[index]
Exercise
Finite Sets
Sets
# sample-set01.rb require "algebra" #intersection p Set[0, 1, 2, 4] & Set[1, 3, 5] == Set[1] p Set[0, 1, 2, 4] & Set[7, 3, 5] == Set.phi #union p Set[0, 1, 2, 4] | Set[1, 3, 5] == Set[0, 1, 2, 3, 4, 5] #membership p Set[1, 3, 2].has? 1 #inclusion p Set[3, 2, 1, 3] < Set[3, 1, 4, 2, 0]
Maps
# sample-map01.rb
require "algebra"
a = Map[0=>2, 1=>2, 2=>0]
b = Map[0=>1, 1=>1, 2=>1]
p a * b #=> {0=>2, 1=>2, 2=>2}
Groups
# sample-group01.rb require "algebra" e = Permutation[0, 1, 2, 3, 4] a = Permutation[1, 0, 3, 4, 2] b = Permutation[0, 2, 1, 3, 4] p a * b #=> [2, 0, 3, 4, 1] g = Group.new(e, a, b) g.complete! p g == PermutationGroup.symmetric(5) #=> true
Calculation of polynomials
# sample-polynomial01.rb require "algebra" P = Polynomial.create(Integer, "x") x = P.var p((x + 1)**100) #=> x^100 + 100x^99 + ... + 100x + 1
Calculation of multi-variate polynomials
# sample-polynomial02.rb require "algebra" P = Polynomial(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
Calculation of multi-variate polynomials (2)
# sample-m-polynomial01.rb
require "algebra"
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
Remainder of the division of a polynomial by polynomials
# sample-divmod01.rb
require "algebra"
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
P.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]
Groebner basis
# sample-groebner01.rb
require "algebra"
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]
Prime field
# sample-primefield01.rb
require "algebra"
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]
Algebraic field
# sample-algebraicfield01.rb
require "algebra"
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
This can be done as following
# sample-algebraicfield02.rb
require "algebra"
F = AlgebraicExtensionField(Rational, "x") {|x| x**2 + x + 1}
x = F.var
p( (x + 1)**100 )
p( (x-1)** 3 / (x**2 - 1) )
H = AlgebraicExtensionField(F, "y") {|y| y**5 + x*y + 1}
y = H.var
p( 1/(x + y + 1)**7 )
Quotient fields
By taking the quotient field of the ring of Integer, rational numbers are obtained.
# sample-quotientfield01.rb require "algebra" 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]
Rational function field
# sample-quotientfield02.rb require "algebra" 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) )
Rational function field over the algebraic extension field
# sample-quotientfield03.rb
require "algebra"
F13 = ResidueClassRing(Integer, 13)
F = AlgebraicExtensionField(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)
Algebraic function field
# sample-quotientfield04.rb
require "algebra"
F13 = ResidueClassRing(Integer, 13)
F = RationalFunctionField(F13, "x")
x = F.var
AF = AlgebraicExtensionField(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)
Linear Algebra
Linear equations
# sample-gaussian-elimination01.rb
require "algebra"
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
Diagonalization of Square Matrix
# sample-diagonalization01.rb
require "algebra"
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, addelms, 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
Elementary Divisors of Matrix
