--------------------^ sample-polynomial01.rb require "polynomial" P = Polynomial.create(Integer, "x") x = P.var p((x + 1)**100) #=> x^100 + 100x^99 + ... + 100x + 1 --------------------$ sample-polynomial01.rb
--------------------^ sample-polynomial02.rb require "polynomial" P = Polynomial.create(Integer, "x", "y", "z") x, y, z = P.vars p((-x + y + z)*(x + y - z)*(x - y + z)) #=> -z^3 + (y + x)z^2 + (y^2 - 2xy + x^2)z - y^3 + xy^2 + x^2y - x^3 --------------------$ sample-polynomial02.rb
--------------------^ sample-m-polynomial01.rb require "m-polynomial" P = MPolynomial(Integer) x, y, z, w = P.vars("xyz") p((-x + y + z)*(x + y - z)*(x - y + z)) #=> -x^3 + x^2y + x^2z + xy^2 - 2xyz + xz^2 - y^3 + y^2z + yz^2 - z^3 --------------------$ sample-m-polynomial01.rb
--------------------^ sample-divmod01.rb require "m-polynomial" require "rational" P = MPolynomial(Rational) x, y, z = P.vars("xyz") f = x**2*y + x*y**2 + y*2 + z**3 g = x*y-z**3 h = y*2-6*z MPolynomial.set_ord(:lex) # lex, grlex, grevlex puts "(#{f}).divmod([#{g}, #{h}]) =>", "#{f.divmod(g, h).inspect}" #=> (x^2y + xy^2 + 2y + z^3).divmod([xy - z^3, 2y - 6z]) => # [[x + y, 1/2z^3 + 1], xz^3 + 3z^4 + z^3 + 6z] # = [[Quotient1,Quotient2], Remainder] --------------------$ sample-divmod01.rb
--------------------^ sample-groebner01.rb require "groebner-basis" require "rational" P = MPolynomial(Rational, "xyz") x, y, z = P.vars("xyz") f1 = x**2 + y**2 + z**2 -1 f2 = x**2 + z**2 - y f3 = x - z p Groebner.basis([f1, f2, f3]) #=> [x - z, y - 2z^2, z^4 + 1/2z^2 - 1/4] --------------------$ sample-groebner01.rb
--------------------^ sample-primefield01.rb require "residue-class-ring" Z13 = ResidueClassRing(Integer, 13) a, b, c, d, e, f, g = Z13 p [e + c, e - c, e * c, e * 2001, 3 + c, 1/c, 1/c * c, d / d, b * 1 / b] #=> [6, 2, 8, 9, 5, 7, 1, 1, 1] p( (1...13).collect{|i| Z13[i]**12} ) #=> [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] --------------------$ sample-primefield01.rb
--------------------^ sample-algebraicfield01.rb require "residue-class-ring" require "polynomial" require "rational" Px = Polynomial(Rational, "x") x = Px.var F = ResidueClassRing(Px, x**2 + x + 1) x = F[x] p( (x + 1)**100 ) #=> -x - 1 p( (x-1)** 3 / (x**2 - 1) ) #=> -3x - 3 G = Polynomial(F, "y") y = G.var p( (x + y + 1)** 7 ) #=> y^7 + (7x + 7)y^6 + 8xy^5 + 4y^4 + (4x + 4)y^3 + 5xy^2 + 7y + x + 1 H = ResidueClassRing(G, y**5 + x*y + 1) y = H[y] p( 1/(x + y + 1)**7 ) #=> (1798/3x + 1825/9)y^4 + (-74x + 5176/9)y^3 + # (-6886/9x - 5917/9)y^2 + (1826/3x - 3101/9)y + 2146/9x + 4702/9 --------------------$ sample-algebraicfield01.rb
--------------------^ sample-algebraicfield02.rb require "residue-class-ring" require "polynomial" require "rational" F = AlgebraicExtentionField(Rational, "x") {|x| x**2 + x + 1} x = F.var p( (x + 1)**100 ) p( (x-1)** 3 / (x**2 - 1) ) H = AlgebraicExtentionField(F, "y") {|y| y**5 + x*y + 1} y = H.var p( 1/(x + y + 1)**7 ) --------------------$ sample-algebraicfield02.rb
