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<TITLE>CLN, a Class Library for Numbers - 9. Univariate polynomials</TITLE>
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Go to the <A HREF="cln_1.html">first</A>, <A HREF="cln_8.html">previous</A>, <A HREF="cln_10.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.
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<H1><A NAME="SEC54" HREF="cln_toc.html#TOC54">9. Univariate polynomials</A></H1>
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<P>
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<A NAME="IDX269"></A>
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<A NAME="IDX270"></A>
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<H2><A NAME="SEC55" HREF="cln_toc.html#TOC55">9.1 Univariate polynomial rings</A></H2>
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<P>
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CLN implements univariate polynomials (polynomials in one variable) over an
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arbitrary ring. The indeterminate variable may be either unnamed (and will be
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printed according to <CODE>cl_default_print_flags.univpoly_varname</CODE>, which
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defaults to <SAMP>`x'</SAMP>) or carry a given name. The base ring and the
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indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
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(accidentally) mix elements of different polynomial rings, e.g.
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<CODE>(a^2+1) * (b^3-1)</CODE> will result in a runtime error. (Ideally this should
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return a multivariate polynomial, but they are not yet implemented in CLN.)
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<P>
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The classes of univariate polynomial rings are
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<PRE>
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Ring
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cl_ring
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<cl_ring.h>
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Univariate polynomial ring
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cl_univpoly_ring
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<cl_univpoly.h>
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+----------------+-------------------+
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Complex polynomial ring | Modular integer polynomial ring
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cl_univpoly_complex_ring | cl_univpoly_modint_ring
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<cl_univpoly_complex.h> | <cl_univpoly_modint.h>
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+----------------+
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Real polynomial ring |
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cl_univpoly_real_ring |
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<cl_univpoly_real.h> |
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+----------------+
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Rational polynomial ring |
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cl_univpoly_rational_ring |
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<cl_univpoly_rational.h> |
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+----------------+
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Integer polynomial ring
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cl_univpoly_integer_ring
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<cl_univpoly_integer.h>
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</PRE>
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<P>
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and the corresponding classes of univariate polynomials are
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<PRE>
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Univariate polynomial
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cl_UP
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<cl_univpoly.h>
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+----------------+-------------------+
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Complex polynomial | Modular integer polynomial
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cl_UP_N | cl_UP_MI
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<cl_univpoly_complex.h> | <cl_univpoly_modint.h>
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+----------------+
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Real polynomial |
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cl_UP_R |
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<cl_univpoly_real.h> |
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+----------------+
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Rational polynomial |
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cl_UP_RA |
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<cl_univpoly_rational.h> |
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+----------------+
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Integer polynomial
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cl_UP_I
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<cl_univpoly_integer.h>
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</PRE>
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<P>
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Univariate polynomial rings are constructed using the functions
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<DL COMPACT>
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<DT><CODE>cl_univpoly_ring cl_find_univpoly_ring (const cl_ring& R)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_ring cl_find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)</CODE>
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<DD>
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This function returns the polynomial ring <SAMP>`R[X]'</SAMP>, unnamed or named.
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<CODE>R</CODE> may be an arbitrary ring. This function takes care of finding out
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about special cases of <CODE>R</CODE>, such as the rings of complex numbers,
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real numbers, rational numbers, integers, or modular integer rings.
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There is a cache table of rings, indexed by <CODE>R</CODE> and <CODE>varname</CODE>.
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This ensures that two calls of this function with the same arguments will
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return the same polynomial ring.
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<DT><CODE>cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R)</CODE>
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<DD>
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<A NAME="IDX271"></A>
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<DT><CODE>cl_univpoly_complex_ring cl_find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_real_ring cl_find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_rational_ring cl_find_univpoly_ring (const cl_rational_ring& R)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_rational_ring cl_find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_integer_ring cl_find_univpoly_ring (const cl_integer_ring& R)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_integer_ring cl_find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_modint_ring cl_find_univpoly_ring (const cl_modint_ring& R)</CODE>
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<DD>
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<DT><CODE>cl_univpoly_modint_ring cl_find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)</CODE>
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<DD>
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These functions are equivalent to the general <CODE>cl_find_univpoly_ring</CODE>,
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only the return type is more specific, according to the base ring's type.
