I just have no idea what np.sum((np.log(AV) + Vlam_est)**2)
could possibly mean. np.log(VA) is an array of floats, that is an array of *numbers*. Vlam_est is a *function*. How you can add numbers and a function I do not know.. Vlam_est will become an array of numbers, once you give it the arguments. NB: it seems, that Vi_est uses the arguments alpha,.., eta, L, K When you lambdify it, you skipped the arguments L and K. Any reason for this? On Thu 18. Aug 2022 at 18:18 Zohreh Karimzadeh <z.karimza...@gmail.com> wrote: > It seems always an expression of parameters and independent variables is > needed to be passed to fit and find parameters. > Zohreh Karimzadeh > *https://www.researchgate.net/profile/Zohreh-Karimzadeh* > <https://www.researchgate.net/profile/Zohreh-Karimzadeh> > Skype Name 49a52224a8b6b38b > Twitter Account @zohrehkarimzad1 > z.karimza...@gmail.com > +989102116325 > ((((((((((((((((Value Water))))))))))))))) > > > On Thu, Aug 18, 2022 at 3:28 PM Peter Stahlecker < > peter.stahlec...@gmail.com> wrote: > >> In your first return statement, where it works, you seem to return a >> number. >> In your second return, your a ‚mixture‘ of numbers and functions: >> Vlam_est is a *function*, which requires four arguments as per its >> definition. Would you not have to return Vlam_est(alpha, beta, gamma, eta) ? >> >> On Thu 18. Aug 2022 at 17:35 Zohreh Karimzadeh <z.karimza...@gmail.com> >> wrote: >> >>> the following code is ok when expression is passed as : >>> >>> import numpy as np >>> from scipy.optimize import minimize, curve_fit >>> from lmfit import Model, Parameters >>> >>> L = np.array([0.299, 0.295, 0.290, 0.284, 0.279, 0.273, 0.268, 0.262, >>> 0.256, 0.250]) >>> K = np.array([2.954, 3.056, 3.119, 3.163, 3.215, 3.274, 3.351, 3.410, >>> 3.446, 3.416]) >>> VA = np.array([0.919, 0.727, 0.928, 0.629, 0.656, 0.854, 0.955, 0.981, >>> 0.908, 0.794]) >>> >>> >>> def f(param): >>> gamma = param[0] >>> alpha = param[1] >>> beta = param[2] >>> eta = param[3] >>> VA_est = gamma - (1 / eta) * np.log(alpha * L ** -eta + beta * K ** >>> -eta) >>> >>> return np.sum((np.log(VA) - VA_est) ** 2) >>> >>> >>> bnds = [(1, np.inf), (0, 1), (0, 1), (-1, np.inf)] >>> x0 = (1, 0.01, 0.98, 1) >>> result = minimize(f, x0, bounds=bnds) >>> print(result.message) >>> print(result.x[0], result.x[1], result.x[2], result.x[3]) >>> >>> but when the expression is passed as the following way: >>> >>> import numpy as np >>> import sympy as sp >>> from scipy.optimize import minimize, curve_fit >>> from lmfit import Model, Parameters >>> >>> L = np.array([0.299, 0.295, 0.290, 0.284, 0.279, 0.273, 0.268, 0.262, >>> 0.256, 0.250]) >>> K = np.array([2.954, 3.056, 3.119, 3.163, 3.215, 3.274, 3.351, 3.410, >>> 3.446, 3.416]) >>> VA = np.array([0.919, 0.727, 0.928, 0.629, 0.656, 0.854, 0.955, 0.981, >>> 0.908, 0.794]) >>> >>> >>> def f(param): >>> gamma, alpha, beta, eta = sp.symbols('gamma, alpha, beta, eta') >>> gamma = param[0] >>> alpha = param[1] >>> beta = param[2] >>> eta = param[3] >>> Vi_est = gamma - (1 / eta) * sp.log(alpha * L ** -eta + beta * K ** >>> -eta) >>> Vlam_est = sp.lambdify((gamma, alpha, beta, eta), Vi_est) >>> >>> return np.sum((np.log(VA) - Vlam_est) ** 2) >>> >>> >>> bnds = [(1, np.inf), (0, 1), (0, 1), (-1, np.inf)] >>> x0 = (1, 0.01, 0.98, 1) >>> >>> result = minimize(f, x0, bounds=bnds) >>> >>> print(result.message) >>> print(result.x[0], result.x[1], result.x[2], result.x[3]) >>> >>> >>> I face difficulty: >>> ********************************************* >>> Traceback (most recent call last): >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\cache.py", >>> line 70, in wrapper >>> retval = cfunc(*args, **kwargs) >>> TypeError: unhashable type: 'numpy.ndarray' >>> >>> During handling of the above exception, another exception occurred: >>> >>> Traceback (most recent call last): >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\cache.py", >>> line 70, in wrapper >>> retval = cfunc(*args, **kwargs) >>> TypeError: unhashable type: 'numpy.ndarray' >>> >>> During handling of the above exception, another exception occurred: >>> >>> Traceback (most recent call last): >>> File >>> "F:\Zohreh\MainZohreh\postdoc-field\CSU\pythonProject\fit_test_2.py", line >>> 26, in <module> >>> result = minimize(f, x0, bounds=bnds) >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_minimize.py", >>> line 692, in minimize >>> res = _minimize_lbfgsb(fun, x0, args, jac, bounds, >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_lbfgsb_py.py", >>> line 308, in _minimize_lbfgsb >>> sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, >>> epsilon=eps, >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_optimize.py", >>> line 263, in _prepare_scalar_function >>> sf = ScalarFunction(fun, x0, args, grad, hess, >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_differentiable_functions.py", >>> line 158, in __init__ >>> self._update_fun() >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_differentiable_functions.py", >>> line 251, in _update_fun >>> self._update_fun_impl() >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_differentiable_functions.py", >>> line 155, in update_fun >>> self.f = fun_wrapped(self.x) >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\scipy\optimize\_differentiable_functions.py", >>> line 137, in fun_wrapped >>> fx = fun(np.copy(x), *args) >>> File >>> "F:\Zohreh\MainZohreh\postdoc-field\CSU\pythonProject\fit_test_2.py", line >>> 17, in f >>> Vi_est = gamma - (1 / eta) * sp.log(alpha * L ** -eta + beta * K ** >>> -eta) >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\cache.py", >>> line 74, in wrapper >>> retval = func(*args, **kwargs) >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\function.py", >>> line 476, in __new__ >>> result = super().