Level 1C: Complex Level 1 - HP -15C Advanced Functions Handbook

Appendix: Accuracy of Numerical Calculations 183
from which the correct total follows.
To understand the error in 3
201
, note that this is calculated as
e
20i in(3) =
e
220.82i...
TQ kegp the final relative error
below one unit in
the 10th significant digit, 201 ln(3) would have to be calculated
with an absolute error rather smaller than 10~
10
, which would
entail carrying at least 14 significant digits for that intermediate
value. The calculator does carry 13 significant digits for certain
intermediate calculations of its own, but a 14th digit would cost
more than it's worth.
Level 1C: Complex Level 1
Most complex arithmetic functions cannot guarantee 9 or 10
correct significant digits in each of a result's real and imaginary
parts separately, although the result will conform to the summary
statement about functions in Level 1 provided f, F, and e are
interpreted as complex numbers. In other words, every complex
function / in Level 1C will produce a calculated complex value
F=(l + t)f whose small complex relative error e must satisfy
|e| < 10~
9
. The complex functions in Level 1C are 0, B, [ill. | L N | ,
I LOG |, | SIN"
1
1, | COS"
1
1, | TAN"
1
1, |SINH"
1
|, | COSH"
1
1, and |TANH"
1
|. Therefore,
a function like X(z) — ln(l + 2) can be calculated accurately for all z
by the same program as given above and with the same
explanation.
To understand why a complex result's real and imaginary parts
might not individually be correct to 9 or 10 significant digits,
consider [x], for example: (a + ib) X (c + id) = (ac — bd) + i(ad+ be)
ideally. Try this with a = c = 9.999999998, 6 = 9.999999999, and
d = 9.999999997; the exact value of the product's real part ( a c — bd)
should then be
(9.99999999S)
2
- (9.999999999) (9.999999997)
= 99.999999980000000004 - 99.999999980000000003
= 10-
18
which requires that at least 20 significant digits be carried during
the intermediate calculation. The HP-15C carries 13 significant
digits for internal intermediate results, and therefore obtains 0
instead of 10~
18
for the real part, but this error is negligible
compared to the imaginary part 199.9999999 .