I'm sorry for the deluge of questions, but I hope you will bear with
me as I come up to speed. This one is more substantive and I think
related to my first question yesterday about simplify() not
simplifying enough. Now, I have two equations with two unknowns I'm
trying to solve and I'm getting 3 solutions. Two are correct, but the
third yields a division by zero error if I try to plug it back into
the equations. And the real problem is that the solution I care about
is missing! These particular two equations are symmetric, so I could
(and have) solved it by subsituting x1 for x2 and solving just one
equation with one unknown. That works and that's how I know the
missing solution. But I want to solve slightly different sets of
equations that are _not_ symmetric and the fact that sage/Maxima fails
me here makes me pessimistic that my results for the asymmetric
versions are correct.
Here's my code:
var('x1, x2')
eqn1 = (1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
/ ((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) ==
0)
eqn2 = (1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
/ ((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) ==
0)
print "eqn1: ", eqn1
print "eqn2: ", eqn1
sols = solve([eqn1, eqn2], x1, x2, solution_dict=True)
print "number of sols: ", len(sols)
print "sols: ", sols
foo = eqn1.lhs().subs(x1=sqrt(5) - 2, x2=sqrt(5) - 2)
print "What about sqrt(5)-2 in eqn1?", foo
bar = eqn2.lhs().subs(x1=sqrt(5) - 2, x2=sqrt(5) - 2)
print "What about sqrt(5)-2 in eqn2?", bar
print "But these both simplify to zero after a simplify_full: ",
foo.simplify_full()
print "But these both simplify to zero after a simplify_full: ",
bar.simplify_full()
print "The third solution I do get yields a divide by zero error, but
I'm willing to forgive this:"
eqn1.lhs().subs(sols[2])
*** And here's what I get when I run it: ***
sage: load problem.sage
eqn1: 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)/((x1^2 +
2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) == 0
eqn2: 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)/((x1^2 +
2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2) == 0
number of sols: 3
sols: [{x2: -sqrt(5), x1: -(sqrt(5) - 10)/(2*sqrt(5) - 1)}, {x2: sqrt
(5), x1: -(sqrt(5) + 10)/(2*sqrt(5) + 1)}, {x2: 0, x1: 0}]
What about sqrt(5)-2 in eqn1? 1/2*((sqrt(5) - 2)^2*((sqrt(5) - 2)^2 +
2*sqrt(5) - 8) + (sqrt(5) - 2)^2 + 2*(sqrt(5) - 2)*((sqrt(5) - 2)^2 +
sqrt(5) - 2))/((sqrt(5) - 2)^2*((sqrt(5) - 2)^2 + 2*sqrt(5) - 3) +
(sqrt(5) - 2)^2 + 2*(sqrt(5) - 2)*((sqrt(5) - 2)^2 + sqrt(5) - 2))
What about sqrt(5)-2 in eqn2? 1/2*((sqrt(5) - 2)^2*((sqrt(5) - 2)^2 +
2*sqrt(5) - 3) - 4*(sqrt(5) - 2)^2 + 2*(sqrt(5) - 2)*((sqrt(5) - 2)^2
+ sqrt(5) - 2))/((sqrt(5) - 2)^2*((sqrt(5) - 2)^2 + 2*sqrt(5) - 3) +
(sqrt(5) - 2)^2 + 2*(sqrt(5) - 2)*((sqrt(5) - 2)^2 + sqrt(5) - 2))
But these both simplify to zero after a simplify_full: 0
But these both simplify to zero after a simplify_full: 0
The third solution I do get yields a divide by zero error, but I'm
willing to forgive this:
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call
last)
...
RuntimeError: power::eval(): division by zero
As usual, any help would be greatly appreciated!
Thank you,
Doug
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