__new__(cls, *args, **options) >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\cache.py", >>> line 74, in wrapper >>> retval = func(*args, **kwargs) >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\core\function.py", >>> line 288, in __new__ >>> evaluated = cls.eval(*args) >>> File >>> "C:\Users\Zohreh\AppData\Roaming\Python\Python310\site-packages\sympy\functions\elementary\exponential.py", >>> line 718, in eval >>> coeff = arg.as_coefficient(I) >>> AttributeError: 'ImmutableDenseNDimArray' object has no attribute >>> 'as_coefficient' >>> >>> >>> >>> >>> >>> >>> >>> >>> Zohreh Karimzadeh >>> https://www.researchgate.net/profile/Zohreh-Karimzadeh >>> Skype Name 49a52224a8b6b38b >>> Twitter Account @zohrehkarimzad1 >>> z.karimza...@gmail.com >>> +989102116325 >>> ((((((((((((((((Value Water))))))))))))))) >>> >>> Zohreh Karimzadeh >>> *https://www.researchgate.net/profile/Zohreh-Karimzadeh* >>> <https://www.researchgate.net/profile/Zohreh-Karimzadeh> >>> Skype Name 49a52224a8b6b38b >>> Twitter Account @zohrehkarimzad1 >>> z.karimza...@gmail.com >>> +989102116325 >>> ((((((((((((((((Value Water))))))))))))))) >>> >>> >>> On Thu, Aug 18, 2022 at 10:42 AM Peter Stahlecker < >>> peter.stahlec...@gmail.com> wrote: >>> >>>> I use lambdify quite a bit, on rather large expressions. >>>> Basically, it always works like this for me: >>>> >>>> import sympy as sm >>>> x1, x2, …, xn = sm.symbols(‚x1, x2, ….., xn‘) >>>> …. >>>> … >>>> expr = some expression of generally with me: sm.sin, sm.cos, sm.exp, >>>> sm.sqrt, >>>> sm.Heaviside, etc.. >>>> This expression may have 50,000 terms, may be an (axb) matrix, whatever. >>>> >>>> expr_lam = sm.lambdify([x1, x2, …,xn], expr) >>>> >>>> Now I can evaluate expr_lam(…) like I would evaluate any numpy function. >>>> >>>> I have no idea, what expr_lam looks like, I would not know how to look >>>> at it. >>>> I assume, it converts sm.sin(..) to np.sin(…), etc >>>> >>>> This is how it works for me. >>>> As I do not really understand your points, like ‚dynamically created‘, >>>> ‚parse and subs‘, this may be of not help at all for you. >>>> >>>> Peter >>>> >>>> >>>> On Thu 18. Aug 2022 at 09:21 Zohreh Karimzadeh <z.karimza...@gmail.com> >>>> wrote: >>>> >>>>> Before run I import sp.sqrt or sp.exp but after run they get >>>>> disappeared. My expression is big and dynamically created and not >>>>> possible to parse and subs np.exp or sp.exp. >>>>> >>>>> Zohreh Karimzadeh >>>>> >>>>> Contact me on >>>>> +989102116325 >>>>> and at >>>>> z.karimza...@gmail.com >>>>> 🌧️🌍🌱 >>>>> >>>>> >>>>> On Thu, 18 Aug 2022, 01:17 Aaron Meurer, <asmeu...@gmail.com> wrote: >>>>> >>>>>> Your expression uses "sqrt" but you haven't imported it from >>>>>> anywhere, since you only did "import sympy as sp". You need to use >>>>>> sp.sqrt. >>>>>> >>>>>> Aaron Meurer >>>>>> >>>>>> On Wed, Aug 17, 2022 at 11:02 AM Zohreh Karimzadeh < >>>>>> z.karimza...