How to build macros for vector-matrix operators?

Question

Can anybody build macros for doing the following vector operations? Or are there already such macros? For example,

1) Example data: \def\a{(5,3,0)}
                 \def\b{(5,1,6)}
                 \def\A{(1,0,1\\0,2,0\\0,0,3)}
2) Vector addition: $\a+\b$ returns (10,4,6)
3) Vector subtraction: $\a-\b$ returns (0,2,-6)
4) Vector matrix multiplication: $\A*\a$ returns (5,6,0)

Thanks to the David's code, I expanded the followings for 1) vector addition, 2) vector subtraction, 3) vector multiplication, 4) vector division. What I'm going to do is to expand the code to 5) \car and \cdr for list processing. The following is the first step.

\documentclass{article}

\makeatletter

%main macros
\def\rvecadd#1#2{\edef\tmp{\noexpand\@rvecadd+#1,\relax#2,\relax}\tmp}
\def\rvecsub#1#2{\edef\tmp{\noexpand\@rvecadd-#1,\relax#2,\relax}\tmp}
\def\rvecmul#1#2{\edef\tmp{\noexpand\@rvecadd*#1,\relax#2,\relax}\tmp}
\def\rvecdiv#1#2{\edef\tmp{\noexpand\@rvecadd/#1,\relax#2,\relax}\tmp}
\def\rcar#1{\edef\tmp{\noexpand\@rcar#1,\relax}\tmp}
\def\rcdr#1{\edef\tmp{\noexpand\@rcdr#1,\relax}\tmp}

%recursive macros
\def\@rvecadd#1#2,#3\relax#4,#5\relax{%
\the\numexpr#2#1#4\relax
\ifx\@#3\@\expandafter\@gobble
\else\expandafter\@firstofone
\fi{,\@rvecadd#1#3\relax#5\relax} }

\def\@rcar#1,#2\relax{%
#1\relax
\ifx\@#2\@\expandafter\@gobble
\else\expandafter\@firstofone
\fi}

\def\@rcdr#1,#2\relax{%
#2\relax
\ifx\@#2\@\expandafter\@gobble
\else\expandafter\@firstofone
\fi}


\begin{document}
%Data
\def\a{5,3,0,100,1}
\def\b{5,1,6,1000,1}
\def\A{1,0,1\\0,2,0\\0,0,3}
\def\aT{5\\3\\0}

%Test
$(\a) + (\b) = (\rvecadd\a\b)$\par
$(\a) - (\b) = (\rvecsub\a\b)$\par
$(\a) * (\b) = (\rvecmul\a\b)$\par
$(\a) / (\b) = (\rvecdiv\a\b)$

%car test:
\rcar\a

%cdr test
(\rcdr\a)

\end{document}

Answer 1

Out of time to do multiplication now, but her are the easy ones vector addition/subtraction.

\documentclass{article}

\def\a{5,3,0}
\def\b{5,1,6}
\def\A{1,0,1\\0,2,0\\0,0,3}
%2) Vector addition: $\a+\b$ returns (10,4,6)
%3) Vector subtraction: $\a-\b$ returns (0,2,-6)
%4) Vector matrix multiplication: $\A*\aT$ returns (5,6,0)

\makeatletter

\def\rvecadd#1#2{\edef\tmp{\noexpand\@rvecadd+#1,\relax#2,\relax}\tmp}
\def\rvecsub#1#2{\edef\tmp{\noexpand\@rvecadd-#1,\relax#2,\relax}\tmp}

\def\@rvecadd#1#2,#3\relax#4,#5\relax{%
\the\numexpr#2#1#4\relax
\ifx\@#3\@\expandafter\@gobble
\else\expandafter\@firstofone
\fi{,\@rvecadd#1#3\relax#5\relax}}

\begin{document}

\def\a{5,3,0}
\def\b{5,1,6}
\def\A{1,0,1\\0,2,0\\0,0,3}
\def\aT{5\\3\\0}


$(\a) + (\b) = (\rvecadd\a\b)$


$(\a) - (\b) = (\rvecsub\a\b)$

\end{document}

Answer 2

I changed the syntax a bit, omitting the parentheses you had in your vectors and matrices. With the code below, you can do

\setvector\a{5,3,0}
\setmatrix\A{1,0,1\\0,2,0\\0,0,3}
\begin{equation}
  \usematrix\A \cdot \usevvector\a
  = \matrixtimesvector\result\A\a  \usevvector\result
\end{equation}

to display the matrix \A, the vecor \a (vertically), compute the \result of multiplying \A by \a, and display the \result as a vertical vector.

