Double precision floating point complex numbers¶
Sage supports arithmetic using double-precision complex numbers. A
double-precision complex number is a complex number x + I*y with
\(x\), \(y\) 64-bit (8 byte) floating point numbers (double precision).
The field ComplexDoubleField implements the field
of all double-precision complex numbers. You can refer to this
field by the shorthand CDF. Elements of this field are of type
ComplexDoubleElement. If \(x\) and \(y\) are coercible to
doubles, you can create a complex double element using
ComplexDoubleElement(x,y). You can coerce more
general objects \(z\) to complex doubles by typing either
ComplexDoubleField(x) or CDF(x).
EXAMPLES:
sage: ComplexDoubleField()
Complex Double Field
sage: CDF
Complex Double Field
sage: type(CDF.0)
<class 'sage.rings.complex_double.ComplexDoubleElement'>
sage: ComplexDoubleElement(sqrt(2), 3) # needs sage.symbolic
1.4142135623730951 + 3.0*I
sage: parent(CDF(-2))
Complex Double Field
>>> from sage.all import *
>>> ComplexDoubleField()
Complex Double Field
>>> CDF
Complex Double Field
>>> type(CDF.gen(0))
<class 'sage.rings.complex_double.ComplexDoubleElement'>
>>> ComplexDoubleElement(sqrt(Integer(2)), Integer(3)) # needs sage.symbolic
1.4142135623730951 + 3.0*I
>>> parent(CDF(-Integer(2)))
Complex Double Field
ComplexDoubleField() CDF type(CDF.0) ComplexDoubleElement(sqrt(2), 3) # needs sage.symbolic parent(CDF(-2))
sage: CC == CDF
False
sage: CDF is ComplexDoubleField() # CDF is the shorthand
True
sage: CDF == ComplexDoubleField()
True
>>> from sage.all import *
>>> CC == CDF
False
>>> CDF is ComplexDoubleField() # CDF is the shorthand
True
>>> CDF == ComplexDoubleField()
True
CC == CDF CDF is ComplexDoubleField() # CDF is the shorthand CDF == ComplexDoubleField()
>>> from sage.all import *
>>> CC == CDF
False
>>> CDF is ComplexDoubleField() # CDF is the shorthand
True
>>> CDF == ComplexDoubleField()
True
CC == CDF CDF is ComplexDoubleField() # CDF is the shorthand CDF == ComplexDoubleField()
The underlying arithmetic of complex numbers is implemented using functions and macros in GSL (the GNU Scientific Library), and should be very fast. Also, all standard complex trig functions, log, exponents, etc., are implemented using GSL, and are also robust and fast. Several other special functions, e.g. eta, gamma, incomplete gamma, etc., are implemented using the PARI C library.
AUTHORS:
William Stein (2006-09): first version
Travis Scrimshaw (2012-10-18): Added doctests to get full coverage
Jeroen Demeyer (2013-02-27): fixed all PARI calls (Issue #14082)
Vincent Klein (2017-11-15) : add __mpc__() to class ComplexDoubleElement. ComplexDoubleElement constructor support and gmpy2.mpc parameter.
- class sage.rings.complex_double.ComplexDoubleElement[source]¶
Bases:
FieldElementAn approximation to a complex number using double precision floating point numbers. Answers derived from calculations with such approximations may differ from what they would be if those calculations were performed with true complex numbers. This is due to the rounding errors inherent to finite precision calculations.
- abs()[source]¶
This function returns the magnitude \(|z|\) of the complex number \(z\).
See also
EXAMPLES:
sage: CDF(2,3).abs() 3.605551275463989
>>> from sage.all import * >>> CDF(Integer(2),Integer(3)).abs() 3.605551275463989
CDF(2,3).abs()
- abs2()[source]¶
This function returns the squared magnitude \(|z|^2\) of the complex number \(z\), otherwise known as the complex norm.
See also
EXAMPLES:
sage: CDF(2,3).abs2() 13.0
>>> from sage.all import * >>> CDF(Integer(2),Integer(3)).abs2() 13.0
CDF(2,3).abs2()
- agm(right, algorithm='optimal')[source]¶
Return the Arithmetic-Geometric Mean (AGM) of
selfandright.INPUT:
right– complex; another complex numberalgorithm– string (default:'optimal'); the algorithm to use (see below)
OUTPUT:
(complex) A value of the AGM of
selfandright. Note that this is a multi-valued function, and the algorithm used affects the value returned, as follows:'pari': Call the pari:agm function from the pari library.'optimal': Use the AGM sequence such that at each stage \((a,b)\) is replaced by \((a_1,b_1)=((a+b)/2,\pm\sqrt{ab})\) where the sign is chosen so that \(|a_1-b_1| \leq |a_1+b_1|\), or equivalently \(\Re(b_1/a_1) \geq 0\). The resulting limit is maximal among all possible values.'principal': Use the AGM sequence such that at each stage \((a,b)\) is replaced by \((a_1,b_1)=((a+b)/2,\pm\sqrt{ab})\) where the sign is chosen so that \(\Re(b_1/a_1) \geq 0\) (the so-called principal branch of the square root).
See Wikipedia article Arithmetic-geometric mean
EXAMPLES:
sage: i = CDF(I) # needs sage.symbolic sage: (1+i).agm(2-i) # rel tol 1e-15 # needs sage.symbolic 1.6278054848727064 + 0.1368275483973686*I
>>> from sage.all import * >>> i = CDF(I) # needs sage.symbolic >>> (Integer(1)+i).agm(Integer(2)-i) # rel tol 1e-15 # needs sage.symbolic 1.6278054848727064 + 0.1368275483973686*I
i = CDF(I) # needs sage.symbolic (1+i).agm(2-i) # rel tol 1e-15 # needs sage.symbolic
An example to show that the returned value depends on the algorithm parameter:
sage: a = CDF(-0.95,-0.65) sage: b = CDF(0.683,0.747) sage: a.agm(b, algorithm='optimal') -0.3715916523517613 + 0.31989466020683*I sage: a.agm(b, algorithm='principal') # rel tol 1e-15 0.33817546298618006 - 0.013532696956540503*I sage: a.agm(b, algorithm='pari') -0.37159165235176134 + 0.31989466020683005*I
>>> from sage.all import * >>> a = CDF(-RealNumber('0.95'),-RealNumber('0.65')) >>> b = CDF(RealNumber('0.683'),RealNumber('0.747')) >>> a.agm(b, algorithm='optimal') -0.3715916523517613 + 0.31989466020683*I >>> a.agm(b, algorithm='principal') # rel tol 1e-15 0.33817546298618006 - 0.013532696956540503*I >>> a.agm(b, algorithm='pari') -0.37159165235176134 + 0.31989466020683005*I
a = CDF(-0.95,-0.65) b = CDF(0.683,0.747) a.agm(b, algorithm='optimal') a.agm(b, algorithm='principal') # rel tol 1e-15 a.agm(b, algorithm='pari')
Some degenerate cases:
sage: CDF(0).agm(a) 0.0 sage: a.agm(0) 0.0 sage: a.agm(-a) 0.0
>>> from sage.all import * >>> CDF(Integer(0)).agm(a) 0.0 >>> a.agm(Integer(0)) 0.0 >>> a.agm(-a) 0.0
CDF(0).agm(a) a.agm(0) a.agm(-a)
- algdep(n)[source]¶
Return a polynomial of degree at most \(n\) which is approximately satisfied by this complex number. Note that the returned polynomial need not be irreducible, and indeed usually won’t be if \(z\) is a good approximation to an algebraic number of degree less than \(n\).
