HP -15C Advanced Functions Handbook page 102

1 00 Section 4: Using Matrix Operations
useful for measuring errors in calculations. A matrix is said to be
ill-conditioned ifK(A) is very large.
If rounding or other errors are present in matrix elements, these
errors will propagate through subsequent matrix calculations.
They can be magnified significantly. For example, suppose that X
and B are nonzero vectors satisfying AX = B for some square
matrix A. Suppose A is perturbed by AA and we compute B + AB =
(A + AA)X. Then
with equality for some perturbation AA. This measures how much
the relative uncertainty in A can be magnified when propagated
into the product.
The condition number also measures how much larger in norm the
relative uncertainty of the solution to a system can be compared to
that of the stored data. Suppose again that X and B are nonzero
vectors satisfying AX = B for some matrix A. Suppose now that
matrix B is perturbed (by rounding errors, for example) by an
amount AB. Let X + AX satisfy A(X + AX) = B + AB. Then
with equality for some perturbation AB.
Suppose instead that matrix A is perturbed by AA. Let X + AX
satisfy (A + AA)(X + AX) = B. If d(A,AA) = #(A)||AA|| / ||A|| < 1,
then
Similarly, if A""
1
+ Z is the inverse of the perturbed matrix A + AA,
then
Moreover, certain perturbations AA cause the inequalities to
become equalities.
All of the preceding relationships show how the relative error of the
result is related to the relative error of matrix A via the condition
number K(A). For each inequality, there are matrices for which