Making Difficult Equations Easier; Scaling - HP -15C Advanced Functions Handbook

1 04 Section 4: Using Matrix Operations
X, in error by E — X — Z. Then E satisfies the linear system AE =
AX — AZ = R, where R is the residual for Z. The next step is to
calculate the residual and then to solve AE = R for E. The
calculated solution, denoted by F, is treated as an approximation to
E = X — Z and is added to Z to obtain a new approximation to X:
F + Z « (X - Z) + Z = X.
In order for F + Z to be a better approximation to X than is Z, the
residual R = B — AZ must be calculated to extended precision. The
HP-ISO's | MATRIX | 6 operation does this. The system matrix A is
used for finding both solutions, Z and F. The LU decomposition
formed while calculating Z can be used for calculating F, thereby
shortening the execution time. The refinement process can be
repeated, but most of the improvement occurs in the first
refinement.
(Refer to Applications at the end of this section for a program that
performs one iteration of refinement.)
Making Difficult Equations Easier
A system of equations EX — B is difficult to numerically solve
accurately if E is ill-conditioned (nearly singular). Even iterative
refinement can fail to improve the calculated solution when E is
sufficiently ill-conditioned. However, instances arise in practice when
a modest extra effort suffices to change difficult equations into others
with the same solution, but which are easier to solve. Scaling and
preconditioning are two processes to do this.
Scaling
Bad scaling is a common cause of poor results from attempts to
numerically invert ill-conditioned matrices or to solve systems of
equations with ill-conditioned system matrices. But it is a cause
that you can easily diagnose and cure.
Suppose a matrix E is obtained from a matrix A by E = LAR,
where L and R are scaling diagonal matrices whose diagonal
elements are all integer powers of 10. Then E is said to be obtained
from A by scaling. L scales the rows of A, and R scales the
columns. Presumably E"
1
= R"
1
A"
1
L"
1
can be obtained either from
A"
1
by scaling or from E by inverting.