.. _guide: ================== SymPy User's Guide ================== .. role:: input(strong) Introduction ============ If you are new to SymPy, start with the :ref:`Tutorial `. If you went through it, now it's time to learn how SymPy works internally, which is what this guide is about. Once you grasp the ideas behind SymPy, you will be able to use it effectively and also know how to extend it and fix it. You may also be just interested in :ref:`SymPy Modules Reference `. Learning SymPy ============== Everyone has different ways of understanding the code written by others. Ondřej's approach ----------------- Let's say I'd like to understand how ``x + y + x`` works and how it is possible that it gets simplified to ``2*x + y``. I write a simple script, I usually call it ``t.py`` (I don't remember anymore why I call it that way):: from sympy.abc import x, y e = x + y + x print e And I try if it works .. parsed-literal:: $ :input:`python t.py` y + 2*x Now I start `winpdb `_ on it (if you've never used winpdb -- it's an excellent multiplatform debugger, works on Linux, Windows and Mac OS X): .. parsed-literal:: $ :input:`winpdb t.py` y + 2*x and a winpdb window will popup, I move to the next line using F6: .. image:: pics/winpdb1.png Then I step into (F7) and after a little debugging I get for example: .. image:: pics/winpdb2.png .. tip:: Make the winpdb window larger on your screen, it was just made smaller to fit in this guide. I see values of all local variables in the left panel, so it's very easy to see what's happening. You can see, that the ``y + 2*x`` is emerging in the ``obj`` variable. Observing that ``obj`` is constructed from ``c_part`` and ``nc_part`` and seeing what ``c_part`` contains (``y`` and ``2*x``). So looking at the line 28 (the whole line is not visible on the screenshot, so here it is):: c_part, nc_part, lambda_args, order_symbols = cls.flatten(map(_sympify, args)) you can see that the simplification happens in ``cls.flatten``. Now you can set the breakpoint on the line 28, quit winpdb (it will remember the breakpoint), start it again, hit F5, this will stop at this breakpoint, hit F7, this will go into the function ``Add.flatten()``:: @classmethod def flatten(cls, seq): """ Takes the sequence "seq" of nested Adds and returns a flatten list. Returns: (commutative_part, noncommutative_part, lambda_args, order_symbols) Applies associativity, all terms are commutable with respect to addition. """ terms = {} # term -> coeff # e.g. x**2 -> 5 for ... + 5*x**2 + ... coeff = S.Zero # standalone term # e.g. 3 + ... lambda_args = None order_factors = [] while seq: o = seq.pop(0) and then you can study how it works. I am going to stop here, this should be enough to get you going -- with the above technique, I am able to understand almost any Python code. SymPy's Architecture ==================== We try to make the sources easily understandable, so you can look into the sources and read the doctests, it should be well documented and if you don't understand something, ask on the mailinglist_. You can find all the decisions archived in the issues_, to see rationale for doing this and that. Basics ------ All symbolic things are implemented using subclasses of the ``Basic`` class. First, you need to create symbols using ``Symbol("x")`` or numbers using ``Integer(5)`` or ``Float(34.3)``. Then you construct the expression using any class from SymPy. For example ``Add(Symbol("a"), Symbol("b"))`` gives an instance of the ``Add`` class. You can call all methods, which the particular class supports. For easier use, there is a syntactic sugar for expressions like: ``cos(x) + 1`` is equal to ``cos(x).__add__(1)`` is equal to ``Add(cos(x), Integer(1))`` or ``2/cos(x)`` is equal to ``cos(x).__rdiv__(2)`` is equal to ``Mul(Rational(2), Pow(cos(x), Rational(-1)))``. So, you can write normal expressions using python arithmetics like this:: a = Symbol("a") b = Symbol("b") e = (a + b)**2 print e but from the SymPy point of view, we just need the classes ``Add``, ``Mul``, ``Pow``, ``Rational``, ``Integer``. Automatic evaluation to canonical form -------------------------------------- For computation, all expressions need to be in a canonical form, this is done during the creation of the particular instance and only inexpensive operations are performed, necessary to put the expression in the canonical form. So the canonical form doesn't mean the simplest possible expression. The exact list of operations performed depend on the implementation. Obviously, the definition of the canonical form is arbitrary, the only requirement is that all equivalent expressions must have the same canonical form. We tried the conversion to a canonical (standard) form to be as fast as possible and also in a way so that the result is what you would write by hand - so for example ``b*a + -4 + b + a*b + 4 + (a + b)**2`` becomes ``2*a*b + b + (a + b)**2``. Whenever you construct an expression, for example ``Add(x, x)``, the ``Add.