Classes for symbolic functions#
To enable their usage as part of symbolic expressions, symbolic function
classes are derived from one of the subclasses of
GinacFunction: the code of these functions is written in C++ and part of the Pynac support library; most elementary functions are of this type
SymbolicFunction: symbolic functions defined on the Sage command line are of this type
GinacFunction for its symbolic builtin
functions. Users can define any other additional
function() factory, see Factory for symbolic functions
Several parameters are supported by the superclass’
Examples follow below.
nargs: the number of arguments
name: the string that is printed on the CLI; the name of the member functions that are attempted for evaluation of Sage element arguments; also the name of the Pynac function that is associated with a
alt_name: the second name of the member functions that are attempted for evaluation of Sage element arguments
latex_name: what is printed when
conversions: a dict containing the function’s name in other CAS
False, when floating-point evaluating the expression do not evaluate function arguments before calling the
_evalf_()member of the function
preserved_arg: if nonzero, the index (starting with
1) of the function argument that determines the return type. Note that, e.g,
atan2()uses both arguments to determine return type, through a different mechanism
Function classes can define the following Python member functions:
_eval_(*args): the only mandatory member function, evaluating the argument and returning the result; if
Noneis returned the expression stays unevaluated
_eval_numpy_(*args): evaluation of
f(args)with arguments of numpy type
_evalf_(*args, **kwds): called when the expression is floating-point evaluated; may receive a
parentkeyword specifying the expected parent of the result. If not defined an attempt is made to convert the result of
_imag_part_(*args): return conjugate, real part, imaginary part of the expression
_derivative_(*args, index): return derivative with respect to the parameter indexed by
index(starting with 0) of
_tderivative_(): same as
_derivative_()but don’t apply chain rule; only one of the two functions may be defined
_power_(*args, expo): return
_series_(*args, **kwds): return the power series at
orderwith respect to
f(args); these three values are received in
kwds. If not defined the series is attempted to be computed by differentiation.
print(*args): return what should be printed on the CLI with
print_latex(*args): return what should be output with
The following examples are intended for Sage developers. Users can define
functions interactively through the
function() factory, see
Factory for symbolic functions.
The simplest example is a function returning nothing, it practically behaves
like a symbol. Setting
nargs=0 allows any number of arguments:
sage: from sage.symbolic.function import BuiltinFunction sage: class Test1(BuiltinFunction): ....: def __init__(self): ....: BuiltinFunction.__init__(self, 'test', nargs=0) ....: def _eval_(self, *args): ....: pass sage: f = Test1() sage: f() test() sage: f(1,2,3)*f(1,2,3) test(1, 2, 3)^2
In the following the
sin function of
CBF(0) is called because with
floating point arguments the
my_sin() member function
is attempted, and after that
sin() which succeeds:
sage: class Test2(BuiltinFunction): ....: def __init__(self): ....: BuiltinFunction.__init__(self, 'my_sin', alt_name='sin', ....: latex_name=r'\SIN', nargs=1) ....: def _eval_(self, x): ....: return 5 ....: def _evalf_(self, x, **kwds): ....: return 3.5 sage: f = Test2() sage: f(0) 5 sage: f(0, hold=True) my_sin(0) sage: f(0, hold=True).n() 3.50000000000000 sage: f(CBF(0)) 0 sage: latex(f(0, hold=True)) \SIN\left(0\right) sage: f(1,2) Traceback (most recent call last): ... TypeError: Symbolic function my_sin takes exactly 1 arguments (2 given)
- class sage.symbolic.function.BuiltinFunction#
This is the base class for symbolic functions defined in Sage.
If a function is provided by the Sage library, we don’t need to pickle the custom methods, since we can just initialize the same library function again. This allows us to use Cython for custom methods.
We assume that each subclass of this class will define one symbolic function. Make sure you use subclasses and not just call the initializer of this class.
- class sage.symbolic.function.Function#
Base class for symbolic functions defined through Pynac in Sage.
This is an abstract base class, with generic code for the interfaces and a
__call__()method. Subclasses should implement the
Return a default variable.
sage: sin.default_variable() x
Return the name of this function.
sage: foo = function("foo", nargs=2) sage: foo.name() 'foo'
Return the number of arguments that this function takes.
sage: foo = function("foo", nargs=2) sage: foo.number_of_arguments() 2 sage: foo(x,x) foo(x, x) sage: foo(x) Traceback (most recent call last): ... TypeError: Symbolic function foo takes exactly 2 arguments (1 given)
Return the variables (of which there are none) present in this function.
sage: sin.variables() ()
- class sage.symbolic.function.GinacFunction#
This class provides a wrapper around symbolic functions already defined in Pynac/GiNaC.
GiNaC provides custom methods for these functions defined at the C++ level. It is still possible to define new custom functionality or override those already defined.
There is also no need to register these functions.
- class sage.symbolic.function.SymbolicFunction#
This is the basis for user defined symbolic functions. We try to pickle or hash the custom methods, so subclasses must be defined in Python not Cython.
Return a pickled version of the function
None, just return
This is a wrapper around
sage: from sage.symbolic.function import pickle_wrapper sage: def f(x): return x*x sage: isinstance(pickle_wrapper(f), bytes) True sage: pickle_wrapper(None) is None True
Return a unpickled version of the function defined by
None, just return
This is a wrapper around
sage: from sage.symbolic.function import pickle_wrapper, unpickle_wrapper sage: def f(x): return x*x sage: s = pickle_wrapper(f) sage: g = unpickle_wrapper(s) sage: g(2) 4 sage: unpickle_wrapper(None) is None True