根据多项式的部分分式或分子分母的系数写出多项式的分式

Question

我需要从以下任一方式稳健地生成显示的方程

1- 分子和分母的系数分别由\NumCoeffMat和定义\DenCoeffMat

2- 或部分分数项定义为\PartialFraction

PS：分母中最高阶的系数始终为正数，因此数组中未给出。

任何零系数都应s从书面表达式中排除与其相关的阶数。

\documentclass{article}

\edef\NumCoeffMat{{4, 32, 62}} % numerator coefficients (i.e. numerator =  4s^2 + 32s + 62)
\edef\DenCoeffMat{{12, 47, 60}} % denominator coefficients (i.e. denominator = s^3 + 12s^2 + 47s + 60)

\def\PartialFraction{1/-3, 2/-4, 1/-5} % gain/coefficient and root (i.e. 1/(s+3) + 2/(s+4) + 1/(s+5) )

\begin{document}
    \[
    \frac{4\,s^2 + 32\,s + 62}{s^3 + 12\,s^2 + 47\,s + 60}
    \]
\end{document}

Answer 1

这是编写多项式的通用宏。第一个参数是不确定的，第二个参数是系数列表

\documentclass{article}

\usepackage{pgffor}

\newcount\degree
\newif\ifzero
\newif\ifcoeff

% \polynomial{<indeterminate>}{<list of coefficients>}
\newcommand*\polynomial[2]{%
  \global\degree=0\relax
  \foreach\x in {#2} {\global\advance\degree 1\relax}%
  \ifnum\degree=0
    0%
  \else
    \global\zerotrue
    \foreach\coeff in {#2} {%
      \global\advance\degree-1
      \coefffalse
      \ifnum\coeff=0 \else
        \ifnum\coeff>0 \ifzero\else+\fi\fi
        \ifnum\coeff=-1
          -\ifnum\degree=0 1\fi
        \else\ifnum\coeff=1
          \ifnum\degree=0 1\fi
        \else
          \coefftrue  
          \coeff
        \fi\fi 
        \ifnum\degree>0\relax\ifcoeff\,\fi{#1}\ifnum\degree>1^{\the\degree}\fi\fi
        \global\zerofalse
      \fi
    }%
    \ifzero 0\fi
  \fi
}

\newcommand\PartialFraction[2][s]{%
  \global\zerotrue
  \foreach \n/\r in {#2} {%
    \ifnum\n=0 \else
      \ifnum\n<0
        -\edef\n{\the\numexpr-\n\relax}%
      \else
        \ifzero\else +\fi
      \fi
      \global\zerofalse
      \frac{\n}{s\ifnum\r>0 -\r\else\ifnum\r<0 +\the\numexpr-\r\relax\fi\fi}%
    \fi}%
  \ifzero 0\fi
}

\begin{document}
\[
    \frac{\polynomial{s}{4,32,62}}{\polynomial{s}{1,12,47,60}}
\]
\[
    \PartialFraction{1/-3, 2/-4, 1/-5}
\]
\end{document}

下一个宏允许你按照你想要的方式给出系数

% \Fraction[<indeterminate> defaults to s]
\newcommand\Fraction[1][s]{%
  \begingroup
    \def\polyaux##1{{1,##1}}%
    \edef\next{\noexpand\frac{\unexpanded{\polynomial{#1}}\NumCoeffMat}{\unexpanded{\polynomial{#1}}\expandafter\polyaux\DenCoeffMat}}%
    \next
  \endgroup
}
\[
  \def\NumCoeffMat{{4, 32, 62}}  % numerator = 4s^2+32s+62 
  \def\DenCoeffMat{{12, 47, 60}} % denominator = s^3+12s^2+47s+60
  \Fraction
\]

Answer 2

使用 expl3 clist....

\documentclass{article}

\def\NumCoeffMat{4, 32, 62} % numerator coefficients (i.e. numerator =  4s^2 + 32s + 62)
\def\DenCoeffMat{12, 47, 60} % denominator coefficients (i.e. denominator = s^3 + 12s^2 + 47s + 60)

\edef\DenCoeffMatX{1,\DenCoeffMat} % add the leading 1 back
\ExplSyntaxOn
\int_new:N\l_tmp_int
\tl_new:N\l_sep
\tl_set:Nn\l_sep{}

\def\z#1{
\int_set:Nn\l_tmp_int{\clist_count:N#1 - 1} % order of polynomial
\clist_map_inline:Nn#1{
\int_compare:nNnT{##1}> {0} % skip 0 terms
  {\l_sep                   % + except first term
   \int_compare:nNnT{##1}> {1}{##1\,} % skip coeff 1
  \int_compare:nNnT{\l_tmp_int}> {0}{
    s\int_compare:nNnT{\l_tmp_int}> {1}{^{\int_use:N\l_tmp_int}}}} % skip power 1
\int_decr:N\l_tmp_int
\tl_set:Nn\l_sep{+}
}}
\ExplSyntaxOff

\begin{document}

\[
\frac{\z\NumCoeffMat}{\z\DenCoeffMatX}
\]
\end{document}

Answer 3

这使用 展开多项式\foreach。

\documentclass{article}
\usepackage{tikz}

\edef\NumCoeffMat{{4, 32, 62}}% numerator coefficients (i.e. numerator =  4s^2 + 32s + 62)

\newcounter{exponent}
\newcommand{\poly}[1]{\bgroup
  \setcounter{exponent}{-1}%
  \def\zero{0}%
  \def\one{1}%
  \expandafter\edef\expandafter\list#1
  \foreach \x in \list {\stepcounter{exponent}}% max exponent
  \foreach \x in \list {\ifnum\value{exponent}>0 
    \ifx\x\zero \else
      \ifx\x\one \else \x \fi
      \ifnum\value{exponent}>1 s^\theexponent +
      \else s +
      \fi
    \fi
  \else \x% last coeficient in polynomial
  \fi
  \addtocounter{exponent}{-1}}%
\egroup}
  

\begin{document}
    
    $\poly{\NumCoeffMat}$

\end{document}

Answer 4

作为练习，我尝试用我的functional包裹：

\documentclass{article}

\usepackage{functional}

\def\NumCoeffMat{4, 32, 62} % numerator coefficients (i.e. numerator =  4s^2 + 32s + 62)
\def\DenCoeffMat{12, 47, 60} % denominator coefficients (i.e. denominator = s^3 + 12s^2 + 47s + 60)

\edef\DenCoeffMatX{1,\DenCoeffMat} % add the leading 1 back

\IgnoreSpacesOn

\PrgNewFunction \Z {M} {
  \IntSet \lTmpaInt {\ClistVarCount #1}
  \IntDecr \lTmpaInt
  \ClistClear \lTmpaClist
  \ClistVarMapInline #1 {
    \IntCompareTF {##1} > {0} {
      \IntCompareTF {##1} > {1} {
        \TlSet \lTmpaTl {##1}
      }{
        \TlSet \lTmpaTl {}
      }
      \IntCompareTF {\lTmpaInt} > {1} {
        \TlPutRight \lTmpaTl {\Expand{s^{\OnlyValue\lTmpaInt}}}
      }{
        \IntCompareTF {\lTmpaInt} = {1} {
          \TlPutRight \lTmpaTl {s}
        } { }
      }
      \ClistPutRight \lTmpaClist {\Value\lTmpaTl}
    } { }
    \IntDecr \lTmpaInt
  }
  \ClistVarJoin \lTmpaClist {+}
}

\IgnoreSpacesOff

\begin{document}

\[ \frac{\Z\NumCoeffMat}{\Z\DenCoeffMatX} \]

\end{document}