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<H1><A NAME="SEC55" HREF="cln_toc.html#TOC55">9. Univariate polynomials</A></H1>
<P>
<A NAME="IDX279"></A>
<A NAME="IDX280"></A>
<H2><A NAME="SEC56" HREF="cln_toc.html#TOC56">9.1 Univariate polynomial rings</A></H2>
<P>
CLN implements univariate polynomials (polynomials in one variable) over an
arbitrary ring. The indeterminate variable may be either unnamed (and will be
printed according to <CODE>cl_default_print_flags.univpoly_varname</CODE>, which
defaults to <SAMP>`x'</SAMP>) or carry a given name. The base ring and the
indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
(accidentally) mix elements of different polynomial rings, e.g.
<CODE>(a^2+1) * (b^3-1)</CODE> will result in a runtime error. (Ideally this should
return a multivariate polynomial, but they are not yet implemented in CLN.)
<P>
The classes of univariate polynomial rings are
<PRE>
Ring
cl_ring
<cl_ring.h>
|
|
Univariate polynomial ring
cl_univpoly_ring
<cl_univpoly.h>
|
+----------------+-------------------+
| | |
Complex polynomial ring | Modular integer polynomial ring
cl_univpoly_complex_ring | cl_univpoly_modint_ring
<cl_univpoly_complex.h> | <cl_univpoly_modint.h>
|
+----------------+
| |
Real polynomial ring |
cl_univpoly_real_ring |
<cl_univpoly_real.h> |
|
+----------------+
| |
Rational polynomial ring |
cl_univpoly_rational_ring |
<cl_univpoly_rational.h> |
|
+----------------+
|
Integer polynomial ring
cl_univpoly_integer_ring
<cl_univpoly_integer.h>
</PRE>
<P>
and the corresponding classes of univariate polynomials are
<PRE>
Univariate polynomial
cl_UP
<cl_univpoly.h>
|
+----------------+-------------------+
| | |
Complex polynomial | Modular integer polynomial
cl_UP_N | cl_UP_MI
<cl_univpoly_complex.h> | <cl_univpoly_modint.h>
|
+----------------+
| |
Real polynomial |
cl_UP_R |
<cl_univpoly_real.h> |
|
+----------------+
| |
Rational polynomial |
cl_UP_RA |
<cl_univpoly_rational.h> |
|
+----------------+
|
Integer polynomial
cl_UP_I
<cl_univpoly_integer.h>
</PRE>
<P>
Univariate polynomial rings are constructed using the functions
<DL COMPACT>
<DT><CODE>cl_univpoly_ring cl_find_univpoly_ring (const cl_ring& R)</CODE>
<DD>
<DT><CODE>cl_univpoly_ring cl_find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)</CODE>
<DD>
This function returns the polynomial ring <SAMP>`R[X]'</SAMP>, unnamed or named.
<CODE>R</CODE> may be an arbitrary ring. This function takes care of finding out
about special cases of <CODE>R</CODE>, such as the rings of complex numbers,
real numbers, rational numbers, integers, or modular integer rings.
There is a cache table of rings, indexed by <CODE>R</CODE> and <CODE>varname</CODE>.
This ensures that two calls of this function with the same arguments will
return the same polynomial ring.
<DT><CODE>cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R)</CODE>
<DD>
<A NAME="IDX281"></A>
<DT><CODE>cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)</CODE>
<DD>
<DT><CODE>cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R)</CODE>
<DD>
<DT><CODE>cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)</CODE>
<DD>
<DT><CODE>cl_univpoly_rational_ring cl_find_univpoly_ring (const cl_rational_ring& R)</CODE>
<DD>
<DT><CODE>cl_univpoly_rational_ring cl_find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)</CODE>
<DD>
<DT><CODE>cl_univpoly_integer_ring cl_find_univpoly_ring (const cl_integer_ring& R)</CODE>
<DD>
<DT><CODE>cl_univpoly_integer_ring cl_find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)</CODE>
<DD>
<DT><CODE>cl_univpoly_modint_ring cl_find_univpoly_ring (const cl_modint_ring& R)</CODE>
<DD>
<DT><CODE>cl_univpoly_modint_ring cl_find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)</CODE>
<DD>
These functions are equivalent to the general <CODE>cl_find_univpoly_ring</CODE>,
only the return type is more specific, according to the base ring's type.
