Determinant Operator - Revision history

WikiSysop at 13:56, 25 August 2022

2022-08-25T13:56:59Z

← Older revision		Revision as of 09:56, 25 August 2022
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	<p>The determinant of a matrix is a value associated with it whose interpretation depends upon the purpose of the matrix. For example, where the matrix represents the coefficients in a system of linear equations, a non-zero determinant indicates that the system has a unique solution, and a zero determinant indicates that the system has either no solutions or many solutions. For the original source document that introduced this operator, see [http://www.jsoftware.com/papers/satn42.htm Determinant-Like Functions Produced by the Dot Operator].</p>		<p>The determinant of a matrix is a value associated with it whose interpretation depends upon the purpose of the matrix. For example, where the matrix represents the coefficients in a system of linear equations, a non-zero determinant indicates that the system has a unique solution, and a zero determinant indicates that the system has either no solutions or many solutions. For the original source document that introduced this operator, see [http://www.jsoftware.com/papers/satn42.htm Determinant-Like Functions Produced by the Dot Operator].</p>

	<p>There are many formulas that ~~many~~ be used to calculate the value of the determinant, one of which (from Wikipedia, due to [https://en.wikipedia.org/wiki/Leibniz_formula_for_determinants Leibniz]) is particularly suited to our purposes:</p>		<p>There are many formulas that may be used to calculate the value of the determinant, one of which (from Wikipedia, due to [https://en.wikipedia.org/wiki/Leibniz_formula_for_determinants Leibniz]) is particularly suited to our purposes:</p>

	[[image:LeibnizFormula.png]]		[[image:LeibnizFormula.png]]

Sudleyplace at 23:51, 15 April 2018

2018-04-15T23:51:06Z

← Older revision		Revision as of 19:51, 15 April 2018
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	<h2>Examples</h2>		<h2>Examples</h2>

	<apll>~~      ~~⎕←a←?4 4⍴10~~<br />~~		<apll><pre>
	1 8 1 8~~<br />~~		⎕←a←?4 4⍴10
	5 1 7 2~~<br />~~		1 8 1 8
	6 3 9 4~~<br />~~		5 1 7 2
	4 9 8 8~~<br />~~		6 3 9 4
	~~      ~~-.×a~~<br />~~		4 9 8 8
	26~~<br />~~		-.×a
	~~      ~~+.×a<br />		26
	~~18886~~</apll>		+.×a
			18886</pre></apll>

	<h2>Alternate Derived Functions</h2>		<h2>Alternate Derived Functions</h2>
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	<p>One such alternate derived function is <apll>⌊.+</apll> which finds the [https://en.wikipedia.org/wiki/Assignment_problem lowest cost assignment] of agents (rows) to tasks (columns) where each agent is assigned one and only one task. For example,</p>		<p>One such alternate derived function is <apll>⌊.+</apll> which finds the [https://en.wikipedia.org/wiki/Assignment_problem lowest cost assignment] of agents (rows) to tasks (columns) where each agent is assigned one and only one task. For example,</p>

	<apll>~~      ~~⌊.+a<br />		<apll><pre>
	10</apll>		⌊.+a
			10</pre></apll>

	<p>This result means that there is a minimal assignment of choices of an entry from each column and each row that sums to <apll>10</apll>, a result easily seen using another pair of operands to the Determinant Operator so as to display the actual choices:</p>		<p>This result means that there is a minimal assignment of choices of an entry from each column and each row that sums to <apll>10</apll>, a result easily seen using another pair of operands to the Determinant Operator so as to display the actual choices:</p>

	<apll>~~      ~~{(+/⍺)<+/⍵:⍺ ⋄ ⍵}.,a~~<br />~~		<apll><pre>
	4 3 1 2</apll>		{(+/⍺)<+/⍵:⍺ ⋄ ⍵}.,a
			4 3 1 2</pre></apll>

	<p>That is, a minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>		<p>That is, a minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>

