PolynomialAsMatrix - Revision history

Nathaniel: now CVX-friendly

2023-08-05T17:52:31Z

now CVX-friendly

← Older revision		Revision as of 17:52, 5 August 2023
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	\|rel=[[CopositivePolynomial]]<br />[[PolynomialOptimize]]<br />[[PolynomialSOS]]		\|rel=[[CopositivePolynomial]]<br />[[PolynomialOptimize]]<br />[[PolynomialSOS]]
	\|cat=[[List of functions#Polynomial_optimization\|Polynomial optimization]]		\|cat=[[List of functions#Polynomial_optimization\|Polynomial optimization]]
	\|upd=~~July 31~~, 2023		\|upd=August 5, 2023
	\|cvx=no}}		\|cvx=yes}}
	<tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions\|function]] that computes a compact form of a fully symmetric matrix representation of an even-degree homogeneous polynomial. More specifically, if p is a homogeneous polynomial of degree 2d then there is a unique fully symmetric matrix <math>M</math> with the property that <math>p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})</math>, where <math>\mathbf{x} = (x_1,x_2,\ldots,x_n)</math> as a column vector. Here, "fully symmetric" means that <math>M^T = M</math>, <math>M</math> is supported on the [[symmetric subspace]] (i.e., <math>PMP = M</math>, where <math>P</math> is the [[SymmetricProjection\|symmetric projection]]), and <math>M</math> equals its own [[PartialTranspose\|partial transpose]] (across any bipartition).		<tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions\|function]] that computes a compact form of a fully symmetric matrix representation of an even-degree homogeneous polynomial. More specifically, if p is a homogeneous polynomial of degree 2d then there is a unique fully symmetric matrix <math>M</math> with the property that <math>p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})</math>, where <math>\mathbf{x} = (x_1,x_2,\ldots,x_n)</math> as a column vector. Here, "fully symmetric" means that <math>M^T = M</math>, <math>M</math> is supported on the [[symmetric subspace]] (i.e., <math>PMP = M</math>, where <math>P</math> is the [[SymmetricProjection\|symmetric projection]]), and <math>M</math> equals its own [[PartialTranspose\|partial transpose]] (across any bipartition).

Nathaniel: /* The Motzkin polynomial */

2023-08-01T13:23:02Z

The Motzkin polynomial

← Older revision		Revision as of 13:23, 1 August 2023
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	The output of this function is always sparse, so we used the <tt>full</tt> function around the output above just to make the output easier to see. When <math>d > 1</math> (like in this example), we can verify that the above matrix represents the given polynomial via the isometry created by the [[SymmetricProjection]] function. The following verification of this fact requires MATLAB's symbolic toolbox:		The output of this function is always sparse, so we used the <tt>full</tt> function around the output above just to make the output easier to see. When <math>d > 1</math> (like in this example), we can verify that the above matrix represents the given polynomial via the isometry created by the [[SymmetricProjection]] function. The following verification of this fact requires MATLAB's symbolic toolbox:
	<syntaxhighlight>		<syntaxhighlight>
			>> P = SymmetricProjection(n,d,1,0);
	>> MF = PMP';		>> MF = PMP';
	>> syms x y z		>> syms x y z

Nathaniel: /* The Motzkin polynomial */ typo

2023-08-01T13:22:39Z

The Motzkin polynomial: typo

← Older revision		Revision as of 13:22, 1 August 2023
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	</syntaxhighlight>		</syntaxhighlight>

	The output of this function is always sparse, so we used the <tt>full</tt> function around the output above just to make the output easier to see. When <math>d > 1</math> (like in this example), we can verify that the above matrix represents the given polynomial via isometry created by the [[SymmetricProjection]] function. The following verification of this fact requires MATLAB's symbolic toolbox:		The output of this function is always sparse, so we used the <tt>full</tt> function around the output above just to make the output easier to see. When <math>d > 1</math> (like in this example), we can verify that the above matrix represents the given polynomial via the isometry created by the [[SymmetricProjection]] function. The following verification of this fact requires MATLAB's symbolic toolbox:
	<syntaxhighlight>		<syntaxhighlight>
	>> MF = PMP';		>> MF = PMP';