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</DL>
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<H2><A NAME="SEC56" HREF="cln_toc.html#TOC56">9.2 Functions on univariate polynomials</A></H2>
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<P>
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Given a univariate polynomial ring <CODE>R</CODE>, the following members can be used.
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<DL COMPACT>
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<DT><CODE>cl_ring R->basering()</CODE>
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<DD>
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<A NAME="IDX272"></A>
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This returns the base ring, as passed to <SAMP>`cl_find_univpoly_ring'</SAMP>.
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<DT><CODE>cl_UP R->zero()</CODE>
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<DD>
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<A NAME="IDX273"></A>
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This returns <CODE>0 in R</CODE>, a polynomial of degree -1.
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<DT><CODE>cl_UP R->one()</CODE>
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<DD>
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<A NAME="IDX274"></A>
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This returns <CODE>1 in R</CODE>, a polynomial of degree <= 0.
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<DT><CODE>cl_UP R->canonhom (const cl_I& x)</CODE>
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<DD>
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<A NAME="IDX275"></A>
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This returns <CODE>x in R</CODE>, a polynomial of degree <= 0.
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<DT><CODE>cl_UP R->monomial (const cl_ring_element& x, uintL e)</CODE>
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<DD>
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<A NAME="IDX276"></A>
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This returns a sparse polynomial: <CODE>x * X^e</CODE>, where <CODE>X</CODE> is the
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indeterminate.
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<DT><CODE>cl_UP R->create (sintL degree)</CODE>
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<DD>
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<A NAME="IDX277"></A>
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Creates a new polynomial with a given degree. The zero polynomial has degree
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<CODE>-1</CODE>. After creating the polynomial, you should put in the coefficients,
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using the <CODE>set_coeff</CODE> member function, and then call the <CODE>finalize</CODE>
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member function.
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</DL>
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<P>
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The following are the only destructive operations on univariate polynomials.
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<DL COMPACT>
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<DT><CODE>void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)</CODE>
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<DD>
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<A NAME="IDX278"></A>
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This changes the coefficient of <CODE>X^index</CODE> in <CODE>x</CODE> to be <CODE>y</CODE>.
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After changing a polynomial and before applying any "normal" operation on it,
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you should call its <CODE>finalize</CODE> member function.
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<DT><CODE>void finalize (cl_UP& x)</CODE>
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<DD>
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<A NAME="IDX279"></A>
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This function marks the endpoint of destructive modifications of a polynomial.
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It normalizes the internal representation so that subsequent computations have
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less overhead. Doing normal computations on unnormalized polynomials may
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produce wrong results or crash the program.
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</DL>
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<P>
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The following operations are defined on univariate polynomials.
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<DL COMPACT>
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<DT><CODE>cl_univpoly_ring x.ring ()</CODE>
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<DD>
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<A NAME="IDX280"></A>
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Returns the ring to which the univariate polynomial <CODE>x</CODE> belongs.
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<DT><CODE>cl_UP operator+ (const cl_UP&, const cl_UP&)</CODE>
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<DD>
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<A NAME="IDX281"></A>
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Returns the sum of two univariate polynomials.
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<DT><CODE>cl_UP operator- (const cl_UP&, const cl_UP&)</CODE>
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<DD>
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<A NAME="IDX282"></A>
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Returns the difference of two univariate polynomials.
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<DT><CODE>cl_UP operator- (const cl_UP&)</CODE>
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<DD>
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Returns the negative of a univariate polynomial.
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<DT><CODE>cl_UP operator* (const cl_UP&, const cl_UP&)</CODE>
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<DD>
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<A NAME="IDX283"></A>
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Returns the product of two univariate polynomials. One of the arguments may
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also be a plain integer or an element of the base ring.
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<DT><CODE>cl_UP square (const cl_UP&)</CODE>
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<DD>
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<A NAME="IDX284"></A>
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Returns the square of a univariate polynomial.
