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WikiSysop: /* Eigenvalues and Eigenvectors */

2024-02-19T00:45:15Z

Eigenvalues and Eigenvectors

← Older revision		Revision as of 20:45, 18 February 2024
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	{\| border="1" cellpadding="5" cellspacing="5" rules="all" summary="" style=""border:1px solid black;"		{\| border="1" cellpadding="5" cellspacing="5" rules="all" summary="" style=""border:1px solid black;"
	\|<apll>Z←⌹⍠1 R</apll>\|\|<apll>Z</apll> is a Complex floating point vector of the Eigenvalues of <apll>R</apll>		\|<apll>Z←⌹⍠1 RA</apll>\|\|<apll>Z</apll> is a Complex floating point vector of the Eigenvalues of <apll>RA</apll>
	\|-		\|-
	\|<apll>Z←⌹⍠2 R</apll>\|\|<apll>Z</apll> is a Complex floating point matrix of the Eigenvectors of <apll>R</apll>, one per column		\|<apll>Z←⌹⍠2 RA</apll>\|\|<apll>Z</apll> is a Complex floating point matrix of the Eigenvectors of <apll>RA</apll>, one per column
	\|-		\|-
	\|<apll>Z←⌹⍠3 R</apll>\|\|<apll>Z</apll> is a two-element nested vector with a Complex floating point vector of the Eigenvalues in the first element and a Complex floating point matrix of the Eigenvectors in the second		\|<apll>Z←⌹⍠3 RA</apll>\|\|<apll>Z</apll> is a two-element nested vector with a Complex floating point vector of the Eigenvalues (<apll>⌹⍠1 RA</apll>) in the first element and a Complex floating point matrix of the Eigenvectors (<apll>⌹⍠2 RA</apll>) in the second
	\|-		\|-
	\|<apll>Z←⌹⍠4 R</apll>\|\|<apll>Z</apll> is a three-element nested vector with a Complex floating point vector of the Eigenvalues in the first element, a Complex floating point matrix of the Eigenvectors in the second, and a Real matrix of the Schur vectors in the third		\|<apll>Z←⌹⍠4 RA</apll>\|\|<apll>Z</apll> is a three-element nested vector with a Complex floating point vector of the Eigenvalues (<apll>⌹⍠1 RA</apll>) in the first element, a Complex floating point matrix of the Eigenvectors (<apll>⌹⍠2 RA</apll>) in the second, and a Real matrix of the Schur vectors in the third
			\|-
			\|<apll>Z←⌹⍠5 RA</apll>\|\|<apll>Z</apll> is a two-element nested vector consisting of the QR Decomposition of <apll>RA</apll>; the first element is <apll>Q</apll>, an orthonormal matrix (<apll>Q≡⍉⌹Q</apll>) and the second element is <apll>R</apll>, an upper-triangular matrix, which, together, satisfy the identity <apll>RA≡Q+.×R</apll>.
	\|}		\|}

WikiSysop: /* Eigenvalues and Eigenvectors */

2024-01-08T00:17:43Z

Eigenvalues and Eigenvectors

← Older revision		Revision as of 20:17, 7 January 2024
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	{\| border="1" cellpadding="5" cellspacing="5" rules="all" summary="" style=""border:1px solid black;"		{\| border="1" cellpadding="5" cellspacing="5" rules="all" summary="" style=""border:1px solid black;"
	\|<apll>~~Z←(⌹⍠1)~~ R</apll>\|\|<apll>Z</apll> is a Complex floating point vector of the Eigenvalues of <apll>R</apll>		\|<apll>Z←⌹⍠1 R</apll>\|\|<apll>Z</apll> is a Complex floating point vector of the Eigenvalues of <apll>R</apll>
	\|-		\|-
	\|<apll>~~Z←(⌹⍠2)~~ R</apll>\|\|<apll>Z</apll> is a Complex floating point matrix of the Eigenvectors of <apll>R</apll>, one per column		\|<apll>Z←⌹⍠2 R</apll>\|\|<apll>Z</apll> is a Complex floating point matrix of the Eigenvectors of <apll>R</apll>, one per column
	\|-		\|-
	\|<apll>~~Z←(⌹⍠3)~~ R</apll>\|\|<apll>Z</apll> is a two-element nested vector with a Complex floating point vector of the Eigenvalues in the first element and a Complex floating point matrix of the Eigenvectors in the second		\|<apll>Z←⌹⍠3 R</apll>\|\|<apll>Z</apll> is a two-element nested vector with a Complex floating point vector of the Eigenvalues in the first element and a Complex floating point matrix of the Eigenvectors in the second
	\|-		\|-
	\|<apll>~~Z←(⌹⍠4)~~ R</apll>\|\|<apll>Z</apll> is a three-element nested vector with a Complex floating point vector of the Eigenvalues in the first element, a Complex floating point matrix of the Eigenvectors in the second, and a Real matrix of the Schur vectors in the third		\|<apll>Z←⌹⍠4 R</apll>\|\|<apll>Z</apll> is a three-element nested vector with a Complex floating point vector of the Eigenvalues in the first element, a Complex floating point matrix of the Eigenvectors in the second, and a Real matrix of the Schur vectors in the third
	\|}		\|}

