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## -*- encoding: utf-8 -*- """ Doctests from French Sage book Test file for chapter "Domaines de calcul" ("Computation Domains")
Tests extracted from ./domaines.tex.
Sage example in ./domaines.tex, line 6::
sage: x = var('x')
Sage example in ./domaines.tex, line 38::
sage: o = 12/35 sage: type(o) <type 'sage.rings.rational.Rational'>
Sage example in ./domaines.tex, line 45::
sage: type(12/35) <type 'sage.rings.rational.Rational'>
Sage example in ./domaines.tex, line 77::
sage: o = 720 sage: o.factor() 2^4 * 3^2 * 5
Sage example in ./domaines.tex, line 85::
sage: type(o).factor(o) 2^4 * 3^2 * 5
Sage example in ./domaines.tex, line 95::
sage: 720.factor() 2^4 * 3^2 * 5
Sage example in ./domaines.tex, line 102::
sage: o = 720 / 133 sage: o.numerator().factor() 2^4 * 3^2 * 5
Sage example in ./domaines.tex, line 140::
sage: 3 * 7 21
Sage example in ./domaines.tex, line 146::
sage: (2/3) * (6/5) 4/5
Sage example in ./domaines.tex, line 151::
sage: (1 + I) * (1 - I) 2
Sage example in ./domaines.tex, line 156::
sage: (x + 2) * (x + 1) (x + 2)*(x + 1) sage: (x + 1) * (x + 2) (x + 2)*(x + 1)
Sage example in ./domaines.tex, line 177::
sage: def puissance_quatre(a): ....: a = a * a ....: a = a * a ....: return a
Sage example in ./domaines.tex, line 185::
sage: puissance_quatre(2) 16 sage: puissance_quatre(3/2) 81/16 sage: puissance_quatre(I) 1 sage: puissance_quatre(x+1) (x + 1)^4 sage: M = matrix([[0,-1],[1,0]]); M [ 0 -1] [ 1 0] sage: puissance_quatre(M) [1 0] [0 1]
Sage example in ./domaines.tex, line 215::
sage: t = type(5/1); t <type 'sage.rings.rational.Rational'> sage: t == type(5) False
Sage example in ./domaines.tex, line 288::
sage: a = 5; a 5 sage: a.is_unit() False
Sage example in ./domaines.tex, line 295::
sage: a = 5/1; a 5 sage: a.is_unit() True
Sage example in ./domaines.tex, line 311::
sage: parent(5) Integer Ring sage: parent(5/1) Rational Field
Sage example in ./domaines.tex, line 318::
sage: ZZ Integer Ring sage: QQ Rational Field
Sage example in ./domaines.tex, line 326::
sage: QQ(5).parent() Rational Field sage: ZZ(5/1).parent() Integer Ring sage: ZZ(1/5) Traceback (most recent call last): ... TypeError: no conversion of this rational to integer
Sage example in ./domaines.tex, line 340::
sage: ZZ(1), QQ(1), RR(1), CC(1) (1, 1, 1.00000000000000, 1.00000000000000)
Sage example in ./domaines.tex, line 355::
sage: cartesian_product([QQ, QQ]) The Cartesian product of (Rational Field, Rational Field)
Sage example in ./domaines.tex, line 360::
sage: ZZ.fraction_field() Rational Field
Sage example in ./domaines.tex, line 365::
sage: ZZ['x'] Univariate Polynomial Ring in x over Integer Ring
Sage example in ./domaines.tex, line 373::
sage: Z5 = GF(5); Z5 Finite Field of size 5 sage: P = Z5['x']; P Univariate Polynomial Ring in x over Finite Field of size 5 sage: M = MatrixSpace(P, 3, 3); M Full MatrixSpace of 3 by 3 dense matrices over Univariate Polynomial Ring in x over Finite Field of size 5
Sage example in ./domaines.tex, line 383::