WikiSysop at 22:06, 21 February 2016

2016-02-21T22:06:03Z

← Older revision		Revision as of 18:06, 21 February 2016
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	<p>Moreover, just as <apll>-.×</apll> uses an optimized algorithm ([http://www.gnu.org/software/gsl/manual/html_node/LU-Decomposition.html#LU-Decomposition LU Decomposition]), so does <apll>⌊.+</apll> ([https://github.com/maandree/hungarian-algorithm-n3/blob/master/hungarian.c Kuhn's Hungarian Algorithm]) as well as <apll>⌈.+</apll> and <apll>{(+/⍺)<+/⍵:⍺ ⋄ ⍵}</apll>, along with a number of variations of the latter anonymous function (e.g. <apll>{(+/⍺)>+/⍵:⍺ ⋄ ⍵}</apll>, <apll>{(+/⍺)<+/⍵:⍵ ⋄ ⍺}</apll>, etc.). Consequently, all of the above derived functions return results very quickly for reasonably large arguments. For example, <apll>⌊.+</apll> on a random <apll>50</apll> by <apll>50</apll> matrix takes only a few milliseconds.</p>		<p>Moreover, just as <apll>-.×</apll> uses an optimized algorithm ([http://www.gnu.org/software/gsl/manual/html_node/LU-Decomposition.html#LU-Decomposition LU Decomposition]), so does <apll>⌊.+</apll> ([https://github.com/maandree/hungarian-algorithm-n3/blob/master/hungarian.c Kuhn's Hungarian Algorithm]) as well as <apll>⌈.+</apll> and <apll>{(+/⍺)<+/⍵:⍺ ⋄ ⍵}</apll>, along with a number of variations of the latter anonymous function (e.g. <apll>{(+/⍺)>+/⍵:⍺ ⋄ ⍵}</apll>, <apll>{(+/⍺)<+/⍵:⍵ ⋄ ⍺}</apll>, etc.). Consequently, all of the above derived functions return results very quickly for reasonably large arguments. For example, <apll>⌊.+</apll> on a random <apll>50</apll> by <apll>50</apll> matrix takes only a few milliseconds.</p>

			<p>Another variant of the lowest cost assignment is <apll>⌊.×</apll> and <apll>⌈.×</apll>. Rather than treat these expressions specially, they can be resolved via the identities <apll>⌊.×R ←→ ⌊.+⍟R</apll> and <apll>⌈.×R ←→ ⌈.+⍟R</apll>.

	<h2>References</h2>		<h2>References</h2>

WikiSysop at 16:20, 26 October 2013

2013-10-26T16:20:02Z

← Older revision		Revision as of 12:20, 26 October 2013
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	[[image:LeibnizFormula.png]]		[[image:LeibnizFormula.png]]

	<p>for an ''n''×''n'' matrix, where sgn is the [https://en.wikipedia.org/wiki/Even_and_odd_permutations sign function] of [https://en.wikipedia.org/wiki/Permutation permutation]s in the [https://en.wikipedia.org/wiki/Permutation_group permutation group] ''S''<sub>''n''</sub>, which returns +1 and −1 for [https://en.wikipedia.org/wiki/Even_and_odd_permutations even and odd permutations], respectively. The latter sum of positive or negative quantities is actually an alternating sum (<apll>-/</apll> in APL) and the righthand part is a product of terms (<apll>×</apll> in APL), hence this formula may be described as an alternating sum of products. This means that the usual determinant of a matrix is obtained via <apll>-.×</apll>. Moreover, the [https://en.wikipedia.org/wiki/Permanent permanent] of a matrix uses <apll>+.×</apll>.</p>		<p>for an ''n''×''n'' matrix, where sgn is the [https://en.wikipedia.org/wiki/Even_and_odd_permutations sign function] of [https://en.wikipedia.org/wiki/Permutation permutation]s in the [https://en.wikipedia.org/wiki/Permutation_group permutation group] ''S''<sub>''n''</sub>, which returns +1 and −1 for [https://en.wikipedia.org/wiki/Even_and_odd_permutations even and odd permutations], respectively. The latter sum of positive or negative quantities is actually an alternating sum (<apll>-/</apll> in APL) and the righthand part is a product of terms (<apll>×/</apll> in APL), hence this formula may be described as an alternating sum of products. This means that the usual determinant of a matrix is obtained via <apll>-.×</apll>. Moreover, the [https://en.wikipedia.org/wiki/Permanent permanent] of a matrix uses <apll>+.×</apll>.</p>