ALGORITHM: Uses the PARI C-library algdep command.
EXAMPLES:
sage: z = (1/2)*(1 + RDF(sqrt(3)) * CDF.0); z # abs tol 1e-16 # needs sage.symbolic 0.5 + 0.8660254037844387*I sage: p = z.algdep(5); p # needs sage.libs.pari sage.symbolic x^2 - x + 1 sage: abs(z^2 - z + 1) < 1e-14 # needs sage.symbolic True
>>> from sage.all import * >>> z = (Integer(1)/Integer(2))*(Integer(1) + RDF(sqrt(Integer(3))) * CDF.gen(0)); z # abs tol 1e-16 # needs sage.symbolic 0.5 + 0.8660254037844387*I >>> p = z.algdep(Integer(5)); p # needs sage.libs.pari sage.symbolic x^2 - x + 1 >>> abs(z**Integer(2) - z + Integer(1)) < RealNumber('1e-14') # needs sage.symbolic True
z = (1/2)*(1 + RDF(sqrt(3)) * CDF.0); z # abs tol 1e-16 # needs sage.symbolic p = z.algdep(5); p # needs sage.libs.pari sage.symbolic abs(z^2 - z + 1) < 1e-14 # needs sage.symbolic
sage: CDF(0,2).algdep(10) # needs sage.libs.pari x^2 + 4 sage: CDF(1,5).algdep(2) # needs sage.libs.pari x^2 - 2*x + 26
>>> from sage.all import * >>> CDF(Integer(0),Integer(2)).algdep(Integer(10)) # needs sage.libs.pari x^2 + 4 >>> CDF(Integer(1),Integer(5)).algdep(Integer(2)) # needs sage.libs.pari x^2 - 2*x + 26
CDF(0,2).algdep(10) # needs sage.libs.pari CDF(1,5).algdep(2) # needs sage.libs.pari
>>> from sage.all import * >>> CDF(Integer(0),Integer(2)).algdep(Integer(10)) # needs sage.libs.pari x^2 + 4 >>> CDF(Integer(1),Integer(5)).algdep(Integer(2)) # needs sage.libs.pari x^2 - 2*x + 26
CDF(0,2).algdep(10) # needs sage.libs.pari CDF(1,5).algdep(2) # needs sage.libs.pari
- arccos()[source]¶
This function returns the complex arccosine of the complex number \(z\), \({\rm arccos}(z)\). The branch cuts are on the real axis, less than -1 and greater than 1.
EXAMPLES:
sage: CDF(1,1).arccos() 0.9045568943023814 - 1.0612750619050357*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arccos() 0.9045568943023814 - 1.0612750619050357*I
CDF(1,1).arccos()
- arccosh()[source]¶
This function returns the complex hyperbolic arccosine of the complex number \(z\), \({\rm arccosh}(z)\). The branch cut is on the real axis, less than 1.
EXAMPLES:
sage: CDF(1,1).arccosh() 1.0612750619050357 + 0.9045568943023814*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arccosh() 1.0612750619050357 + 0.9045568943023814*I
CDF(1,1).arccosh()
- arccot()[source]¶
This function returns the complex arccotangent of the complex number \(z\), \({\rm arccot}(z) = {\rm arctan}(1/z).\)
EXAMPLES:
sage: CDF(1,1).arccot() # rel tol 1e-15 0.5535743588970452 - 0.4023594781085251*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arccot() # rel tol 1e-15 0.5535743588970452 - 0.4023594781085251*I
CDF(1,1).arccot() # rel tol 1e-15
- arccoth()[source]¶
This function returns the complex hyperbolic arccotangent of the complex number \(z\), \({\rm arccoth}(z) = {\rm arctanh(1/z)}\).
EXAMPLES:
sage: CDF(1,1).arccoth() # rel tol 1e-15 0.4023594781085251 - 0.5535743588970452*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arccoth() # rel tol 1e-15 0.4023594781085251 - 0.5535743588970452*I
CDF(1,1).arccoth() # rel tol 1e-15
- arccsc()[source]¶
This function returns the complex arccosecant of the complex number \(z\), \({\rm arccsc}(z) = {\rm arcsin}(1/z)\).
EXAMPLES:
sage: CDF(1,1).arccsc() # rel tol 1e-15 0.45227844715119064 - 0.5306375309525178*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arccsc() # rel tol 1e-15 0.45227844715119064 - 0.5306375309525178*I
CDF(1,1).arccsc() # rel tol 1e-15
- arccsch()[source]¶
This function returns the complex hyperbolic arccosecant of the complex number \(z\), \({\rm arccsch}(z) = {\rm arcsin}(1/z)\).
EXAMPLES:
sage: CDF(1,1).arccsch() # rel tol 1e-15 0.5306375309525178 - 0.45227844715119064*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arccsch() # rel tol 1e-15 0.5306375309525178 - 0.45227844715119064*I
CDF(1,1).arccsch() # rel tol 1e-15
- arcsec()[source]¶
This function returns the complex arcsecant of the complex number \(z\), \({\rm arcsec}(z) = {\rm arccos}(1/z)\).
EXAMPLES:
sage: CDF(1,1).arcsec() # rel tol 1e-15 1.118517879643706 + 0.5306375309525178*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arcsec() # rel tol 1e-15 1.118517879643706 + 0.5306375309525178*I
CDF(1,1).arcsec() # rel tol 1e-15
- arcsech()[source]¶
This function returns the complex hyperbolic arcsecant of the complex number \(z\), \({\rm arcsech}(z) = {\rm arccosh}(1/z)\).
EXAMPLES:
sage: CDF(1,1).arcsech() # rel tol 1e-15 0.5306375309525176 - 1.118517879643706*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arcsech() # rel tol 1e-15 0.5306375309525176 - 1.118517879643706*I
CDF(1,1).arcsech() # rel tol 1e-15
- arcsin()[source]¶
This function returns the complex arcsine of the complex number \(z\), \({\rm arcsin}(z)\). The branch cuts are on the real axis, less than -1 and greater than 1.
EXAMPLES:
sage: CDF(1,1).arcsin() 0.6662394324925152 + 1.0612750619050357*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arcsin() 0.6662394324925152 + 1.0612750619050357*I
CDF(1,1).arcsin()
- arcsinh()[source]¶
This function returns the complex hyperbolic arcsine of the complex number \(z\), \({\rm arcsinh}(z)\). The branch cuts are on the imaginary axis, below \(-i\) and above \(i\).
EXAMPLES:
sage: CDF(1,1).arcsinh() 1.0612750619050357 + 0.6662394324925152*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arcsinh() 1.0612750619050357 + 0.6662394324925152*I
CDF(1,1).arcsinh()
- arctan()[source]¶
This function returns the complex arctangent of the complex number \(z\), \({\rm arctan}(z)\). The branch cuts are on the imaginary axis, below \(-i\) and above \(i\).