# sample-elementary-divisor01.rb require "algebra" M = SquareMatrix(Rational, 4) a = M[ [2, 0, 0, 0], [0, 2, 0, 0], [0, 0, 2, 0], [5, 0, 0, 2] ] P = Polynomial(Rational, "x") MP = SquareMatrix(P, 4) ac = a.char_matrix(MP) ac.display; puts #=> #x - 2, 0, 0, 0 # 0, x - 2, 0, 0 # 0, 0, x - 2, 0 # -5, 0, 0, x - 2 p ac.elementary_divisor #=> [1, x - 2, x - 2, x^2 - 4x + 4] require "matrix-algebra-triplet" at = ac.to_triplet.e_diagonalize at.body.display; puts #=> # 1, 0, 0, 0 # 0, x - 2, 0, 0 # 0, 0, x - 2, 0 # 0, 0, 0, x^2 - 4x + 4 at.left.display; puts #=> # 0, 0, 0, -1/5 # 0, 1, 0, 0 # 0, 0, 1, 0 # 5, 0, 0, x - 2 at.right.display; puts #=> # 1, 0, 0, 1/5x - 2/5 # 0, 1, 0, 0 # 0, 0, 1, 0 # 0, 0, 0, 1 p at.left * ac * at.right == at.body #=> true
Jordan Canonical Form of Matrix
# sample-jordan-form01.rb
require "algebra"
M4 = SquareMatrix(Rational, 4)
m = M4[
[-1, 1, 2, -1],
[-5, 3, 4, -2],
[3, -1, 0, 1],
[5, -2, -2, 3]
]
m.jordan_form.display; #=>
# 2, 0, 0, 0
# 0, 1, 1, 0
# 0, 0, 1, 1
# 0, 0, 0, 1
puts
#-----------------------------------
m = M4[
[3, 1, -1, 1],
[-3, -1, 3, -1],
[-2, -2, 0, 0],
[0, 0, -4, 2]
]
jf, pt, qt, field, modulus = m.jordan_form_info
p modulus #=> [a^2 + 4]
jf.display; puts #=>
# 2, 1, 0, 0
# 0, 2, 0, 0
# 0, 0, a, 0
# 0, 0, 0, -a
m = m.convert_to(jf.type)
p jf == pt * m * qt #=> true
#-----------------------------------
m = M4[
[-1, 1, 2, -1],
[-5, 3, 4, -2],
[3, -1, 0, 1],
[5, -2, -2, 0]
]
jf, pt, qt, field, modulus = m.jordan_form_info
p modulus #=> [a^3 + 3a - 1, b^2 + ab + a^2 + 3]
jf.display; puts #=>
# 2, 0, 0, 0
# 0, a, 0, 0
# 0, 0, b, 0
# 0, 0, 0, -b - a
m = m.convert_to(jf.type)
p jf == pt * m * qt #=> true
The proofs of the theorem of Cayley-Hamilton
# sample-cayleyhamilton01.rb
require "algebra"
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
The expression of Groebner basis by original generators
# sample-groebner02.rb require "algebra" 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
The quotients and the remainder by arbitrary basis
# sample-groebner03.rb
require "algebra"
F5 = ResidueClassRing(Integer, 2)
F = AlgebraicExtensionField(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
Factorization
Factorization of Integer coefficient polynomial
# sample-factorize01.rb require "algebra" 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
Factorization of Zp coefficient polynomial
# sample-factorize02.rb require "algebra" 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
Factorization of the algebraic extension of Rational polynomial
# sample-factorize03.rb
require "algebra"
A = AlgebraicExtensionField(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)
Factorization of the algebraic extension of the algebraic extension of Rational
# sample-factorize04.rb
require "algebra"
A = AlgebraicExtensionField(Rational, "a") {|a| a**2 - 2}
B = AlgebraicExtensionField(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)
Factorization of x^4 + 10x^2 + 1
# sample-factorize05.rb
require "algebra"
def show(f, mod = 0)
if mod > 0
zp = ResidueClassRing(Integer, mod)
pzp = Polynomial(zp, f.variable)
f = f.convert_to(pzp)
end
fact = f.factorize
printf "mod %2d: %-15s => %s\n", mod, f, fact
end
Px = Polynomial(Integer, "x")
x = Px.var
f = x**4 + 10*x**2 + 1
