问题
I want to develop a GUI application which displays a given mathematical equation. When you click upon a particular variable in the equation to signify that it is the unknown variable ie., to be calculated, the equation transforms itself to evaluate the required unknown variable.
For example:
a = (b+c*d)/e
Let us suppose that I click upon "d" to signify that it is the unknown variable. Then the equation should be re-structured to:
d = (a*e - b)/c
As of now, I just want to know how I can go about rearranging the given equation based on user input. One suggestion I got from my brother was to use pre-fix/post-fix notational representation in back end to evaluate it.
Is that the only way to go or is there any simpler suggestion? Also, I will be using not only basic mathematical functions but also trignometric and calculus (basic I think. No partial differential calculus and all that) as well. I think that the pre/post-fix notation evaluation might not be helpful in evaluation higher mathematical functions.
But that is just my opinion, so please point out if I am wrong. Also, I will be using SymPy for mathematical evaluation so evaluation of a given mathematical equation is not a problem, creating a specific equation from a given generic one is my main problem.
回答1:
Using SymPy, your example would go something like this:
>>> import sympy
>>> a,b,c,d,e = sympy.symbols('abcde')
>>> r = (b+c*d)/e
>>> l = a
>>> r = sympy.solve(l-r,d)
>>> l = d
>>> r
[(-b + a*e)/c]
>>>
It seems to work for trigonometric functions too:
>>> l = a
>>> r = b*sympy.sin(c)
>>> sympy.solve(l-r,c)
[asin(a/b)]
>>>
And since you are working with a GUI, you'll (probably) want to convert back and forth from strings to expressions:
>>> r = '(b+c*d)/e'
>>> sympy.sympify(r)
(b + c*d)/e
>>> sympy.sstr(_)
'(b + c*d)/e'
>>>
or you may prefer to display them as rendered LaTeX or MathML.
回答2:
If you want to do this out of the box, without relying on librairies, I think that the problems you will find are not Python related. If you want to find such equations, you have to describe the heuristics necessary to solve these equations.
First, you have to represent your equation. What about separating:
- operands:
- symbolic operands (a,b)
- numeric operands (1,2)
- operators:
- unary operators (-, trig functions)
- binary operators (+,-,*,/)
Unary operators will obviously enclose one operand, binary ops will enclose two.
What about types?
I think that all of these components should derivate from a single common expression type.
And this class would have a getsymbols method to locate quickly symbols in your expressions.
And then distinguish between unary and binary operators, add a few basic complement/reorder primitives...
Something like:
class expression(object):
def symbols(self):
if not hasattr(self, '_symbols'):
self._symbols = self._getsymbols()
return self._symbols
def _getsymbols(self):
"""
return type: list of strings
"""
raise NotImplementedError
class operand(expression): pass
class symbolicoperand(operand):
def __init__(self, name):
self.name = name
def _getsymbols(self):
return [self.name]
def __str__(self):
return self.name
class numericoperand(operand):
def __init__(self, value):
self.value = value
def _getsymbols(self):
return []
def __str__(self):
return str(self.value)
class operator(expression): pass
class binaryoperator(operator):
def __init__(self, lop, rop):
"""
@type lop, rop: expression
"""
self.lop = lop
self.rop = rop
def _getsymbols(self):
return self.lop._getsymbols() + self.rop._getsymbols()
@staticmethod
def complementop():
"""
Return complement operator:
op.complementop()(op(a,b), b) = a
"""
raise NotImplementedError
def reorder():
"""
for op1(a,b) return op2(f(b),g(a)) such as op1(a,b) = op2(f(a),g(b))
"""
raise NotImplementedError
def _getstr(self):
"""
string representing the operator alone
"""
raise NotImplementedError
def __str__(self):
lop = str(self.lop)
if isinstance(self.lop, operator):
lop = '(%s)' % lop
rop = str(self.rop)
if isinstance(self.rop, operator):
rop = '(%s)' % rop
return '%s%s%s' % (lop, self._getstr(), rop)
class symetricoperator(binaryoperator):
def reorder(self):
return self.__class__(self.rop, self.lop)
class asymetricoperator(binaryoperator):
@staticmethod
def _invert(operand):
"""
div._invert(a) -> 1/a
sub._invert(a) -> -a
"""
raise NotImplementedError
def reorder(self):
return self.complementop()(self._invert(self.rop), self.lop)
class div(asymetricoperator):
@staticmethod
def _invert(operand):
if isinstance(operand, div):
return div(self.rop, self.lop)
else:
return div(numericoperand(1), operand)
@staticmethod
def complementop():
return mul
def _getstr(self):
return '/'
class mul(symetricoperator):
@staticmethod
def complementop():
return div
def _getstr(self):
return '*'
class add(symetricoperator):
@staticmethod
def complementop():
return sub
def _getstr(self):
return '+'
class sub(asymetricoperator):
@staticmethod
def _invert(operand):
if isinstance(operand, min):
return operand.op
else:
return min(operand)
@staticmethod
def complementop():
return add
def _getstr(self):
return '-'
class unaryoperator(operator):
def __init__(self, op):
"""
@type op: expression
"""
self.op = op
@staticmethod
def complement(expression):
raise NotImplementedError
def _getsymbols(self):
return self.op._getsymbols()
class min(unaryoperator):
@staticmethod
def complement(expression):
if isinstance(expression, min):
return expression.op
else:
return min(expression)
def __str__(self):
return '-' + str(self.op)
With this basic structure set up, you should be able to describe a simple heuristic to solve very simple equations. Just think of the simple rules you learned to solve equations, and write them down. That should work :)
And then a very naive solver:
def solve(left, right, symbol):
"""
@type left, right: expression
@type symbol: string
"""
if symbol not in left.symbols():
if symbol not in right.symbols():
raise ValueError('%s not in expressions' % symbol)
left, right = right, left
solved = False
while not solved:
if isinstance(left, operator):
if isinstance(left, unaryoperator):
complementor = left.complement
right = complementor(right)
left = complementor(left)
elif isinstance(left, binaryoperator):
if symbol in left.rop.symbols():
left = left.reorder()
else:
right = left.complementop()(right, left.rop)
left = left.lop
elif isinstance(left, operand):
assert isinstance(left, symbolicoperand)
assert symbol==left.name
solved = True
print symbol,'=',right
a,b,c,d,e = map(symbolicoperand, 'abcde')
solve(a, div(add(b,mul(c,d)),e), 'd') # d = ((a*e)-b)/c
solve(numericoperand(1), min(min(a)), 'a') # a = 1
回答3:
Things have sure changed since 2009. I don't know how your GUI application is going, but this is now possible directly in IPython qtconsole (which one could embed inside a custom PyQt/PySide application, and keep track of all the defined symbols, to allow GUI interaction in a separate listbox, etc.)
(Uses the sympyprt extension for IPython)
回答4:
What you want to do isn't easy. Some equations are quite straight forward to rearrange (like make b the subject of a = b*c+d, which is b = (a-d)/c), while others are not so obvious (like make x the subject of y = x*x + 4*x + 4), while others are not possible (especially when you trigonometric functions and other complications).
As other people have said, check out Sage. It does what you want:
You can solve equations for one variable in terms of others:
sage: x, b, c = var('x b c')
sage: solve([x^2 + b*x + c == 0],x)
[x == -1/2*b - 1/2*sqrt(b^2 - 4*c), x == -1/2*b + 1/2*sqrt(b^2 - 4*c)]
回答5:
Sage has support for symbolic math. You could just use some of the equation manipulating functions built-in:
http://sagemath.org/
来源：https://stackoverflow.com/questions/1010583/mathematical-equation-manipulation-in-python