VasCalc Documentation - vectorSpace - Revision history

Sankaran at 20:59, 31 May 2006

2006-05-31T20:59:12Z

← Older revision		Revision as of 16:59, 31 May 2006
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	The vectorSpace package is a small ~~package~~ used to set up quotients of finite dimensional vector spaces, with an interface through Mathematica. By default, vectorSpace works over arbitrarily-sized rationals, using the BigRational implementation written by Eric Laroche (see [[VasCalc - Bibliography]]). See below for instructions on using other fields. The main advantage in using this package is that by being less general, it stands to perform much faster than similar functions in Mathematica.		The vectorSpace package is a small part of the [[VasCalc]] project, used to set up quotients of finite dimensional vector spaces, with an interface through Mathematica. By default, vectorSpace works over arbitrarily-sized rationals, using the BigRational implementation written by Eric Laroche (see [[VasCalc - Bibliography]]). See below for instructions on using other fields. The main advantage in using this package is that by being less general, it stands to perform much faster than similar functions in Mathematica.

	==Installation==		==Installation==
Line 13:		Line 13:
	Also, download [http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace.m vectorSpace.m] (the Mathematica interface) to any location.		Also, download [http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace.m vectorSpace.m] (the Mathematica interface) to any location.

		⚫	==Usage==
	Now open vectorSpace.m in a text editor. Near the top, there is a line that reads
	:<code>AddToClassPath["some_path"]</code>

	Change <i>some_path</i> to the location in which the directory <i>vectorSpace</i> resides (ie. the parent directory of <i>vectorSpace</i>). Make sure you keep the quotes around the path name.

⚫	==Usage==
	In Mathematica, type		In Mathematica, type
	: ~~<code>~~ ~~In[~~ ]:= << /~~path_to_~~/vectorSpace.m </~~code~~>		{{In \| n=1\| in=<nowiki><</path_to/vectorSpace.m</nowiki>}}
	to load the definitions.		to load the definitions.
			We also need to specify where the vectorSpace package lives. Replace "/some_path" with the parent directory of the vectorSpace folder created earlier, retaining the quotes, in the following command:
			{{In\| n=2\|in = SetVectorSpacePath["/some_path"];}}

	We need to start by setting up a finite-dim vector space to work with. The command NewVS[n] returns a reference to a Java instance of the class QuotientSpace. This class represents a v.s. of dimension n, along with any relations between the basis vectors (described below). Here we create a vector space of dimension 5:		We need to start by setting up a finite-dim vector space to work with. The command NewVS[n] returns a reference to a Java instance of the class QuotientSpace. This class represents a v.s. of dimension n, along with any relations between the basis vectors (described below). Here we create a vector space of dimension 5:

	~~:<code>~~ In~~[ ] :~~= qs = ~~NewVS~~[5]; ~~</code>~~		{{In\|n=3\|in= qs = NewVectorSpace[5];}}

	In the vector space package, vectors are stored as lists of ordered pairs {index, coeff}, where the index refers to a fixed implicit original basis. For convenience, the Mathematica interface allows one to label the basis elements with Mathematica symbols by passing a Basis[..] object to the operations described below, and hence you can use all of Mathematica's symbolic manipulations on these expressions. Note, however, that this labelling does not change the internal representation of the vectors; if you change the basis you use, it's unlikely you'll get meaningful results, as you may have already defined relations in terms of another basis.		In the vector space package, vectors are stored as lists of ordered pairs {index, coeff}, where the index refers to a fixed implicit original basis. For convenience, the Mathematica interface allows one to label the basis elements with Mathematica symbols by passing a Basis[..] object to the operations described below, and hence you can use all of Mathematica's symbolic manipulations on these expressions. Note, however, that this labelling does not change the internal representation of the vectors; if you change the basis you use, it's unlikely you'll get meaningful results, as you may have already defined relations in terms of another basis.

	To set up an initial basis, say of elements e0, ... , e4:		To set up an initial basis, say of elements e0, ... , e4:
		⚫	{{In\|n=4\| in = b = Basis[e0,e1,e2,e3,e4];}}

⚫	~~:<code>~~ In[] := b= Basis[e0,e1,e2,e3,e4]; ~~</code>~~

	Here are the operations one can perform:		Here are the operations one can perform:
Line 46:		Line 43:
	Here are some examples, using the earlier setup:		Here are some examples, using the earlier setup:
	Let's introduce a relation		Let's introduce a relation

	~~:<code>~~In[ ] := ModBy[3e0 + e3 - 2e4, qs, b] ~~</code>~~		{{In\|n=5\| in = ModBy[3e0 + e3 - 2e4, qs, b];}}

	Now the quotient space has one less dimension:		Now the quotient space has one less dimension:
	~~:<code>In[~~ ] := GetDimension[qs]~~</code>~~		{{InOut\|n=6\| in = GetDimension[qs]\|out = 4}}
	:<code>Out[ ] = 4</code>