EXAMPLES:
sage: CDF(1,1).arctan() 1.0172219678978514 + 0.4023594781085251*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arctan() 1.0172219678978514 + 0.4023594781085251*I
CDF(1,1).arctan()
- arctanh()[source]¶
This function returns the complex hyperbolic arctangent of the complex number \(z\), \({\rm arctanh} (z)\). The branch cuts are on the real axis, less than -1 and greater than 1.
EXAMPLES:
sage: CDF(1,1).arctanh() 0.4023594781085251 + 1.0172219678978514*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).arctanh() 0.4023594781085251 + 1.0172219678978514*I
CDF(1,1).arctanh()
- arg()[source]¶
This function returns the argument of
self, the complex number \(z\), denoted by \(\arg(z)\), where \(-\pi < \arg(z) <= \pi\).EXAMPLES:
sage: CDF(1,0).arg() 0.0 sage: CDF(0,1).arg() 1.5707963267948966 sage: CDF(0,-1).arg() -1.5707963267948966 sage: CDF(-1,0).arg() 3.141592653589793
>>> from sage.all import * >>> CDF(Integer(1),Integer(0)).arg() 0.0 >>> CDF(Integer(0),Integer(1)).arg() 1.5707963267948966 >>> CDF(Integer(0),-Integer(1)).arg() -1.5707963267948966 >>> CDF(-Integer(1),Integer(0)).arg() 3.141592653589793
CDF(1,0).arg() CDF(0,1).arg() CDF(0,-1).arg() CDF(-1,0).arg()
- argument()[source]¶
This function returns the argument of the
self, the complex number \(z\), in the interval \(-\pi < arg(z) \leq \pi\).EXAMPLES:
sage: CDF(6).argument() 0.0 sage: CDF(i).argument() # needs sage.symbolic 1.5707963267948966 sage: CDF(-1).argument() 3.141592653589793 sage: CDF(-1 - 0.000001*i).argument() # needs sage.symbolic -3.1415916535897934
>>> from sage.all import * >>> CDF(Integer(6)).argument() 0.0 >>> CDF(i).argument() # needs sage.symbolic 1.5707963267948966 >>> CDF(-Integer(1)).argument() 3.141592653589793 >>> CDF(-Integer(1) - RealNumber('0.000001')*i).argument() # needs sage.symbolic -3.1415916535897934
CDF(6).argument() CDF(i).argument() # needs sage.symbolic CDF(-1).argument() CDF(-1 - 0.000001*i).argument() # needs sage.symbolic
- conj()[source]¶
This function returns the complex conjugate of the complex number \(z\):
\[\overline{z} = x - i y.\]EXAMPLES:
sage: z = CDF(2,3); z.conj() 2.0 - 3.0*I
>>> from sage.all import * >>> z = CDF(Integer(2),Integer(3)); z.conj() 2.0 - 3.0*I
z = CDF(2,3); z.conj()
- conjugate()[source]¶
This function returns the complex conjugate of the complex number \(z\):
\[\overline{z} = x - i y.\]EXAMPLES:
sage: z = CDF(2,3); z.conjugate() 2.0 - 3.0*I
>>> from sage.all import * >>> z = CDF(Integer(2),Integer(3)); z.conjugate() 2.0 - 3.0*I
z = CDF(2,3); z.conjugate()
- cos()[source]¶
This function returns the complex cosine of the complex number \(z\):
\[\cos(z) = \frac{e^{iz} + e^{-iz}}{2}\]EXAMPLES:
sage: CDF(1,1).cos() # abs tol 1e-16 0.8337300251311491 - 0.9888977057628651*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).cos() # abs tol 1e-16 0.8337300251311491 - 0.9888977057628651*I
CDF(1,1).cos() # abs tol 1e-16
- cosh()[source]¶
This function returns the complex hyperbolic cosine of the complex number \(z\):
\[\cosh(z) = \frac{e^z + e^{-z}}{2}.\]EXAMPLES:
sage: CDF(1,1).cosh() # abs tol 1e-16 0.8337300251311491 + 0.9888977057628651*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).cosh() # abs tol 1e-16 0.8337300251311491 + 0.9888977057628651*I
CDF(1,1).cosh() # abs tol 1e-16
- cot()[source]¶
This function returns the complex cotangent of the complex number \(z\):
\[\cot(z) = \frac{1}{\tan(z)}.\]EXAMPLES:
sage: CDF(1,1).cot() # rel tol 1e-15 0.21762156185440268 - 0.8680141428959249*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).cot() # rel tol 1e-15 0.21762156185440268 - 0.8680141428959249*I
CDF(1,1).cot() # rel tol 1e-15
- coth()[source]¶
This function returns the complex hyperbolic cotangent of the complex number \(z\):
\[\coth(z) = \frac{1}{\tanh(z)}.\]EXAMPLES:
sage: CDF(1,1).coth() # rel tol 1e-15 0.8680141428959249 - 0.21762156185440268*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).coth() # rel tol 1e-15 0.8680141428959249 - 0.21762156185440268*I
CDF(1,1).coth() # rel tol 1e-15
- csc()[source]¶
This function returns the complex cosecant of the complex number \(z\):
\[\csc(z) = \frac{1}{\sin(z)}.\]EXAMPLES:
sage: CDF(1,1).csc() # rel tol 1e-15 0.6215180171704284 - 0.30393100162842646*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).csc() # rel tol 1e-15 0.6215180171704284 - 0.30393100162842646*I
CDF(1,1).csc() # rel tol 1e-15
- csch()[source]¶
This function returns the complex hyperbolic cosecant of the complex number \(z\):
\[{\rm csch}(z) = \frac{1}{{\rm sinh}(z)}.\]EXAMPLES:
sage: CDF(1,1).csch() # rel tol 1e-15 0.30393100162842646 - 0.6215180171704284*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).csch() # rel tol 1e-15 0.30393100162842646 - 0.6215180171704284*I
CDF(1,1).csch() # rel tol 1e-15
- dilog()[source]¶
Return the principal branch of the dilogarithm of \(x\), i.e., analytic continuation of the power series
\[\log_2(x) = \sum_{n \ge 1} x^n / n^2.\]EXAMPLES:
sage: CDF(1,2).dilog() # needs sage.libs.pari -0.059474798673809476 + 2.0726479717747566*I sage: CDF(10000000,10000000).dilog() # needs sage.libs.pari -134.411774490731 + 38.79396299904504*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(2)).dilog() # needs sage.libs.pari -0.059474798673809476 + 2.0726479717747566*I >>> CDF(Integer(10000000),Integer(10000000)).dilog() # needs sage.libs.pari -134.411774490731 + 38.79396299904504*I
CDF(1,2).dilog() # needs sage.libs.pari CDF(10000000,10000000).dilog() # needs sage.libs.pari
- eta(omit_frac=0)[source]¶
Return the value of the Dedekind \(\eta\) function on
self.INPUT:
self– element of the upper half plane (if not, raises a ValueError)omit_frac– boolean (default:False); ifTrue, omit the \(e^{\pi i z / 12}\) factor
OUTPUT: a complex double number
ALGORITHM: Uses the PARI C library.