Nathaniel: /* Examples */ Motzkin verification

2023-08-01T13:22:08Z

Examples: Motzkin verification

@@ Line 69: / Line 69: @@
    -0.1000         0         0         0         0         0         0         0         0
          0         0         0         0         0         0         0         0    1.0000
 </syntaxhighlight>

Nathaniel: /* The Motzkin polynomial */ finish

2023-08-01T13:16:54Z

The Motzkin polynomial: finish

@@ Line 51: / Line 51: @@
 >> d = 3;% half the degree of the polynomial
 >> p = zeros(nchoosek(n+2*d-1,2*d),1);% initialize p to have the correct size
->> Coming soon.
+>> p(exp2ind([4 2 0])) = 1;% the "4 2 0" here are the exponents in the term x^4y^2z^0, while the "1" is the coefficient of this term
 </syntaxhighlight>

Nathaniel: /* Examples */ expanded examples

2023-08-01T12:47:19Z

Examples: expanded examples

← Older revision		Revision as of 12:47, 1 August 2023
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	</syntaxhighlight>		</syntaxhighlight>
	The output of this function is always sparse, so we used the <tt>full</tt> function around the output above just to make the output easier to see. Since <math>d = 1</math>, the above matrix is the usual symmetric matrix representation of the quadratric form <math>p</math>. The following verification of this fact requires MATLAB's symbolic toolbox:		The output of this function is always sparse, so we used the <tt>full</tt> function around the output above just to make the output easier to see. Since <math>d = 1</math>, the above matrix is the usual symmetric matrix representation of the quadratric form <math>p</math>. The following verification of this fact requires MATLAB's symbolic toolbox:
			<syntaxhighlight>
	~~Coming soon.~~		>> syms x y z
			>> expand([x y z]M[x;y;z])

			ans =

			x^2 + 2xy + 3xz + 4y^2 + 5yz + 6z^2
			</syntaxhighlight>

	===The Motzkin polynomial===		===The Motzkin polynomial===
	Coming soon.		The Motzkin polynomial is the following degree-6 homogeneous polynomial: <math>p(x,y,z) = x^4y^2 + x^2y^4 - 3x^2y^2z^2 + z^6</math>. To use this <tt>PolynomialAsMatrix</tt> function, we have to first arrange the monomials of p (even the ones with coefficient 0) in lexicographical order and list them in a vector. To make this task (much!) easier, we use the <tt>[[exp2ind]]</tt> helper function:
			<syntaxhighlight>
			>> n = 3;% number of variables
			>> d = 3;% half the degree of the polynomial
			>> p = zeros(nchoosek(n+2d-1,2d),1);% initialize p to have the correct size
			>> Coming soon.
			</syntaxhighlight>