	<h2>Examples</h2>		<h2>Examples</h2>
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	<p>That is, a minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>		<p>That is, a minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>

	<p>Moreover, just as <apll>-.×</apll> uses ~~a specialized~~ algorithm ([http://www.gnu.org/software/gsl/manual/html_node/LU-Decomposition.html#LU-Decomposition LU Decomposition]), so does <apll>⌊.+</apll> ([https://github.com/maandree/hungarian-algorithm-n3/blob/master/hungarian.c Kuhn's Hungarian Algorithm]) as well as <apll>⌈.+</apll> and <apll>{(+/⍺)<+/⍵:⍺ ⋄ ⍵}</apll>, along with a number of variations of the latter anonymous function (e.g. <apll>{(+/⍺)>+/⍵:⍺ ⋄ ⍵}</apll>, <apll>{(+/⍺)<+/⍵:⍵ ⋄ ⍺}</apll>, etc.). Consequently, all of the above derived functions return results very quickly for reasonably large arguments. For example, <apll>⌊.+</apll> on a random <apll>50</apll> by <apll>50</apll> matrix takes only a few milliseconds.</p>		<p>Moreover, just as <apll>-.×</apll> uses an optimized algorithm ([http://www.gnu.org/software/gsl/manual/html_node/LU-Decomposition.html#LU-Decomposition LU Decomposition]), so does <apll>⌊.+</apll> ([https://github.com/maandree/hungarian-algorithm-n3/blob/master/hungarian.c Kuhn's Hungarian Algorithm]) as well as <apll>⌈.+</apll> and <apll>{(+/⍺)<+/⍵:⍺ ⋄ ⍵}</apll>, along with a number of variations of the latter anonymous function (e.g. <apll>{(+/⍺)>+/⍵:⍺ ⋄ ⍵}</apll>, <apll>{(+/⍺)<+/⍵:⍵ ⋄ ⍺}</apll>, etc.). Consequently, all of the above derived functions return results very quickly for reasonably large arguments. For example, <apll>⌊.+</apll> on a random <apll>50</apll> by <apll>50</apll> matrix takes only a few milliseconds.</p>

	<h2>References</h2>		<h2>References</h2>

WikiSysop at 14:06, 26 October 2013

2013-10-26T14:06:28Z

← Older revision		Revision as of 10:06, 26 October 2013
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	<p>That is, a minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>		<p>That is, a minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>

	<p>Moreover, just as <apll>-.×</apll> uses a [https://github.com/maandree/hungarian-algorithm-n3/blob/master/hungarian.c ~~specialized algorithm~~]~~, so does <apll>⌊.+</apll>~~ as well as <apll>⌈.+</apll> and <apll>{(+/⍺)<+/⍵:⍺ ⋄ ⍵}</apll>, along with a number of variations of the latter anonymous function (e.g. <apll>{(+/⍺)>+/⍵:⍺ ⋄ ⍵}</apll>, <apll>{(+/⍺)<+/⍵:⍵ ⋄ ⍺}</apll>, etc.). Consequently, all of the above derived functions return results very quickly for reasonably large arguments. For example, <apll>⌊.+</apll> on a random <apll>50</apll> by <apll>50</apll> matrix takes only a few milliseconds.</p>		<p>Moreover, just as <apll>-.×</apll> uses a specialized algorithm ([http://www.gnu.org/software/gsl/manual/html_node/LU-Decomposition.html#LU-Decomposition LU Decomposition]), so does <apll>⌊.+</apll> ([https://github.com/maandree/hungarian-algorithm-n3/blob/master/hungarian.c Kuhn's Hungarian Algorithm]) as well as <apll>⌈.+</apll> and <apll>{(+/⍺)<+/⍵:⍺ ⋄ ⍵}</apll>, along with a number of variations of the latter anonymous function (e.g. <apll>{(+/⍺)>+/⍵:⍺ ⋄ ⍵}</apll>, <apll>{(+/⍺)<+/⍵:⍵ ⋄ ⍺}</apll>, etc.). Consequently, all of the above derived functions return results very quickly for reasonably large arguments. For example, <apll>⌊.+</apll> on a random <apll>50</apll> by <apll>50</apll> matrix takes only a few milliseconds.</p>