@gmail.com> wrote: >>>>>> >>>>>>> Here is my code: >>>>>>> >>>>>>> import matplotlib.pyplot as plt >>>>>>> import numpy as np >>>>>>> import sympy as sp >>>>>>> import pandas as pd >>>>>>> #exp_NaCl path: F:\Zohreh\MainZohreh\postdoc-field\CSU\Duplicat_Pure >>>>>>> df = >>>>>>> pd.read_excel(r'F:\Zohreh\MainZohreh\postdoc-field\CSU\Duplicat_Pure\data.xlsx', >>>>>>> sheet_name='NaCl_exp') >>>>>>> XNa = df['XNa'] >>>>>>> XCl = df['XCl'] >>>>>>> Xwater = df['Xwater'] >>>>>>> Y = df['gama_x'] >>>>>>> L=['WwaterNaCl', 'UwaterNaCl', 'VwaterNaCl', 'XCl', 'XNa', 'Xwater', >>>>>>> 'BNaCl'] >>>>>>> for j in range(len(L)): >>>>>>> locals()[L[j]] = sp.symbols(L[j]) >>>>>>> expr = >>>>>>> -0.0118343195266272*BNaCl*XCl*XNa*(-2*(9.19238815542512*sqrt(XNa) + >>>>>>> 9.19238815542512*sqrt(XCl + XNa) + 1)*exp(-9.19238815542512*sqrt(XNa) - >>>>>>> 9.19238815542512*sqrt(XCl + XNa)) + 2)/((XCl + XNa)*(sqrt(XNa) + >>>>>>> sqrt(XCl + XNa))**2) + >>>>>>> 0.00591715976331361*BNaCl*XCl*(-2*(9.19238815542512*sqrt(XNa) + >>>>>>> 9.19238815542512*sqrt(XCl + XNa) + 1)*exp(-9.19238815542512*sqrt(XNa) - >>>>>>> 9.19238815542512*sqrt(XCl + XNa)) + 2)/(sqrt(XNa) + sqrt(XCl + XNa))**2 >>>>>>> + 0.00591715976331361*BNaCl*XNa*(-2*(9.19238815542512*sqrt(XNa) + >>>>>>> 9.19238815542512*sqrt(XCl + XNa) + 1)*exp(-9.19238815542512*sqrt(XNa) - >>>>>>> 9.19238815542512*sqrt(XCl + XNa)) + 2)/(sqrt(XNa) + sqrt(XCl + XNa))**2 >>>>>>> - 1.0*Cl*WwaterNaCl*Xwater*(0.5*XCl + 0.5*XNa + 0.5)/XCl - >>>>>>> 0.5*Cl*WwaterNaCl/XCl - 4.0*UwaterNaCl*XCl*XNa*Xwater + >>>>>>> 2.0*UwaterNaCl*XCl*Xwater + 2.0*UwaterNaCl*XNa*Xwater - >>>>>>> 4.0*UwaterNaCl*XNa - 6.0*VwaterNaCl*XCl*XNa*Xwater**2 - >>>>>>> 4.0*VwaterNaCl*XCl*Xwater**2 + 2.0*VwaterNaCl*XNa*Xwater**2 - >>>>>>> 1.0*WwaterNaCl*Xwater*(0.5*XCl + 0.5*XNa + 0.5) + 2.0*WwaterNaCl*Xwater >>>>>>> - 0.5*WwaterNaCl - 1.45739430799067*(0.707106781186548*sqrt(XNa) + >>>>>>> 0.707106781186548*sqrt(XCl + XNa))*(-XCl - XNa + >>>>>>> 1)/(9.19238815542512*sqrt(XNa) + 9.19238815542512*sqrt(XCl + XNa) + 1) >>>>>>> - 1.45739430799067*(0.707106781186548*sqrt(XNa) + >>>>>>> 0.707106781186548*sqrt(XCl + XNa))*(-1.4142135623731*sqrt(XNa) - >>>>>>> 1.4142135623731*sqrt(XCl + XNa) + 1)/(9.19238815542512*sqrt(XNa) + >>>>>>> 9.19238815542512*sqrt(XCl + XNa) + 1) - >>>>>>> 0.448429017843282*log(9.19238815542512*sqrt(XNa) + >>>>>>> 9.19238815542512*sqrt(XCl + XNa) + 1) >>>>>>> model_func = sp.lambdify(L, expr ) >>>>>>> >>>>>>> def f(param): >>>>>>> BNaCl = param[0] >>>>>>> UwaterNaCl = param[1] >>>>>>> VwaterNaCl = param[2] >>>>>>> WwaterNaCl = param[3] >>>>>>> Y_est = model_func >>>>>>> return np.sum((np.log(Y) - Y_est)**2) >>>>>>> >>>>>>> >>>>>>> bnds = [(1, np.inf), (0, 1), (0, 1), (-1, np.inf)] >>>>>>> x0 = (1, 0.01, 0.98, 1) >>>>>>> con = {"type": "eq", "fun": c} >>>>>>> >>>>>>> result = minimize(f, x0, bounds=bnds) >>>>>>> >>>>>>> print(result.fun) >>>>>>> print(result.message) >>>>>>> print(result.x[0], result.x[1], result.x[2], result.x[3]) >>>>>>> >>>>>>> while I got : >>>>>>> NameError: name 'sqrt' is not defined >>>>>>> >>>>>>> Zohreh Karimzadeh >>>>>>> *https://www.researchgate.net/profile/Zohreh-Karimzadeh* >>>>>>> <https://www.researchgate.net/profile/Zohreh-Karimzadeh> >>>>>>> Skype Name 49a52224a8b6b38b >>>>>>> Twitter Account @zohrehkarimzad1 >>>>>>> z.karimza...