Aaron,
Thanks for your response! I did find a detour by using juse lambda (not
taking sqrt). Appreciate your help!
Shawn
On Tuesday, February 3, 2015 at 7:26:35 PM UTC-5, Aaron Meurer wrote:
>
> You've discovered that SymPy is really bad at dealing with algebraic
> expressions. I would recommend just sticking to what you did the first
> time, instead of dealing with sqrt(lambda), just use lambda.
>
> By the way, I don't think simplify() uses the new assumptions. You have to
> literally use refine() if you want to simplify things with them.
>
> Aaron Meurer
>
> On Mon, Feb 2, 2015 at 9:52 PM, Yuxiang Wang <wangyux...@gmail.com
> <javascript:>> wrote:
>
>> Hi Aaron,
>>
>> Thank you for your response!
>>
>> 1) You are right that it is a bug. If I do this:
>>
>> from sympy import symbols, sqrt, simplify, Matrix, Q
>> from sympy.assumptions.assume import global_assumptions
>>
>> b11, b22, b33, b12, b13, b23 = symbols('b11, b22, b33, b12, b13, b23',
>> real=True)
>> b = Matrix([[b11, b12, b13], [b12, b22, b23], [b13, b23, b33]])
>> lambda1sq, lambda2sq, lambda3sq = b.eigenvals()
>>
>> Then an error would pop up. Instead, if I do this:
>>
>> from sympy import symbols, sqrt, simplify, Matrix, Q
>> from sympy.assumptions.assume import global_assumptions
>>
>> b11, b22, b33, b12, b13, b23 = symbols('b11, b22, b33, b12, b13, b23')
>> b = Matrix([[b11, b12, b13], [b12, b22, b23], [b13, b23, b33]])
>> for elem in b:
>> global_assumptions.add(Q.real(elem))
>> lambda1sq, lambda2sq, lambda3sq = b.eigenvals()
>>
>> It would calculate alright.
>>
>>
>> 2) I still have a problem, as can be viewed more clearly on here:
>>
>>
>> http://nbviewer.ipython.org/github/yw5aj/ipynb/blob/master/SymbolicOgden.ipynb
>>
>> A brief description is - the sqrt(matrix determinant) is not equal to the
>> product of the eigenvalues of matrix**(1/2).
>>
>> Could you please help a little bit in there...? Would what I am doing
>> even be mathematically correct?
>>
>> Thanks!
>>
>> Shawn
>>
>> On Monday, February 2, 2015 at 2:49:19 PM UTC-5, Aaron Meurer wrote:
>>>
>>>
>>>
>>> On Mon, Feb 2, 2015 at 1:14 PM, Yuxiang Wang <wangyux...@gmail.com>
>>> wrote:
>>>
>>>> Dear all,
>>>>
>>>> I am currently trying to do solid mechanics (finite deformation) in
>>>> SymPy. There are a lot of matrices that are positive definite, but I do
>>>> not
>>>> know whether there is a way to define this property in SymPy. Any help
>>>> would be deeply appreciated!
>>>>
>>>> Take this code snippet for example:
>>>>
>>>>
>>>> ```
>>>> from sympy import symbols, init_printing, sqrt, simplify, Matrix
>>>> init_printing()
>>>>
>>>> # Define a positive-definite real symmetric matrix
>>>> b11, b22, b33, b12, b13, b23 = symbols('b11, b22, b33, b12, b13, b23')
>>>> b = Matrix([[b11, b12, b13], [b12, b22, b23], [b13, b23, b33]])
>>>>
>>>> J = sqrt(b.det())
>>>> lambda1sq, lambda2sq, lambda3sq = b.eigenvals()
>>>> lambda1, lambda2, lambda3 = sqrt(lambda1sq), sqrt(lambda2sq),
>>>> sqrt(lambda3sq)
>>>>
>>>> simplify(lambda1 * lambda2 * lambda3 - J)
>>>> ```
>>>>
>>>> 1) Supposedly, the result of the simplify() should be zero. However it
>>>> is not, because we are not sure whether the eigenvalues of the matrix b
>>>> are
>>>> positive! They are because the matrix should be positive definite, but I
>>>> do
>>>> not know how to define that.
>>>>
>>>
>>> Maybe refine(lambda1, Q.positive(lambda1sq)).
>>>
>>>
>>>>
>>>> 2) Also, I know that I could've used real=True in where I defined the
>>>> symbols of b11, b22, b33.... But I didn't do that. The reason is, if I use
>>>> real=True, then I cannot get the eigenvalues any more - it will say
>>>>
>>>> TypeError: cannot determine truth value of-b11*b22*b33 + b11*b23**2 +
>>>> b12**2*b33 - 2*b12*b13*b23 + b13**2*b22 + 2*(-b11 - b22 - b33)**3/27 -
>>>> (-b11 - b22 - b33)*(b11*b22 + b11*b33 - b12**2 - b13**2 + b22*b33 -
>>>> b23**2)/3 < 0
>>>>
>>>
>>> This error almost always indicates a bug.
>>>
>>> Aaron Meurer
>>>
>>>
>>>>
>>>> Any way to tighten that up as well?
>>>>
>>>> Thanks,
>>>>
>>>> Shawn
>>>>
>>>>
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