	<h2>References</h2>		<h2>References</h2>

WikiSysop at 17:30, 24 October 2013

2013-10-24T17:30:50Z

← Older revision		Revision as of 13:30, 24 October 2013
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	<p>for an ''n''×''n'' matrix, where sgn is the [https://en.wikipedia.org/wiki/Even_and_odd_permutations sign function] of [https://en.wikipedia.org/wiki/Permutation permutation]s in the [https://en.wikipedia.org/wiki/Permutation_group permutation group] ''S''<sub>''n''</sub>, which returns +1 and −1 for [https://en.wikipedia.org/wiki/Even_and_odd_permutations even and odd permutations], respectively. The latter sum of positive or negative quantities is actually an alternating sum (<apll>-/</apll> in APL) and the righthand part is a product of terms (<apll>×</apll> in APL), hence this formula may be described as an alternating sum of products. This means that the usual determinant of a matrix is obtained via <apll>-.×</apll>. Moreover, the [https://en.wikipedia.org/wiki/Permanent permanent] of a matrix uses <apll>+.×</apll>.</p>		<p>for an ''n''×''n'' matrix, where sgn is the [https://en.wikipedia.org/wiki/Even_and_odd_permutations sign function] of [https://en.wikipedia.org/wiki/Permutation permutation]s in the [https://en.wikipedia.org/wiki/Permutation_group permutation group] ''S''<sub>''n''</sub>, which returns +1 and −1 for [https://en.wikipedia.org/wiki/Even_and_odd_permutations even and odd permutations], respectively. The latter sum of positive or negative quantities is actually an alternating sum (<apll>-/</apll> in APL) and the righthand part is a product of terms (<apll>×</apll> in APL), hence this formula may be described as an alternating sum of products. This means that the usual determinant of a matrix is obtained via <apll>-.×</apll>. Moreover, the [https://en.wikipedia.org/wiki/Permanent permanent] of a matrix uses <apll>+.×</apll>.</p>

	<p>~~For example,~~</p>		<h2>Examples</h2>

	<apll>      ⎕←a←?4 4⍴10<br />		<apll>      ⎕←a←?4 4⍴10<br />
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	18886</apll>		18886</apll>

	<p>The Leibniz formula ~~also~~ allows us to substitute other functions for (<apll>-</apll>) and (<apll>×</apll>) to obtain alternate derived functions in order to solve different problems.</p>		<h2>Alternate Derived Functions</h2>

			<p>The Leibniz formula nicely allows us to substitute other functions for (<apll>-</apll>) and (<apll>×</apll>) to obtain alternate derived functions in order to solve different problems, while <b>still using the same basic algorithm</b>.</p>

	<p>One such alternate derived function is <apll>⌊.+</apll> which finds the [https://en.wikipedia.org/wiki/Assignment_problem lowest cost assignment] of agents (rows) to tasks (columns) where each agent is assigned one and only one task. For example,</p>		<p>One such alternate derived function is <apll>⌊.+</apll> which finds the [https://en.wikipedia.org/wiki/Assignment_problem lowest cost assignment] of agents (rows) to tasks (columns) where each agent is assigned one and only one task. For example,</p>
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	4 3 1 2</apll>		4 3 1 2</apll>

	<p>That is, ~~the~~ minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>		<p>That is, a minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>