</DL>
<H2><A NAME="SEC57" HREF="cln_toc.html#TOC57">9.2 Functions on univariate polynomials</A></H2>
<P>
Given a univariate polynomial ring <CODE>R</CODE>, the following members can be used.
<DL COMPACT>
<DT><CODE>cl_ring R->basering()</CODE>
<DD>
<A NAME="IDX282"></A>
This returns the base ring, as passed to <SAMP>`cl_find_univpoly_ring'</SAMP>.
<DT><CODE>cl_UP R->zero()</CODE>
<DD>
<A NAME="IDX283"></A>
This returns <CODE>0 in R</CODE>, a polynomial of degree -1.
<DT><CODE>cl_UP R->one()</CODE>
<DD>
<A NAME="IDX284"></A>
This returns <CODE>1 in R</CODE>, a polynomial of degree <= 0.
<DT><CODE>cl_UP R->canonhom (const cl_I& x)</CODE>
<DD>
<A NAME="IDX285"></A>
This returns <CODE>x in R</CODE>, a polynomial of degree <= 0.
<DT><CODE>cl_UP R->monomial (const cl_ring_element& x, uintL e)</CODE>
<DD>
<A NAME="IDX286"></A>
This returns a sparse polynomial: <CODE>x * X^e</CODE>, where <CODE>X</CODE> is the
indeterminate.
<DT><CODE>cl_UP R->create (sintL degree)</CODE>
<DD>
<A NAME="IDX287"></A>
Creates a new polynomial with a given degree. The zero polynomial has degree
<CODE>-1</CODE>. After creating the polynomial, you should put in the coefficients,
using the <CODE>set_coeff</CODE> member function, and then call the <CODE>finalize</CODE>
member function.
</DL>
<P>
The following are the only destructive operations on univariate polynomials.
<DL COMPACT>
<DT><CODE>void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)</CODE>
<DD>
<A NAME="IDX288"></A>
This changes the coefficient of <CODE>X^index</CODE> in <CODE>x</CODE> to be <CODE>y</CODE>.
After changing a polynomial and before applying any "normal" operation on it,
you should call its <CODE>finalize</CODE> member function.
<DT><CODE>void finalize (cl_UP& x)</CODE>
<DD>
<A NAME="IDX289"></A>
This function marks the endpoint of destructive modifications of a polynomial.
It normalizes the internal representation so that subsequent computations have
less overhead. Doing normal computations on unnormalized polynomials may
produce wrong results or crash the program.
</DL>
<P>
The following operations are defined on univariate polynomials.
<DL COMPACT>
<DT><CODE>cl_univpoly_ring x.ring ()</CODE>
<DD>
<A NAME="IDX290"></A>
Returns the ring to which the univariate polynomial <CODE>x</CODE> belongs.
<DT><CODE>cl_UP operator+ (const cl_UP&, const cl_UP&)</CODE>
<DD>
<A NAME="IDX291"></A>
Returns the sum of two univariate polynomials.
<DT><CODE>cl_UP operator- (const cl_UP&, const cl_UP&)</CODE>
<DD>
<A NAME="IDX292"></A>
Returns the difference of two univariate polynomials.
<DT><CODE>cl_UP operator- (const cl_UP&)</CODE>
<DD>
Returns the negative of a univariate polynomial.
<DT><CODE>cl_UP operator* (const cl_UP&, const cl_UP&)</CODE>
<DD>
<A NAME="IDX293"></A>
Returns the product of two univariate polynomials. One of the arguments may
also be a plain integer or an element of the base ring.
