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F I H ) F E 'B A 8 @ 8 & 5

Block relaxation schemes are generalizations of the “point” relaxation schemes described

above. They update a whole set of components at each time, typically a subvector of the

§

solution vector, instead of only one component. The matrix and the right-hand side and

solution vectors are partitioned as follows:

§ § § §

£ F S

™W £

W ¤W£ WVUSW£

TTT R XYW£ W£ W£

§ § § §

£

„ F „ S

„ X r±Q—w°i ¯®

W TWVUR

TT

§ ¬§ § § §

£

( `R ( R F ( R S

¬‚

© c7

¬¨

. .

QR

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. . . .

..

. .

. . . .

‚ ‚ ‚

. . .

. . . .

§ WX § §

£ F S

X X

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TTT TT

X bX

X

©¨ RS RF

in which the partitionings of and into subvectors and are identical and compatible

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with the partitioning of . Thus, for any vector partitioned as in (4.15),

T

Xf

¬ R X t§

© § ( u F vR

u

u

T

W

RX b

in which denotes the -th component of the vector according to the above partitioning. E E

§

The diagonal blocks in are square and assumed nonsingular.

Now de¬ne, similarly to the scalar case, the splitting

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C³

³

£ q¦ ¡k q`¤¨¤¦tst ¢ V£¤

s©| | §£¥ j

£¡¨ !p

§

¢

with

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W § ¥£

£

a±!w°i ¯®

( „

¬ ..

#

‚ .

§ XbX

§ §

£

VWT

TT

W W£

X

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§‚ ² ¬ £

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..

W ..

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With these de¬nitions, it is easy to generalize the previous three iterative procedures de-

¬ned earlier, namely, Jacobi, Gauss-Seidel, and SOR. For example, the block Jacobi it-

eration is now de¬ned as a technique in which the new subvectors are all replaced H GR

PVIF

according to rT

T

X R S w RX

¬ rP"QUIHG R F IRR § ©

'w%

W

H

or, rT

T

©X w RX

§ ¬ rP"QUIHG R F § vE

¬

( R S W I“ R

'w%

W d“ R

R R "¢V(

( ¡(

W

H y