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
    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
    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
    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 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
    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
    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
    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
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
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































    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































      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














        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 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 1 hour ago









                      Michael E2Michael E2

                      146k11195466




                      146k11195466























                          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 2 hours ago









                              m_goldbergm_goldberg

                              84.5k872196




                              84.5k872196























                                  1














                                  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














                                    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












                                      1








                                      1







                                      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






























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                                          Zagreb

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                                          This article is about the Croatian capital city. For other uses, see Zagreb (disambiguation). City in City of Zagreb, Croatia Zagreb City Grad Zagreb City of Zagreb Clockwise, from top: St. Mark's Square, Croatian State Archives, Zagreb Tram, Art Pavilion, Cibona & HOTO towers and Croatian National Theatre. Flag Coat of arms Zagreb Location of Zagreb in Croatia Show map of Croatia Zagreb Zagreb (Europe) Show map of Europe Coordinates: 45°49′N 15°59′E  /  45.817°N 15.983°E  / 45.817; 15.983 Coordinates: 45°49′N 15°59′E  /  45.817°N 15.983°E  / 45.817; 15.983 Country   Croatia County City of Zagreb RC diocese 1094 Free royal city 1242 Unified 1850 Subdivisions 17 city districts 218 local committees (70 settlements) Government  • Type Mayor-Council  • Mayor Milan Bandić (BM 365)  • City Assembly 51 members BM 365, ZL, NS-R (14) SDP, GLAS, HNS, HSS, NH-PS (13) ...
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                                          Understanding the information contained in the Deep Space Network XML data?

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                                          up vote 1 down vote favorite NASA's Deep Space Network activity can be viewed conveniently at https://eyes.nasa.gov/dsn/dsn.html but if you would like data in a more raw XML form, it can be accessed as described in this answer Below is a small Python script that reads the XML into a list of dictionaries, and saves it to disk as a JSON. The dictionaries are organized at the higher level by dish, then by spacecraft per dish. Question: What are the most-likely meanings of all the data field contained in the Deep Space Network XML data? This is the example url from the other answer, used in the script below: https://eyes.nasa.gov/dsn/data/1365107113.xml def dictify(r,root=True): # https://stackoverflow.com/questions/2148119/how-to-convert-an-xml-string-to-a-dictionary-in-python/30923963#30923963 if root: ret...
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