¨ ©

¨¦

§ §

where the constant is equal to 10. The domain and coef¬cients for this problem are shown

§

˜ ˜

is Figure 3.11. If the number of points in each direction is 34, then there are

ud

¡ ¤§ ¥ §¢

¡

¡© © ¢ ¡ ¥ ¥ © ¡ ¥

§

6 l ˜ ˜ ¢ Y 7

˜

interior points in each direction and a matrix of size is

obtained. In this test example, as well as the other ones described below, the right-hand

side is generated as

¤

1C i)1)‘1Y•B q

¤

in which . The initial guess is always taken to be a vector of pseudo-

G

random values.

Problem 2: F2DB. The second test problem is similar to the previous one but involves

˜ ‚ ˜ ‚

discontinuous coef¬cient functions and . Here, and the functions

y¡ ¤ p9

are also de¬ned by (3.4). However, the functions and now both take the value

00 0

1,000 inside the subsquare of width centered at ( ), and one elsewhere in the domain,

6

66

i.e.,

¢ $ ¢ & £ 7 £ C x B C x B ‚

g0

¡

§

¨¦

¨©

¦ ¨©

¦ )¡

1 )%

£ ˜ ˜ ˜

Problem 3: F3D. The third test problem is three-dimensional with ¢£

£¤¢ ˜

¥ 7

internal mesh points in each direction leading to a problem of size . In this case,

we take £ £ £

C ™ x B C ™ g B C ™ g B ‚

¨¦ ©"

¨¦ ¨© ¦

£

£

C y g B C ™ g B 9

¤

¤ &¤

©"

¨¦ "

¨¦

§ §

and £ £

p p C ™ x B C ™ x B ¡

78

7

©

¨¦ © ¦

¨

The constant is taken to be equal to 10.0 as before.

§

The Harwell-Boeing collection is a large data set consisting of test matrices which

have been contributed by researchers and engineers from many different disciplines. These

have often been used for test purposes in the literature [78]. The collection provides a data

structure which constitutes an excellent medium for exchanging matrices. The matrices are

stored as ASCII ¬les with a very speci¬c format consisting of a four- or ¬ve-line header.

Then, the data containing the matrix is stored in CSC format together with any right-

hand sides, initial guesses, or exact solutions when available. The SPARSKIT library also

provides routines for reading and generating matrices in this format.

Only one matrix from the collection was selected for testing the algorithms described

in the coming chapters. The matrices in the last two test examples are both irregularly

structured.

Problem 4: ORS The matrix selected from the Harwell-Boeing collection is ORSIRR1.

7 ˜

£ 7

This matrix arises from a reservoir engineering problem. Its size is and it has

z

u

a total of 6,858 nonzero elements. The original problem is based on a

¢

irregular grid. In this case and the next one, the matrices are preprocessed by scaling their

rows and columns.

d¥ © g¢ ¤© ©

n¥ u¡ § £ ¥

© ¡

Problem 5: FID This test matrix is extracted from the well known ¬‚uid ¬‚ow simulation

package FIDAP [84]. It is actually the test example number 36 from this package and

£

features a two-dimensional Chemical Vapor Deposition in a Horizontal Reactor. The matrix

˜

¥¤7 ¤ ¢

¢

has a size of and has nonzero elements. It has a symmetric pattern

and few diagonally dominant rows or columns. The rows and columns are prescaled in the

same way as in the previous example. Figure 3.12 shows the patterns of the matrices ORS

and FID.

„

¥ ¢ T©

D ¡

Patterns of the matrices ORS (left) and FID

(right).

) ") ¨ ¢

£ §