Reconstructing a polynomial from its coefficient array












1















A polynomial coefficient matrix: 



mat = 

  CoefficientList[3 + 5 x^3 + 4 y^3 + 2 x + 6 x^2 y + 7 x y^2 + 8 x y, {x, y}];


begin{equation}
left(
begin{array}{cccc}
3 & 0 & 0 & 4 \
2 & 8 & 7 & 0 \
0 & 6 & 0 & 0 \
5 & 0 & 0 & 0 \
end{array}
right)
end{equation}



Another matrix:



list = 

  {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1,o1, p1}};


whose matrix form is:
begin{equation}
left(
begin{array}{cccc}
a1 & b1 & c1 & d1 \
e1 & f1 & g1 & h1 \
i1 & j1 & k1 & l1 \
m1 & n1 & o1 & p1 \
end{array}
right)
end{equation}



How can I generate the following polynomial automatically?



$text{a1}+text{d1} y^3+text{e1} x+text{f1} x y+text{g1} x y^2+text{j1} x^2 y+text{m1} x^3$






list-manipulation algebraic-manipulation







share|improve this question




















  • 1





    Why are some entries of the matrix ignored? Maybe this, if that is a mistake: {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1, o1, p1}}.y^Range[0, 3].x^Range[0, 3]

    – Michael E2
    3 hours ago











  • There's an example in the docs for CoefficientList for how to recover the polynomial from the matrix: Look for Fold[FromDigits[Reverse[#1], #2] &, %, {x, y}].

    – Michael E2
    3 hours ago











  • @MichaelE2 I am ignoring only those entries for which the constant is zero as in coefficientlist matrix.

    – Chandan Sharma
    3 hours ago






  • 1





    Do you mean Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat * list, {x, y}]?

    – Michael E2
    2 hours ago













  • @MichaelE2 Exactly.

    – Chandan Sharma
    2 hours ago
















1















A polynomial coefficient matrix: 



mat = 

  CoefficientList[3 + 5 x^3 + 4 y^3 + 2 x + 6 x^2 y + 7 x y^2 + 8 x y, {x, y}];


begin{equation}
left(
begin{array}{cccc}
3 & 0 & 0 & 4 \
2 & 8 & 7 & 0 \
0 & 6 & 0 & 0 \
5 & 0 & 0 & 0 \
end{array}
right)
end{equation}



Another matrix:



list = 

  {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1,o1, p1}};


whose matrix form is:
begin{equation}
left(
begin{array}{cccc}
a1 & b1 & c1 & d1 \
e1 & f1 & g1 & h1 \
i1 & j1 & k1 & l1 \
m1 & n1 & o1 & p1 \
end{array}
right)
end{equation}



How can I generate the following polynomial automatically?



$text{a1}+text{d1} y^3+text{e1} x+text{f1} x y+text{g1} x y^2+text{j1} x^2 y+text{m1} x^3$






list-manipulation algebraic-manipulation







share|improve this question




















  • 1





    Why are some entries of the matrix ignored? Maybe this, if that is a mistake: {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1, o1, p1}}.y^Range[0, 3].x^Range[0, 3]

    – Michael E2
    3 hours ago











  • There's an example in the docs for CoefficientList for how to recover the polynomial from the matrix: Look for Fold[FromDigits[Reverse[#1], #2] &, %, {x, y}].

    – Michael E2
    3 hours ago











  • @MichaelE2 I am ignoring only those entries for which the constant is zero as in coefficientlist matrix.

    – Chandan Sharma
    3 hours ago






  • 1





    Do you mean Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat * list, {x, y}]?

    – Michael E2
    2 hours ago













  • @MichaelE2 Exactly.

    – Chandan Sharma
    2 hours ago














1












1








1








A polynomial coefficient matrix: 



mat = 

  CoefficientList[3 + 5 x^3 + 4 y^3 + 2 x + 6 x^2 y + 7 x y^2 + 8 x y, {x, y}];


begin{equation}
left(
begin{array}{cccc}
3 & 0 & 0 & 4 \
2 & 8 & 7 & 0 \
0 & 6 & 0 & 0 \
5 & 0 & 0 & 0 \
end{array}
right)
end{equation}



Another matrix:



list = 

  {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1,o1, p1}};


whose matrix form is:
begin{equation}
left(
begin{array}{cccc}
a1 & b1 & c1 & d1 \
e1 & f1 & g1 & h1 \
i1 & j1 & k1 & l1 \
m1 & n1 & o1 & p1 \
end{array}
right)
end{equation}



How can I generate the following polynomial automatically?



