Matrix Inverse/Divide - Revision history

WikiSysop at 14:58, 8 October 2023

2023-10-08T14:58:54Z

← Older revision		Revision as of 10:58, 8 October 2023
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	13<pn>r</pn>36 1<pn>r</pn>18 ¯7<pn>r</pn>36		13<pn>r</pn>36 1<pn>r</pn>18 ¯7<pn>r</pn>36
	</pre></apll>		</pre></apll>

			<h2>Variations On Matrix Divide</h2>

			<p>For variations on this function, see the [[Variant#Eigenvalues_and_Eigenvectors\|Variant Operator]].</p>

Paul Robinson: t/c

2019-10-16T16:28:54Z

t/c

← Older revision		Revision as of 12:28, 16 October 2019
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	<br><hr width=35%>		<br><hr width=35%>

	{{Article footer\|1\|Matrix}}		{{Article footer\|2\|Matrix}}

Paul Robinson: add explanation

2019-10-16T16:25:25Z

add explanation

← Older revision		Revision as of 12:25, 16 October 2019
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	<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">		'''Caution:''' Be careful not to confuse this symbol, which is <apll>⌹</apll>, with <apll>⍠</apll> which is [[Variant]].
			<hr /><table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
	<tr>		<tr>
	<td>		<td>
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	<br><hr width=35%>		<br><hr width=35%>

	{{~~Language Toolbar~~}}		{{Article footer\|1\|Matrix}}

WikiSysop at 23:15, 27 September 2019

2019-09-27T23:15:11Z

← Older revision		Revision as of 19:15, 27 September 2019
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	</tr>		</tr>
	<tr>		<tr>
	<td><apll>R</apll> is a numeric scalar, vector or matrix; otherwise signal a <apll>RANK ~~ERROR</apll>. If <apll>R</apll> is a matrix, <apll>≥/⍴R</apll> must be true; otherwise signal a <apll>LENGTH~~ ERROR</apll>.</td>		<td><apll>R</apll> is a numeric scalar, vector or matrix; otherwise signal a <apll>RANK ERROR</apll>.</td>
	</tr>		</tr>
	<tr>		<tr>
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	</table>		</table>
	<br />		<br />
	<p>This feature implements matrix inversion on integer or floating point arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.</p>		<p>This feature implements matrix inversion on '''Boolean''', '''integer''', or '''floating point''' arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.</p>

	<p>Matrix inverse (<apll>⌹R</apll>) and matrix division (<apll>L⌹R</apll>) on Rational or ~~VFP~~ arguments each have two limitations above and beyond that of normal conformability:</p>		<p>Matrix inverse (<apll>⌹R</apll>) and matrix division (<apll>L⌹R</apll>) on '''Rational''', '''VFP''', or '''Ball''' arguments each have two limitations above and beyond that of normal conformability:</p>
	<ul>		<ul>
	<li><p>for a square right argument that it be non-singular, and</p></li>		<li><p>for a '''square''' (<apll>=/⍴R</apll>) right argument that it be non-singular, and</p></li>
	<li><p>for an overdetermined (<apll>>/⍴R</apll>) right argument that the symmetric matrix <apll>(⍉R)+.×R</apll> be non-singular.</p></li>		<li><p>for an '''overdetermined''' (<apll>>/⍴R</apll>) or '''underdetermined''' (<apll></⍴R</apll>) right argument that the symmetric matrix <apll>(⍉R)+.×R</apll> be non-singular.</p></li>
	</ul>		</ul>
	<p>These limitations are due to the algorithm ([http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination Gauss-Jordan Elimination]) used to implement Matrix Inverse/Divide on Rational and VFP numbers.</p>		<p>These limitations are due to the algorithm ([http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination Gauss-Jordan Elimination]) used to implement Matrix Inverse/Divide on Rational and VFP numbers.</p>

			<p>Overdetermined matrices are evaluated equivalently to the expression <apll>(⌹(⍉R)+.×R)+.×⍉R</apll>.</p>
			<p>Underdetermined matrices are evaluated equivalently to the expression <apll>(⍉R)+.×⌹R+.×⍉R</apll>.</p>

	<p>For example,</p>		<p>For example,</p>
Line 74:		Line 77:
	</table>		</table>
	<br />		<br />
	<p>This feature implements matrix division on ~~integers~~ and floating point arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any two numeric arrays meeting the rank and shape requirements above are divisible.</p>		<p>This feature implements matrix division on '''Boolean''', '''integer''' and '''floating point''' arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any two numeric arrays meeting the rank and shape requirements above are divisible.</p>

			<p>As noted above, Matrix Division on '''Rational''', '''VFP''', or '''Ball''' arguments uses a different algorithm and has slightly more restrictive conformability requirements.</p>
			<ul>
			<li><p>for a '''square''' (<apll>=/⍴R</apll>) right argument that it be non-singular, and</p></li>
			<li><p>for an '''overdetermined''' (<apll>>/⍴R</apll>) or '''underdetermined''' (<apll></⍴R</apll>) right argument that the symmetric matrix <apll>(⍉R)+.×R</apll> be non-singular.</p></li>
			</ul>