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	&ensp; 1 3		&ensp; 1 3
	&ensp; 2 1		&ensp; 2 1
	&ensp; MatFact←{(Eval Evec)~~←(⌹⍠3)~~ ⍵		&ensp; MatFact←{(Eval Evec)←⌹⍠3 ⍵
	<span style="color:blue;">➥</span>⋄ Diag←∘⌻!Eval ⍝ Factorial of Eigenvalues on diagonal		<span style="color:blue;">➥</span>⋄ Diag←∘⌻!Eval ⍝ Factorial of Eigenvalues on diagonal
	<span style="color:blue;">➥</span>⋄ Evec+.×Diag+.×⌹Evec}		<span style="color:blue;">➥</span>⋄ Evec+.×Diag+.×⌹Evec}

WikiSysop: /* Eigenvalues and Eigenvectors */

2020-08-23T13:11:56Z

Eigenvalues and Eigenvectors

← Older revision		Revision as of 09:11, 23 August 2020
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	<span style="color:blue;">➥</span>⋄ ⎕IO←1 ⋄ <9○1⌷[2] MatFact M ⍝ Factorial ...		<span style="color:blue;">➥</span>⋄ ⎕IO←1 ⋄ <9○1⌷[2] MatFact M ⍝ Factorial ...
	<span style="color:blue;">➥</span>}		<span style="color:blue;">➥</span>}
	&ensp; HyperFact ~~1i2~~		&ensp; HyperFact 1<_i/>2
	&ensp; 0.~~11229J0~~.32361		&ensp; 0.11229<_i/>0.32361
	&ensp; !~~1i2~~		&ensp; !1<_i/>2
	&ensp; 0.~~11229J0~~.32361		&ensp; 0.11229<_i/>0.32361
	</pre></apll>		</pre></apll>

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	<apll><pre>		<apll><pre>
	&ensp; HyperFact <⍳4		&ensp; HyperFact <⍳4
	&ensp; 0.~~0060975i¯0~~.~~0010787j¯0~~.~~001618k¯0~~.0021573		&ensp; 0.0060975<_i/>¯0.0010787<_j/>¯0.001618<_k/>¯0.0021573
	&ensp; !<⍳4		&ensp; !<⍳4
	&ensp; 0.~~0060975i¯0~~.~~0010787j¯0~~.~~001618k¯0~~.0021573		&ensp; 0.0060975<_i/>¯0.0010787<_j/>¯0.001618<_k/>¯0.0021573
	</pre></apll>		</pre></apll>

	{{Article footer\|9\|Variant}}		{{Article footer\|9\|Variant}}
	[[Category:Operators]]		[[Category:Operators]]

WikiSysop: /* Combining Variants */

2020-08-23T13:08:12Z

Combining Variants

← Older revision		Revision as of 09:08, 23 August 2020
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	<p>Dyadic Iota allows several other Variants such as setting the Index Origin and Comparison Tolerance. They may be combined into one right operand, for example, any of the following expressions</p>		<p>Dyadic Iota allows several other Variants such as setting the Index Origin and Comparison Tolerance. They may be combined into one right operand, for example, any of the following expressions</p>

	<apll> L⍳⍠(0 ~~1E¯10~~ 'd') R</apll>   or<br />		<apll> L⍳⍠(0 1<_E/>¯10 'd') R</apll>   or<br />
	<apll> L⍳⍠(0 'd' ~~1E¯10~~) R</apll>   or<br />		<apll> L⍳⍠(0 'd' 1<_E/>¯10) R</apll>   or<br />
	<apll> L⍳⍠('d' 0 ~~1E¯10~~) R</apll><br />		<apll> L⍳⍠('d' 0 1<_E/>¯10) R</apll><br />

