HP -15C Advanced Functions Handbook page 84

82 SectionS: Calculating in Complex Mode
z = ln(-8/(2 + 9)) ± i2mr for n = 0, 1, 2, ...
This equation has only two complex conjugate roots z for each
integer n. Therefore use the equivalent function
f(z) = 2 - ln(-8/(2 + 9)) ± i2nir for n = 0, I, 2, ....
and apply Newton's iteration
z
k + l
=z
k
- (z
k
- ln(-8/(2* + 9)) ± /2mr)/(l + l/(z
k
+ 9)).
As a first guess, choose z
0
as A(n), the approximation given earlier.
A bit of algebraic rearrangement using the fact that ln(±^ = ±jV/2
leads to this formula:
= A(n) + ((z
k
-A(n)) + (z
k
+ 9)ln(ilm(A(n))/(z
k
+ 9)))/(2
A
+ 10) .
In the program below, He(A(n)) is stored in R
0
and Im(A(n)) is
stored in R^ Note that only one of each conjugate pair of roots is
calculated for each n.
Keystrokes
|T1CLEAR|PRGM|
Display
Program mode.
000-
001-42,21
002-43, 5
003-
004-
005-
006-
007-
008- 43
009-
010-
011- 44
012-
013-
014- 43
015-
016- 44
017-
018- 42
,11
, 8
36
40
48
5
40
26
20
36
1
8
10
12
16
0
34
25
Program for A (n).
Specifies real arithmetic.
Calculates (2n + V^TT.
Calculates
-ln((2n + Vz)TT/8).
Forms complex A (n).