The \(\eta\) function is
\[\eta(z) = e^{\pi i z / 12} \prod_{n=1}^{\infty} (1 - e^{2\pi inz})\]EXAMPLES:
We compute a few values of
eta():sage: CDF(0,1).eta() # needs sage.libs.pari 0.7682254223260566 sage: CDF(1,1).eta() # needs sage.libs.pari 0.7420487758365647 + 0.1988313702299107*I sage: CDF(25,1).eta() # needs sage.libs.pari 0.7420487758365647 + 0.1988313702299107*I
>>> from sage.all import * >>> CDF(Integer(0),Integer(1)).eta() # needs sage.libs.pari 0.7682254223260566 >>> CDF(Integer(1),Integer(1)).eta() # needs sage.libs.pari 0.7420487758365647 + 0.1988313702299107*I >>> CDF(Integer(25),Integer(1)).eta() # needs sage.libs.pari 0.7420487758365647 + 0.1988313702299107*I
CDF(0,1).eta() # needs sage.libs.pari CDF(1,1).eta() # needs sage.libs.pari CDF(25,1).eta() # needs sage.libs.pari
eta()works even if the inputs are large:sage: CDF(0, 10^15).eta() 0.0 sage: CDF(10^15, 0.1).eta() # abs tol 1e-10 # needs sage.libs.pari -0.115342592727 - 0.19977923088*I
>>> from sage.all import * >>> CDF(Integer(0), Integer(10)**Integer(15)).eta() 0.0 >>> CDF(Integer(10)**Integer(15), RealNumber('0.1')).eta() # abs tol 1e-10 # needs sage.libs.pari -0.115342592727 - 0.19977923088*I
CDF(0, 10^15).eta() CDF(10^15, 0.1).eta() # abs tol 1e-10 # needs sage.libs.pari
We compute a few values of
eta(), but with the fractional power of \(e\) omitted:sage: CDF(0,1).eta(True) # needs sage.libs.pari 0.9981290699259585
>>> from sage.all import * >>> CDF(Integer(0),Integer(1)).eta(True) # needs sage.libs.pari 0.9981290699259585
CDF(0,1).eta(True) # needs sage.libs.pari
We compute
eta()to low precision directly from the definition:sage: z = CDF(1,1); z.eta() # needs sage.libs.pari 0.7420487758365647 + 0.1988313702299107*I sage: i = CDF(0,1); pi = CDF(pi) # needs sage.symbolic sage: exp(pi * i * z / 12) * prod(1 - exp(2*pi*i*n*z) # needs sage.libs.pari sage.symbolic ....: for n in range(1, 10)) 0.7420487758365647 + 0.19883137022991068*I
>>> from sage.all import * >>> z = CDF(Integer(1),Integer(1)); z.eta() # needs sage.libs.pari 0.7420487758365647 + 0.1988313702299107*I >>> i = CDF(Integer(0),Integer(1)); pi = CDF(pi) # needs sage.symbolic >>> exp(pi * i * z / Integer(12)) * prod(Integer(1) - exp(Integer(2)*pi*i*n*z) # needs sage.libs.pari sage.symbolic ... for n in range(Integer(1), Integer(10))) 0.7420487758365647 + 0.19883137022991068*I
z = CDF(1,1); z.eta() # needs sage.libs.pari i = CDF(0,1); pi = CDF(pi) # needs sage.symbolic exp(pi * i * z / 12) * prod(1 - exp(2*pi*i*n*z) # needs sage.libs.pari sage.symbolic for n in range(1, 10))The optional argument allows us to omit the fractional part:
sage: z.eta(omit_frac=True) # needs sage.libs.pari 0.9981290699259585 sage: pi = CDF(pi) # needs sage.symbolic sage: prod(1 - exp(2*pi*i*n*z) for n in range(1,10)) # abs tol 1e-12 # needs sage.libs.pari sage.symbolic 0.998129069926 + 4.59084695545e-19*I
>>> from sage.all import * >>> z.eta(omit_frac=True) # needs sage.libs.pari 0.9981290699259585 >>> pi = CDF(pi) # needs sage.symbolic >>> prod(Integer(1) - exp(Integer(2)*pi*i*n*z) for n in range(Integer(1),Integer(10))) # abs tol 1e-12 # needs sage.libs.pari sage.symbolic 0.998129069926 + 4.59084695545e-19*I
z.eta(omit_frac=True) # needs sage.libs.pari pi = CDF(pi) # needs sage.symbolic prod(1 - exp(2*pi*i*n*z) for n in range(1,10)) # abs tol 1e-12 # needs sage.libs.pari sage.symbolic
We illustrate what happens when \(z\) is not in the upper half plane:
sage: z = CDF(1) sage: z.eta() Traceback (most recent call last): ... ValueError: value must be in the upper half plane
>>> from sage.all import * >>> z = CDF(Integer(1)) >>> z.eta() Traceback (most recent call last): ... ValueError: value must be in the upper half plane
z = CDF(1) z.eta()
You can also use functional notation:
sage: z = CDF(1,1) sage: eta(z) # needs sage.libs.pari 0.7420487758365647 + 0.1988313702299107*I
>>> from sage.all import * >>> z = CDF(Integer(1),Integer(1)) >>> eta(z) # needs sage.libs.pari 0.7420487758365647 + 0.1988313702299107*I
z = CDF(1,1) eta(z) # needs sage.libs.pari
- exp()[source]¶
This function returns the complex exponential of the complex number \(z\), \(\exp(z)\).
EXAMPLES:
sage: CDF(1,1).exp() # abs tol 4e-16 1.4686939399158851 + 2.2873552871788423*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).exp() # abs tol 4e-16 1.4686939399158851 + 2.2873552871788423*I
CDF(1,1).exp() # abs tol 4e-16
We numerically verify a famous identity to the precision of a double:
sage: z = CDF(0, 2*pi); z # needs sage.symbolic 6.283185307179586*I sage: exp(z) # rel tol 1e-4 # needs sage.symbolic 1.0 - 2.4492935982947064e-16*I
>>> from sage.all import * >>> z = CDF(Integer(0), Integer(2)*pi); z # needs sage.symbolic 6.283185307179586*I >>> exp(z) # rel tol 1e-4 # needs sage.symbolic 1.0 - 2.4492935982947064e-16*I
z = CDF(0, 2*pi); z # needs sage.symbolic exp(z) # rel tol 1e-4 # needs sage.symbolic
- gamma()[source]¶
Return the gamma function \(\Gamma(z)\) evaluated at
self, the complex number \(z\).EXAMPLES:
sage: # needs sage.libs.pari sage: CDF(5,0).gamma() 24.0 sage: CDF(1,1).gamma() 0.49801566811835607 - 0.15494982830181067*I sage: CDF(0).gamma() Infinity sage: CDF(-1,0).gamma() Infinity
>>> from sage.all import * >>> # needs sage.libs.pari >>> CDF(Integer(5),Integer(0)).gamma() 24.0 >>> CDF(Integer(1),Integer(1)).gamma() 0.49801566811835607 - 0.15494982830181067*I >>> CDF(Integer(0)).gamma() Infinity >>> CDF(-Integer(1),Integer(0)).gamma() Infinity
# needs sage.libs.pari CDF(5,0).gamma() CDF(1,1).gamma() CDF(0).gamma() CDF(-1,0).gamma()
- gamma_inc(t)[source]¶
Return the incomplete gamma function evaluated at this complex number.
