Appendix A: Functions and Instructions 171
Complex zeros can include both real and non-real
zeros, as in the example to the right.
Each row of the resulting matrix represents an
alternate zero, with the components ordered the
same as the
varOrGuess
list. To extract a row,
index the matrix by [
row
].
cZeros({u_ù v_ì u_ì v_,v_^2+u_},
{u_,v_}) ¸
1/2 ì
3
2
øi 1/2 +
3
2
øi
1/2 +
3
2
øi 1/2 ì
3
2
øi
0
0
Extract row 2:
ans(1)[2] ¸
1/2 +
3
2
øi 1/2 ì
3
2
øi
Simultaneous
polynomials
can have extra
variables that have no values, but represent given
numeric values that could be substituted later.
cZeros({u_ùv_ìu_ì(c_ùv_),
v_^2+u_},{u_,v_}) ¸
ë (
1ì 4øc_+1)
2
4
1ì 4øc_+1
2
ë (
1ì 4øc_ì 1)
2
4
ë (
1ì 4øc_ì 1)
2
0 0
You can also include unknown variables that do
not appear in the expressions. These zeros show
how families of zeros might contain arbitrary
constants of the form @
k
, where
k
is an integer
suffix from 1 through 255. The suffix resets to 1
when you use
ClrHome or ƒ 8:Clear Home.
For polynomial systems, computation time or
memory exhaustion may depend strongly on the
order in which you list unknowns. If your initial
choice exhausts memory or your patience, try
rearranging the variables in the expressions
and/or
varOrGuess
list.
cZeros({u_ù v_ì u_ì v_,v_^2+u_},
{u_,v_,w_}) ¸
1/2 ì
3
2
øi 1/2 +
3
2
øi @1
1/2 +
3
2
øi 1/2 ì
3
2
øi @1
0
0 @1
If you do not include any guesses and if any
expression is non-polynomial in any variable but
all expressions are linear in all unknowns,
cZeros() uses Gaussian elimination to attempt to
determine all zeros.
cZeros({u_+v_ì
e
^(w_),u_ì v_ì
i
},
{u_,v_}) ¸
e
w_
2
+1/2øi
e
w_
ì
i
2
If a system is neither polynomial in all of its
variables nor linear in its unknowns,
cZeros()
determines at most one zero using an
approximate iterative method. To do so, the
number of unknowns must equal the number of
expressions, and all other variables in the
expressions must simplify to numbers.
cZeros({
e
^(z_)ì w_,w_ì z_^2},
{w_,z_}) ¸
[]
.494… ë.703…
A non-real guess is often necessary to determine
a non-real zero. For convergence, a guess might
have to be rather close to a zero.
cZeros({
e
^(z_)ì w_,w_ì z_^2},
{w_,z_=1+
i
}) ¸
[]
.149…+4.89…øi 1.588…+1.540…øi
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