	<p>~~As noted above, Matrix Division on Rational or VFP arguments uses a different algorithm and has slightly more restrictive conformability requirements~~.</p>		<p>Overdetermined matrices are evaluated equivalently to the expression <apll>(⌹(⍉R)+.×R)+.×(⍉R)+.×L</apll>.</p>
			<p>Underdetermined matrices are evaluated equivalently to the expression <apll>(⍉R)+.×⌹R+.×(⍉R)+.×L</apll>.</p>

	<p>For example,</p>		<p>For example,</p>

Sudleyplace at 22:56, 15 April 2018

2018-04-15T22:56:55Z

← Older revision		Revision as of 18:56, 15 April 2018
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	<p>For example,</p>		<p>For example,</p>

	<apll>		<apll><pre>
	~~      ~~⌹3 3{rho}0~~<br />~~		⌹3 3{rho}0
	~~ ~~0 0 0~~<br />~~		0 0 0
	~~ ~~0 0 0~~<br />~~		0 0 0
	~~ ~~0 0 0~~<br />~~		0 0 0
	~~      ~~⌹3 3⍴1 2 3 4~~<br />~~		⌹3 3⍴1 2 3 4
	¯0.1944444444~~   ~~0.2777777778~~   ~~0.02777777778~~<br />~~		¯0.1944444444 0.2777777778 0.02777777778
	~~ ~~0.05555555556~~ ~~¯0.2222222222~~   ~~0.2777777778~~<br />~~		0.05555555556 ¯0.2222222222 0.2777777778
	~~ ~~0.3611111111~~   ~~0.05555555556~~ ~~¯0.1944444444~~<br />~~		0.3611111111 0.05555555556 ¯0.1944444444
	~~      ~~⌹3 3⍴1 2 3 4x<br />		⌹3 3⍴1 2 3 4<pn>x</pn>
	~~¯7r36  5r18  1r36~~<br />		¯7<pn>r</pn>36 5<pn>r</pn>18 1<pn>r</pn>36
	~~ 1r18 ¯2r9   5r18~~<br />		1<pn>r</pn>18 ¯2<pn>r</pn>9 5<pn>r</pn>18
	~~13r36  1r18 ¯7r36~~<br />		13<pn>r</pn>36 1<pn>r</pn>18 ¯7<pn>r</pn>36
	</~~apll~~>		</pre></apll>
	<br />
	<br />
	<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">		<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
	<tr>		<tr>
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	<p>For example,</p>		<p>For example,</p>

	<apll>		<apll><pre>
	~~      ~~a←3 3⍴0~~<br />~~		a←3 3⍴0
	~~      ~~a⌹a~~<br />~~		a⌹a
	~~ ~~0 0 0~~<br />~~		0 0 0
	~~ ~~0 0 0~~<br />~~		0 0 0
	~~ ~~0 0 0~~<br />~~		0 0 0
	~~      ~~1 2 3⌹3 3⍴1 2 3 4~~<br />~~		1 2 3⌹3 3⍴1 2 3 4
	0.4444444444 0.4444444444 ¯0.1111111111~~<br />~~		0.4444444444 0.4444444444 ¯0.1111111111
	~~      ~~1 2 3⌹3 3⍴1 2 3 4x<br />		1 2 3⌹3 3⍴1 2 3 4<pn>x</pn>
	~~4r9 4r9 ¯1r9~~</apll>		4<pn>r</pn>9 4<pn>r</pn>9 ¯1<pn>r</pn>9</pre></apll>

	<hr width=35%><br>		<hr width=35%><br>

Robert Wallick: added link to Domino wiki page

2015-07-31T01:37:45Z

added link to Domino wiki page

← Older revision		Revision as of 21:37, 30 July 2015
Line 91:		Line 91:
	4r9 4r9 ¯1r9</apll>		4r9 4r9 ¯1r9</apll>

			<hr width=35%><br>

			<big>For several more everyday problem-solving examples using Matrix Inverse see also '''[[Symbol_Domino\|Domino Symbol ⌹]]'''.</big>

			<br><hr width=35%>

	{{Language Toolbar}}		{{Language Toolbar}}

WikiSysop at 23:42, 30 July 2015

2015-07-30T23:42:04Z

← Older revision		Revision as of 19:42, 30 July 2015
Line 20:		Line 20:
	</table>		</table>
	<br />		<br />
	<p>This feature implements matrix inversion using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.</p>		<p>This feature implements matrix inversion on integer or floating point arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.</p>