EXAMPLES:
sage: CDF(1,1).gamma_inc(CDF(2,3)) # needs sage.libs.pari 0.0020969148636468277 - 0.059981913655449706*I sage: CDF(1,1).gamma_inc(5) # needs sage.libs.pari -0.001378130936215849 + 0.006519820023119819*I sage: CDF(2,0).gamma_inc(CDF(1,1)) # needs sage.libs.pari 0.7070920963459381 - 0.4203536409598115*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).gamma_inc(CDF(Integer(2),Integer(3))) # needs sage.libs.pari 0.0020969148636468277 - 0.059981913655449706*I >>> CDF(Integer(1),Integer(1)).gamma_inc(Integer(5)) # needs sage.libs.pari -0.001378130936215849 + 0.006519820023119819*I >>> CDF(Integer(2),Integer(0)).gamma_inc(CDF(Integer(1),Integer(1))) # needs sage.libs.pari 0.7070920963459381 - 0.4203536409598115*I
CDF(1,1).gamma_inc(CDF(2,3)) # needs sage.libs.pari CDF(1,1).gamma_inc(5) # needs sage.libs.pari CDF(2,0).gamma_inc(CDF(1,1)) # needs sage.libs.pari
- imag()[source]¶
Return the imaginary part of this complex double.
EXAMPLES:
sage: a = CDF(3,-2) sage: a.imag() -2.0 sage: a.imag_part() -2.0
>>> from sage.all import * >>> a = CDF(Integer(3),-Integer(2)) >>> a.imag() -2.0 >>> a.imag_part() -2.0
a = CDF(3,-2) a.imag() a.imag_part()
- imag_part()[source]¶
Return the imaginary part of this complex double.
EXAMPLES:
sage: a = CDF(3,-2) sage: a.imag() -2.0 sage: a.imag_part() -2.0
>>> from sage.all import * >>> a = CDF(Integer(3),-Integer(2)) >>> a.imag() -2.0 >>> a.imag_part() -2.0
a = CDF(3,-2) a.imag() a.imag_part()
- is_NaN()[source]¶
Check if
selfis not-a-number.EXAMPLES:
sage: CDF(1, 2).is_NaN() False sage: CDF(NaN).is_NaN() # needs sage.symbolic True sage: (1/CDF(0, 0)).is_NaN() True
>>> from sage.all import * >>> CDF(Integer(1), Integer(2)).is_NaN() False >>> CDF(NaN).is_NaN() # needs sage.symbolic True >>> (Integer(1)/CDF(Integer(0), Integer(0))).is_NaN() True
CDF(1, 2).is_NaN() CDF(NaN).is_NaN() # needs sage.symbolic (1/CDF(0, 0)).is_NaN()
- is_infinity()[source]¶
Check if
selfis \(\infty\).EXAMPLES:
sage: CDF(1, 2).is_infinity() False sage: CDF(0, oo).is_infinity() True
>>> from sage.all import * >>> CDF(Integer(1), Integer(2)).is_infinity() False >>> CDF(Integer(0), oo).is_infinity() True
CDF(1, 2).is_infinity() CDF(0, oo).is_infinity()
- is_integer()[source]¶
Return
Trueif this number is a integer.EXAMPLES:
sage: CDF(0.5).is_integer() False sage: CDF(I).is_integer() # needs sage.symbolic False sage: CDF(2).is_integer() True
>>> from sage.all import * >>> CDF(RealNumber('0.5')).is_integer() False >>> CDF(I).is_integer() # needs sage.symbolic False >>> CDF(Integer(2)).is_integer() True
CDF(0.5).is_integer() CDF(I).is_integer() # needs sage.symbolic CDF(2).is_integer()
- is_negative_infinity()[source]¶
Check if
selfis \(-\infty\).EXAMPLES:
sage: CDF(1, 2).is_negative_infinity() False sage: CDF(-oo, 0).is_negative_infinity() True sage: CDF(0, -oo).is_negative_infinity() False
>>> from sage.all import * >>> CDF(Integer(1), Integer(2)).is_negative_infinity() False >>> CDF(-oo, Integer(0)).is_negative_infinity() True >>> CDF(Integer(0), -oo).is_negative_infinity() False
CDF(1, 2).is_negative_infinity() CDF(-oo, 0).is_negative_infinity() CDF(0, -oo).is_negative_infinity()
- is_positive_infinity()[source]¶
Check if
selfis \(+\infty\).EXAMPLES:
sage: CDF(1, 2).is_positive_infinity() False sage: CDF(oo, 0).is_positive_infinity() True sage: CDF(0, oo).is_positive_infinity() False
>>> from sage.all import * >>> CDF(Integer(1), Integer(2)).is_positive_infinity() False >>> CDF(oo, Integer(0)).is_positive_infinity() True >>> CDF(Integer(0), oo).is_positive_infinity() False
CDF(1, 2).is_positive_infinity() CDF(oo, 0).is_positive_infinity() CDF(0, oo).is_positive_infinity()
- is_square()[source]¶
This function always returns
Trueas \(\CC\) is algebraically closed.EXAMPLES:
sage: CDF(-1).is_square() True
>>> from sage.all import * >>> CDF(-Integer(1)).is_square() True
CDF(-1).is_square()
- log(base=None)[source]¶
This function returns the complex natural logarithm to the given base of the complex number \(z\), \(\log(z)\). The branch cut is the negative real axis.
INPUT:
base– (default: \(e\)) the base of the natural logarithm
EXAMPLES:
sage: CDF(1,1).log() 0.34657359027997264 + 0.7853981633974483*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).log() 0.34657359027997264 + 0.7853981633974483*I
CDF(1,1).log()
This is the only example different from the GSL:
sage: CDF(0,0).log() -infinity
>>> from sage.all import * >>> CDF(Integer(0),Integer(0)).log() -infinity
CDF(0,0).log()
- log10()[source]¶
This function returns the complex base-10 logarithm of the complex number \(z\), \(\log_{10}(z)\).
The branch cut is the negative real axis.
EXAMPLES:
sage: CDF(1,1).log10() 0.15051499783199057 + 0.3410940884604603*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).log10() 0.15051499783199057 + 0.3410940884604603*I
CDF(1,1).log10()
- log_b(b)[source]¶
This function returns the complex base-\(b\) logarithm of the complex number \(z\), \(\log_b(z)\). This quantity is computed as the ratio \(\log(z)/\log(b)\).
The branch cut is the negative real axis.
EXAMPLES:
sage: CDF(1,1).log_b(10) # rel tol 1e-15 0.15051499783199057 + 0.3410940884604603*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).log_b(Integer(10)) # rel tol 1e-15 0.15051499783199057 + 0.3410940884604603*I
CDF(1,1).log_b(10) # rel tol 1e-15
- logabs()[source]¶
This function returns the natural logarithm of the magnitude of the complex number \(z\), \(\log|z|\).