--------------------^ sample-quotientfield01.rb require "localized-ring" require "rational" Q = LocalizedRing(Integer) a = Q.new(3, 5) b = Q.new(5, 3) p [a + b, a - b, a * b, a / b, a + 3, 1 + a] #=> [34/15, -16/15, 15/15, 9/25, 18/5, 8/5] --------------------$ sample-quotientfield01.rb
--------------------^ sample-quotientfield02.rb require "localized-ring" require "polynomial" require "residue-class-ring" F13 = ResidueClassRing(Integer, 13) P = Polynomial(F13, "x") Q = LocalizedRing(P) x = Q[P.var] p ( 1 / (x**2 - 1) - 1 / (x**3 - 1) ) #This is equivalent to the following F = RationalFunctionField(F13, "x") x = F.var p ( 1 / (x**2 - 1) - 1 / (x**3 - 1) ) --------------------$ sample-quotientfield02.rb
--------------------^ sample-quotientfield03.rb require "localized-ring" require "polynomial" require "residue-class-ring" F13 = ResidueClassRing(Integer, 13) F = AlgebraicExtentionField(F13, "a") {|a| a**2 - 2} a = F.var RF = RationalFunctionField(F, "x") x = RF.var p( (a/4*x + RF.unity/2)/(x**2 + a*x + 1) + (-a/4*x + RF.unity/2)/(x**2 - a*x + 1) ) #=> 1/(x**4 + 1) --------------------$ sample-quotientfield03.rb
--------------------^ sample-quotientfield04.rb require "localized-ring" require "polynomial" require "residue-class-ring" F13 = ResidueClassRing(Integer, 13) F = RationalFunctionField(F13, "x") x = F.var AF = AlgebraicExtentionField(F, "a") {|a| a**2 - 2*x} a = AF.var p( (a/4*x + AF.unity/2)/(x**2 + a*x + 1) + (-a/4*x + AF.unity/2)/(x**2 - a*x + 1) ) #=> (-x^3 + x^2 + 1)/(x^4 + 11x^3 + 2x^2 + 1) --------------------$ sample-quotientfield04.rb
--------------------^ sample-gaussian-elimination01.rb require "matrix-algebra" require "mathn" M = MatrixAlgebra(Rational, 5, 4) a = M.matrix{|i, j| i + j} a.display #=> #[0, 1, 2, 3] #[1, 2, 3, 4] #[2, 3, 4, 5] #[3, 4, 5, 6] #[4, 5, 6, 7] a.kernel_basis.each do |v| puts "a * #{v} = #{a * v}" #=> a * [1, -2, 1, 0] = [0, 0, 0, 0, 0] #=> a * [2, -3, 0, 1] = [0, 0, 0, 0, 0] end --------------------$ sample-gaussian-elimination01.rb
--------------------^ sample-diagonalization01.rb require "linear-algebra" require "rational" class Rational < Numeric def inspect; to_s; end end M = SquareMatrix(Rational, 3) a = M[[1,-1,-1], [-1,1,-1], [2,1,-1]] puts "A = "; a.display; puts #A = # 1, -1, -1 # -1, 1, -1 # 2, 1, -1 extfield, roots, tmatrix, evalues, evectors, espaces, chpoly, facts = a.diagonalize puts "Charactoristic Poly.: #{chpoly} => #{facts}" #Charactoristic Poly.: t^3 - t^2 + t - 6 => (t - 2)(t^2 + t + 3) puts "Algebraic Numbers:" roots.each do |po, rs| puts "#{rs.join(', ')} : roots of #{po} == 0" end puts #Algebraic Numbers: #a, -a - 1 : roots of t^2 + t + 3 == 0 puts "EigenSpaces: " evalues.uniq.each do |ev| puts "W_{#{ev}} = <#{espaces[ev].join(', ')}>" end puts #EigenSpaces: #W_{2} = <[4, -5, 1]> #W_{a} = <[1/3a + 1/3, 1/3a + 1/3, 1]> #W_{-a - 1} = <[-1/3a, -1/3a, 1]> puts "P = "; tmatrix.display; puts puts "P^-1 * A * P = "; (tmatrix.inverse * a * tmatrix).display; puts #P = # 4, 1/3a + 1/3, -1/3a # -5, 1/3a + 1/3, -1/3a # 1, 1, 1 # #P^-1 * A * P = # 2, 0, 0 # 0, a, 0 # 0, 0, -a - 1 --------------------$ sample-diagonalization01.rb