sage: M.random_element() # random [2*x^2 + 3*x + 4 4*x^2 + 2*x + 2 4*x^2 + 2*x] [ 3*x 2*x^2 + x + 3 3*x^2 + 4*x] [ 4*x^2 + 3 3*x^2 + 2*x + 4 2*x + 4]
Sage example in ./domaines.tex, line 415::
sage: QQ.category() Join of Category of number fields and Category of quotient fields and Category of metric spaces
Sage example in ./domaines.tex, line 421::
sage: QQ in Fields() True
Sage example in ./domaines.tex, line 427::
sage: QQ in CommutativeAdditiveGroups() True
Sage example in ./domaines.tex, line 432::
sage: QQ['x'] in EuclideanDomains() True
Sage example in ./domaines.tex, line 514::
sage: 5.parent() Integer Ring
Sage example in ./domaines.tex, line 521::
sage: type(factor(4)) <class 'sage.structure.factorization_integer.IntegerFactorization'>
Sage example in ./domaines.tex, line 532::
sage: int(5) 5 sage: type(int(5)) <... 'int'>
Sage example in ./domaines.tex, line 539::
sage: Integer(5) 5 sage: type(Integer(5)) <type 'sage.rings.integer.Integer'>
Sage example in ./domaines.tex, line 551::
sage: factorial(99) / factorial(100) - 1 / 50 -1/100
Sage example in ./domaines.tex, line 574::
sage: 72/53-5/3*2.7 -3.14150943396227
Sage example in ./domaines.tex, line 580::
sage: cos(1), cos(1.) (cos(1), 0.540302305868140)
Sage example in ./domaines.tex, line 589::
sage: pi.n(digits=50) # N(pi,digits=10^6) aussi possible 3.1415926535897932384626433832795028841971693993751
Sage example in ./domaines.tex, line 600::
sage: z = CC(1,2); z.arg() 1.10714871779409
Sage example in ./domaines.tex, line 608::
sage: I.parent() Symbolic Ring
Sage example in ./domaines.tex, line 613::
sage: (1.+2.*I).parent() Symbolic Ring sage: CC(1.+2.*I).parent() Complex Field with 53 bits of precision
Sage example in ./domaines.tex, line 623::
sage: z = 3 * exp(I*pi/4) sage: z.real(), z.imag(), z.abs().canonicalize_radical() (3/2*sqrt(2), 3/2*sqrt(2), 3)
Sage example in ./domaines.tex, line 679::
sage: x, y = var('x, y') sage: bool( (x-y)*(x+y) == x^2-y^2 ) True
Sage example in ./domaines.tex, line 697::
sage: Z4 = IntegerModRing(4); Z4 Ring of integers modulo 4 sage: m = Z4(7); m 3
Sage example in ./domaines.tex, line 706::
sage: 3 * m + 1 2
Sage example in ./domaines.tex, line 712::
sage: Z3 = GF(3); Z3 Finite Field of size 3
Sage example in ./domaines.tex, line 740::
sage: a = matrix(QQ, [[1,2,3],[2,4,8],[3,9,27]]) sage: (a^2 + 1) * a^(-1) [ -5 13/2 7/3] [ 7 1 25/3] [ 2 19/2 27]
Sage example in ./domaines.tex, line 752::
sage: M = MatrixSpace(QQ,3,3) sage: M Full MatrixSpace of 3 by 3 dense matrices over Rational Field sage: a = M([[1,2,3],[2,4,8],[3,9,27]]) sage: (a^2 + 1) * a^(-1) [ -5 13/2 7/3] [ 7 1 25/3] [ 2 19/2 27]
Sage example in ./domaines.tex, line 771::
sage: P = ZZ['x']; P Univariate Polynomial Ring in x over Integer Ring sage: F = P.fraction_field(); F Fraction Field of Univariate Polynomial Ring in x over Integer Ring sage: p = P(x+1) * P(x); p x^2 + x sage: p + 1/p (x^4 + 2*x^3 + x^2 + 1)/(x^2 + x) sage: parent(p + 1/p) Fraction Field of Univariate Polynomial Ring in x over Integer Ring
Sage example in ./domaines.tex, line 826::
