Root
The roots (sometimes also called "zeros") of an equation
 |
are the values of 
for which the equation is satisfied.
Roots 
which belong to certain sets
are usually preceded by a modifier to indicate such, e.g., 
is called
a rational root, 
is called
a real root, and 
is called
a complex root.
The fundamental theorem of algebra states that every polynomial equation of degree
has exactly 
complex roots, where some roots may have
a multiplicity greater than 1 (in which case they are said to be degenerate). In
the Wolfram Language, the expression
Root[p(x),
k] represents the 
th root of the polynomial 
, where 
, ..., 
is an index indicating
the root number in the Wolfram Language's
ordering.
The similar concept of the "
th root" 
of a complex number 
is known as an
nth root.
The roots of a complex function can be obtained by separating it into its real and imaginary plots and plotting these curves (which are related by the Cauchy-Riemann
equations) separately. Their intersections give the complex roots of the original
function. For example, the plot above shows the curves representing the real and
imaginary parts of 
, with the three roots indicated
as black points.
Householder (1970) gives an algorithm for constructing root-finding algorithms with an arbitrary order of convergence. Special root-finding techniques can often be applied when the function in question is a polynomial.
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