DSS Structurally Symmetric Matrix Storage
Direct sparse solvers can also solve symmetrically structured systems of
equations. A symmetrically structured system of equations is one where the pattern of non-zero
elements is symmetric. That is, a matrix has a symmetric structure if is not zero if and only if
is not zero. From the point of
view of the solver software, a "non-zero" element of a matrix is any element stored in the
a
j
,i
a
i
, j
values
array, even if its value is
equal to 0. In that sense, any non-symmetric matrix can be turned into a symmetrically
structured matrix by carefully adding zeros to the values
array. For example, the above matrix
B
can be turned into a
symmetrically structured matrix by adding two non-zero entries: 
The matrix
B
can
be considered to be symmetrically structured with 15 non-zero elements and represented as: one-based
indexing | ||||||||||||||||
values | = | (1 | -1 | -3 | -2 | 5 | 0 | 4 | 6 | 4 | -4 | 2 | 7 | 8 | 0 | -5) |
columns | = | (1 | 2 | 4 | 1 | 2 | 5 | 3 | 4 | 5 | 1 | 3 | 4 | 2 | 3 | 5) |
rowIndex | = | (1 | 4 | 7 | 10 | 13 | 16) | |||||||||
zero-based
indexing | ||||||||||||||||
values | = | (1 | -1 | -3 | -2 | 5 | 0 | 4 | 6 | 4 | -4 | 2 | 7 | 8 | 0 | -5) |
columns | = | (0 | 1 | 3 | 0 | 1 | 4 | 2 | 3 | 4 | 0 | 2 | 3 | 1 | 2 | 4) |
rowIndex | = | (0 | 3 | 6 | 9 | 12 | 15) |
Storage Format Restrictions
The storage
format for the sparse solver must conform to two important restrictions:
- the non-zero values in a given row must be placed into thevaluesarray in the order in which they occur in the row (from left to right);
- no diagonal element can be omitted from thevaluesarray for any symmetric or structurally symmetric matrix.
The second restriction implies that if symmetric or structurally
symmetric matrices have zero diagonal elements, then they must be explicitly represented in
the
values
array.