calculus -- bugs in solve

[williamstein]

We get a "solution" from solve() that isn't actually a solution:

sage: f(x) = (sin(x) - 8*cos(x)*sin(x))*(sin(x)^2 + cos(x)) - (2*cos(x)*sin(x) - sin(x))*(-2*sin(x)^2 + 2*cos(x)^2 - cos(x))
sage: solve(f(x), x)
[x == pi, x == 1/2*pi, x == 0]
sage: f(pi/2)
-1

The following is correct:

sage: solve(f(x).simplify_trig(), x)
[x == 0, x == pi - arccos(1/3), x == pi]

Reduced example (after manually removing the factor sin(x)):

sage: g(x) = (1 - 8*cos(x))*(sin(x)^2 + cos(x)) - (2*cos(x) - 1)*(-2*sin(x)^2 + 2*cos(x)^2 - cos(x))
sage: solve(g(x), x)
[x == pi, x == 1/2*pi]
sage: g(pi/2)
-1

Upstream: Reported upstream. No feedback yet.

Component: calculus

Stopgaps: todo

Issue created by migration from https://trac.sagemath.org/ticket/3745

[rlmill]

comment:1

Also:

sage: var('a b c')
sage: eqn1 = a - exp((-pi*b)/sqrt(1-b)) == 0
sage: eqn2 = c - atan(2*b*sqrt(1/(sqrt(4*b^4+1) - 2*b^2)))==0
sage: solve([eqn1,eqn2,a==0.1975],b,c,a) 
[]

[rlmill]

comment:3

And even:

sage: var('a,b,c,d')
sage: m = matrix(2,[a,b,c,d])
sage: i2=identity_matrix(SR,2)
sage: eqlist=[(m*m).list()[i] - i2.list()[i] for i in range(4)]
sage: solve(eqlist,a,b,c,d) 
Traceback (most recent call last):
...
ValueError: Unable to solve [b*c + a^2 - 1, b*d + a*b, c*d + a*c, d^2 + b*c - 1] for (a, b, c, d)

[kcrisman]

comment:4

Note this particular bug is still in Maxima as of 5.19.1. More bugs (but also lots more correct answers) have been introduced in the last year, and other bugs have been fixed. Writing a solve from scratch still looks very hard.

[kcrisman]

comment:5

Update from Maxima 5.20.1 in Sage:

sage: sage: var('a,b,c,d')
(a, b, c, d)
sage: sage: m = matrix(2,[a,b,c,d])
sage: sage: i2=identity_matrix(SR,2)
sage: sage: eqlist=[(m*m).list()[i] - i2.list()[i] for i in range(4)]
sage: sage: solve(eqlist,a,b,c,d) 
[a^2 + b*c - 1, a*b + b*d, a*c + c*d, b*c + d^2 - 1]

so this one seems to be working now, at least in the sense that it doesn't throw an error.

The second one now causes a hang.

And the first one is still there :(

[williamstein]

comment:6

We should write our own native solve from scratch. Depending on maxima for solve is silly, for several reasons:

• For algebraic systems we could do much better using Groebner basis and singular.

• Coefficients for symbolic expressions can be in many base rings that don't even make sense to Maxima.

</irrelevant rant>

[kcrisman]

comment:7

It could be worth trying these with #13364.