\documentclass{article}
\usepackage{expl3, xparse}
\usepackage{amsmath}
\ExplSyntaxOn
\seq_new:N \l__mypkg_a_seq
\seq_new:N \l__mypkg_b_seq
\seq_new:N \l__mypkg_result_seq
\int_new:N \l__mypkg_entries_int

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Declaring vectors and matrices
%
\cs_new_protected:Npn \mypkg_set_vector:Nnn #1#2#3
  {
    \seq_set_split:Nnn #1 {#2} {#3} % Split into entries.
    \seq_set_map:NNn #1#1 { \fp_to_tl:n {##1} } % Evaluate each entry.
  }
\NewDocumentCommand { \setvector } { m O{,} m }
  {
    \mypkg_set_vector:Nnn \l__mypkg_result_seq {#2} {#3}
    \cs_set_eq:NN #1 \l__mypkg_result_seq
  }
\NewDocumentCommand { \setmatrix } { m O{\\} O{,} m }
  {
    \seq_clear:N \l__mypkg_result_seq
    \int_set_eq:NN \l__mypkg_entries_int \c_minus_one
    \seq_set_split:Nnn \l__mypkg_a_seq {#2} {#4} % Split into lines.
    \seq_map_inline:Nn \l__mypkg_a_seq
      {
        % Split each line |##1| into entries.  For the first line,
        % |\l__mypkg_entries_int| is -1 still; we set it to the
        % number of entries in the first line.  Then check that the line
        % has the right number of entries: if so, put it in the result
        % as a comma-delimited list, otherwise complain loudly and
        % ignore the line.
        %
        \mypkg_set_vector:Nnn \l__mypkg_b_seq {#3} {##1}
        \int_compare:nT { \l__mypkg_entries_int = \c_minus_one }
          {
            \int_set:Nn \l__mypkg_entries_int
              { \seq_count:N \l__mypkg_b_seq }
          }
        \int_compare:nTF
          { \seq_count:N \l__mypkg_b_seq = \l__mypkg_entries_int }
          {
            \seq_put_right:Nx \l__mypkg_result_seq
              { \seq_use:Nnnn \l__mypkg_b_seq { & } { & } { & } }
          }
          {
            \msg_error:nnxxx { mypkg } { mismatched-lines }
              { \token_to_str:N #1 }
              { \int_use:N \l__mypkg_entries_int }
              { \seq_count:N \l__mypkg_b_seq }
          }
      }
    \cs_set_eq:NN #1 \l__mypkg_result_seq
  }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Adding two vectors
%
\tl_new:N \l__mypkg_operation_tl
\NewDocumentCommand { \addvectors } { m m m }
  {
    \tl_set:Nn \l__mypkg_operation_tl { + }
    \mypkg_add_vectors:NN #2#3
    \cs_set_eq:NN #1 \l__mypkg_result_seq
  }
\NewDocumentCommand { \subvectors } { m m m }
  {
    \tl_set:Nn \l__mypkg_operation_tl { - }
    \mypkg_add_vectors:NN #2#3
    \cs_set_eq:NN #1 \l__mypkg_result_seq
  }
\cs_new_protected:Npn \mypkg_add_vectors:NN #1#2
  {
    \int_compare:nNnTF { \seq_count:N #1 } = { \seq_count:N #2 }
      {
        \seq_clear:N \l__mypkg_result_seq
        \seq_mapthread_function:NNN #1#2 \mypkg_add_vectors:nn
      }
      {
        \msg_error:nnxxxx { mypkg } { mismatched-vectors }
          { \token_to_str:N #1 } { \seq_count:N #1 }
          { \token_to_str:N #2 } { \seq_count:N #2 }
      }
  }
\cs_new_protected:Npn \mypkg_add_vectors:nn #1#2
  {
    \seq_put_right:Nx \l__mypkg_result_seq
      { \fp_to_tl:n { #1 \l__mypkg_operation_tl #2 } }
  }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Scalar product of two vectors
%
\NewDocumentCommand { \scalarproduct } { mmm }
  {
    \tl_set:Nx #1 { \mypkg_scalar_product:NN #2 #3 }
  }
\cs_new:Npn \mypkg_scalar_product:NN #1#2
  {
    \fp_to_tl:n
      { \seq_mapthread_function:NNN #1 #2 \mypkg_scalar_product_aux:nn }
  }
\cs_new:Npn \mypkg_scalar_product_aux:nn #1#2 { + #1 * #2 }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Multiplying matrix times vector
%
\NewDocumentCommand { \matrixtimesvector } { mmm }
  {
    \seq_clear:N \l__mypkg_result_seq
    \seq_map_inline:Nn #2
      {
        \seq_set_split:Nnn \l__mypkg_a_seq { & } {##1}
        \seq_put_right:Nx \l__mypkg_result_seq
          { \mypkg_scalar_product:NN \l__mypkg_a_seq #3 }
      }
    \cs_set_eq:NN #1 \l__mypkg_result_seq
  }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Displaying vectors and matrices
%
\NewDocumentCommand { \usehvector } { m }
  {
    \mode_if_math:F { \msg_warning:nn { mypkg } { text-mode } }
    \ensuremath
      {
        \begin{pmatrix}
          \seq_use:Nnnn #1 { , } { , } { , }
        \end{pmatrix}
      }
  }
\NewDocumentCommand { \usevvector } { m }
  {
    \mode_if_math:F { \msg_warning:nn { mypkg } { text-mode } }
    \ensuremath
      {
        \begin{pmatrix}
          \seq_use:Nnnn #1 { \\ } { \\ } { \\ }
        \end{pmatrix}
      }
  }
\NewDocumentCommand { \usematrix } { m }
  {
    \mode_if_math:F { \msg_warning:nn { mypkg } { text-mode } }
    \ensuremath
      {
        \begin{pmatrix}
          \seq_use:Nnnn #1 { \\ } { \\ } { \\ }
        \end{pmatrix}
      }
  }