This allows for an accurate evaluation of \(\log|z|\) when \(|z|\) is close to \(1\). The direct evaluation of
log(abs(z))would lead to a loss of precision in this case.EXAMPLES:
sage: CDF(1.1,0.1).logabs() 0.09942542937258267 sage: log(abs(CDF(1.1,0.1))) 0.09942542937258259
>>> from sage.all import * >>> CDF(RealNumber('1.1'),RealNumber('0.1')).logabs() 0.09942542937258267 >>> log(abs(CDF(RealNumber('1.1'),RealNumber('0.1')))) 0.09942542937258259
CDF(1.1,0.1).logabs() log(abs(CDF(1.1,0.1)))
sage: log(abs(ComplexField(200)(1.1,0.1))) 0.099425429372582595066319157757531449594489450091985182495705
>>> from sage.all import * >>> log(abs(ComplexField(Integer(200))(RealNumber('1.1'),RealNumber('0.1')))) 0.099425429372582595066319157757531449594489450091985182495705
log(abs(ComplexField(200)(1.1,0.1)))
>>> from sage.all import * >>> log(abs(ComplexField(Integer(200))(RealNumber('1.1'),RealNumber('0.1')))) 0.099425429372582595066319157757531449594489450091985182495705
log(abs(ComplexField(200)(1.1,0.1)))
- norm()[source]¶
This function returns the squared magnitude \(|z|^2\) of the complex number \(z\), otherwise known as the complex norm. If \(c = a + bi\) is a complex number, then the norm of \(c\) is defined as the product of \(c\) and its complex conjugate:
\[\text{norm}(c) = \text{norm}(a + bi) = c \cdot \overline{c} = a^2 + b^2.\]The norm of a complex number is different from its absolute value. The absolute value of a complex number is defined to be the square root of its norm. A typical use of the complex norm is in the integral domain \(\ZZ[i]\) of Gaussian integers, where the norm of each Gaussian integer \(c = a + bi\) is defined as its complex norm.
EXAMPLES:
sage: CDF(2,3).norm() 13.0
>>> from sage.all import * >>> CDF(Integer(2),Integer(3)).norm() 13.0
CDF(2,3).norm()
- nth_root(n, all=False)[source]¶
The
n-th root function.INPUT:
all– boolean (default:False); ifTrue, return a list of alln-th roots
EXAMPLES:
sage: a = CDF(125) sage: a.nth_root(3) 5.000000000000001 sage: a = CDF(10, 2) sage: [r^5 for r in a.nth_root(5, all=True)] # rel tol 1e-14 [9.999999999999998 + 2.0*I, 9.999999999999993 + 2.000000000000002*I, 9.999999999999996 + 1.9999999999999907*I, 9.999999999999993 + 2.0000000000000004*I, 9.999999999999998 + 1.9999999999999802*I] sage: abs(sum(a.nth_root(111, all=True))) # rel tol 0.1 1.1057313523818259e-13
>>> from sage.all import * >>> a = CDF(Integer(125)) >>> a.nth_root(Integer(3)) 5.000000000000001 >>> a = CDF(Integer(10), Integer(2)) >>> [r**Integer(5) for r in a.nth_root(Integer(5), all=True)] # rel tol 1e-14 [9.999999999999998 + 2.0*I, 9.999999999999993 + 2.000000000000002*I, 9.999999999999996 + 1.9999999999999907*I, 9.999999999999993 + 2.0000000000000004*I, 9.999999999999998 + 1.9999999999999802*I] >>> abs(sum(a.nth_root(Integer(111), all=True))) # rel tol 0.1 1.1057313523818259e-13
a = CDF(125) a.nth_root(3) a = CDF(10, 2) [r^5 for r in a.nth_root(5, all=True)] # rel tol 1e-14 abs(sum(a.nth_root(111, all=True))) # rel tol 0.1
- prec()[source]¶
Return the precision of this number (to be more similar to
ComplexNumber). Always returns 53.EXAMPLES:
sage: CDF(0).prec() 53
>>> from sage.all import * >>> CDF(Integer(0)).prec() 53
CDF(0).prec()
- real()[source]¶
Return the real part of this complex double.
EXAMPLES:
sage: a = CDF(3,-2) sage: a.real() 3.0 sage: a.real_part() 3.0
>>> from sage.all import * >>> a = CDF(Integer(3),-Integer(2)) >>> a.real() 3.0 >>> a.real_part() 3.0
a = CDF(3,-2) a.real() a.real_part()
- real_part()[source]¶
Return the real part of this complex double.
EXAMPLES:
sage: a = CDF(3,-2) sage: a.real() 3.0 sage: a.real_part() 3.0
>>> from sage.all import * >>> a = CDF(Integer(3),-Integer(2)) >>> a.real() 3.0 >>> a.real_part() 3.0
a = CDF(3,-2) a.real() a.real_part()
- sec()[source]¶
This function returns the complex secant of the complex number \(z\):
\[{\rm sec}(z) = \frac{1}{\cos(z)}.\]EXAMPLES:
sage: CDF(1,1).sec() # rel tol 1e-15 0.4983370305551868 + 0.591083841721045*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).sec() # rel tol 1e-15 0.4983370305551868 + 0.591083841721045*I
CDF(1,1).sec() # rel tol 1e-15
- sech()[source]¶
This function returns the complex hyperbolic secant of the complex number \(z\):
\[{\rm sech}(z) = \frac{1}{{\rm cosh}(z)}.\]EXAMPLES:
sage: CDF(1,1).sech() # rel tol 1e-15 0.4983370305551868 - 0.591083841721045*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).sech() # rel tol 1e-15 0.4983370305551868 - 0.591083841721045*I
CDF(1,1).sech() # rel tol 1e-15
- sin()[source]¶
This function returns the complex sine of the complex number \(z\):
\[\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}.\]EXAMPLES:
sage: CDF(1,1).sin() 1.2984575814159773 + 0.6349639147847361*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).sin() 1.2984575814159773 + 0.6349639147847361*I
CDF(1,1).sin()
- sinh()[source]¶
This function returns the complex hyperbolic sine of the complex number \(z\):
\[\sinh(z) = \frac{e^z - e^{-z}}{2}.\]EXAMPLES:
sage: CDF(1,1).sinh() 0.6349639147847361 + 1.2984575814159773*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).sinh() 0.6349639147847361 + 1.2984575814159773*I
CDF(1,1).sinh()
- sqrt(all=False, **kwds)[source]¶
The square root function.
