Appendix A: Functions and Instructions 781
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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