With the default 'complex' domain, we don't simplify this::
- sage: (abs(x)^2).simplify()
- abs(x)^2
+ sage: sqrt(x^2).simplify()
+ sqrt(x^2)
But in the 'real' domain, we do::
sage: set_simplification_domain('real')
'real'
- sage: (abs(x)^2).simplify()
- x^2
+ sage: sqrt(x^2).simplify()
+ abs(x)
sage: set_simplification_domain('complex')
'complex'
expr = expr.simplify_factorial()
expr = expr.simplify_log()
return expr
-
-
-def matrix_subs_expr(m, *equations):
- """
- Symbolic matrices have a `subs()` method, but no `subs_expr()`.
- This makes it diffucult to substitute in a list of solutions obtained
- with `solve()`.
-
- INPUT:
-
- - ``m`` -- A symbolic matrix.
-
- - ``equations`` - One or more symbolic equations, presumably for
- the entries of `m`.
-
- OUTPUT:
-
- The result of substituting each equation into `m`, one after another.
-
- EXAMPLES::
-
- sage: w,x,y,z = SR.var('w,x,y,z')
- sage: A = matrix(SR, [[w,x],[y,z]])
- sage: matrix_subs_expr(A, w == 1, x == 2, y == 3, z == 4)
- [1 2]
- [3 4]
-
- """
- from sage.symbolic.expression import is_SymbolicEquation
-
- if not m.base_ring() == SR:
- raise TypeError, 'the matrix "m" must be symbolic'
-
- if isinstance(equations[0], dict):
- eq_dict = equations[0]
- equations = [ x == eq_dict[x] for x in eq_dict.keys() ]
-
- if not all([is_SymbolicEquation(eq) for eq in equations]):
- raise TypeError, "each expression must be an equation"
-
- d = dict([(eq.lhs(), eq.rhs()) for eq in equations])
- return m.subs(d)
-
-
-def matrix_simplify_full(A):
- """
- Simplify each entry of a symbolic matrix using the
- Expression.simplify_full() method.
-
- INPUT:
-
- - ``A`` - The matrix whose entries we should simplify.
-
- OUTPUT:
-
- A copy of ``A`` with all of its entries simplified.
-
- EXAMPLES:
-
- Symbolic matrices (examples stolen from Expression.simplify_full())
- will have their entries simplified::
-
- sage: a,n,k = SR.var('a,n,k')
- sage: f1 = sin(x)^2 + cos(x)^2
- sage: f2 = sin(x/(x^2 + x))
- sage: f3 = binomial(n,k)*factorial(k)*factorial(n-k)
- sage: f4 = x*sin(2)/(x^a)
- sage: A = matrix(SR, [[f1,f2],[f3,f4]])
- sage: matrix_simplify_full(A)
- [ 1 sin(1/(x + 1))]
- [ factorial(n) x^(-a + 1)*sin(2)]
-
- But an exception will be raised if ``A`` is not symbolic::
-
- sage: A = matrix(QQ, [[1,2],[3,4]])
- sage: matrix_simplify_full(A)
- Traceback (most recent call last):
- ...
- ValueError: The base ring of `A` must be the Symbolic Ring.
-
- """
- if not A.base_ring() == SR:
- raise ValueError('The base ring of `A` must be the Symbolic Ring.')
-
- M = A.matrix_space()
- return M(map(lambda x: x.simplify_full(), A))