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 <[email protected]
> <javascript:>> 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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