	<p>may be used to find a Descending Subsequence in origin-0 using a Comparison Tolerance of <apll>~~1E¯10~~</apll>.</p>		<p>may be used to find a Descending Subsequence in origin-0 using a Comparison Tolerance of <apll>1<_E/>¯10</apll>.</p>

	<p>The order of the two numeric values is important: if there is only one number in the right operand, it is applied to the Index Origin; if there are two numbers, the first is applied to the Index Origin and the second to the Comparison Tolerance. The character value may appear anywhere in the right operand — however if a second character-valued Variant were defined for Dyadic Iota, then the two character values would also be order-dependent.</p>		<p>The order of the two numeric values is important: if there is only one number in the right operand, it is applied to the Index Origin; if there are two numbers, the first is applied to the Index Origin and the second to the Comparison Tolerance. The character value may appear anywhere in the right operand — however if a second character-valued Variant were defined for Dyadic Iota, then the two character values would also be order-dependent.</p>

Paul Robinson: add cat

2019-10-16T16:15:35Z

add cat

← Older revision		Revision as of 12:15, 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 [[Matrix Inverse/Divide]].
			<hr /><table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
	<tr>		<tr>
	<td>		<td>
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	</pre></apll>		</pre></apll>

	{{~~Language Toolbar~~}}		{{Article footer\|9\|Variant}}
			[[Category:Operators]]

WikiSysop: /* Shorthand Notation */

2019-08-19T17:10:13Z

Shorthand Notation

← Older revision		Revision as of 13:10, 19 August 2019
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	<td><apll>'i'</apll></td>		<td><apll>'i'</apll></td>
	<td><b>I</b>nterior product</td>		<td><b>I</b>nterior product</td>
	<td><apll>(((L×R)+R×L)÷2</apll></td>		<td><apll>((L×R)+R×L)÷2</apll></td>
	<td><apll>L×R</apll></td>		<td><apll>L×R</apll></td>
	</tr>		</tr>
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	<td><apll>'e'</apll></td>		<td><apll>'e'</apll></td>
	<td><b>E</b>xterior product</td>		<td><b>E</b>xterior product</td>
	<td><apll>(((L×R)-R×L)÷2</apll></td>		<td><apll>((L×R)-R×L)÷2</apll></td>
	<td><apll>0</apll></td>		<td><apll>0</apll></td>
	</tr>		</tr>

WikiSysop: /* Rising and Falling Factorials */

2019-08-06T22:29:23Z

Rising and Falling Factorials

← Older revision		Revision as of 18:29, 6 August 2019
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	<apll><pre>		<apll><pre>
	(!~~⍠¯6~~ 2)6 ⍝ ×/6 4 2 0 ¯2 ¯4		(!⍠(¯6 2))6 ⍝ ×/6 4 2 0 ¯2 ¯4
	0		0
	(!~~⍠¯5~~ 2)6 ⍝ ×/6 4 2 0 ¯2		(!⍠(¯5 2))6 ⍝ ×/6 4 2 0 ¯2
	0		0
	(!~~⍠¯4~~ 2)6 ⍝ ×/6 4 2 0		(!⍠(¯4 2))6 ⍝ ×/6 4 2 0
	0		0
	(!~~⍠¯3~~ 2)6 ⍝ ×/6 4 2		(!⍠(¯3 2))6 ⍝ ×/6 4 2
	48		48
	(!~~⍠¯2~~ 2)6 ⍝ ×/6 4		(!⍠(¯2 2))6 ⍝ ×/6 4
	24		24
	(!~~⍠¯1~~ 2)6 ⍝ ×/6		(!⍠(¯1 2))6 ⍝ ×/6
	6		6
	(!~~⍠¯0~~ 2)6 ⍝ ×/⍬		(!⍠(¯0 2))6 ⍝ ×/⍬
	1		1
	</pre></apll>		</pre></apll>
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	<apll><pre>		<apll><pre>
	(!⍠6 2)6 ⍝ ×/6 8 10 12 14 16		(!⍠(6 2))6 ⍝ ×/6 8 10 12 14 16
	1290240		1290240
	(!⍠5 2)6 ⍝ ×/6 8 10 12 14		(!⍠(5 2))6 ⍝ ×/6 8 10 12 14
	80640		80640
	(!⍠4 2)6 ⍝ ×/6 8 10 12		(!⍠(4 2))6 ⍝ ×/6 8 10 12
	5760		5760
	(!⍠3 2)6 ⍝ ×/6 8 10		(!⍠(3 2))6 ⍝ ×/6 8 10
	480		480
	(!⍠2 2)6 ⍝ ×/6 8		(!⍠(2 2))6 ⍝ ×/6 8
	48		48
	(!⍠1 2)6 ⍝ ×/6		(!⍠(1 2))6 ⍝ ×/6
	6		6
	(!⍠0 2)6 ⍝ ×/⍬		(!⍠(0 2))6 ⍝ ×/⍬
	1		1
	</pre></apll>		</pre></apll>
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	</tr>		</tr>
	</table>		</table>