<DT><CODE>cl_UP square (const cl_UP&)</CODE>
<DD>
<A NAME="IDX294"></A>
Returns the square of a univariate polynomial.
<DT><CODE>cl_UP expt_pos (const cl_UP& x, const cl_I& y)</CODE>
<DD>
<A NAME="IDX295"></A>
<CODE>y</CODE> must be > 0. Returns <CODE>x^y</CODE>.
<DT><CODE>bool operator== (const cl_UP&, const cl_UP&)</CODE>
<DD>
<A NAME="IDX296"></A>
<DT><CODE>bool operator!= (const cl_UP&, const cl_UP&)</CODE>
<DD>
<A NAME="IDX297"></A>
Compares two univariate polynomials, belonging to the same univariate
polynomial ring, for equality.
<DT><CODE>cl_boolean zerop (const cl_UP& x)</CODE>
<DD>
<A NAME="IDX298"></A>
Returns true if <CODE>x</CODE> is <CODE>0 in R</CODE>.
<DT><CODE>sintL degree (const cl_UP& x)</CODE>
<DD>
<A NAME="IDX299"></A>
Returns the degree of the polynomial. The zero polynomial has degree <CODE>-1</CODE>.
<DT><CODE>cl_ring_element coeff (const cl_UP& x, uintL index)</CODE>
<DD>
<A NAME="IDX300"></A>
Returns the coefficient of <CODE>X^index</CODE> in the polynomial <CODE>x</CODE>.
<DT><CODE>cl_ring_element x (const cl_ring_element& y)</CODE>
<DD>
<A NAME="IDX301"></A>
Evaluation: If <CODE>x</CODE> is a polynomial and <CODE>y</CODE> belongs to the base ring,
then <SAMP>`x(y)'</SAMP> returns the value of the substitution of <CODE>y</CODE> into
<CODE>x</CODE>.
<DT><CODE>cl_UP deriv (const cl_UP& x)</CODE>
<DD>
<A NAME="IDX302"></A>
Returns the derivative of the polynomial <CODE>x</CODE> with respect to the
indeterminate <CODE>X</CODE>.
</DL>
<P>
The following output functions are defined (see also the chapter on
input/output).
<DL COMPACT>
<DT><CODE>void fprint (cl_ostream stream, const cl_UP& x)</CODE>
<DD>
<A NAME="IDX303"></A>
<DT><CODE>cl_ostream operator<< (cl_ostream stream, const cl_UP& x)</CODE>
<DD>
<A NAME="IDX304"></A>
Prints the univariate polynomial <CODE>x</CODE> on the <CODE>stream</CODE>. The output may
depend on the global printer settings in the variable
<CODE>cl_default_print_flags</CODE>.
</DL>
<H2><A NAME="SEC58" HREF="cln_toc.html#TOC58">9.3 Special polynomials</A></H2>
<P>
The following functions return special polynomials.
<DL COMPACT>
<DT><CODE>cl_UP_I cl_tschebychev (sintL n)</CODE>
<DD>
<A NAME="IDX305"></A>
<A NAME="IDX306"></A>
Returns the n-th Tchebychev polynomial (n >= 0).
<DT><CODE>cl_UP_I cl_hermite (sintL n)</CODE>
<DD>
<A NAME="IDX307"></A>
<A NAME="IDX308"></A>
Returns the n-th Hermite polynomial (n >= 0).
<DT><CODE>cl_UP_RA cl_legendre (sintL n)</CODE>
<DD>
<A NAME="IDX309"></A>
<A NAME="IDX310"></A>
Returns the n-th Legendre polynomial (n >= 0).
<DT><CODE>cl_UP_I cl_laguerre (sintL n)</CODE>
<DD>
<A NAME="IDX311"></A>
<A NAME="IDX312"></A>
Returns the n-th Laguerre polynomial (n >= 0).
</DL>
<P>
Information how to derive the differential equation satisfied by each
of these polynomials from their definition can be found in the
<CODE>doc/polynomial/</CODE> directory.
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