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\title{\$SPAD/src/algebra cmplxrt.spad}
\author{Patrizia Gianni}
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\begin{abstract}
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\section{package CMPLXRT ComplexRootPackage}
<<package CMPLXRT ComplexRootPackage>>=
)abbrev package CMPLXRT ComplexRootPackage
++ Author: P. Gianni
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: Complex, Float, Fraction, UnivariatePolynomial
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++   This package provides functions complexZeros
++ for finding the complex zeros
++ of univariate polynomials with complex rational number coefficients.
++ The results are to any user specified precision and are returned
++ as either complex rational number or complex floating point numbers
++ depending on the type of the second argument which specifies the
++ precision.

-- Packages for the computation of complex roots of
-- univariate polynomials with rational or gaussian coefficients.

-- Simplified version, the old original based on Gebauer's solver is
-- in ocmplxrt spad
RN     ==> Fraction Integer
I      ==> Integer
NF     ==> Float

ComplexRootPackage(UP,Par) : T == C where
   UP   :   UnivariatePolynomialCategory Complex Integer
   Par  :   Join(Field, OrderedRing) -- will be Float or RN
   CP   ==> Complex Par
   PCI  ==> Polynomial Complex Integer

   T == with
        complexZeros:(UP,Par)  -> List CP
          ++ complexZeros(poly, eps) finds the complex zeros of the
          ++ univariate polynomial poly to precision eps with
          ++ solutions returned as complex floats or rationals
          ++ depending on the type of eps.

   C == add
    complexZeros(p:UP,eps:Par):List CP ==
      x1:Symbol():=new()
      x2:Symbol():=new()
      vv:Symbol():=new()
      lpf:=factors factor(p)$ComplexFactorization(I,UP)
      ris:List CP:=empty()
      for pf in lpf repeat
          pp:=pf.factor pretend SparseUnivariatePolynomial Complex Integer
          q:PCI :=multivariate(pp,vv)
          q:=eval(q,vv,x1::PCI+complex(0,1)*(x2::PCI))
          p1:=map(real,q)$PolynomialFunctions2(Complex I,I)
          p2:=map(imag,q)$PolynomialFunctions2(Complex I,I)
          lz:=innerSolve([p1,p2],[],[x1,x2],
                          eps)$InnerNumericFloatSolvePackage(I,Par,Par)
          ris:=append([complex(first z,second z) for z in lz],ris)
      ris

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

<<package CMPLXRT ComplexRootPackage>>

@
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