The Accuracy Of Numerical Solutions To Linear Systems - HP -15C Advanced Functions Handbook

Section 4: Using Matrix Operations 1 03
The Accuracy of Numerical Solutions
to Linear Systems
The preceding discussion dealt with how uncertainties in the data
are reflected in the solutions of systems of linear equations and in
matrix inverses. But even when data is exact, uncertainties are
introduced in numerically calculated solutions and inverses.
Consider solving the linear system AX = B for the theoretical
solution X. Because of rounding errors during the calculations, the
calculated solution Z is in general not the solution to the original
system AX = B, but rather the solution to the perturbed system
(A + AA)Z = B. The perturbation AA satisfies || AA|| s$ e \\A\\, where
e is usually a very small number. In many cases, AA will amount to
less than one in the 10th digit of each element of A.
For a calculated solution Z, the residual is R = B — AZ. Then
||R|| < e||A||||Z||. So the expected residual for a calculated solution is
small. But although the residual R is usually small, the error Z — X
may not be small if A is ill-conditioned:
A useful rule-of-thumb for the accuracy of the computed solution is
/ number of correct \ numberof \ / n » n i l A - I I K i /-m \ • u- •,.
\l digits / \s carried /
where n is the dimension of A. For the HP-15C, which carries 10
accurate digits,
(number of correct decimal digits) ^ 9 — log(|| A|| HA'^I) — log(n).
In many applications, this accuracy may be adequate. When
additional accuracy is desired, the computed solution Z can usually
be improved by iterative refinement (also known as residual
correction).
Iterative refinement involves calculating a solution to a system of
equations, then improving its accuracy using the residual
associated with the solution to modify that solution.
To use iterative refinement, first calculate a solution Z to the
original system AX = B. Z is then treated as an approximation to