#f = x**4 - 10*x**2 + 1
show(f)
Primes.new.each do |mod|
break if mod > 100
show(f, mod)
end
#mod 0: x^4 + 10x^2 + 1 => x^4 + 10x^2 + 1
#mod 2: x^4 + 1 => (x + 1)^4
#mod 3: x^4 + x^2 + 1 => (x + 2)^2(x + 1)^2
#mod 5: x^4 + 1 => (x^2 + 3)(x^2 + 2)
#mod 7: x^4 + 3x^2 + 1 => (x^2 + 4x + 6)(x^2 + 3x + 6)
#mod 11: x^4 - x^2 + 1 => (x^2 + 5x + 1)(x^2 + 6x + 1)
#mod 13: x^4 + 10x^2 + 1 => (x^2 - x + 12)(x^2 + x + 12)
#mod 17: x^4 + 10x^2 + 1 => (x^2 + 3x + 1)(x^2 + 14x + 1)
#mod 19: x^4 + 10x^2 + 1 => (x + 17)(x + 10)(x + 9)(x + 2)
#mod 23: x^4 + 10x^2 + 1 => (x^2 + 6)(x^2 + 4)
#mod 29: x^4 + 10x^2 + 1 => (x^2 + 21)(x^2 + 18)
#mod 31: x^4 + 10x^2 + 1 => (x^2 + 22x + 30)(x^2 + 9x + 30)
#mod 37: x^4 + 10x^2 + 1 => (x^2 + 32x + 36)(x^2 + 5x + 36)
#mod 41: x^4 + 10x^2 + 1 => (x^2 + 19x + 1)(x^2 + 22x + 1)
#mod 43: x^4 + 10x^2 + 1 => (x + 40)(x + 29)(x + 14)(x + 3)
#mod 47: x^4 + 10x^2 + 1 => (x^2 + 32)(x^2 + 25)
#mod 53: x^4 + 10x^2 + 1 => (x^2 + 41)(x^2 + 22)
#mod 59: x^4 + 10x^2 + 1 => (x^2 + 13x + 1)(x^2 + 46x + 1)
#mod 61: x^4 + 10x^2 + 1 => (x^2 + 54x + 60)(x^2 + 7x + 60)
#mod 67: x^4 + 10x^2 + 1 => (x + 55)(x + 39)(x + 28)(x + 12)
#mod 71: x^4 + 10x^2 + 1 => (x^2 + 43)(x^2 + 38)
#mod 73: x^4 + 10x^2 + 1 => (x + 68)(x + 44)(x + 29)(x + 5)
#mod 79: x^4 + 10x^2 + 1 => (x^2 + 64x + 78)(x^2 + 15x + 78)
#mod 83: x^4 + 10x^2 + 1 => (x^2 + 18x + 1)(x^2 + 65x + 1)
#mod 89: x^4 + 10x^2 + 1 => (x^2 + 9x + 1)(x^2 + 80x + 1)
#mod 97: x^4 + 10x^2 + 1 => (x + 88)(x + 54)(x + 43)(x + 9)
Factorization of Integer or Rational coefficient multi-variate polynomial
# sample-m-factorize01.rb
require "algebra"
P = MPolynomial(Integer)
x, y, z = P.vars("xyz")
f = x**3 + y**3 + z**3 - 3*x*y*z
p f.factorize #=> (x + y + z)(x^2 - xy - xz + y^2 - yz + z^2)
PQ = MPolynomial(Rational)
x, y, z = PQ.vars("xyz")
f = x**3 + y**3/8 + z**3 - 3*x*y*z/2
p f.factorize #=> (1/8)(2x + y + 2z)(4x^2 - 2xy - 4xz + y^2 - 2yz + 4z^2)
Factorization of Zp coefficient multi-variate polynomial
# sample-m-factorize02.rb
require "algebra"
Z7 = ResidueClassRing(Integer, 7)
P = MPolynomial(Z7)
x, y, z = P.vars("xyz")
f = x**3 + y**3 + z**3 - 3*x*y*z
p f.factorize #=> (x + 4y + 2z)(x + 2y + 4z)(x + y + z)
Algebraic Equations
Minimal Polynomial
# sample-algebraic-equation01.rb
require "algebra"
PQ = MPolynomial(Rational)
a, b, c = PQ.vars("abc")
p AlgebraicEquation.minimal_polynomial(a + b + c, a**2-2, b**2-3, c**2-5)
#=> x^8 - 40x^6 + 352x^4 - 960x^2 + 576
Minimal Splitting Field
# sample-splitting-field01.rb
require "algebra"
PQ = Polynomial(Rational, "x")
x = PQ.var
f = x**4 + 2
p f #=> x^4 + 2
field, modulus, facts = f.decompose
p modulus #=> [a^4 + 2, b^2 + a^2]
p facts #=> (x - a)(x + a)(x - b)(x + b)
fp = Polynomial(field, "x")
x = fp.var
facts = Factors.new(facts.collect{|g, n| [g.evaluate(x), n]})
p facts.pi == f.convert_to(fp) #=> true
Galois Group of polynomials
# sample-galois-group01.rb require "rational" require "polynomial" P = Algebra.Polynomial(Rational, "x") x = P.var (x**3 - 3*x + 1).galois_group.each do |g| p g end #=> [0, 1, 2] # [1, 2, 0] # [2, 0, 1]] (x**3 - x + 1).galois_group.each do |g| p g end #=> [0, 1, 2] # [1, 0, 2] # [2, 0, 1] # [0, 2, 1] # [1, 2, 0] # [2, 1, 0]