Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: e.numerical_approx(20) [ 2.2071 0.39645] [ 1.5858 2.2071] sage: (e-I).numerical_approx(20) [2.2071 - 1.0000*I 0.39645] [ 1.5858 2.2071 - 1.0000*I] sage: M=matrix(QQ,4,[i/(i+1) for i in range(12)]);M [ 0 1/2 2/3] [ 3/4 4/5 5/6] [ 6/7 7/8 8/9] [ 9/10 10/11 11/12] sage: M.numerical_approx() [0.000000000000000 0.500000000000000 0.666666666666667] [0.750000000000000 0.800000000000000 0.833333333333333] [0.857142857142857 0.875000000000000 0.888888888888889] [0.900000000000000 0.909090909090909 0.916666666666667] sage: matrix(SR, 2, 2, range(4)).n() [0.000000000000000 1.00000000000000] [ 2.00000000000000 3.00000000000000] sage: numerical_approx(M) [0.000000000000000 0.500000000000000 0.666666666666667] [0.750000000000000 0.800000000000000 0.833333333333333] [0.857142857142857 0.875000000000000 0.888888888888889] [0.900000000000000 0.909090909090909 0.916666666666667] permanent() Calculate and return the permanent of the m × n matrix self using Ryser’s algorithm. Let A = (a i,j ) be an m × n matrix over any commutative ring, with m ≤ n. The permanent of A is per(A) = ∑ π a 1,π(1) a 2,π(2) · · · a m,π(m) where the summation extends over all one-to-one functions π from {1, . . . , m} to {1, . . . , n}. The product a 1,π(1) a 2,π(2) · · · a m,π(m) is called diagonal product. So the permanent of an m × n matrix A is the sum of all the diagonal products of A. INPUT: •A – matrix of size m × n with m ≤ n OUTPUT: permanent of the matrix A ALGORITHM: Modification of theorem 7.1.1. from Brualdi and Ryser: Combinatorial Matrix Theory. Instead of deleting columns from A, we choose columns from A and calculate the product of the row sums of the selected submatrix. EXAMPLES: sage: M = MatrixSpace(ZZ,4,4) sage: A = M([1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]) 218 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: A.permanent() 24 sage: M = MatrixSpace(QQ,3,6) sage: A = M([1,1,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,1]) sage: A.permanent() 36 sage: M = MatrixSpace(RR,3,6) sage: A = M([1.0,1.0,1.0,1.0,0,0,0,1.0,1.0,1.0,1.0,0,0,0,1.0,1.0,1.0,1.0]) sage: A.permanent() 36.0000000000000 See Sloane’s sequence OEIS A079908(3) = 36, “The Dancing School Problems” sage: oeis(79908) # optional -- internet A079908: Solution to the Dancing School Problem with 3 girls and n+3 boys: f(3,n). sage: _(3) # optional -- internet 36 sage: M = MatrixSpace(ZZ,4,5) sage: A = M([1,1,0,1,1,0,1,1,1,1,1,0,1,0,1,1,1,0,1,0]) sage: A.permanent() 32 See Minc: Permanents, Example 2.1, p. 5. sage: M = MatrixSpace(QQ,2,2) sage: A = M([1/5,2/7,3/2,4/5]) sage: A.permanent() 103/175 sage: R. = PolynomialRing(ZZ) sage: A = MatrixSpace(R,2)([[a,1], [a,a+1]]) sage: A.permanent() a^2 + 2*a sage: R. = PolynomialRing(ZZ,2) sage: A = MatrixSpace(R,2)([x, y, x^2, y^2]) sage: A.permanent() x^2*y + x*y^2 AUTHORS: •Jaap Spies (2006-02-16) •Jaap Spies (2006-02-21): added definition of permanent permanental_minor(k) Return the permanental k-minor of an m × n matrix. This is the sum of the permanents of all possible k by k submatrices of A. See Brualdi and Ryser: Combinatorial Matrix Theory, p. 203. Note the typo p 0 (A) = 0 in that reference! For applications see Theorem 7.2.1 and Theorem 7.2.4. Note that the permanental m-minor equals per(A) if m = n. For a (0,1)-matrix A the permanental k-minor counts the number of different selections of k 1’s of A with no two of the 1’s on the same row and no two of the 1’s on the same column. INPUT: 219
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