Ill conditioning arises while solving linear equations of the type
for given loads and stiffness in (say) linear
static analysis, approximations are introduced in the solution because
all calculations are carried out in finite precision arithmetic. This
becomes important when is ill-conditioned because there is
a possibility of these approximations leading to large errors in the
displacements. The extent of these errors can be quantified by the
'condition number' of the stiffness matrix.
The condition number of a matrix (with respect to inversion) measures
worst-case of changes in corresponding to small changes in
or . It can be calculated using the product of
norm of the matrix times the norm of its inverse.
where is a subordinate matrix
If is a symmetric matrix, the condition number can be shown to
the ratio of its maximum and minimum eigenvalues and
The minimum value of is 1 and the
maximum value is infinity. If the condition number is small, the
computed solution is reliable (i.e. a reliable approximation
to the true solution of ). If the condition
number is large, (i.e. if the matrix is almost singular) the results
cannot be trusted.
GSA computes a lower bound approximation to the 1-norm condition number
of and this is reported as part of the solver output. This can be used
to evaluate the accuracy of the solution both qualitatively and
quantitatively. The (qualitative) rule of thumb for accuracy is – the
number of digits of accuracy in is
In general any stiffness matrix with condition number above
1015 can produce results with no accuracy at all. Any results
produced from matrices with condition number greater than
1010 must be treated with caution.
Where a model is ill-conditioned, Model Stability analysis can help
detect the causes of ill conditioning.
For a given condition number, we can also compute the maximum relative
error in . The max. relative error in is
defined as the maximum ratio of norms of error in to
Given a matrix with condition number , the maximum
relative error in when solving is
where is the constant 'unit-roundoff' and is equal to
1.11e-16 for double precision floating point numbers. The maximum
relative error is computed and reported as part of solver output.
Ideally, this should be small , since a small relative error
indicates a reliable solution but as
, the relative error grows
GSA calculates the condition number using Higham and Tisseur's block
1-Norm condition number estimation algorithm.
These are reported in the Analysis Details output.