	{{SourceCode\|name=PolynomialAsMatrix}}		{{SourceCode\|name=PolynomialAsMatrix}}

Nathaniel: Expanded

2023-08-01T12:06:20Z

Expanded

← Older revision		Revision as of 12:06, 1 August 2023
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	\|name=PolynomialAsMatrix		\|name=PolynomialAsMatrix
	\|desc=Creates a compact fully symmetric matrix representation of a polynomial		\|desc=Creates a compact fully symmetric matrix representation of a polynomial
	\|rel=[[PolynomialOptimize]]<br />[[PolynomialSOS]]		\|rel=[[CopositivePolynomial]]<br />[[PolynomialOptimize]]<br />[[PolynomialSOS]]
	\|cat=[[List of functions#Polynomial_optimization\|Polynomial optimization]]		\|cat=[[List of functions#Polynomial_optimization\|Polynomial optimization]]
	\|upd=July 31, 2023		\|upd=July 31, 2023
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	<tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions\|function]] that computes a compact form of a fully symmetric matrix representation of an even-degree homogeneous polynomial. More specifically, if p is a homogeneous polynomial of degree 2d then there is a unique fully symmetric matrix <math>M</math> with the property that <math>p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})</math>, where <math>\mathbf{x} = (x_1,x_2,\ldots,x_n)</math> as a column vector. Here, "fully symmetric" means that <math>M^T = M</math>, <math>M</math> is supported on the [[symmetric subspace]] (i.e., <math>PMP = M</math>, where <math>P</math> is the [[SymmetricProjection\|symmetric projection]]), and <math>M</math> equals its own [[PartialTranspose\|partial transpose]] (across any bipartition).		<tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions\|function]] that computes a compact form of a fully symmetric matrix representation of an even-degree homogeneous polynomial. More specifically, if p is a homogeneous polynomial of degree 2d then there is a unique fully symmetric matrix <math>M</math> with the property that <math>p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})</math>, where <math>\mathbf{x} = (x_1,x_2,\ldots,x_n)</math> as a column vector. Here, "fully symmetric" means that <math>M^T = M</math>, <math>M</math> is supported on the [[symmetric subspace]] (i.e., <math>PMP = M</math>, where <math>P</math> is the [[SymmetricProjection\|symmetric projection]]), and <math>M</math> equals its own [[PartialTranspose\|partial transpose]] (across any bipartition).

	This function returns this fully symmetric matrix <math>M</math>, in symmetric coordinates. That is, there is an isometry <math>V</math> from the symmetric subspace of <math>(\mathbb{C}^n)^{\otimes d}</math> to <math>(\mathbb{C}^n)^{\otimes d}</math> itself with the property that the output of this function equals <math>V^*MV</math>.		This function returns this fully symmetric matrix <math>M</math>, in symmetric coordinates. That is, there is an isometry <math>V</math> from the symmetric subspace of <math>(\mathbb{C}^n)^{\otimes d}</math> to <math>(\mathbb{C}^n)^{\otimes d}</math> itself with the property that the output of this function equals <math>V^*MV</math>. This isometry <math>V</math> is computed by <tt>[[SymmetricProjection]](N,D,1,0)</tt>.

	==Syntax==		==Syntax==
	* ~~Coming soon.~~		* <tt>M = PolynomialAsMatrix(P,N,D)</tt>
			* <tt>M = PolynomialAsMatrix(P,N,D,K)</tt>

	==Argument descriptions==		==Argument descriptions==
	* ~~Coming soon~~.		* <tt>P</tt>: The polynomial, represented as a vector of coefficients of its monomials in lexicographic order.
			* <tt>N</tt>: The number of variables in the polynomial.
			* <tt>D</tt>: Half the degree of the polynomial.
			* <tt>K</tt> (optional, default 0): A non-negative integer that indicates the level of the SOS or SOS-type hierarchy. More specifically, this input argument causes the output matrix to represent the polynomial (S(x))^K * P(x) instead of P(x) itself, where S(x) = x1^2 + x2^2 + ... + xN^2.

	==Examples==		==Examples==
			===A simple low-degree polynomial===
			Let's turn the homogeneous polynomial <math>p(x,y,z) = x^2 + 2xy + 3xz + 4y^2 + 5yz + 6z^2</math> into its fully symmetric matrix representation.
			<syntaxhighlight>
			>> n = 3;% number of variables
			>> d = 1;% half the degree of the polynomial
			>> p = [1;2;3;4;5;6];% coefficients of the polynomial, in lexicographical order of the monomials
			>> M = full(PolynomialAsMatrix(p,n,d))

			M =

			1.0000 1.0000 1.5000
			1.0000 4.0000 2.5000
			1.5000 2.5000 6.0000
			</syntaxhighlight>
			The output of this function is always sparse, so we used the <tt>full</tt> function around the output above just to make the output easier to see. Since <math>d = 1</math>, the above matrix is the usual symmetric matrix representation of the quadratric form <math>p</math>. The following verification of this fact requires MATLAB's symbolic toolbox:

			Coming soon.

			===The Motzkin polynomial===
	Coming soon.		Coming soon.

