Solving An Equation For Its Complex Roots - HP -15C Advanced Functions Handbook

80 Section 3: Calculating in Complex Mode
3. To calculate higher number roots z/,:
• Press | R/S | to calculate each successive higher-number
root. Each root z
k
is placed in the complex X-registers and
its number k is placed in the Y-register. Between root
calculations, you can perform other calculations without
disturbing this program (if R
2
, RS, R4, and the Index
register aren't changed).
• Store the number of the root k in the Index register (using
[STO| Q]), then press |R/S| to calculate z
k
. The complex root
and its number are placed in the X- and Y-registers,
respectively. (By pressing |R/S| again, you can continue
calculating higher-number roots.)
Example: Use the previous program to compute (1)
1/10
°. Calculate
ZQ, Zi, and 2
50
for this expression.
Keystrokes
rglfp/Rl
1001 ENTER 11
EDQjjjfhold)
fR/sl
EGHKhold)
fR/Sl
E M (hold)
Display
Run mode.
1 Enters n = 100 and 2 = 1
(purely real).
1.0000 Calculates 2
0
(real part).
0.0000 Imaginary part of 2
0
.
0.9980 Calculates z
1
(real part).
0.0628 Imaginary part of Z]_.
50.0000 Stores root number in
Index register.
-1.0000 Calculates 2
50
(real part).
0.0000 Imaginary part of 2
50
.
Solving an Equation for Its Complex Roots
A common method for solving the complex equation f ( z ) — 0
numerically is Newton's iteration. This method starts with an
approximation ZQ to a root and repeatedly calculates
** + !=**-/(**)//*(**)
until z
k
converges.
The following example shows how | SOLVE | can be used with
Newton's iteration to estimate complex roots. (A different