@gmail.com >>>>>>> +989102116325 >>>>>>> >>>>>>> ((((((((((((((((Value Water))))))))))))))) >>>>>>> >>>>>>> >>>>>>> On Wed, Aug 17, 2022 at 7:46 PM Peter Stahlecker < >>>>>>> peter.stahlec...@gmail.com> wrote: >>>>>>> >>>>>>>> I use lambdify(....) a lot, but always like this: >>>>>>>> >>>>>>>> x = sympy.symbols('x') >>>>>>>> expr = symy.S(10.) * sympy.sqrt(x) >>>>>>>> expr_lam = sympy.lambdify([x], expr) >>>>>>>> >>>>>>>> a = expr_lam(10.) >>>>>>>> >>>>>>>> This seems to work for me. >>>>>>>> >>>>>>>> On Wed 17. Aug 2022 at 20:38, Zohreh Karimzadeh < >>>>>>>> z.karimza...@gmail.com> wrote: >>>>>>>> >>>>>>>>> Dear sympy group >>>>>>>>> Thanks for your sympy. >>>>>>>>> >>>>>>>>> I am working on a code, after creating my big expression using >>>>>>>>> sympy it includes sqrt. >>>>>>>>> >>>>>>>>> I need to lambdify my expression to make it consistent with numpy >>>>>>>>> and other suffs. >>>>>>>>> >>>>>>>>> expr =10 * sp.sqrt(sp.symbols('x')) >>>>>>>>> >>>>>>>>> model_func = sp.lambdify('x', expr) >>>>>>>>> >>>>>>>>> But I found my expression after lambdifying becomes somethings >>>>>>>>> like this: >>>>>>>>> >>>>>>>>> 10*sqrt(x) >>>>>>>>> >>>>>>>>> while I need : >>>>>>>>> >>>>>>>>> 10*numpy.sqrt(x) >>>>>>>>> >>>>>>>>> Could possibly let me know how get sqrt to work with numpy? >>>>>>>>> >>>>>>>>> Regards, >>>>>>>>> >>>>>>>>> Zohreh >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> -- >>>>>>>>> You received this message because you are subscribed to the Google >>>>>>>>> Groups "sympy" group. >>>>>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>>>>> send an email to sympy+unsubscr...@googlegroups.com. >>>>>>>>> To view this discussion on the web visit >>>>>>>>> https://groups.google.com/d/msgid/sympy/1f0b313f-31c5-402e-991e-142a556016f4n%40googlegroups.com >>>>>>>>> <https://groups.google.com/d/msgid/sympy/1f0b313f-31c5-402e-991e-142a556016f4n%40googlegroups.com?utm_medium=email&utm_source=footer> >>>>>>>>> . >>>>>>>>> >>>>>>>> -- >>>>>>>> Best regards, >>>>>>>> >>>>>>>> Peter Stahlecker >>>>>>>> >>>>>>>> -- >>>>>>>> You received this message because you are subscribed to the Google >>>>>>>> Groups "sympy" group. >>>>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>>>> send an email to sympy+unsubscr...@googlegroups.com. >>>>>>>> To view this discussion on the web visit >>>>>>>> https://groups.google.com/d/msgid/sympy/CABKqA0ZoGwsadsk4SWCbJVMbCDwXcO_gNGumJH00GAeEFod7Cw%40mail.gmail.com >>>>>>>> <https://groups.google.com/d/msgid/sympy/CABKqA0ZoGwsadsk4SWCbJVMbCDwXcO_gNGumJH00GAeEFod7Cw%40mail.gmail.com?utm_medium=email&utm_source=footer> >>>>>>>> . >>>>>>>> >>>>>>> -- >>>>>>> You received this message because you are subscribed to the Google >>>>>>> Groups "sympy" group. >>>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>>> send an email to sympy+unsubscr...