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
    2 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
    2 hours ago











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

    – Chandan Sharma
    2 hours ago






  • 1





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

    – Michael E2
    1 hour ago













  • @MichaelE2 Exactly.

    – Chandan Sharma
    1 hour 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
    2 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
    2 hours ago











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

    – Chandan Sharma
    2 hours ago






  • 1





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

    – Michael E2
    1 hour ago













  • @MichaelE2 Exactly.

    – Chandan Sharma
    1 hour 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 1 hour ago









m_goldberg

84.5k872196




84.5k872196










asked 2 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
    2 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
    2 hours ago











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

    – Chandan Sharma
    2 hours ago






  • 1





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

    – Michael E2
    1 hour ago













  • @MichaelE2 Exactly.

    – Chandan Sharma
    1 hour 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
    2 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
    2 hours ago











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

    – Chandan Sharma
    2 hours ago






  • 1





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

    – Michael E2
    1 hour ago













  • @MichaelE2 Exactly.

    – Chandan Sharma
    1 hour 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
2 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
2 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
2 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
2 hours ago













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

– Chandan Sharma
2 hours ago





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

– Chandan Sharma
2 hours ago




1




1





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

– Michael E2
1 hour ago







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

– Michael E2
1 hour ago















@MichaelE2 Exactly.

– Chandan Sharma
1 hour ago





@MichaelE2 Exactly.

– Chandan Sharma
1 hour ago










4 Answers
4






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














    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














      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































        0














        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























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          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          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 1 hour ago









              John DotyJohn Doty

              6,6641924




              6,6641924























                  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 1 hour ago









                      m_goldbergm_goldberg

                      84.5k872196




                      84.5k872196























                          1














                          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




























                            1














                            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


























                              1












                              1








                              1







                              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 55 mins ago









                              Michael E2Michael E2

                              146k11195466




                              146k11195466























                                  0














                                  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




























                                    0














                                    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


























                                      0












                                      0








                                      0







                                      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 21 mins ago









                                      kglrkglr

                                      178k9198409




                                      178k9198409






























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                                          Fotorealismo

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                                          Guerrino Boatto, Venice's St. Barnaba Church and canal Il fotorealismo è un genere di pittura basato sull'uso di una o più fotografie dalle quali l'artista prende le informazioni necessarie da utilizzare poi nel processo della creazione dell'opera, in genere ad olio o ad acrilico, al fine di avere un aspetto finale il più simile possibile alla fotografia. Indice 1 Storia 2 Lista dei fotorealisti 3 Note 4 Voci correlate 5 Altri progetti Storia | Il fotorealismo è un movimento che nasce negli Stati Uniti d'America alla fine degli anni sessanta [1] sulla scia della Pop Art [2] [3] [4] , e insieme condividono il carattere reazionario verso la travolgente ascesa dei media, che alla metà del ventesimo secolo si trasforma in un fenomeno di massa talmente imponente da minacciare il valore dell'immagine nell'arte. [5] [6] Ne consegue così che, colto il gesto d'inizio che fu della Pop Art, i fotorealisti aprono ad un ...
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                                          Sidney Franklin

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                                          Questa voce sull'argomento registi statunitensi è solo un abbozzo . Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Sidney Franklin, foto di Carl Van Vechten Oscar alla memoria Irving G. Thalberg 1943 Sidney Franklin , all'anagrafe Sidney Arnold Franklin , (San Francisco, 21 marzo 1893 – Santa Monica, 18 maggio 1972), è stato un regista, produttore cinematografico e attore statunitense. Il 20 giugno 1963, si sposò con l'attrice australiana Enid Bennett. Il matrimonio durò fino alla morte dell'attrice, il 14 maggio 1969 [1] Indice 1 Filmografia 1.1 Regista 1.2 Aiuto regia (parziale) 1.3 Produttore 1.4 Attore (parziale) 2 Note 3 Altri progetti 4 Collegamenti esterni Filmografia | Regista | The Baby , co-regia Chester M. Franklin (1915) The Rivals , co-regia Chester M. Franklin (1915) Little Dick's First Case , co-regia Chester M. Franklin (1915) Her Filmland Hero , co-...
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