__new__()`` is called and it determines what to return. In this case:: >>> from sympy import Add >>> from sympy.abc import x >>> e = Add(x, x) >>> e 2*x >>> type(e) ``e`` is actually an instance of ``Mul(2, x)``, because ``Add.__new__()`` returned ``Mul``. Comparisons ----------- Expressions can be compared using a regular python syntax:: >>> from sympy.abc import x, y >>> x + y == y + x True >>> x + y == y - x False We made the following decision in SymPy: ``a = Symbol("x")`` and another ``b = Symbol("x")`` (with the same string "x") is the same thing, i.e ``a == b`` is ``True``. We chose ``a == b``, because it is more natural - ``exp(x) == exp(x)`` is also ``True`` for the same instance of ``x`` but different instances of ``exp``, so we chose to have ``exp(x) == exp(x)`` even for different instances of ``x``. Sometimes, you need to have a unique symbol, for example as a temporary one in some calculation, which is going to be substituted for something else at the end anyway. This is achieved using ``Dummy("x")``. So, to sum it up:: >>> from sympy import Symbol, Dummy >>> Symbol("x") == Symbol("x") True >>> Dummy("x") == Dummy("x") False Debugging --------- Starting with 0.6.4, you can turn on/off debug messages with the environment variable ``SYMPY_DEBUG``, which is expected to have the values True or False. For example, to turn on debugging, you would issue:: [user@localhost]: SYMPY_DEBUG=True ./bin/isympy Functionality ------------- There are no given requirements on classes in the library. For example, if they don't implement the ``fdiff()`` method and you construct an expression using such a class, then trying to use the ``Basic.series()`` method will raise an exception of not finding the ``fdiff()`` method in your class. This "duck typing" has an advantage that you just implement the functionality which you need. You can define the ``cos`` class like this:: class cos(Function): pass and use it like ``1 + cos(x)``, but if you don't implement the ``fdiff()`` method, you will not be able to call ``(1 + cos(x)).series()``. The symbolic object is characterized (defined) by the things which it can do, so implementing more methods like ``fdiff()``, ``subs()`` etc., you are creating a "shape" of the symbolic object. Useful things to implement in new classes are: ``hash()`` (to use the class in comparisons), ``fdiff()`` (to use it in series expansion), ``subs()`` (to use it in expressions, where some parts are being substituted) and ``series()`` (if the series cannot be computed using the general ``Basic.series()`` method). When you create a new class, don't worry about this too much - just try to use it in your code and you will realize immediately which methods need to be implemented in each situation. All objects in sympy are immutable - in the sense that any operation just returns a new instance (it can return the same instance only if it didn't change). This is a common mistake to change the current instance, like ``self.arg = self.arg + 1`` (wrong!). Use ``arg = self.arg + 1; return arg`` instead. The object is immutable in the sense of the symbolic expression it represents. It can modify itself to keep track of, for example, its hash. Or it can recalculate anything regarding the expression it contains. But the expression cannot be changed. So you can pass any instance to other objects, because you don't have to worry that it will change, or that this would break anything. Conclusion ---------- Above are the main ideas behind SymPy that we try to obey. The rest depends on the current implementation and may possibly change in the future. The point of all of this is that the interdependencies inside SymPy should be kept to a minimum. If one wants to add new functionality to SymPy, all that is necessary is to create a subclass of ``Basic`` and implement what you want. Functions --------- How to create a new function with one variable:: class sign(Function): nargs = 1 @classmethod def eval(cls, arg): if isinstance(arg, Basic.NaN): return S.NaN if isinstance(arg, Basic.Zero): return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if isinstance(arg, Basic.Mul): coeff, terms = arg.as_coeff_mul() if not isinstance(coeff, Basic.One): return cls(coeff) * cls(Basic.Mul(*terms)) is_finite = True def _eval_conjugate(self): return self def _eval_is_zero(self): return isinstance(self[0], Basic.Zero) and that's it. The ``_eval_*`` functions are called when something is needed. The ``eval`` is called when the class is about to be instantiated and it should return either some simplified instance of some other class or if the class should be unmodified, return ``None`` (see ``core/function.py`` in ``Function.