	Reducing a vector in the quotient:		Reducing a vector in the quotient:
	~~:<code>In[ ] :~~= ReduceQuotient[e0 + (1/3)e1 - e4, qs, b]</~~code>~~		{{InOut\|n=7\|in= ReduceQuotient[e0 + (1/3)e1 - e4, qs, b] //Simplify\|
	~~:<code>Out[ ]~~ = e1/4 - (2*e3~~)/3~~ + e4~~/3</code>~~		out =(1/3)(e1 - e3 - e4)}}

	==Changing the Coefficient Field==		==Changing the Coefficient Field==
			By default, vectorSpace works over an implementation of the rationals that allows for arbitrarily large numbers. If you wish to use a different field, you must
			# Download the source file [http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/Coefficient.java Coefficient.java], whose sole purpose is to provide a wrapper to the field. This file contains comments detailing the necessary modifications for a different field; basically one needs to define the arithmetic operations, and return the additive and multiplicative identities in a standard manner.
			# If you wish to use vectorSpace in Mathematica, you will also need to edit vectorSpace.m in order to manipulate your expressions properly in Mathematica. In particular, you will have to rewrite <code> ToJCoeff[c] </code> to create a instance of the Coefficient class from the Mathematica expression c; also, <code> ToMCoeff[c] </code> for the other direction.

			If you succeed in getting another interesting field to work, feel free to add to this page.

Sankaran at 23:38, 29 May 2006

2006-05-29T23:38:04Z

← Older revision		Revision as of 19:38, 29 May 2006
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	:<code>Out[ ] = 4</code>		:<code>Out[ ] = 4</code>

			Reducing a vector in the quotient:

	:<code>In[ ] := ~~ResolveQuotient~~[e0 + (1/3)e1 - e4, qs, b]</code>		:<code>In[ ] := ReduceQuotient[e0 + (1/3)e1 - e4, qs, b]</code>
	:<code>Out[ ] = e1/4 - (2*e3)/3 + e4/3</code>		:<code>Out[ ] = e1/4 - (2*e3)/3 + e4/3</code>


	:<code>In[ ]:= ResolveVector[3e1 +2/3 e0 - 2 e2, b] </code>
	:<code>Out[ ] = { {0,2/3}, {1,3}, {2,-2} } </code>

	==Changing the Coefficient Field==		==Changing the Coefficient Field==

Sankaran at 23:27, 29 May 2006

2006-05-29T23:27:00Z

← Older revision		Revision as of 19:27, 29 May 2006
Line 60:		Line 60:
	:<code>Out[ ] = { {0,2/3}, {1,3}, {2,-2} } </code>		:<code>Out[ ] = { {0,2/3}, {1,3}, {2,-2} } </code>

	===Changing the Coefficient Field===		==Changing the Coefficient Field==

70.29.23.228 at 23:26, 29 May 2006

2006-05-29T23:26:18Z

@@ Line 1: / Line 1: @@
-The vectorSpace package is a small package used to set up quotients of finite dimensional vector spaces, with an interface through Mathematica. By default, vectorSpace works over arbitrarily-sized rationals, using the BigRational implementation written by Eric Laroche (see [[VasCalc - Bibliography]]). See below for instructions on using other fields.
+The vectorSpace package is a small package used to set up quotients of finite dimensional vector spaces, with an interface through Mathematica. By default, vectorSpace works over arbitrarily-sized rationals, using the BigRational implementation written by Eric Laroche (see [[VasCalc - Bibliography]]).  See below for instructions on using other fields. The main advantage in using this package is that by being less general, it stands to perform much faster than similar functions in Mathematica.
 ==Installation==
@@ Line 27: / Line 27: @@
 :<code> In[ ] := qs = NewVS[5]; </code>
-In the vector space package, vectors are stored as lists of ordered pairs {index, coeff}, where the index refers to a fixed implicit original basis. For convenience, the Mathematica interface allows one to label the basis elements with Mathematica symbols by passing a Basis[..] object to the operations described below, and hence you can use all of Mathematica's symbolic manipulations on these expressions. Note, however, that this labelling does not change the internal representation of the vectors; if you change the basis you use, it's unlikely you'll get meaningful results.
+In the vector space package, vectors are stored as lists of ordered pairs {index, coeff}, where the index refers to a fixed implicit original basis. For convenience, the Mathematica interface allows one to label the basis elements with Mathematica symbols by passing a Basis[..] object to the operations described below, and hence you can use all of Mathematica's symbolic manipulations on these expressions. Note, however, that this labelling does not change the internal representation of the vectors; if you change the basis you use, it's unlikely you'll get meaningful results, as you may have already defined relations in terms of another basis.
 To set up an initial basis, say of elements e0, ... , e4:
@@ Line 34: / Line 34: @@
 Here are the operations one can perform:
 :<code>In[ ]:= ResolveVector[3e1 +2/3 e0 - 2 e2, b] </code>
 :<code>Out[ ] = { {0,2/3}, {1,3}, {2,-2} } </code>

