zeros(Expr, Var) ⇒ list

zeros(Expr, Var=Guess) ⇒ list

Returns a list of candidate real values of Var that make Expr=0. zeros() does this by computing exp►list(solve(Expr=0,Var),Var).

For some purposes, the result form for zeros() is more convenient than that of solve(). However, the result form of zeros() cannot express implicit solutions, solutions that require inequalities, or solutions that do not involve Var.

Note: See also cSolve(), cZeros(), and solve().

zeros({Expr1, Expr2},
{VarOrGuess1, VarOrGuess2 [, … ]}) ⇒ matrix

Returns candidate real zeros of the simultaneous algebraic expressions, where 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.

If all of the expressions are polynomials and if you do NOT specify any initial guesses, zeros() uses the lexical Gröbner/Buchberger elimination method to attempt to determine all real zeros.

For example, suppose you have a circle of radius r at the origin and another circle of radius r centered where the first circle crosses the positive x-axis. Use zeros() to find the intersections.

As illustrated by r in the example to the right, simultaneous polynomial expressions can have extra variables that have no values, but represent given numeric values that could be substituted later.

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].

You can also (or instead) include unknowns that do not appear in the expressions. For example, you can include z as an unknown to extend the previous example to two parallel intersecting cylinders of radius r. The cylinder zeros illustrate how families of zeros might contain arbitrary constants in the form ck, where k is an integer suffix from 1 through 255.

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.

If you do not include any guesses and if any expression is non-polynomial in any variable but all expressions are linear in the unknowns, zeros() uses Gaussian elimination to attempt to determine all real zeros.

If a system is neither polynomial in all of its variables nor linear in its unknowns, zeros() 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.

Each unknown starts at its guessed value if there is one; otherwise, it starts at 0.0.