is the smallest value for which

ρq < ǫ

giving

q ≤ qd =log ǫ

log ρ.





CONVERGENCE RATE


||e(k+1)|| = ρ||e(k)||



ρq < ǫ

giving

q ≤ qd =log ǫ

log ρ.





CONVERGENCE RATE


||e(k+1)|| = ρ||e(k)||



ρq < ǫ

giving

q ≤ qd =log ǫ

log ρ.





SPECTRAL RADIUS - MODEL PROBLEM

Thus iteration matrices where the spectral radius is close to1 will converge slowly.

For the model problem it can be shown that for Jacobiiteration

ρ = ρ(PJ) =cos

(

πn

)

+ β2 cos(

πm

)

1+ β2,

and for Gauss-Seidel

ρ = ρ(PG) = [ρ(PJ)]2.





CONVERGENCE PROPERTIES

If we take n = m, dx = dy and n >> 1 then for Jacobiiteration we have

qd =log ǫ

log(

1− π2

2n2

) = −2n2

π2log ǫ

For Gauss-Seidel we have that ρ(PG) = [ρ(PJ)]2 so that

qd = − n2

π2log ǫ

Gauss-Seidel converges twice as fast as Jacobi.





CONVERGENCE PROPERTIES

If we take n = m, dx = dy and n >> 1 then for Jacobiiteration we have

qd =log ǫ

log(

1− π2

2n2

) = −2n2

π2log ǫ

For Gauss-Seidel we have that ρ(PG) = [ρ(PJ)]2 so that

qd = − n2

π2log ǫ

Gauss-Seidel converges twice as fast as Jacobi.





OPTIMAL SOR

We can relate the eigenvalues λ of the point SOR iterationmatrix to the eigenvalues µ of the point Jacobi iterationmatrix by the equation

(λ + ω − 1)2 = λω2µ2

From this equation it can be proved that the optimumvalue of omega is

ωb =2

1+√

1− [ρ(PJ)]2

There exist methods to estimate ρ(PJ), but for the modelproblem we have a formula.





OPTIMAL SOR

We can relate the eigenvalues λ of the point SOR iterationmatrix to the eigenvalues µ of the point Jacobi iterationmatrix by the equation

(λ + ω − 1)2 = λω2µ2

From this equation it can be proved that the optimumvalue of omega is

ωb =2

1+√

1− [ρ(PJ)]2

There exist methods to estimate ρ(PJ), but for the modelproblem we have a formula.





CONVERGENCE RATES FOR POINT SOR

For point SOR on the model problem with optimum valuesand n = m it can be shown that

ρ =1− sinπ

n

1+ sinπn

giving

qd = − n

2πlog ǫ.





CONVERGENCE RATES FOR LINE SOR

For line SOR on the model problem with optimum valuesand n = m it can be shown that

ρ =

(

1− sinπn

1+ sinπn

)2

giving

qd = − n

2√2π

log ǫ.

Line SOR converges√2 times faster than point SOR for the

model problem.

When choosing a method we must decide whether fasterconvergence compensates for extra computations required.





CONVERGENCE RATES FOR LINE SOR

For line SOR on the model problem with optimum valuesand n = m it can be shown that

ρ =

(

1− sinπn

1+ sinπn

)2

giving

qd = − n

2√2π

log ǫ.

Line SOR converges√2 times faster than point SOR for the

model problem.

When choosing a method we must decide whether fasterconvergence compensates for extra computations required.




A linear system may be expressed as

Lx+Dx+Ux = b.

so that in a general iterative scheme may be written

x(k+1) = Px(k) +Q,

The condition for convergence on the iteration matrix P is

||P|| < 1.


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