	==Eigenvalues and Eigenvectors==		==Eigenvalues and Eigenvectors==

WikiSysop: /* Shorthand Notation */

2019-03-07T20:15:19Z

Shorthand Notation

← Older revision		Revision as of 16:15, 7 March 2019
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	<th><i>C</i></th>		<th><i>C</i></th>
	<th>Name</th>		<th>Name</th>
	<th>Quaternion/Octonion — Non-commutative ~~(<apll>D∊3 7</apll>)~~</th>		<th>Quaternion/Octonion — Non-commutative</th>
	<th>Real/Complex — Commutative ~~(<apll>D∊0 1</apll>)~~</th>		<th>Real/Complex — Commutative</th>
	</tr>		</tr>
	<tr>		<tr>

WikiSysop at 19:37, 20 November 2018

2018-11-20T19:37:24Z

← Older revision		Revision as of 15:37, 20 November 2018
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	<td><apll>L×R</apll></td>		<td><apll>L×R</apll></td>
	<td><i>C</i></td>		<td><i>C</i></td>
	<td>Except for <b>Dot Product</b>, these variants are non-trivial only when multiplying <b>Non-commutative</b> numbers.<br />~~Thes~~ definitions ~~are for~~ scalar <apll>L</apll> and <apll>R</apll> ~~only.<br /><apll>D</apll> is~~ the ~~# of <b>imaginary</b> coefficients (<apll>D←¯1+(=L)⌈=R</apll>)~~.<br /><br />		<td>Except for <b>Dot Product</b>, these variants are non-trivial only when multiplying <b>Non-commutative</b> numbers.<br />The following definitions on scalar <apll>L</apll> and <apll>R</apll> extend to higher rank arrays in the usual way.<br /><br />
	<table border="1" cellpadding="10" cellspacing="0" rules="all" summary="">		<table border="1" cellpadding="10" cellspacing="0" rules="all" summary="">
	<tr>		<tr>
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	<td><apll>'x'</apll></td>		<td><apll>'x'</apll></td>
	<td><b>Cross</b> product</td>		<td><b>Cross</b> product</td>
	<td><apll>-.~~×⍉(~~<~~0,∘.=⍨⍳D),D↑1↓~~>~~[⎕IO]L~~ R</apll></td>		<td><apll>¯9○<(∘⌻¯9○L)+.×>¯9○R</apll>   <b>Quaternions only</b><br />where <apll>¯9○R</apll> returns <apll>(R-+R)÷2</apll>, i.e. as a pure imaginary number</td>
	<td><apll>0</apll></td>		<td><apll>0</apll></td>
	</tr>		</tr>

WikiSysop: /* Shorthand Notation */

2018-11-15T17:03:20Z

Shorthand Notation

← Older revision		Revision as of 13:03, 15 November 2018
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	<td><apll>L×R</apll></td>		<td><apll>L×R</apll></td>
	<td><i>C</i></td>		<td><i>C</i></td>
	<td>Except for <b>Dot Product</b>, these variants are ~~of use~~ only when multiplying <b>Non-commutative</b> numbers.<br /><apll>D</apll> is the # of imaginary coefficients (<apll>D←¯1+(=L)⌈=R</apll>).<br /><br />		<td>Except for <b>Dot Product</b>, these variants are non-trivial only when multiplying <b>Non-commutative</b> numbers.<br />Thes definitions are for scalar <apll>L</apll> and <apll>R</apll> only.<br /><apll>D</apll> is the # of <b>imaginary</b> coefficients (<apll>D←¯1+(=L)⌈=R</apll>).<br /><br />
	<table border="1" cellpadding="10" cellspacing="0" rules="all" summary="">		<table border="1" cellpadding="10" cellspacing="0" rules="all" summary="">
	<tr>		<tr>