Difference between revisions of "Determinant Operator"
Line 26:  Line 26:  
[[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>  
Line 56:  Line 56:  
<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  +  <p>Moreover, just as <apll>.×</apll> uses an optimized algorithm ([http://www.gnu.org/software/gsl/manual/html_node/LUDecomposition.html#LUDecomposition LU Decomposition]), so does <apll>⌊.+</apll> ([https://github.com/maandree/hungarianalgorithmn3/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> 
Revision as of 16:20, 26 October 2013


R is a numeric scalar, vector, or matrix.  
f and g are functions. 
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 nonzero 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 DeterminantLike Functions Produced by the Dot Operator.
There are many formulas that many be used to calculate the value of the determinant, one of which (from Wikipedia, due to Leibniz) is particularly suited to our purposes:
for an n×n matrix, where sgn is the sign function of permutations in the permutation group S_{n}, which returns +1 and −1 for even and odd permutations, respectively. The latter sum of positive or negative quantities is actually an alternating sum (/ in APL) and the righthand part is a product of terms (×/ 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 .×. Moreover, the permanent of a matrix uses +.×.
Examples
⎕←a←?4 4⍴10
1 8 1 8
5 1 7 2
6 3 9 4
4 9 8 8
.×a
26
+.×a
18886
Alternate Derived Functions
The Leibniz formula nicely allows us to substitute other functions for () and (×) to obtain alternate derived functions in order to solve different problems, while still using the same basic algorithm.
One such alternate derived function is ⌊.+ which finds the lowest cost assignment of agents (rows) to tasks (columns) where each agent is assigned one and only one task. For example,
⌊.+a
10
This result means that there is a minimal assignment of choices of an entry from each column and each row that sums to 10, a result easily seen using another pair of operands to the Determinant Operator so as to display the actual choices:
{(+/⍺)<+/⍵:⍺ ⋄ ⍵}.,a
4 3 1 2
That is, a minimal set of choices is the 4 in column 1, the 3 in column 2, the 1 in column 3, and the 2 in column 4, which, by construction, are all in different rows.
Moreover, just as .× uses an optimized algorithm (LU Decomposition), so does ⌊.+ (Kuhn's Hungarian Algorithm) as well as ⌈.+ and {(+/⍺)<+/⍵:⍺ ⋄ ⍵}, along with a number of variations of the latter anonymous function (e.g. {(+/⍺)>+/⍵:⍺ ⋄ ⍵}, {(+/⍺)<+/⍵:⍵ ⋄ ⍺}, etc.). Consequently, all of the above derived functions return results very quickly for reasonably large arguments. For example, ⌊.+ on a random 50 by 50 matrix takes only a few milliseconds.
References
 DeterminantLike Functions Produced by the Dot Operator, SATN42 (Sharp APL Technical Notes), 19820401, by K.E. Iverson
 Leibniz Formula, from Wikipedia
 Permanent, from Wikipedia
 Assignment Problem, from Wikipedia