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<DT><CODE>cl_UP expt_pos (const cl_UP& x, const cl_I& y)</CODE>
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<DD>
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<A NAME="IDX285"></A>
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<CODE>y</CODE> must be > 0. Returns <CODE>x^y</CODE>.
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<DT><CODE>bool operator== (const cl_UP&, const cl_UP&)</CODE>
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<DD>
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<A NAME="IDX286"></A>
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<DT><CODE>bool operator!= (const cl_UP&, const cl_UP&)</CODE>
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<DD>
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<A NAME="IDX287"></A>
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Compares two univariate polynomials, belonging to the same univariate
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polynomial ring, for equality.
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<DT><CODE>cl_boolean zerop (const cl_UP& x)</CODE>
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<DD>
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<A NAME="IDX288"></A>
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Returns true if <CODE>x</CODE> is <CODE>0 in R</CODE>.
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<DT><CODE>sintL degree (const cl_UP& x)</CODE>
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<DD>
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<A NAME="IDX289"></A>
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Returns the degree of the polynomial. The zero polynomial has degree <CODE>-1</CODE>.
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<DT><CODE>cl_ring_element coeff (const cl_UP& x, uintL index)</CODE>
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<DD>
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<A NAME="IDX290"></A>
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Returns the coefficient of <CODE>X^index</CODE> in the polynomial <CODE>x</CODE>.
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<DT><CODE>cl_ring_element x (const cl_ring_element& y)</CODE>
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<DD>
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<A NAME="IDX291"></A>
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Evaluation: If <CODE>x</CODE> is a polynomial and <CODE>y</CODE> belongs to the base ring,
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then <SAMP>`x(y)'</SAMP> returns the value of the substitution of <CODE>y</CODE> into
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<CODE>x</CODE>.
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<DT><CODE>cl_UP deriv (const cl_UP& x)</CODE>
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<DD>
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<A NAME="IDX292"></A>
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Returns the derivative of the polynomial <CODE>x</CODE> with respect to the
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indeterminate <CODE>X</CODE>.
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</DL>
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<P>
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The following output functions are defined (see also the chapter on
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input/output).
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<DL COMPACT>
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<DT><CODE>void fprint (cl_ostream stream, const cl_UP& x)</CODE>
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<DD>
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<A NAME="IDX293"></A>
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<DT><CODE>cl_ostream operator<< (cl_ostream stream, const cl_UP& x)</CODE>
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<DD>
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<A NAME="IDX294"></A>
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Prints the univariate polynomial <CODE>x</CODE> on the <CODE>stream</CODE>. The output may
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depend on the global printer settings in the variable
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<CODE>cl_default_print_flags</CODE>.
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</DL>
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<H2><A NAME="SEC57" HREF="cln_toc.html#TOC57">9.3 Special polynomials</A></H2>
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<P>
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The following functions return special polynomials.
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<DL COMPACT>
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<DT><CODE>cl_UP_I cl_tschebychev (sintL n)</CODE>
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<DD>
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<A NAME="IDX295"></A>
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<A NAME="IDX296"></A>
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Returns the n-th Tchebychev polynomial (n >= 0).
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<DT><CODE>cl_UP_I cl_hermite (sintL n)</CODE>
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<DD>
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<A NAME="IDX297"></A>
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<A NAME="IDX298"></A>
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Returns the n-th Hermite polynomial (n >= 0).
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<DT><CODE>cl_UP_RA cl_legendre (sintL n)</CODE>
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<DD>
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<A NAME="IDX299"></A>
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<A NAME="IDX300"></A>
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Returns the n-th Legendre polynomial (n >= 0).
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<DT><CODE>cl_UP_I cl_laguerre (sintL n)</CODE>
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<DD>
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<A NAME="IDX301"></A>
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<A NAME="IDX302"></A>
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Returns the n-th Laguerre polynomial (n >= 0).
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</DL>
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<P>
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Information how to derive the differential equation satisfied by each
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of these polynomials from their definition can be found in the
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<CODE>doc/polynomial/</CODE> directory.
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<P><HR><P>
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Go to the <A HREF="cln_1.html">first</A>, <A HREF="cln_8.html">previous</A>, <A HREF="cln_10.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.
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