Sankaran at 22:14, 29 May 2006

2006-05-29T22:14:32Z

← Older revision		Revision as of 18:14, 29 May 2006
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	Here are the operations one can perform:		Here are the operations one can perform:
	*The command ResolveVector[v, basis] converts a vector as a formal sum into {index, coeff} pairs:		*The command ResolveVector[v, basis] converts a vector as a formal sum into {index, coeff} pairs:
	:<code>In[ ]:= ResolveVector[3e1 +2/3 e0 - 2 e2] </code>		:<code>In[ ]:= ResolveVector[3e1 +2/3 e0 - 2 e2, b] </code>
	:<code>Out[ ] = { {0,2/3}, {1,3}, {2,-2} } </code>		:<code>Out[ ] = { {0,2/3}, {1,3}, {2,-2} } </code>

Sankaran at 22:13, 29 May 2006

2006-05-29T22:13:59Z

← Older revision		Revision as of 18:13, 29 May 2006
Line 1:		Line 1:
	The vectorSpace package is a small package used to set up quotients of finite dimensional vector spaces. ~~The~~ ~~source~~ ~~files~~ for ~~this~~ ~~package~~ ~~are~~		The vectorSpace package is a small package used to set up quotients of finite dimensional vector spaces, with an interface through Mathematica. By default, vectorSpace works over arbitrarily-sized rationals, using the BigRational implementation written by Eric Laroche (see [[VasCalc - Bibliography]]). See below for instructions on using other fields.

			==Installation==
⚫	*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/Coefficient.~~java~~ Coefficient.~~java~~],
			If you wish to use this package on its own, follow the directions below.
⚫	*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/QuotientSpace.~~java~~ QuotientSpace.~~java~~],

⚫	*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/AlgebraVector.~~java~~ AlgebraVector.~~java~~].
			Create a directory on your file system named <i>vectorSpace</i>, and download the following four files from the repository into it:

		⚫	*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/Coefficient.class Coefficient.class],
		⚫	*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/QuotientSpace.class QuotientSpace.class],
		⚫	*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/AlgebraVector.class AlgebraVector.class],
			*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/BigRational.class BigRational.class].

			Also, download [http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace.m vectorSpace.m] (the Mathematica interface) to any location.

			Now open vectorSpace.m in a text editor. Near the top, there is a line that reads
			:<code>AddToClassPath["some_path"]</code>

			Change <i>some_path</i> to the location in which the directory <i>vectorSpace</i> resides (ie. the parent directory of <i>vectorSpace</i>). Make sure you keep the quotes around the path name.

	The class contained in [http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/BigRational.java BigRational.java] contains Eric Laroche's implementation of arithemetic involving arbitrarily large rationals (see [[VasCalc - Bibliography]]), which is used as the default base field in vectorSpace. Additionally, there are two small test programs:[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/testQuotSp.java testQuotSp.java] and [http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/testAlgVect.java testAlgVect.java].
	==Usage==		==Usage==
			In Mathematica, type
			: <code> In[ ]:= << /path_to_/vectorSpace.m </code>
			to load the definitions.

			We need to start by setting up a finite-dim vector space to work with. The command NewVS[n] returns a reference to a Java instance of the class QuotientSpace. This class represents a v.s. of dimension n, along with any relations between the basis vectors (described below). Here we create a vector space of dimension 5:

			:<code> In[ ] := qs = NewVS[5]; </code>

			In the vector space package, vectors are stored as lists of ordered pairs {index, coeff}, where the index refers to a fixed implicit original basis. For convenience, the Mathematica interface allows one to label the basis elements with Mathematica symbols by passing a Basis[..] object to the operations described below, and hence you can use all of Mathematica's symbolic manipulations on these expressions. Note, however, that this labelling does not change the internal representation of the vectors; if you change the basis you use, it's unlikely you'll get meaningful results.

			To set up an initial basis, say of elements e0, ... , e4:

			:<code> In[] := b= Basis[e0,e1,e2,e3,e4]; </code>

			Here are the operations one can perform:
			*The command ResolveVector[v, basis] converts a vector as a formal sum into {index, coeff} pairs:
			:<code>In[ ]:= ResolveVector[3e1 +2/3 e0 - 2 e2] </code>
			:<code>Out[ ] = { {0,2/3}, {1,3}, {2,-2} } </code>

Drorbn: Documentation - vectorSpace moved to VasCalc Documentation - vectorSpace

2006-05-25T15:50:12Z

Documentation - vectorSpace moved to VasCalc Documentation - vectorSpace

← Older revision	Revision as of 11:50, 25 May 2006
(No difference)

142.150.137.199 at 13:58, 25 May 2006

2006-05-25T13:58:32Z

New page

The vectorSpace package is a small package used to set up quotients of finite dimensional vector spaces. The source files for this package are

*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/Coefficient.java Coefficient.java],
*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/QuotientSpace.java QuotientSpace.java],
*[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/AlgebraVector.java AlgebraVector.java].

The class contained in [http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/BigRational.java BigRational.java] contains Eric Laroche's implementation of arithemetic involving arbitrarily large rationals (see [[VasCalc - Bibliography]]), which is used as the default base field in vectorSpace. Additionally, there are two small test programs:[http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/testQuotSp.java testQuotSp.java] and [http://katlas.math.toronto.edu/svn/VasCalc/trunk/vectorSpace/testAlgVect.java testAlgVect.java].
==Usage==