®s„D¬BB% G H

%¬ ¬

1. For Do:

S£ S

2.

¨i% ¬B¬BD% G m¥

¬

3. For , Do:

¦

¨ E U

G

a ¥ 0 ` ©&

S %

4. Compute

!&

a %S `

¡¨ S S S

5. Compute .

6. EndDo

u§ S E z SS

«§

7.

ª SS

8. If then Stop; else Compute .

SS

S S

9. EndDo

When is nonsingular, the same result as before is obtained if no dropping is used on ¡

, namely, that the factorization will exist and be exact in that . Regarding the

implementation, if the zero pattern is known in advance, the computation of the inner

products in line 4 does not pose a particular problem. Without any dropping in , this

algorithm may be too costly in terms of storage. It is interesting to see that this algorithm

˜

has a connection with ICNE, the incomplete Cholesky applied to the matrix . The

following result is stated without proof.

‚n)|¡w¤

¥ ¦ ¦ ¥ ˜ U#§

"¢ §

±w®

¯

Let be an matrix and let . Consider a zero-pattern

® ²n ¥ ²m

set which is such that for any , with and , the following

%

3 3

% ¥

YW G

holds: X

¦ ¦ ¬ ¦ ¡ % ¥ `

¡ % 7`

a ¥ 07` a a

%

%

Then the matrix obtained from Algorithm 10.15 with the zero-pattern set is identi-

cal with the factor that would be obtained from the Incomplete Cholesky factorization

applied to with the zero-pattern set .

§

For a proof, see [222]. This result shows how a zero-pattern can be de¬ned which guaran-

˜ U

tees the existence of an Incomplete Cholesky factorization on .

8˜ H ˜ ¤ ¨°¡ $ H&’ ”8 ” &’ ¤ ©¦§

©8 $ $

¨ ¡¨ ¨¦ ¤

§ © §¥ c#&) $

©1

¢

¡

ª“t˜•wS¡

˜

–

1 Assume that is the Symmetric Positive De¬nite matrix arising from the 5-point ¬nite differ-

ence discretization of the Laplacean on a given mesh. We reorder the matrix using the red-black

ordering and obtain the reordered matrix

4 7 ¤

R W£ ¤

e

£ A© £

We then form the Incomplete Cholesky factorization on this matrix.

³

e A

&'% Show the ¬ll-in pattern for the IC(0) factorization for a matrix of size associated

¢$

¡

with a mesh.

&1 Show the nodes associated with these ¬ll-ins on the 5-point stencil in the ¬nite difference

mesh.

&32 Give an approximate count of the total number of ¬ll-ins when the original mesh is square,

with the same number of mesh points in each direction. How does this compare with the

natural ordering? Any conclusions?

–

£$£

2 Consider a tridiagonal nonsingular matrix .

&'% What can be said about its ILU(0) factorization (when it exists)?

&1 Suppose that the matrix is permuted (symmetrically, i.e., both rows and columns) using the

permutation

e¤ A 3§3§ ¥¦

¨¡

£

©

& Show the pattern of the permuted matrix.

&

Show the locations of the ¬ll-in elements in the ILU(0) factorization.

&

Show the pattern of the ILU(1) factorization as well as the ¬ll-ins generated.

& Show the level of ¬ll of each element at the end of the ILU(1) process (including the

¬ll-ins).

& What can be said of the ILU(2) factorization for this permuted matrix?

–

3 Assume that is the matrix arising from the 5-point ¬nite difference discretization of an elliptic

operator on a given mesh. We reorder the original linear system using the red-black ordering and

obtain the reordered linear system

d h

7 7 7

¤

R£ e

R R

d h

A© £ A© A©

d

&'% Show how to obtain a system (called the reduced system) which involves the variable ©

only.

&1 Show that this reduced system is also a sparse matrix. Show the stencil associated with

the reduced system matrix on the original ¬nite difference mesh and give a graph-theory

interpretation of the reduction process. What is the maximum number of nonzero elements