Nathaniel: fix links

2023-08-01T02:56:04Z

fix links

← Older revision		Revision as of 02:56, 1 August 2023
Line 6:		Line 6:
	\|upd=July 31, 2023		\|upd=July 31, 2023
	\|cvx=no}}		\|cvx=no}}
	<tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions\|function]] that computes a compact form of a fully symmetric matrix representation of an even-degree homogeneous polynomial. More specifically, if p is a homogeneous polynomial of degree 2d then there is a unique fully symmetric matrix <math>M</math> with the property that <math>p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})</math>, where <math>\mathbf{x} = (x_1,x_2,\ldots,x_n)</math> as a column vector. Here, "fully symmetric" means that <math>M^T = M</math>, <math>M</math> is supported on the [[symmetric subspace]] (i.e., <math>PMP = M</math>, where <math>P</math> is the [[symmetric projection~~\|SymmetricProjection~~]]), and <math>M</math> equals its own [[partial transpose~~\|PartialTranspose~~]] (across any bipartition).		<tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions\|function]] that computes a compact form of a fully symmetric matrix representation of an even-degree homogeneous polynomial. More specifically, if p is a homogeneous polynomial of degree 2d then there is a unique fully symmetric matrix <math>M</math> with the property that <math>p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})</math>, where <math>\mathbf{x} = (x_1,x_2,\ldots,x_n)</math> as a column vector. Here, "fully symmetric" means that <math>M^T = M</math>, <math>M</math> is supported on the [[symmetric subspace]] (i.e., <math>PMP = M</math>, where <math>P</math> is the [[SymmetricProjection\|symmetric projection]]), and <math>M</math> equals its own [[PartialTranspose\|partial transpose]] (across any bipartition).

	This function returns this fully symmetric matrix <math>M</math>, in symmetric coordinates. That is, there is an isometry <math>V</math> from the symmetric subspace of <math>(\mathbb{C}^n)^{\otimes d}</math> to <math>(\mathbb{C}^n)^{\otimes d}</math> itself with the property that the output of this function equals <math>V^*MV</math>.		This function returns this fully symmetric matrix <math>M</math>, in symmetric coordinates. That is, there is an isometry <math>V</math> from the symmetric subspace of <math>(\mathbb{C}^n)^{\otimes d}</math> to <math>(\mathbb{C}^n)^{\otimes d}</math> itself with the property that the output of this function equals <math>V^*MV</math>.

Nathaniel: Started page

2023-08-01T02:53:17Z

Started page

New page

{{Function
|name=PolynomialAsMatrix
|desc=Creates a compact fully symmetric matrix representation of a polynomial
|rel=[[PolynomialOptimize]]<br />[[PolynomialSOS]]
|cat=[[List of functions#Polynomial_optimization|Polynomial optimization]]
|upd=July 31, 2023
|cvx=no}}
<tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions|function]] that computes a compact form of a fully symmetric matrix representation of an even-degree homogeneous polynomial. More specifically, if p is a homogeneous polynomial of degree 2d then there is a unique fully symmetric matrix <math>M</math> with the property that <math>p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})</math>, where <math>\mathbf{x} = (x_1,x_2,\ldots,x_n)</math> as a column vector. Here, "fully symmetric" means that <math>M^T = M</math>, <math>M</math> is supported on the [[symmetric subspace]] (i.e., <math>PMP = M</math>, where <math>P</math> is the [[symmetric projection|SymmetricProjection]]), and <math>M</math> equals its own [[partial transpose|PartialTranspose]] (across any bipartition).

This function returns this fully symmetric matrix <math>M</math>, in symmetric coordinates. That is, there is an isometry <math>V</math> from the symmetric subspace of <math>(\mathbb{C}^n)^{\otimes d}</math> to <math>(\mathbb{C}^n)^{\otimes d}</math> itself with the property that the output of this function equals <math>V^*MV</math>.

==Syntax==
* Coming soon.

==Argument descriptions==
* Coming soon.

==Examples==
Coming soon.

{{SourceCode|name=PolynomialAsMatrix}}

==References==
Coming soon.