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Appendix A: Categorical Listing of Operations 801
cZeros() MATH/Algebra/Complex menu
cZeros( expression , var ) ⇒
⇒⇒
⇒ list
Returns a list of candidate real and non-real
values of
var that make expression =0. cZeros()
does this by computing
exp8list(cSolve( expression =0, var ), var ).
Otherwise,
cZeros() is similar to zeros().
Note: See also
cSolve(), solve(), and zeros().
Display Digits mode in
Fix 3:
cZeros(x^5+4x^4+5x^3ì 6xì 3,x)
¸
{ë 2.125 ë.612 .965
ë 1.114 ì 1.073ø i
ë 1.114 + 1.073ø i }
Note: If expression is non-polynomial with
functions such as
abs(), angle(), conj(), real(),
or
imag(), you should place an underscore _
(
@ ¥ , H 2  ) at the end of var . By
default, a variable is treated as a real value. If
you use
var _ , the variable is treated as complex.
You should also use
var _ for any other variables
in
expression that might have unreal values.
Otherwise, you may receive unexpected results.
z is treated as real:
cZeros(conj(z)ì 1ì i ,z) ¸
{1+ i }
z_ is treated as complex:
cZeros(conj(z_)ì 1ì i ,z_) ¸
{1ì i }
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 _ (
@ ¥ , H 2 ) so
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
ii
i 1/2 +
3
2
ø i
ii
i
1/2 +
3
2
ø i
ii
i 1/2 ì
3
2
ø i
ii
i
0 0
Extract row 2:
ans(1)[2] ¸
⎣
⎣⎣
⎣
⎡
⎡⎡
⎡
⎦
⎦⎦
⎦
⎤
⎤⎤
⎤
1/2 +
3
2
ø i
ii
i 1/2 ì
3
2
ø i
ii
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