Reconstructing a polynomial from its coefficient array
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
edited 2 hours ago
m_goldberg
84.5k872196
asked 3 hours ago
Chandan SharmaChandan Sharma
1075
add a comment |
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
edited 2 hours ago
m_goldberg
84.5k872196
asked 3 hours ago
Chandan SharmaChandan Sharma
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 forCoefficientListfor how to recover the polynomial from the matrix: Look forFold[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 meanFold[FromDigits[Reverse[#1], #2] &, Unitize@mat * list, {x, y}]?
– Michael E2
2 hours ago
@MichaelE2 Exactly.
– Chandan Sharma
2 hours ago
add a comment |
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
edited 2 hours ago
m_goldberg
84.5k872196
asked 3 hours ago
Chandan SharmaChandan Sharma
1075
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
edited 2 hours ago
m_goldberg
84.5k872196
asked 3 hours ago
Chandan SharmaChandan Sharma
1075
edited 2 hours ago
m_goldberg
84.5k872196
asked 3 hours ago
Chandan SharmaChandan Sharma
1075
edited 2 hours ago
m_goldberg
84.5k872196
edited 2 hours ago
m_goldberg
84.5k872196
edited 2 hours ago
m_goldberg
84.5k872196
84.5k872196
asked 3 hours ago
Chandan SharmaChandan Sharma
1075
asked 3 hours ago
Chandan SharmaChandan Sharma
1075
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 forCoefficientListfor how to recover the polynomial from the matrix: Look forFold[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 meanFold[FromDigits[Reverse[#1], #2] &, Unitize@mat * list, {x, y}]?
– Michael E2
2 hours ago
@MichaelE2 Exactly.
– Chandan Sharma
2 hours ago
add a comment |
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 forCoefficientListfor how to recover the polynomial from the matrix: Look forFold[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 meanFold[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
add a comment |
5 Answers
5
active
oldest
votes
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 *)
answered 2 hours ago
John DotyJohn Doty
6,6641924
add a comment |
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 *)
answered 2 hours ago
Michael E2Michael E2
146k11195466
add a comment |
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
answered 1 hour ago
kglrkglr
178k9198409
add a comment |
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.
answered 3 hours ago
m_goldbergm_goldberg
84.5k872196
add a comment |
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
answered 3 mins ago
Mr.Wizard♦Mr.Wizard
230k294741040
add a comment |
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 *)
answered 2 hours ago
John DotyJohn Doty
6,6641924
add a comment |
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 *)
answered 2 hours ago
John DotyJohn Doty
6,6641924
add a comment |
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 *)
answered 2 hours ago
John DotyJohn Doty
6,6641924
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 *)
answered 2 hours ago
John DotyJohn Doty
6,6641924
answered 2 hours ago
John DotyJohn Doty
6,6641924
answered 2 hours ago
John DotyJohn Doty
6,6641924
answered 2 hours ago
John DotyJohn Doty
6,6641924
6,6641924
add a comment |
add a comment |
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 *)
answered 2 hours ago
Michael E2Michael E2
146k11195466
add a comment |
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 *)
answered 2 hours ago
Michael E2Michael E2
146k11195466
add a comment |
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 *)
answered 2 hours ago
Michael E2Michael E2
146k11195466
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 *)
answered 2 hours ago
Michael E2Michael E2
146k11195466
answered 2 hours ago
Michael E2Michael E2
146k11195466
answered 2 hours ago
Michael E2Michael E2
146k11195466
answered 2 hours ago
Michael E2Michael E2
146k11195466
146k11195466
add a comment |
add a comment |
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
answered 1 hour ago
kglrkglr
178k9198409
add a comment |
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
answered 1 hour ago
kglrkglr
178k9198409
add a comment |
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
answered 1 hour ago
kglrkglr
178k9198409
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
answered 1 hour ago
kglrkglr
178k9198409
answered 1 hour ago
kglrkglr
178k9198409
answered 1 hour ago
kglrkglr
178k9198409
answered 1 hour ago
kglrkglr
178k9198409
178k9198409
add a comment |
add a comment |
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.
answered 3 hours ago
m_goldbergm_goldberg
84.5k872196
add a comment |
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.
answered 3 hours ago
m_goldbergm_goldberg
84.5k872196
add a comment |
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.
answered 3 hours ago
m_goldbergm_goldberg
84.5k872196
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.
answered 3 hours ago
m_goldbergm_goldberg
84.5k872196
answered 3 hours ago
m_goldbergm_goldberg
84.5k872196
answered 3 hours ago
m_goldbergm_goldberg
84.5k872196
answered 3 hours ago
m_goldbergm_goldberg
84.5k872196
84.5k872196
add a comment |
add a comment |
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
answered 3 mins ago
Mr.Wizard♦Mr.Wizard
230k294741040
add a comment |
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
answered 3 mins ago
Mr.Wizard♦Mr.Wizard
230k294741040
add a comment |
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
answered 3 mins ago
Mr.Wizard♦Mr.Wizard
230k294741040
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
answered 3 mins ago
Mr.Wizard♦Mr.Wizard
230k294741040
answered 3 mins ago
Mr.Wizard♦Mr.Wizard
230k294741040
answered 3 mins ago
Mr.Wizard♦Mr.Wizard
230k294741040
answered 3 mins ago
Mr.Wizard♦Mr.Wizard
230k294741040
230k294741040
add a comment |
add a comment |
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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
CoefficientListfor how to recover the polynomial from the matrix: Look forFold[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