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One of the ¬rst divide-and-conquer ideas used in structural analysis exploited the partition-

ing (13.1) in a direct solution framework. This approach, which is covered in this section,

introduces the Schur complement and explains some of its properties.

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Consider the linear system written in the form (13.2), in which is assumed to be nonsin-

gular. From the ¬rst equation the unknown can be expressed as |

un¦ ¦ j

˜

}

¥x c 5

D| h

v

Upon substituting this into the second equation, the following reduced system is obtained:

2n1 ¦ j

a

˜

5c v £ x }

¥c

£

‚

D h

v

The matrix

un4 ¦ j

a

˜

5c v £

£

D

is called the Schur complement matrix associated with the variable. If this matrix can be

formed and the linear system (13.4) can be solved, all the interface variables will become

available. Once these variables are known, the remaining unknowns can be computed, via

(13.3). Because of the particular structure of , observe that any linear system solution

with it decouples in separate systems. The parallelism in this situation arises from this

natural decoupling.

A solution method based on this approach involves four steps:

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{ {˜p p { p

˜

§§

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¢

! hC §¤ !

¤ …S !

§

¥

¤ ¤

¢

¡

Obtain the right-hand side of the reduced system (13.4).

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Form the Schur complement matrix (13.5).

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Solve the reduced system (13.4).

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Back-substitute using (13.3) to obtain the other unknowns.

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One linear system solution with the matrix can be saved by reformulating the algorithm

in a more elegant form. De¬ne

5c v ¥c

and ¥

D8 5 8 D h

v

The matrix and the vector are needed in steps (1) and (2). Then rewrite step (4) as

¥

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¥c 5c v ¥¦9 8 85 t

D| D

v

which gives the following algorithm.

vx¤ … ¢

¡¦ ™

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1. Solve , and for and , respectively

¥ ¥ ¥

D8 5 8 85 8

D

2. Compute ¥

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… D©

a8 8

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