Data update
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Ingy döt Net
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12390 changed files with 318560 additions and 27248 deletions
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@ -1,3 +1,25 @@
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;Related tasks:
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* [[Arrays]]
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* [[Vector]]
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* [[Matrices]]
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** [[Determinant and permanent]]
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*** [[wp:Laplace_expansion|Laplace expansion]] <math>O\bigl(n!\bigr)</math>
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*** [[wp:Leibniz_formula_for_determinants|Leibniz fomula]] <math>\Omega\bigl(n!\cdot n\bigr)</math>
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*** [[wp:Bareiss_algorithm|Bareiss algorithm]] <math>O\bigl(n^3\bigr)</math>
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*** [[wp:LU_decomposition|LU-decomposition]] <math>O\bigl(n^3\bigr)</math>
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*** [[wp:Strassen_algorithm|Strassen algorithm]] <math>O\bigl(n^{2.807}\bigr)</math>
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*** [[wp:Coppersmith%E2%80%93Winograd_algorithm|Coppersmith-Winograd algorithm]] <math>O\bigl(n^{2.376}\bigr)</math>
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*** [[wp:Jean-Fran%C3%A7ois_Le_Gall|Le Gall algorithm]]
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*** [https://codegolf.stackexchange.com/questions/236835/birds-algorithm-for-computing-determinants Bird's algorithm]
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* [[Bivector]]
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* [[Antivector]]
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* [[Tensor]]
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* [[Quaternion]]
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* [[Rotor]]
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* [[Motor]]
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* [[Sedenion]]
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* [[Octonion]]
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<br>
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For a given matrix, return the [[wp:Determinant|determinant]] and the [[wp:Permanent_(mathematics)|permanent]] of the matrix.
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For a matrix of [[wp:Orthonormal_basis|orthonormal basis]] vectors, return the [[wp:Levi-Civita_symbol|Levi-Civita symbol]] of the orthonormal basis vector permutation.
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@ -6,24 +28,10 @@ The determinant is given by
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while the permanent is given by
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:: <big><math> \operatorname{perm}(A)=\sum_\sigma\prod_{i=1}^n M_{i,\sigma_i}</math></big>
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In both cases the sum is over the permutations <math>\sigma</math> of the permutations of 1, 2, ..., ''n''. (A permutation's sign is 1 if there are an even number of inversions and -1 otherwise; see [[wp:Parity of a permutation|parity of a permutation]].)
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;Complexity of known algorithms:
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* [[wp:Laplace_expansion|Laplace expansion]] <math>O\bigl(n!\bigr)</math>
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* [[wp:Leibniz_formula_for_determinants|Leibniz fomula]] <math>\Omega\bigl(n!\cdot n\bigr)</math>
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* [[wp:Bareiss_algorithm|Bareiss algorithm]] <math>O\bigl(n^3\bigr)</math>
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* [[wp:LU_decomposition|LU-decomposition]] <math>O\bigl(n^3\bigr)</math>
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* [[wp:Strassen_algorithm|Strassen algorithm]] <math>O\bigl(n^{2.807}\bigr)</math>
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* [[wp:Coppersmith%E2%80%93Winograd_algorithm|Coppersmith-Winograd algorithm]] <math>O\bigl(n^{2.376}\bigr)</math>
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* [[wp:Jean-Fran%C3%A7ois_Le_Gall|Le Gall algorithm]]
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* [https://codegolf.stackexchange.com/questions/236835/birds-algorithm-for-computing-determinants Bird's algorithm]
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;Related task:
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<br><br>
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;C.f.