$text{a1}+text{d1} y^3+text{e1} x+text{f1} x y+text{g1} x y^2+text{j1} x^2 y+text{m1} x^3$






list-manipulation algebraic-manipulation







share|improve this question
















A polynomial coefficient matrix: 



mat = 

  CoefficientList[3 + 5 x^3 + 4 y^3 + 2 x + 6 x^2 y + 7 x y^2 + 8 x y, {x, y}];


begin{equation}
left(
begin{array}{cccc}
3 & 0 & 0 & 4 \
2 & 8 & 7 & 0 \
0 & 6 & 0 & 0 \
5 & 0 & 0 & 0 \
end{array}
right)
end{equation}



Another matrix:



list = 

  {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1,o1, p1}};


whose matrix form is:
begin{equation}
left(
begin{array}{cccc}
a1 & b1 & c1 & d1 \
e1 & f1 & g1 & h1 \
i1 & j1 & k1 & l1 \
m1 & n1 & o1 & p1 \
end{array}
right)
end{equation}



How can I generate the following polynomial automatically?



$text{a1}+text{d1} y^3+text{e1} x+text{f1} x y+text{g1} x y^2+text{j1} x^2 y+text{m1} x^3$






list-manipulation algebraic-manipulation




list-manipulation algebraic-manipulation






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 2 hours ago









m_goldberg

84.5k872196




84.5k872196










asked 3 hours ago









Chandan SharmaChandan Sharma

1075




1075








  • 1





    Why are some entries of the matrix ignored? Maybe this, if that is a mistake: {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1, o1, p1}}.y^Range[0, 3].x^Range[0, 3]

    – Michael E2
    3 hours ago











  • There's an example in the docs for CoefficientList for how to recover the polynomial from the matrix: Look for Fold[FromDigits[Reverse[#1], #2] &, %, {x, y}].

    – Michael E2
    3 hours ago











  • @MichaelE2 I am ignoring only those entries for which the constant is zero as in coefficientlist matrix.

    – Chandan Sharma
    3 hours ago






  • 1





    Do you mean Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat * list, {x, y}]?

    – Michael E2
    2 hours ago













  • @MichaelE2 Exactly.

    – Chandan Sharma
    2 hours ago














  • 1





    Why are some entries of the matrix ignored? Maybe this, if that is a mistake: {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1, o1, p1}}.y^Range[0, 3].x^Range[0, 3]

    – Michael E2
    3 hours ago











  • There's an example in the docs for CoefficientList for how to recover the polynomial from the matrix: Look for Fold[FromDigits[Reverse[#1], #2] &, %, {x, y}].

    – Michael E2
    3 hours ago











  • @MichaelE2 I am ignoring only those entries for which the constant is zero as in coefficientlist matrix.

    – Chandan Sharma
    3 hours ago






  • 1





    Do you mean Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat * list, {x, y}]?

    – Michael E2
    2 hours ago













  • @MichaelE2 Exactly.

    – Chandan Sharma
    2 hours ago








1




1





Why are some entries of the matrix ignored? Maybe this, if that is a mistake: {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1, o1, p1}}.y^Range[0, 3].x^Range[0, 3]

– Michael E2
3 hours ago





Why are some entries of the matrix ignored? Maybe this, if that is a mistake: {{a1, b1, c1, d1}, {e1, f1, g1, h1}, {i1, j1, k1, l1}, {m1, n1, o1, p1}}.y^Range[0, 3].x^Range[0, 3]

– Michael E2
3 hours ago













There's an example in the docs for CoefficientList for how to recover the polynomial from the matrix: Look for Fold[FromDigits[Reverse[#1], #2] &, %, {x, y}].

– Michael E2
3 hours ago





There's an example in the docs for CoefficientList for how to recover the polynomial from the matrix: Look for Fold[FromDigits[Reverse[#1], #2] &, %, {x, y}].

– Michael E2
3 hours ago













@MichaelE2 I am ignoring only those entries for which the constant is zero as in coefficientlist matrix.

– Chandan Sharma
3 hours ago





@MichaelE2 I am ignoring only those entries for which the constant is zero as in coefficientlist matrix.

– Chandan Sharma
3 hours ago




1




1





Do you mean Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat * list, {x, y}]?

– Michael E2
2 hours ago







Do you mean Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat * list, {x, y}]?

– Michael E2
2 hours ago















@MichaelE2 Exactly.

