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is the -th row of the matrix in sparse format and is the vector of the

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components of gathered into a short vector which is consistent with the column indices

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of the elements in the row . £ © §

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Most direct methods for sparse linear systems perform an LU factorization of the original

matrix and try to reduce cost by minimizing ¬ll-ins, i.e., nonzero elements introduced

during the elimination process in positions which were initially zeros. The data structures

employed are rather complicated. The early codes relied heavily on linked lists which are

convenient for inserting new nonzero elements. Linked-list data structures were dropped

in favor of other more dynamic schemes that leave some initial elbow room in each row

for the insertions, and then adjust the structure as more ¬ll-ins are introduced.

A typical sparse direct solution solver for positive de¬nite matrices consists of four

phases. First, preordering is applied to minimizing ¬ll-in. Two popular methods are used:

minimal degree ordering and nested-dissection ordering. Second, a symbolic factorization

is performed. This means that the factorization is processed only symbolically, i.e., without

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numerical values. Third, the numerical factorization, in which the actual factors and ¥

are formed, is processed. Finally, the forward and backward triangular sweeps are executed

for each different right-hand side. In a code where numerical pivoting is necessary, the

symbolic phase cannot be separated from the numerical factorization.

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For comparison purposes it is important to use a common set of test matrices that represent

a wide spectrum of applications. There are two distinct ways of providing such data sets.

The ¬rst approach is to collect sparse matrices in a well-speci¬ed standard format from

various applications. This approach is used in the Harwell-Boeing collection of test matri-

ces. The second approach is to generate these matrices with a few sample programs such

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as those provided in the SPARSKIT library [179]. The coming chapters will use exam-

ples from these two sources. In particular, ¬ve test problems will be emphasized for their

varying degrees of dif¬culty.

The SPARSKIT package can generate matrices arising from the discretization of the

two- or three-dimensional Partial Differential Equations

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on rectangular regions with general mixed-type boundary conditions. In the test problems, 6 )z B

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the regions are the square , or the cube ; the Dirichlet condition

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is always used on the boundary. Only the discretized matrix is of importance, since

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the right-hand side will be created arti¬cially. Therefore, the right-hand side, , is not

relevant.

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Problem 1: F2DA. In the ¬rst test problem which will be labeled F2DA, the domain is

two-dimensional, with

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