			<p>Moreover, just as <apll>-.×</apll> uses a [https://github.com/maandree/hungarian-algorithm-n3/blob/master/hungarian.c specialized algorithm], so does <apll>⌊.+</apll> as well as <apll>⌈.+</apll> and <apll>{(+/⍺)<+/⍵:⍺ ⋄ ⍵}</apll>, along with a number of variations of the latter anonymous function (e.g. <apll>{(+/⍺)>+/⍵:⍺ ⋄ ⍵}</apll>, <apll>{(+/⍺)<+/⍵:⍵ ⋄ ⍺}</apll>, etc.). Consequently, all of the above derived functions return results very quickly for reasonably large arguments. For example, <apll>⌊.+</apll> on a random <apll>50</apll> by <apll>50</apll> matrix takes only a few milliseconds.</p>

			<h2>References</h2>

	<p>~~Moreover, just as~~ <~~apll~~>-.×</~~apll> uses a specialized algorithm~~, ~~so does <apll>⌊~~.+</~~apll~~> ~~as well as~~ <~~apll~~>⌈.+</~~apll~~> ~~and~~ <~~apll~~>~~{(+~~/~~⍺)<+~~/~~⍵:⍺ ⋄ ⍵}~~.,</~~apll~~> ~~along with several of its variations, so those derived functions quickly return results for large arguments. For example,~~ <~~apll~~>⌊.+</~~apll> on a <apll>50<~~/~~apll> by <apll>50~~</~~apll~~> ~~matrix takes only a few milliseconds.~~</p>		<ol>
			<li>[http://www.jsoftware.com/papers/satn42.htm Determinant-Like Functions Produced by the Dot Operator], SATN-42 (Sharp APL Technical Notes), 1982-04-01, by K.E. Iverson</li>
			<li>[https://en.wikipedia.org/wiki/Leibniz_formula_for_determinants Leibniz Formula], from Wikipedia</li>
			<li>[https://en.wikipedia.org/wiki/Permanent Permanent], from Wikipedia</li>
			<li>[https://en.wikipedia.org/wiki/Assignment_problem Assignment Problem], from Wikipedia</li>
			</ol>

WikiSysop: WikiSysop moved page Determinant to Determinant Operator

2013-10-24T16:46:03Z

WikiSysop moved page Determinant to Determinant Operator

← Older revision	Revision as of 12:46, 24 October 2013
(No difference)

WikiSysop at 16:45, 24 October 2013

2013-10-24T16:45:35Z

← Older revision		Revision as of 12:45, 24 October 2013
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	18886</apll>		18886</apll>

	<p>The Leibniz formula also allows us to substitute other functions for (<apll>-</apll>) and (<apll>×</apll>) to obtain alternate derived functions.</p>		<p>The Leibniz formula also allows us to substitute other functions for (<apll>-</apll>) and (<apll>×</apll>) to obtain alternate derived functions in order to solve different problems.</p>

	<p>One such alternate ~~form~~ is <apll>⌊.+</apll> which finds the lowest cost assignment of ~~people~~ (rows) to tasks (columns) where each ~~person~~ is assigned one and only one task. For example,</p>		<p>One such alternate derived function is <apll>⌊.+</apll> which finds the [https://en.wikipedia.org/wiki/Assignment_problem lowest cost assignment] of agents (rows) to tasks (columns) where each agent is assigned one and only one task. For example,</p>

	<apll>      ⌊.+a<br />		<apll>      ⌊.+a<br />
	10</apll>		10</apll>

	<p>This result means that there is a minimal assignment of choices of an entry from each column and each row that sums to <apll>10</apll>, a result easily seen using another pair of ~~functions~~ to display the actual choices:</p>		<p>This result means that there is a minimal assignment of choices of an entry from each column and each row that sums to <apll>10</apll>, a result easily seen using another pair of operands to the Determinant Operator so as to display the actual choices:</p>

	<apll>      {(+/⍺)<+/⍵:⍺ ⋄ ⍵}.,a<br />		<apll>      {(+/⍺)<+/⍵:⍺ ⋄ ⍵}.,a<br />
	4 3 1 2</apll>		4 3 1 2</apll>

	<p>That is, the minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>.</p>		<p>That is, the minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>, which, by construction, are all in different rows.</p>