– Chandan Sharma
2 hours ago





@MichaelE2 Exactly.

– Chandan Sharma
2 hours ago










5 Answers
5






active

oldest

votes


















1














Using mat as the template:



Plus @@ Flatten[Table[If[mat[[i, j]] == 0, 0, list[[i, j]] x^(i - 1) y^(j - 1)], 

   {i, 1, 4}, {j, 1, 4}]]

(* a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3 *)





share|improve this answer































    2














    Adapting an example from the documentation for CoefficientList:



    Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat*list, {x, y}]
    
(*  a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3  *)





    share|improve this answer































      2














      Internal`FromCoefficientList[mat, {x, y}]



      3 + 2 x + 5 x^3 + 8 x y + 6 x^2 y + 7 x y^2 + 4 y^3




      Internal`FromCoefficientList[list Unitize[mat], {x, y}]



      a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3







      share|improve this answer































        1














        You will have to tell Mathematica where the zero coefficients are, but if you do that it can be done like so:



        list = {{a1, 0, 0, d1}, {e1, f1, g1, 0}, {0, 0, 0, l1}, {m1, 0, 0, 0}}; 
        
Fold[FromDigits[Reverse[#1], #2] &, list, {x, y}] // Expand



        a1 + e1 x + m1 x^3 + f1 x y + g1 x y^2 + d1 y^3 + l1 x^2 y^3




        This is discussed in the documentation of CoefficientList in the section Properties & Relations.






        share|improve this answer































          0














          Terse:



          Total[Array[x^# y^#2 &, {4, 4}, 0] list Unitize@mat, 2]
          




          a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3





          share























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            5 Answers
            5






            active

            oldest

            votes








            5 Answers
            5






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1














            Using mat as the template:



            Plus @@ Flatten[Table[If[mat[[i, j]] == 0, 0, list[[i, j]] x^(i - 1) y^(j - 1)], 
            
   {i, 1, 4}, {j, 1, 4}]]
            
(* a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3 *)





            share|improve this answer




























              1














              Using mat as the template:



              Plus @@ Flatten[Table[If[mat[[i, j]] == 0, 0, list[[i, j]] x^(i - 1) y^(j - 1)], 
              
   {i, 1, 4}, {j, 1, 4}]]
              
(* a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3 *)





              share|improve this answer


























                1












                1








                1







                Using mat as the template:



                Plus @@ Flatten[Table[If[mat[[i, j]] == 0, 0, list[[i, j]] x^(i - 1) y^(j - 1)], 
                
   {i, 1, 4}, {j, 1, 4}]]
                
(* a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3 *)





                share|improve this answer













                Using mat as the template:



                Plus @@ Flatten[Table[If[mat[[i, j]] == 0, 0, list[[i, j]] x^(i - 1) y^(j - 1)], 
                
   {i, 1, 4}, {j, 1, 4}]]
                
(* a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3 *)






                share|improve this answer












                share|improve this answer



                share|improve this answer










                answered 2 hours ago









                John DotyJohn Doty

                6,6641924




                6,6641924























                    2














                    Adapting an example from the documentation for CoefficientList:



                    Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat*list, {x, y}]
                    
(*  a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3  *)





                    share|improve this answer




























                      2














                      Adapting an example from the documentation for CoefficientList:



                      Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat*list, {x, y}]
                      
(*  a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3  *)





                      share|improve this answer


























                        2












                        2








                        2







                        Adapting an example from the documentation for CoefficientList:



                        Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat*list, {x, y}]
                        
(*  a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3  *)





                        share|improve this answer













                        Adapting an example from the documentation for CoefficientList:



                        Fold[FromDigits[Reverse[#1], #2] &, Unitize@mat*list, {x, y}]
                        
(*  a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3  *)






                        share|improve this answer












                        share|improve this answer



                        share|improve this answer










                        answered 2 hours ago









                        Michael E2Michael E2

                        146k11195466




                        146k11195466























                            2














                            Internal`FromCoefficientList[mat, {x, y}]



                            3 + 2 x + 5 x^3 + 8 x y + 6 x^2 y + 7 x y^2 + 4 y^3




                            Internal`FromCoefficientList[list Unitize[mat], {x, y}]



                            a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3







                            share|improve this answer




























                              2














                              Internal`FromCoefficientList[mat, {x, y}]



                              3 + 2 x + 5 x^3 + 8 x y + 6 x^2 y + 7 x y^2 + 4 y^3




                              Internal`FromCoefficientList[list Unitize[mat], {x, y}]