@googlegroups.com. >>>>>>> To view this discussion on the web visit >>>>>>> https://groups.google.com/d/msgid/sympy/CA%2B1XYLPRvXZ6jiJbUS_xpWNKqMuUH7Kt5evue%2BwKEwDMvGekBQ%40mail.gmail.com >>>>>>> <https://groups.google.com/d/msgid/sympy/CA%2B1XYLPRvXZ6jiJbUS_xpWNKqMuUH7Kt5evue%2BwKEwDMvGekBQ%40mail.gmail.com?utm_medium=email&utm_source=footer> >>>>>>> . >>>>>> >>>>>> >>>>>>> -- >>>>>> You received this message because you are subscribed to the Google >>>>>> Groups "sympy" group. >>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>> send an email to sympy+unsubscr...@googlegroups.com. >>>>>> To view this discussion on the web visit >>>>>> https://groups.google.com/d/msgid/sympy/CAKgW%3D6JfUmU7Uu%2BSrcA1STxVvWWm7bGWE%3Dit8CTchksTC0Qk7g%40mail.gmail.com >>>>>> <https://groups.google.com/d/msgid/sympy/CAKgW%3D6JfUmU7Uu%2BSrcA1STxVvWWm7bGWE%3Dit8CTchksTC0Qk7g%40mail.gmail.com?utm_medium=email&utm_source=footer> >>>>>> . >>>>>> >>>>> -- >>>>> You received this message because you are subscribed to the Google >>>>> Groups "sympy" group. >>>>> To unsubscribe from this group and stop receiving emails from it, send >>>>> an email to sympy+unsubscr...@googlegroups.com. >>>>> To view this discussion on the web visit >>>>> https://groups.google.com/d/msgid/sympy/CA%2B1XYLPiCR%3DS2Fac3FZtjMpspqB7BRKtYEi45BVWPjkizVbNvw%40mail.gmail.com >>>>> <https://groups.google.com/d/msgid/sympy/CA%2B1XYLPiCR%3DS2Fac3FZtjMpspqB7BRKtYEi45BVWPjkizVbNvw%40mail.gmail.com?utm_medium=email&utm_source=footer> >>>>> . >>>>> >>>> -- >>>> Best regards, >>>> >>>> Peter 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<https://groups.google.com/d/msgid/sympy/CA%2B1XYLMK-fgpxc71GYzue5gJvd%3Dfj2sV6Dvhj8zrmVpPhiVk%2Bw%40mail.gmail.com?utm_medium=email&utm_source=footer> >>> . >>> >> -- >> Best regards, >> >> Peter Stahlecker >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to sympy+unsubscr...@googlegroups.com. >> > To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/CABKqA0aBsD2WhpTuQqzZGUK1pfkUyH4q3Om9DdBQOpoaaO4rqQ%40mail.gmail.com >> <https://groups.google.com/d/msgid/sympy/CABKqA0aBsD2WhpTuQqzZGUK1pfkUyH4q3Om9DdBQOpoaaO4rqQ%40mail.gmail.com?utm_medium=email&utm_source=footer> >> . >> > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sympy+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/CA%2B1XYLNiQNVa_hg25e-_f8xs%2B2w88p7JC4ntneBrqO4YFajTgA%40mail.gmail.com > <https://groups.google.com/d/msgid/sympy/CA%2B1XYLNiQNVa_hg25e-_f8xs%2B2w88p7JC4ntneBrqO4YFajTgA%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > -- Best regards, Peter Stahlecker -- You received this message because you are subscribed to the Google Groups "sympy" group. 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