Fixed and Arbitrary Precision Numerical Fields - Sage

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Sage Reference Manual: Fixed and Arbitrary Precision Numerical Fields, Release 6.2Note: PARI is vastly more efficient at computing the Riemann zeta function. See the example below forhow to use it.EXAMPLES:sage: RDF(2).zeta()1.64493406685sage: RDF.pi()^2/61.64493406685sage: RDF(-2).zeta()-2.37378795339e-18sage: RDF(1).zeta()+infinity# slightly random-ish arch dependent outputsage.rings.real_double.RealDoubleField()Return the unique instance of the real double field.EXAMPLES:sage: RealDoubleField() is RealDoubleField()Trueclass sage.rings.real_double.RealDoubleField_classBases: sage.rings.ring.FieldAn approximation to the field of real numbers using double precision floating point numbers. Answers derivedfrom calculations in this approximation may differ from what they would be if those calculations were performedin the true field of real numbers. This is due to the rounding errors inherent to finite precision calculations.EXAMPLES:sage: RR == RDFFalsesage: RDF == RealDoubleField()Truesage: RDF(1)1.0sage: RDF(2/3)0.666666666667A TypeError is raised if the coercion doesn’t make sense:# RDF is the shorthandsage: RDF(QQ[’x’].0)Traceback (most recent call last):...TypeError: cannot coerce nonconstant polynomial to floatsage: RDF(QQ[’x’](3))3.0One can convert back and forth between double precision real numbers and higher-precision ones, though ofcourse there may be loss of precision:sage: a = RealField(200)(2).sqrt(); a1.4142135623730950488016887242096980785696718753769480731767sage: b = RDF(a); b1.41421356237sage: a.parent()(b)1.414213562373095145474621858738828450441360473632812500000018 Chapter 1. Double Precision Real Numbers
Sage Reference Manual: Fixed and Arbitrary Precision Numerical Fields, Release 6.2sage: a.parent()(b) == bTruesage: b == RR(a)TrueNaN()Return Not-a-Number NaN.EXAMPLES:sage: RDF.NaN()NaNalgebraic_closure()Return the algebraic closure of self, i.e., the complex double field.EXAMPLES:sage: RDF.algebraic_closure()Complex Double Fieldcharacteristic()Returns 0, since the field of real numbers has characteristic 0.EXAMPLES:sage: RDF.characteristic()0complex_field()Return the complex field with the same precision as self, i.e., the complex double field.EXAMPLES:sage: RDF.complex_field()Complex Double Fieldconstruction()Returns the functorial construction of self, namely, completion of the rational numbers with respect tothe prime at ∞.Also preserves other information that makes this field unique (i.e. the Real Double Field).EXAMPLES:sage: c, S = RDF.construction(); SRational Fieldsage: RDF == c(S)Trueeuler_constant()Return Euler’s gamma constant to double precision.EXAMPLES:sage: RDF.euler_constant()0.577215664902factorial(n)Return the factorial of the integer n as a real number.EXAMPLES:19
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CHAPTERFOURFIELD OF ARBITRARY PRECI
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CHAPTERFIVEFILE:SAGE/RINGS/REAL_INT
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CHAPTERSIXTHESE CLASSES ARE VERY LA
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CHAPTERTHIRTEENINDICES AND TABLES
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PYTHON MODULE INDEXrsage.rings.comp
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INDEXAabs() (sage.rings.complex_dou
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Fixed and Arbitrary Precision Numerical Fields - Sage

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