--------------------^ sample-cayleyhamilton01.rb require "matrix-algebra" require "m-polynomial" require "polynomial" n = 4 R = MPolynomial(Integer) MR = SquareMatrix(R, n) m = MR.matrix{|i, j| R.var("x#{i}#{j}") } Rx = Polynomial(R, "x") ch = m.char_polynomial(Rx) p ch.evaluate(m) #=> 0 --------------------$ sample-cayleyhamilton01.rb
--------------------^ sample-groebner02.rb require "groebner-basis" require "rational" P = MPolynomial(Rational) x, y, z = P.vars "xyz" f1 = x**2 + y**2 + z**2 -1 f2 = x**2 + z**2 - y f3 = x - z coeff, basis = Groebner.basis_coeff([f1, f2, f3]) basis.each_with_index do |b, i| p [coeff[i].inner_product([f1, f2, f3]), b] p coeff[i].inner_product([f1, f2, f3]) == b #=> true end --------------------$ sample-groebner02.rb
--------------------^ sample-groebner03.rb require "groebner-basis" require "residue-class-ring" F5 = ResidueClassRing(Integer, 2) F = AlgebraicExtentionField(F5, "a") {|a| a**3 + a + 1} a = F.var P = MPolynomial(F) x, y, z = P.vars("xyz") f1 = x + y**2 + z**2 - 1 f2 = x**2 + z**2 - y * a f3 = x - z - a f = x**3 + y**3 + z**3 q, r = f.divmod_s(f1, f2, f3) p f == q.inner_product([f1, f2, f3]) + r #=> true --------------------$ sample-groebner03.rb
--------------------^ sample-factorize01.rb require "polynomial" require "polynomial-factor" P = Polynomial(Integer, "x") x = P.var f = 8*x**7 - 20*x**6 + 6*x**5 - 11*x**4 + 44*x**3 - 9*x**2 - 27 p f.factorize #=> (2x - 3)^3(x^2 + x + 1)^2 --------------------$ sample-factorize01.rb
--------------------^ sample-factorize02.rb require "polynomial" require "polynomial-factor" require "residue-class-ring" Z7 = ResidueClassRing(Integer, 7) P = Polynomial(Z7, "x") x = P.var f = 8*x**7 - 20*x**6 + 6*x**5 - 11*x**4 + 44*x**3 - 9*x**2 - 27 p f.factorize #=> (x + 5)^2(x + 3)^2(x + 2)^3 --------------------$ sample-factorize02.rb
--------------------^ sample-factorize03.rb require "polynomial" require "polynomial-factor" require "residue-class-ring" require "rational" A = AlgebraicExtentionField(Rational, "a") {|a| a**2 + a + 1} a = A.var P = Polynomial(A, "x") x = P.var f = x**4 + (2*a + 1)*x**3 + 3*a*x**2 + (-3*a - 5)*x - a + 1 p f.factorize #=> (x + a)^3(x - a + 1) --------------------$ sample-factorize03.rb
--------------------^ sample-factorize04.rb require "polynomial" require "polynomial-factor" require "residue-class-ring" require "rational" A = AlgebraicExtentionField(Rational, "a") {|a| a**2 - 2} B = AlgebraicExtentionField(A, "b"){|b| b**2 + 1} P = Polynomial(B, "x") x = P.var f = x**4 + 1 p f.factorize #=> (x - 1/2ab - 1/2a)(x + 1/2ab - 1/2a)(x + 1/2ab + 1/2a)(x - 1/2ab + 1/2a) --------------------$ sample-factorize04.rb