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* [[wp:Computational_complexity_of_matrix_multiplication#Matrix_inversion,_determinant_and_Gaussian_elimination|Computational complexity of matrix multiplication]]
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* [https://dai.fmph.uniba.sk/courses/FPRO/bird_pearls.pdf Richard Bird Pearls]
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* [https://dai.fmph.uniba.sk/courses/FPRO/ Funkcionálne programovanie]
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* [[Permutations by swapping]]
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<br><br>
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@ -0,0 +1,50 @@
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#include "mat.hpp"
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template<size_t N, size_t M = N, enum alg A = alg::std>
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using i32 = mat<int,N,M,A>;
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template<typename T, size_t N, size_t M = N, enum alg A = alg::std>
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constexpr inline bool test(mat<T,N,M,A>& src);
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int main()
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{
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i32<2,2> A =
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{
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{ 1, 2},
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{ 3, 4}
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};
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i32<3,3> B =
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{
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{-2, 2,-3},
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{-1, 1, 3},
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{ 2, 0,-1}
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};
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i32<4,4> C =
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{
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{ 1, 2, 3, 4},
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{ 4, 5, 6, 7},
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{ 7, 8, 9,10},
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{10,11,12,13}
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};
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i32<5,5> D =
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{
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{ 0, 1, 2, 3, 4},
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{ 5, 6, 7, 8, 9},
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{10, 11, 12, 13, 14},
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{15, 16, 17, 18, 19},
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{20, 21, 22, 23, 24}
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};
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exit(test(A) && test(B) && test(C) && test(D) ? EXIT_SUCCESS : EXIT_FAILURE);
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}
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template<typename T, size_t N, size_t M, enum alg A>
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constexpr inline bool test(mat<T,N,M,A>& src)
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{
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src.println();
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printf(" permanent: %+e\n", (flt)src.perm());
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printf("determinant: %+e\n", (flt)src.det());
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printf(" align/size: %zu/%zu\n", alignof(src), sizeof(src));
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puts("");
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return true;
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}
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@ -0,0 +1,29 @@
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/* generalized Kronecker delta - antisymmetrizer */
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template<ari T, size_t N>
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constexpr int8_t gkd(const std::array<T,N>& v, const std::array<T,N>& orig)
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{
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bool trans = 0;
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for(size_t i = 0; i < orig.size(); i++)
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{
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size_t cnt = 0;
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for(size_t j = 0; j < std::min<size_t>(i + 1,v.size()); j++)
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if(orig[i] == v[j]) ++cnt;
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for(size_t j = i + 1; j < v.size(); j++)
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{
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if(orig[i] == v[j]) ++cnt;
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if( v[i] > v[j]) trans += -1;
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}
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if(cnt != 1) return 0;
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}
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return trans ? -1 : 1;
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}
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/* generalized permanent delta - complete symmetrizer *
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* equivalent to abs(gKd(v,orig)) or is_index_sequence */
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template<ari T, size_t N>
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inline constexpr int8_t gpd(const std::array<T,N>& v, const std::array<T,N>& orig)
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{
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for(auto& i : orig)
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if(std::count(v.begin(), v.end(), i) != 1) return 0;
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return 1;
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}
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@ -0,0 +1,28 @@
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require "matrix"
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local arrays = {
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{ {1, 2},
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{3, 4} },
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{ {-2, 2, -3},
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{-1, 1, 3},
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{ 2, 0, -1} },
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{ { 1, 2, 3, 4},
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{ 4, 5, 6, 7},
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{ 7, 8, 9, 10},
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{10, 11, 12, 13} },
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{ { 0, 1, 2, 3, 4},
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{ 5, 6, 7, 8, 9},
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{10, 11, 12, 13, 14},
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{15, 16, 17, 18, 19},
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{20, 21, 22, 23, 24} }
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}
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for arrays as array do
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local m = matrix.from(array)
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print(m)
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print($"\nDeterminant: {m:det()}")
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print($"Permanent : {m:perm()}\n")
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end
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function det-perm ($array) {
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if($array) {
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$size = $array.Count
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function prod($A) {
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$prod = 1
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if($A) { $A | foreach{$prod *= $_} }
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$prod
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}
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function generate($sign, $n, $A) {
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if($n -eq 1) {
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$i = 0
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$prod = prod @($A | foreach{$array[$i++][$_]})
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[pscustomobject]@{det = $sign*$prod; perm = $prod}
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}
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else{
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for($i = 0; $i -lt ($n - 1); $i += 1) {
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generate $sign ($n - 1) $A
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if($n % 2 -eq 0){
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$i1, $i2 = $i, ($n-1)
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$A[$i1], $A[$i2] = $A[$i2], $A[$i1]
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}
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else{
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$i1, $i2 = 0, ($n-1)
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$A[$i1], $A[$i2] = $A[$i2], $A[$i1]
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}
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$sign *= -1
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}
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generate $sign ($n - 1) $A
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}
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}
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$det = $perm = 0
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generate 1 $size @(0..($size-1)) | foreach{
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$det += $_.det
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$perm += $_.perm
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}
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[pscustomobject]@{det = "$det"; perm = "$perm"}
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} else {Write-Error "empty array"}
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}
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det-perm 5
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det-perm @(@(1,0,0),@(0,1,0),@(0,0,1))
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det-perm @(@(0,0,1),@(0,1,0),@(1,0,0))
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det-perm @(@(4,3),@(2,5))
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det-perm @(@(2,5),@(4,3))
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det-perm @(@(4,4),@(2,2))
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