Greetings,
I've found that SageMath (8.3 and 8.6) gives me incorrect results when
integrating the simple integrand sqrt(1-x^2-y^2). The code that follows
illustrates the issue:
banner()
(x,y) = var('x y')
eps = var('eps',latex_name='\\epsilon')
assume(abs(x)<0.1, abs(y) < 0.1, abs(eps) < 0.1, eps > 0)
z = sqrt(1-x^2-y^2)
print(z)
print('Integrating directly')
int1 = z.integrate(x,-eps,eps).integrate(y,-eps,eps)
print(N(int1.substitute(eps==1e-4))) # Should be about 4e-8
print(taylor(int1,(eps,0),4))
print('Integrating via Taylor series')
int2 = taylor(z,(x,0),(y,0),8).integrate(x,-eps,eps).integrate(y,-eps,eps)
print(N(int2.substitute(eps==1e-4)))
print(taylor(int2,(eps,0),4))
Which prints (on sagecell.sagemath.org, to ensure a new version is being
tested):
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 8.6, Release Date: 2019-01-15 │
│ Using Python 2.7.15. Type "help()" for help. │
└────────────────────────────────────────────────────────────────────┘
sqrt(-x^2 - y^2 + 1)
Integrating directly
1.99999998666667e-8
-4/3*eps^4 + 2*eps^2
Integrating via Taylor series
3.99999998666667e-8
-4/3*eps^4 + 4*eps^2
Note the factor of 2 difference in the lowest-order term.
This may be an issue in one of the libraries called by Sage, but I'm not
experienced enough with the ecosystem to know which one (if any) could be at
fault.
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