
Section 9.7 Multiple precision arithmetics 347
(a) an approximation x
.0/
for the correct solution x
is computed in floating-point
arithmetic by forward substitution.
(b) an enclosure E of the error e D x
x
.0/
is computed.
This leads to an enclosure for the solution:
x
2 x
.0/
C E.
Formulas for E are obtained by use of the mean value theorem. They are evaluated
by the exact scalar product.
If the quality of the enclosure x
2 x
.0/
CE is not sufficient, the midpoint x
.1/
:D
m.E/ is used to improve the approximationx
.0/
to x
.0/
Cx
.1/
. Then step (b) is repeated
and so on.
A detailed description and executable algorithms and programs can be found in
Chapter 8 of the ...