INPUT:
all– boolean (default:False); ifTrue, return a list of all square roots
If all is
False, the branch cut is the negative real axis. The result always lies in the right half of the complex plane.EXAMPLES:
We compute several square roots:
sage: a = CDF(2,3) sage: b = a.sqrt(); b # rel tol 1e-15 1.6741492280355401 + 0.8959774761298381*I sage: b^2 # rel tol 1e-15 2.0 + 3.0*I sage: a^(1/2) # abs tol 1e-16 1.6741492280355401 + 0.895977476129838*I
>>> from sage.all import * >>> a = CDF(Integer(2),Integer(3)) >>> b = a.sqrt(); b # rel tol 1e-15 1.6741492280355401 + 0.8959774761298381*I >>> b**Integer(2) # rel tol 1e-15 2.0 + 3.0*I >>> a**(Integer(1)/Integer(2)) # abs tol 1e-16 1.6741492280355401 + 0.895977476129838*I
a = CDF(2,3) b = a.sqrt(); b # rel tol 1e-15 b^2 # rel tol 1e-15 a^(1/2) # abs tol 1e-16
We compute the square root of -1:
sage: a = CDF(-1) sage: a.sqrt() 1.0*I
>>> from sage.all import * >>> a = CDF(-Integer(1)) >>> a.sqrt() 1.0*I
a = CDF(-1) a.sqrt()
We compute all square roots:
sage: CDF(-2).sqrt(all=True) [1.4142135623730951*I, -1.4142135623730951*I] sage: CDF(0).sqrt(all=True) [0.0]
>>> from sage.all import * >>> CDF(-Integer(2)).sqrt(all=True) [1.4142135623730951*I, -1.4142135623730951*I] >>> CDF(Integer(0)).sqrt(all=True) [0.0]
CDF(-2).sqrt(all=True) CDF(0).sqrt(all=True)
- tan()[source]¶
This function returns the complex tangent of the complex number \(z\):
\[\tan(z) = \frac{\sin(z)}{\cos(z)}.\]EXAMPLES:
sage: CDF(1,1).tan() 0.27175258531951174 + 1.0839233273386946*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).tan() 0.27175258531951174 + 1.0839233273386946*I
CDF(1,1).tan()
- tanh()[source]¶
This function returns the complex hyperbolic tangent of the complex number \(z\):
\[\tanh(z) = \frac{\sinh(z)}{\cosh(z)}.\]EXAMPLES:
sage: CDF(1,1).tanh() 1.0839233273386946 + 0.27175258531951174*I
>>> from sage.all import * >>> CDF(Integer(1),Integer(1)).tanh() 1.0839233273386946 + 0.27175258531951174*I
CDF(1,1).tanh()
- zeta()[source]¶
Return the Riemann zeta function evaluated at this complex number.
EXAMPLES:
sage: z = CDF(1, 1) sage: z.zeta() # needs sage.libs.pari 0.5821580597520036 - 0.9268485643308071*I sage: zeta(z) # needs sage.libs.pari 0.5821580597520036 - 0.9268485643308071*I sage: zeta(CDF(1)) # needs sage.libs.pari Infinity
>>> from sage.all import * >>> z = CDF(Integer(1), Integer(1)) >>> z.zeta() # needs sage.libs.pari 0.5821580597520036 - 0.9268485643308071*I >>> zeta(z) # needs sage.libs.pari 0.5821580597520036 - 0.9268485643308071*I >>> zeta(CDF(Integer(1))) # needs sage.libs.pari Infinity
z = CDF(1, 1) z.zeta() # needs sage.libs.pari zeta(z) # needs sage.libs.pari zeta(CDF(1)) # needs sage.libs.pari
- sage.rings.complex_double.ComplexDoubleField()[source]¶
Return the field of double precision complex numbers.
EXAMPLES:
sage: ComplexDoubleField() Complex Double Field sage: ComplexDoubleField() is CDF True
>>> from sage.all import * >>> ComplexDoubleField() Complex Double Field >>> ComplexDoubleField() is CDF True
ComplexDoubleField() ComplexDoubleField() is CDF
- class sage.rings.complex_double.ComplexDoubleField_class[source]¶
Bases:
ComplexDoubleFieldAn approximation to the field of complex numbers using double precision floating point numbers. Answers derived from calculations in this approximation may differ from what they would be if those calculations were performed in the true field of complex numbers. This is due to the rounding errors inherent to finite precision calculations.
ALGORITHM:
Arithmetic is done using GSL (the GNU Scientific Library).
- algebraic_closure()[source]¶
Return the algebraic closure of
self, i.e., the complex double field.EXAMPLES:
sage: CDF.algebraic_closure() Complex Double Field
>>> from sage.all import * >>> CDF.algebraic_closure() Complex Double Field
CDF.algebraic_closure()
- characteristic()[source]¶
Return the characteristic of the complex double field, which is 0.
EXAMPLES:
sage: CDF.characteristic() 0
>>> from sage.all import * >>> CDF.characteristic() 0
CDF.characteristic()
- construction()[source]¶
Return the functorial construction of
self, namely, algebraic closure of the real double field.EXAMPLES:
sage: c, S = CDF.construction(); S Real Double Field sage: CDF == c(S) True
>>> from sage.all import * >>> c, S = CDF.construction(); S Real Double Field >>> CDF == c(S) True
c, S = CDF.construction(); S CDF == c(S)
- gen(n=0)[source]¶
Return the generator of the complex double field.
EXAMPLES:
sage: CDF.0 1.0*I sage: CDF.gen(0) 1.0*I
>>> from sage.all import * >>> CDF.gen(0) 1.0*I >>> CDF.gen(Integer(0)) 1.0*I
CDF.0 CDF.gen(0)
- is_exact()[source]¶
Return whether or not this field is exact, which is always
False.EXAMPLES:
sage: CDF.is_exact() False
>>> from sage.all import * >>> CDF.is_exact() False
CDF.is_exact()
- ngens()[source]¶
The number of generators of this complex field as an \(\RR\)-algebra.
There is one generator, namely
sqrt(-1).EXAMPLES:
sage: CDF.ngens() 1
>>> from sage.all import * >>> CDF.ngens() 1
CDF.ngens()
- pi()[source]¶
Return \(\pi\) as a double precision complex number.
EXAMPLES:
sage: CDF.pi() 3.141592653589793
>>> from sage.all import * >>> CDF.pi() 3.141592653589793
CDF.pi()
- prec()[source]¶
Return the precision of this complex double field (to be more similar to
ComplexField). Always returns 53.EXAMPLES:
sage: CDF.prec() 53
>>> from sage.all import * >>> CDF.prec() 53
CDF.prec()
- precision()[source]¶
Return the precision of this complex double field (to be more similar to
ComplexField). Always returns 53.EXAMPLES:
sage: CDF.prec() 53
>>> from sage.all import * >>> CDF.prec() 53
CDF.prec()
- random_element(xmin=-1, xmax=1, ymin=-1, ymax=1)[source]¶
Return a random element of this complex double field with real and imaginary part bounded by
xmin,xmax,ymin,ymax.EXAMPLES:
sage: CDF.random_element().parent() is CDF True sage: re, im = CDF.random_element() sage: -1 <= re <= 1, -1 <= im <= 1 (True, True) sage: re, im = CDF.random_element(-10,10,-10,10) sage: -10 <= re <= 10, -10 <= im <= 10 (True, True) sage: re, im = CDF.random_element(-10^20,10^20,-2,2) sage: -10^20 <= re <= 10^20, -2 <= im <= 2 (True, True)
>>> from sage.all import * >>> CDF.random_element().parent() is CDF True >>> re, im = CDF.random_element() >>> -Integer(1) <= re <= Integer(1), -Integer(1) <= im <= Integer(1) (True, True) >>> re, im = CDF.random_element(-Integer(10),Integer(10),-Integer(10),Integer(10)) >>> -Integer(10) <= re <= Integer(10), -Integer(10) <= im <= Integer(10) (True, True) >>> re, im = CDF.random_element(-Integer(10)**Integer(20),Integer(10)**Integer(20),-Integer(2),Integer(2)) >>> -Integer(10)**Integer(20) <= re <= Integer(10)**Integer(20), -Integer(2) <= im <= Integer(2) (True, True)
CDF.random_element().parent() is CDF re, im = CDF.random_element() -1 <= re <= 1, -1 <= im <= 1 re, im = CDF.random_element(-10,10,-10,10) -10 <= re <= 10, -10 <= im <= 10 re, im = CDF.random_element(-10^20,10^20,-2,2) -10^20 <= re <= 10^20, -2 <= im <= 2
- real_double_field()[source]¶
The real double field, which you may view as a subfield of this complex double field.