			<p>Moreover, just as <apll>-.×</apll> uses a specialized algorithm, so does <apll>⌊.+</apll> as well as <apll>⌈.+</apll> and <apll>{(+/⍺)<+/⍵:⍺ ⋄ ⍵}.,</apll> along with several of its variations, so those derived functions quickly return results for large arguments. For example, <apll>⌊.+</apll> on a <apll>50</apll> by <apll>50</apll> matrix takes only a few milliseconds.</p>

WikiSysop: Created page with "

2013-10-22T00:48:02Z

Created page with "<table border="1" cellpadding="5" cellspacing="0" rules="none" summary=""> <tr> <td> <table border="0" cellpadding="5" cellspacing="0" summary=""> <tr> <td val..."

New page
<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
<tr>
<td>
<table border="0" cellpadding="5" cellspacing="0" summary="">
<tr>
<td valign="top"><apll>Z←f.g R</apll></td>
<td></td>
<td></td>
<td>returns the generalized determinant of <apll>R</apll>.</td>
</tr>
</table>
</td>
</tr>
<tr>
<td><apll>R</apll> is a numeric scalar, vector, or matrix.</td>
</tr>
<tr>
<td><apll>f</apll> and <apll>g</apll> are functions.</td>
</tr>
</table>

<p>The determinant of a matrix is a value associated with it whose interpretation depends upon the purpose of the matrix. For example, where the matrix represents the coefficients in a system of linear equations, a non-zero determinant indicates that the system has a unique solution, and a zero determinant indicates that the system has either no solutions or many solutions. For the original source document that introduced this operator, see [http://www.jsoftware.com/papers/satn42.htm Determinant-Like Functions Produced by the Dot Operator].</p>

<p>There are many formulas that many be used to calculate the value of the determinant, one of which (from Wikipedia, due to [https://en.wikipedia.org/wiki/Leibniz_formula_for_determinants Leibniz]) is particularly suited to our purposes:</p>

[[image:LeibnizFormula.png]]

<p>for an ''n''×''n'' matrix, where sgn is the [https://en.wikipedia.org/wiki/Even_and_odd_permutations sign function] of [https://en.wikipedia.org/wiki/Permutation permutation]s in the [https://en.wikipedia.org/wiki/Permutation_group permutation group] ''S''<sub>''n''</sub>, which returns +1 and −1 for [https://en.wikipedia.org/wiki/Even_and_odd_permutations even and odd permutations], respectively. The latter sum of positive or negative quantities is actually an alternating sum (<apll>-/</apll> in APL) and the righthand part is a product of terms (<apll>×</apll> in APL), hence this formula may be described as an alternating sum of products. This means that the usual determinant of a matrix is obtained via <apll>-.×</apll>. Moreover, the [https://en.wikipedia.org/wiki/Permanent permanent] of a matrix uses <apll>+.×</apll>.</p>

<p>For example,</p>

<apll>      ⎕←a←?4 4⍴10<br />
1 8 1 8<br />
5 1 7 2<br />
6 3 9 4<br />
4 9 8 8<br />
      -.×a<br />
26<br />
      +.×a<br />
18886</apll>

<p>The Leibniz formula also allows us to substitute other functions for (<apll>-</apll>) and (<apll>×</apll>) to obtain alternate derived functions.</p>

<p>One such alternate form is <apll>⌊.+</apll> which finds the lowest cost assignment of people (rows) to tasks (columns) where each person is assigned one and only one task. For example,</p>

<apll>      ⌊.+a<br />
10</apll>

<p>This result means that there is a minimal assignment of choices of an entry from each column and each row that sums to <apll>10</apll>, a result easily seen using another pair of functions to display the actual choices:</p>

<apll>      {(+/⍺)<+/⍵:⍺ ⋄ ⍵}.,a<br />
4 3 1 2</apll>

<p>That is, the minimal set of choices is the <apll>4</apll> in column <apll>1</apll>, the <apll>3</apll> in column <apll>2</apll>, the <apll>1</apll> in column <apll>3</apll>, and the <apll>2</apll> in column <apll>4</apll>.</p>