sage: k.<a> = NumberField(x^3 + x + 1); a^3; a^4+3*a -a - 1 -a^2 + 2*a
Sage example in ./domaines.tex, line 845::
sage: parent(sin(x)) Symbolic Ring
Sage example in ./domaines.tex, line 850::
sage: SR Symbolic Ring
Sage example in ./domaines.tex, line 855::
sage: SR.category() Category of commutative rings
Sage example in ./domaines.tex, line 884::
sage: R = QQ['x1,x2,x3,x4']; R Multivariate Polynomial Ring in x1, x2, x3, x4 over Rational Field sage: x1, x2, x3, x4 = R.gens()
Sage example in ./domaines.tex, line 890::
sage: x1 * (x2 - x3) x1*x2 - x1*x3
Sage example in ./domaines.tex, line 895::
sage: (x1+x2)*(x1-x2) - (x1^2 -x2^2) 0
Sage example in ./domaines.tex, line 902::
sage: prod( (a-b) for (a,b) in Subsets([x1,x2,x3,x4],2) ) x1^3*x2^2*x3 - x1^2*x2^3*x3 - x1^3*x2*x3^2 + x1*x2^3*x3^2 + x1^2*x2*x3^3 - x1*x2^2*x3^3 - x1^3*x2^2*x4 + x1^2*x2^3*x4 + x1^3*x3^2*x4 - x2^3*x3^2*x4 - x1^2*x3^3*x4 + x2^2*x3^3*x4 + x1^3*x2*x4^2 - x1*x2^3*x4^2 - x1^3*x3*x4^2 + x2^3*x3*x4^2 + x1*x3^3*x4^2 - x2*x3^3*x4^2 - x1^2*x2*x4^3 + x1*x2^2*x4^3 + x1^2*x3*x4^3 - x2^2*x3*x4^3 - x1*x3^2*x4^3 + x2*x3^2*x4^3
Sage example in ./domaines.tex, line 914::
# example slightly modified with respect to the book, since on some # machines we get a negative sign sage: x1, x2, x3, x4 = SR.var('x1, x2, x3, x4') sage: p = prod( (a-b) for (a,b) in Subsets([x1,x2,x3,x4],2) ) sage: bool(p == (x1 - x2)*(x1 - x3)*(x1 - x4)*(x2 - x3)*(x2 - x4)*(x3 - x4)) or bool(p == -(x1 - x2)*(x1 - x3)*(x1 - x4)*(x2 - x3)*(x2 - x4)*(x3 - x4)) True
Sage example in ./domaines.tex, line 938::
sage: x = var('x') sage: p = 54*x^4+36*x^3-102*x^2-72*x-12 sage: factor(p) 6*(x^2 - 2)*(3*x + 1)^2
Sage example in ./domaines.tex, line 963::
sage: R = ZZ['x']; R Univariate Polynomial Ring in x over Integer Ring
Sage example in ./domaines.tex, line 968::
sage: q = R(p); q 54*x^4 + 36*x^3 - 102*x^2 - 72*x - 12
Sage example in ./domaines.tex, line 974::
sage: parent(q) Univariate Polynomial Ring in x over Integer Ring
Sage example in ./domaines.tex, line 979::
sage: factor(q) 2 * 3 * (3*x + 1)^2 * (x^2 - 2)
Sage example in ./domaines.tex, line 985::
sage: R = QQ['x']; R Univariate Polynomial Ring in x over Rational Field sage: q = R(p); q 54*x^4 + 36*x^3 - 102*x^2 - 72*x - 12 sage: factor(q) (54) * (x + 1/3)^2 * (x^2 - 2)
Sage example in ./domaines.tex, line 1001::
sage: R = ComplexField(16)['x']; R Univariate Polynomial Ring in x over Complex Field with 16 bits of precision sage: q = R(p); q 54.00*x^4 + 36.00*x^3 - 102.0*x^2 - 72.00*x - 12.00 sage: factor(q) (54.00) * (x - 1.414) * (x + 0.3333)^2 * (x + 1.414)
Sage example in ./domaines.tex, line 1012::
sage: R = QQ[sqrt(2)]['x']; R Univariate Polynomial Ring in x over Number Field in sqrt2 with defining polynomial x^2 - 2 sage: q = R(p); q 54*x^4 + 36*x^3 - 102*x^2 - 72*x - 12 sage: factor(q) (54) * (x - sqrt2) * (x + sqrt2) * (x + 1/3)^2
Sage example in ./domaines.tex, line 1025::
sage: R = GF(5)['x']; R Univariate Polynomial Ring in x over Finite Field of size 5 sage: q = R(p); q 4*x^4 + x^3 + 3*x^2 + 3*x + 3 sage: factor(q) (4) * (x + 2)^2 * (x^2 + 3)
""" |