Appendix A: Functions and Instructions 803
cZeros({
expression1
,
expression2
[,
…
] },
{
varOrGuess1
,
varOrGuess2
[,
…
] }) ⇒
⇒⇒
⇒
matrix
Returns candidate positions where the
expressions are zero simultaneously. Each
varOrGuess
specifies an unknown whose value you
seek.
Optionally, you can specify an initial guess for a
variable. Each
varOrGuess
must have the form:
variable
– or –
variable
=
real or non-real number
For example,
x is valid and so is x=3+
i
.
If all of the expressions are polynomials and you
do NOT specify any initial guesses,
cZeros() uses
the lexical Gröbner/Buchberger elimination
method to attempt to determine all complex
zeros.
Note: The following examples use an underscore _ s
that the variables will be treated as complex.
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
ø
i
1/2 +
3
2
ø
i
1/2 +
3
2
ø
i
1/2 ì
3
2
ø
i
0 0
Extract row 2:
ans(1)[2] ¸
1/2 +
3
2
ø
i
1/2 ì
3
2
ø
i
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
ø
i
1/2 +
3
2
ø
i
@1
1/2 +
3
2
ø
i
1/2 ì
3
2
ø
i
@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
+1/2ø
i
e
w_
ì i
2
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