			<p>Matrix inverse (<apll>⌹R</apll>) and matrix division (<apll>L⌹R</apll>) on Rational or VFP arguments each have two limitations above and beyond that of normal conformability:</p>
			<ul>
			<li><p>for a square right argument that it be non-singular, and</p></li>
			<li><p>for an overdetermined (<apll>>/⍴R</apll>) right argument that the symmetric matrix <apll>(⍉R)+.×R</apll> be non-singular.</p></li>
			</ul>
			<p>These limitations are due to the algorithm ([http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination Gauss-Jordan Elimination]) used to implement Matrix Inverse/Divide on Rational and VFP numbers.</p>

	<p>For example,</p>		<p>For example,</p>
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	0 0 0<br />		0 0 0<br />
	0 0 0<br />		0 0 0<br />
	0 0 0</~~apll~~><br />		0 0 0<br />
			⌹3 3⍴1 2 3 4<br />
			¯0.1944444444   0.2777777778   0.02777777778<br />
			0.05555555556 ¯0.2222222222   0.2777777778<br />
			0.3611111111   0.05555555556 ¯0.1944444444<br />
			⌹3 3⍴1 2 3 4x<br />
			¯7r36  5r18  1r36<br />
			1r18 ¯2r9   5r18<br />
			13r36  1r18 ¯7r36<br />
			</apll>
	<br />		<br />
	<br />		<br />
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	</table>		</table>
	<br />		<br />
	<p>This feature implements matrix division using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any two numeric arrays meeting the rank and shape requirements above are divisible.</p>		<p>This feature implements matrix division on integers and floating point arguments using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any two numeric arrays meeting the rank and shape requirements above are divisible.</p>

			<p>As noted above, Matrix Division on Rational or VFP arguments uses a different algorithm and has slightly more restrictive conformability requirements.</p>

	<p>For example,</p>		<p>For example,</p>
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	0 0 0<br />		0 0 0<br />
	0 0 0<br />		0 0 0<br />
	0 0 0</apll>		0 0 0<br />
			1 2 3⌹3 3⍴1 2 3 4<br />
			0.4444444444 0.4444444444 ¯0.1111111111<br />
			1 2 3⌹3 3⍴1 2 3 4x<br />
			4r9 4r9 ¯1r9</apll>


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Paul Robinson: add toolbar footer

2014-11-24T14:38:58Z

add toolbar footer

← Older revision		Revision as of 10:38, 24 November 2014
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	0 0 0<br />		0 0 0<br />
	0 0 0</apll>		0 0 0</apll>


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Sudleyplace at 21:01, 7 May 2008

2008-05-07T21:01:42Z

@@ Line 4: / Line 4: @@
      <table border="0" cellpadding="5" cellspacing="0" summary="">
      <tr>
-       <td><apll>Z←⌹R</apll></td>
+       <td valign="top"><apll>Z←⌹R</apll></td>
        <td></td>
        <td></td>
@@ Line 13: / Line 13: @@
 </tr>
 <tr>
-   <td><apll>R</apll> is a numeric scalar, vector or matrix.  If <apll>R</apll> is a matrix, <apll>≥/⍴R</apll> must be true.</td>
+   <td><apll>R</apll> is a numeric scalar, vector or matrix; otherwise signal a <apll>RANK ERROR</apll>.  If <apll>R</apll> is a matrix, <apll>≥/⍴R</apll> must be true; otherwise signal a <apll>LENGTH ERROR</apll>.</td>
 </tr>
 <tr>
@@ Line 25: / Line 25: @@
 <apll>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{domino}3 3{rho}0<br />
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⌹3 3{rho}0<br />
-&nbsp;0 0 0<br>
+&nbsp;0 0 0<br />
 &nbsp;0 0 0<br />
 &nbsp;0 0 0</apll><br />

Sudleyplace: New page:

Z←⌹R
2008-05-07T20:07:15Z

New page: <table border="1" cellpadding="5" cellspacing="0" rules="none" summary=""> <tr> <td> <table border="0" cellpadding="5" cellspacing="0" summary=""> <tr> <td><apll>Z←⌹R</...

New page
<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
<tr>
<td>
<table border="0" cellpadding="5" cellspacing="0" summary="">
<tr>
<td><apll>Z←⌹R</apll></td>
<td></td>
<td></td>
<td>returns the inverse of the right argument.</td>
</tr>
</table>
</td>
</tr>
<tr>
<td><apll>R</apll> is a numeric scalar, vector or matrix. If <apll>R</apll> is a matrix, <apll>≥/⍴R</apll> must be true.</td>
</tr>
<tr>
<td><apll>Z</apll> is a numeric array of rank <apll>⍴⍴R</apll>, and shape <apll>⌽⍴R</apll>.</td>
</tr>
</table>
<br />
<p>This feature implements matrix inversion using [http://www.gnu.org/software/gsl/manual/html_node/Singular-Value-Decomposition.html Singular Value Decomposition]. In particular, this means that any numeric array meeting the rank and shape requirements above is invertible.</p>

<p>For example,</p>

<apll>
      {domino}3 3{rho}0<br />
 0 0 0<br>
 0 0 0<br />
 0 0 0</apll><br />