__new__`` for implementation details). See also tests in `sympy/functions/elementary/tests/test_interface.py `_ that test this interface. You can use them to create your own new functions. The applied function ``sign(x)`` is constructed using :: sign(x) both inside and outside of SymPy. Unapplied functions ``sign`` is just the class itself:: sign both inside and outside of SymPy. This is the current structure of classes in SymPy:: class BasicType(type): pass class MetaBasicMeths(BasicType): ... class BasicMeths(AssumeMeths): __metaclass__ = MetaBasicMeths ... class Basic(BasicMeths): ... class FunctionClass(MetaBasicMeths): ... class Function(Basic, RelMeths, ArithMeths): __metaclass__ = FunctionClass ... The exact names of the classes and the names of the methods and how they work can be changed in the future. This is how to create a function with two variables:: class chebyshevt_root(Function): nargs = 2 @classmethod def eval(cls, n, k): if not 0 <= k < n: raise ValueError("must have 0 <= k < n") return cos(S.Pi*(2*k + 1)/(2*n)) .. note:: the first argument of a @classmethod should be ``cls`` (i.e. not ``self``). Here it's how to define a derivative of the function:: >>> from sympy import Function, sympify, cos >>> class my_function(Function): ... nargs = 1 ... ... def fdiff(self, argindex = 1): ... return cos(self.args[0]) ... ... @classmethod ... def eval(cls, arg): ... arg = sympify(arg) ... if arg == 0: ... return sympify(0) So guess what this ``my_function`` is going to be? Well, it's derivative is ``cos`` and the function value at 0 is 0, but let's pretend we don't know:: >>> from sympy import pprint >>> pprint(my_function(x).series(x, 0, 10)) 3 5 7 9 x x x x / 10\ x - -- + --- - ---- + ------ + O\x / 6 120 5040 362880 Looks familiar indeed:: >>> from sympy import sin >>> pprint(sin(x).series(x, 0, 10)) 3 5 7 9 x x x x / 10\ x - -- + --- - ---- + ------ + O\x / 6 120 5040 362880 Let's try a more complicated example. Let's define the derivative in terms of the function itself:: >>> class what_am_i(Function): ... nargs = 1 ... ... def fdiff(self, argindex = 1): ... return 1 - what_am_i(self.args[0])**2 ... ... @classmethod ... def eval(cls, arg): ... arg = sympify(arg) ... if arg == 0: ... return sympify(0) So what is ``what_am_i``? Let's try it:: >>> pprint(what_am_i(x).series(x, 0, 10)) 3 5 7 9 x 2*x 17*x 62*x / 10\ x - -- + ---- - ----- + ----- + O\x / 3 15 315 2835 Well, it's ``tanh``:: >>> from sympy import tanh >>> pprint(tanh(x).series(x, 0, 10)) 3 5 7 9 x 2*x 17*x 62*x / 10\ x - -- + ---- - ----- + ----- + O\x / 3 15 315 2835 The new functions we just defined are regular SymPy objects, you can use them all over SymPy, e.g.:: >>> from sympy import limit >>> limit(what_am_i(x)/x, x, 0) 1 Common tasks ------------ Please use the same way as is shown below all across SymPy. **accessing parameters**:: >>> from sympy import sign, sin >>> from sympy.abc import x, y, z >>> e = sign(x**2) >>> e.args (x**2,) >>> e.args[0] x**2 Number arguments (in Adds and Muls) will always be the first argument; other arguments might be in arbitrary order: >>> (1 + x + y*z).args[0] 1 >>> (1 + x + y*z).args[1] in (x, y*z) True >>> (y*z).args (y, z) >>> sin(y*z).args (y*z,) Never use internal methods or variables, prefixed with "``_``" (example: don't use ``_args``, use ``.args`` instead). **testing the structure of a SymPy expression** Applied functions:: >>> from sympy import sign, exp, Function >>> e = sign(x**2) >>> isinstance(e, sign) True >>> isinstance(e, exp) False >>> isinstance(e, Function) True So ``e`` is a ``sign(z)`` function, but not ``exp(z)`` function. Unapplied functions:: >>> from sympy import sign, exp, FunctionClass >>> e = sign >>> f = exp >>> g = Add >>> isinstance(e, FunctionClass) True >>> isinstance(f, FunctionClass) True >>> isinstance(g, FunctionClass) False >>> g is Add True So ``e`` and ``f`` are functions, ``g`` is not a function. Contributing ============ We welcome every SymPy user to participate in it's development. Don't worry if you've never contributed to any open source project, we'll help you learn anything necessary, just ask on our mailinglist_. Don't be afraid to ask anything and don't worry that you are wasting our time if you are new to SymPy and ask questions that maybe most of the people know the answer to -- you are not, because that's exactly what the mailinglist_ is for and people answer your emails because they want to. Also we try hard to answer every email, so you'll always get some feedback and pointers what to do next. Improving the code ------------------ Go to issues_ that are sorted by priority and simply find something that you would like to get fixed and fix it. If you find something odd, please report it into issues_ first before fixing it. Feel free to consult with us on the mailinglist_. Then send your patch either to the issues_ or the mailinglist_. Please read our excellent `SymPy Patches Tutorial `_ at our wiki for a guide on how to write patches to SymPy, how to work with Git, and how to make your life easier as you get started with SymPy. .. _issues: https://github.com/sympy/sympy/issues .. _mailinglist: https://groups.google.com/forum/#!forum/sympy Improving the docs ------------------ Please see :ref:`the documentation ` how to fix and improve SymPy's documentation. All contribution is very welcome.