                              a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3







                              share|improve this answer


























                                2












                                2








                                2







                                Internal`FromCoefficientList[mat, {x, y}]



                                3 + 2 x + 5 x^3 + 8 x y + 6 x^2 y + 7 x y^2 + 4 y^3




                                Internal`FromCoefficientList[list Unitize[mat], {x, y}]



                                a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3







                                share|improve this answer













                                Internal`FromCoefficientList[mat, {x, y}]



                                3 + 2 x + 5 x^3 + 8 x y + 6 x^2 y + 7 x y^2 + 4 y^3




                                Internal`FromCoefficientList[list Unitize[mat], {x, y}]



                                a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3








                                share|improve this answer












                                share|improve this answer



                                share|improve this answer










                                answered 1 hour ago









                                kglrkglr

                                178k9198409




                                178k9198409























                                    1














                                    You will have to tell Mathematica where the zero coefficients are, but if you do that it can be done like so:



                                    list = {{a1, 0, 0, d1}, {e1, f1, g1, 0}, {0, 0, 0, l1}, {m1, 0, 0, 0}}; 
                                    
Fold[FromDigits[Reverse[#1], #2] &, list, {x, y}] // Expand



                                    a1 + e1 x + m1 x^3 + f1 x y + g1 x y^2 + d1 y^3 + l1 x^2 y^3




                                    This is discussed in the documentation of CoefficientList in the section Properties & Relations.






                                    share|improve this answer




























                                      1














                                      You will have to tell Mathematica where the zero coefficients are, but if you do that it can be done like so:



                                      list = {{a1, 0, 0, d1}, {e1, f1, g1, 0}, {0, 0, 0, l1}, {m1, 0, 0, 0}}; 
                                      
Fold[FromDigits[Reverse[#1], #2] &, list, {x, y}] // Expand



                                      a1 + e1 x + m1 x^3 + f1 x y + g1 x y^2 + d1 y^3 + l1 x^2 y^3




                                      This is discussed in the documentation of CoefficientList in the section Properties & Relations.






                                      share|improve this answer


























                                        1












                                        1








                                        1







                                        You will have to tell Mathematica where the zero coefficients are, but if you do that it can be done like so:



                                        list = {{a1, 0, 0, d1}, {e1, f1, g1, 0}, {0, 0, 0, l1}, {m1, 0, 0, 0}}; 
                                        
Fold[FromDigits[Reverse[#1], #2] &, list, {x, y}] // Expand



                                        a1 + e1 x + m1 x^3 + f1 x y + g1 x y^2 + d1 y^3 + l1 x^2 y^3




                                        This is discussed in the documentation of CoefficientList in the section Properties & Relations.






                                        share|improve this answer













                                        You will have to tell Mathematica where the zero coefficients are, but if you do that it can be done like so:



                                        list = {{a1, 0, 0, d1}, {e1, f1, g1, 0}, {0, 0, 0, l1}, {m1, 0, 0, 0}}; 
                                        
Fold[FromDigits[Reverse[#1], #2] &, list, {x, y}] // Expand



                                        a1 + e1 x + m1 x^3 + f1 x y + g1 x y^2 + d1 y^3 + l1 x^2 y^3




                                        This is discussed in the documentation of CoefficientList in the section Properties & Relations.







                                        share|improve this answer












                                        share|improve this answer



                                        share|improve this answer










                                        answered 3 hours ago









                                        m_goldbergm_goldberg

                                        84.5k872196




                                        84.5k872196























                                            0














                                            Terse:



                                            Total[Array[x^# y^#2 &, {4, 4}, 0] list Unitize@mat, 2]
                                            




                                            a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3





                                            share




























                                              0














                                              Terse:



                                              Total[Array[x^# y^#2 &, {4, 4}, 0] list Unitize@mat, 2]
                                              




                                              a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3





                                              share


























                                                0












                                                0








                                                0







                                                Terse:



                                                Total[Array[x^# y^#2 &, {4, 4}, 0] list Unitize@mat, 2]
                                                




                                                a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3





                                                share













                                                Terse:



                                                Total[Array[x^# y^#2 &, {4, 4}, 0] list Unitize@mat, 2]
                                                




                                                a1 + e1 x + m1 x^3 + f1 x y + j1 x^2 y + g1 x y^2 + d1 y^3






                                                share











                                                share


                                                share










                                                answered 3 mins ago









                                                Mr.Wizard♦Mr.Wizard

                                                230k294741040




                                                230k294741040






























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