EXAMPLES:
sage: CDF.real_double_field() Real Double Field
>>> from sage.all import * >>> CDF.real_double_field() Real Double Field
CDF.real_double_field()
- to_prec(prec)[source]¶
Return the complex field to the specified precision. As doubles have fixed precision, this will only return a complex double field if prec is exactly 53.
EXAMPLES:
sage: CDF.to_prec(53) Complex Double Field sage: CDF.to_prec(250) Complex Field with 250 bits of precision
>>> from sage.all import * >>> CDF.to_prec(Integer(53)) Complex Double Field >>> CDF.to_prec(Integer(250)) Complex Field with 250 bits of precision
CDF.to_prec(53) CDF.to_prec(250)
- zeta(n=2)[source]¶
Return a primitive \(n\)-th root of unity in this CDF, for \(n \geq 1\).
INPUT:
n– positive integer (default: 2)
OUTPUT: a complex \(n\)-th root of unity
EXAMPLES:
sage: CDF.zeta(7) # rel tol 1e-15 0.6234898018587336 + 0.7818314824680298*I sage: CDF.zeta(1) 1.0 sage: CDF.zeta() -1.0 sage: CDF.zeta() == CDF.zeta(2) True
>>> from sage.all import * >>> CDF.zeta(Integer(7)) # rel tol 1e-15 0.6234898018587336 + 0.7818314824680298*I >>> CDF.zeta(Integer(1)) 1.0 >>> CDF.zeta() -1.0 >>> CDF.zeta() == CDF.zeta(Integer(2)) True
CDF.zeta(7) # rel tol 1e-15 CDF.zeta(1) CDF.zeta() CDF.zeta() == CDF.zeta(2)
sage: CDF.zeta(0.5) Traceback (most recent call last): ... ValueError: n must be a positive integer sage: CDF.zeta(0) Traceback (most recent call last): ... ValueError: n must be a positive integer sage: CDF.zeta(-1) Traceback (most recent call last): ... ValueError: n must be a positive integer
>>> from sage.all import * >>> CDF.zeta(RealNumber('0.5')) Traceback (most recent call last): ... ValueError: n must be a positive integer >>> CDF.zeta(Integer(0)) Traceback (most recent call last): ... ValueError: n must be a positive integer >>> CDF.zeta(-Integer(1)) Traceback (most recent call last): ... ValueError: n must be a positive integer
CDF.zeta(0.5) CDF.zeta(0) CDF.zeta(-1)
>>> from sage.all import * >>> CDF.zeta(RealNumber('0.5')) Traceback (most recent call last): ... ValueError: n must be a positive integer >>> CDF.zeta(Integer(0)) Traceback (most recent call last): ... ValueError: n must be a positive integer >>> CDF.zeta(-Integer(1)) Traceback (most recent call last): ... ValueError: n must be a positive integer
CDF.zeta(0.5) CDF.zeta(0) CDF.zeta(-1)
- class sage.rings.complex_double.ComplexToCDF[source]¶
Bases:
MorphismFast morphism for anything such that the elements have attributes
.realand.imag(e.g. numpy complex types).EXAMPLES:
sage: # needs numpy sage: import numpy sage: f = CDF.coerce_map_from(numpy.complex128) sage: f(numpy.complex128(I)) 1.0*I sage: f(numpy.complex128(I)).parent() Complex Double Field
>>> from sage.all import * >>> # needs numpy >>> import numpy >>> f = CDF.coerce_map_from(numpy.complex128) >>> f(numpy.complex128(I)) 1.0*I >>> f(numpy.complex128(I)).parent() Complex Double Field
# needs numpy import numpy f = CDF.coerce_map_from(numpy.complex128) f(numpy.complex128(I)) f(numpy.complex128(I)).parent()
- class sage.rings.complex_double.FloatToCDF[source]¶
Bases:
MorphismFast morphism from anything with a
__float__method to a CDF element.EXAMPLES:
sage: f = CDF.coerce_map_from(ZZ); f Native morphism: From: Integer Ring To: Complex Double Field sage: f(4) 4.0 sage: f = CDF.coerce_map_from(QQ); f Native morphism: From: Rational Field To: Complex Double Field sage: f(1/2) 0.5 sage: f = CDF.coerce_map_from(int); f Native morphism: From: Set of Python objects of class 'int' To: Complex Double Field sage: f(3r) 3.0 sage: f = CDF.coerce_map_from(float); f Native morphism: From: Set of Python objects of class 'float' To: Complex Double Field sage: f(3.5) 3.5
>>> from sage.all import * >>> f = CDF.coerce_map_from(ZZ); f Native morphism: From: Integer Ring To: Complex Double Field >>> f(Integer(4)) 4.0 >>> f = CDF.coerce_map_from(QQ); f Native morphism: From: Rational Field To: Complex Double Field >>> f(Integer(1)/Integer(2)) 0.5 >>> f = CDF.coerce_map_from(int); f Native morphism: From: Set of Python objects of class 'int' To: Complex Double Field >>> f(3) 3.0 >>> f = CDF.coerce_map_from(float); f Native morphism: From: Set of Python objects of class 'float' To: Complex Double Field >>> f(RealNumber('3.5')) 3.5
f = CDF.coerce_map_from(ZZ); f f(4) f = CDF.coerce_map_from(QQ); f f(1/2) f = CDF.coerce_map_from(int); f f(3r) f = CDF.coerce_map_from(float); f f(3.5)
- sage.rings.complex_double.is_ComplexDoubleElement(x)[source]¶
Return
Trueifxis aComplexDoubleElement.EXAMPLES:
sage: from sage.rings.complex_double import is_ComplexDoubleElement sage: is_ComplexDoubleElement(0) doctest:warning... DeprecationWarning: The function is_ComplexDoubleElement is deprecated; use 'isinstance(..., ComplexDoubleElement)' instead. See https://github.com/sagemath/sage/issues/38128 for details. False sage: is_ComplexDoubleElement(CDF(0)) True
>>> from sage.all import * >>> from sage.rings.complex_double import is_ComplexDoubleElement >>> is_ComplexDoubleElement(Integer(0)) doctest:warning... DeprecationWarning: The function is_ComplexDoubleElement is deprecated; use 'isinstance(..., ComplexDoubleElement)' instead. See https://github.com/sagemath/sage/issues/38128 for details. False >>> is_ComplexDoubleElement(CDF(Integer(0))) True
from sage.rings.complex_double import is_ComplexDoubleElement is_ComplexDoubleElement(0) is_ComplexDoubleElement(CDF(0))