\msg_new:nnn { mypkg } { mismatched-lines }
  { The~lines~of~#1~should~have~#2~entries,~but~this~one~has~#3. }
\msg_new:nnn { mypkg } { mismatched-vectors }
  { The~vector~#1~has~#2~entries,~but~the~vector~#3~has~#4~entries. }
\msg_new:nnn { mypkg } { text-mode }
  { The~\iow_char:N\\use...~commands~should~only~be~used~in~math~mode. }

\ExplSyntaxOff
\begin{document}
\setvector\a{5,3,0}
\setvector\b{5,1,6}
\setmatrix\A{1,0,1\\0,2,0\\0,0,3}

Vector addition: $\usehvector\a+\usehvector\b$ returns
$\addvectors\result\a\b \usehvector\result$.  Subtraction:
\begin{equation}
  \usevvector\a - \usevvector\b
  = \subvectors\result\a\b \usevvector\result .
\end{equation}

Scalar product:
\begin{equation}
  \usehvector\a \cdot \usevvector\b
  = \scalarproduct\result\a\b \result
\end{equation}
Matrix-vector product
\begin{equation}
  \usematrix\A \cdot \usevvector\a
  = \matrixtimesvector\result\A\a  \usevvector\result
\end{equation}

% 2) Vector addition: $\a+\b$ returns (10,4,6)
% 3) Vector subtraction: $\a-\b$ returns (0,2,-6)
% 4) Vector matrix multiplication: $\A*\a$ returns (5,6,0)
\end{document}