#20019: Wrong approximate real roots of cubic equation and matrix eigenvalues
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Reporter: dlaurain | Owner:
Type: PLEASE CHANGE | Status: new
Priority: minor | Milestone: sage-7.1
Component: numerical | Resolution:
Keywords: eigenvalues cubic | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by dlaurain):
Thanks jdemyer.
It's rather confusing, because with the following code (it is the
classical cubic roots computing) :
{{{
# MrcEigenValues returns eigenvalues of matrix mq, roots x of det(xI - mq)
= 0
def MrcEigenValues(mq):
f(x) = mq.charpoly('x') #; print "f : ",f
cf = f.list() #; print " cf = ",cf
a = cf[3] ; b = cf[2] ; c = cf[1] ; d = cf[0] # equation ax^3 + bx^2 +
cx + d = 0
Q = (3*a*c - b^2)/(9*a^2)
R = (9*a*b*c - 27*a^2*d - 2*b^3)/(54*a^3)
U = Q^3 + R^2 ; V = 1
if U < 0:
U = -U ; V = I
S = (R + V*sqrt(U))^(1/3)
T = (R - V*sqrt(U))^(1/3)
eig1 = S + T -b/(3*a)
eig2 = -(S + T)/2 - b/(3*a) + I*(sqrt(3)/2)*(S - T)
eig3 = -(S + T)/2 - b/(3*a) - I*(sqrt(3)/2)*(S - T)
#
return [eig1,eig2,eig3]
m = matrix(SR,3,3, [ [ -0.5898846112288165, -1.2780028056044432,
0.19570118737659], [ -0.6573758889893906, 0.7138094643959463,
0.5958093145901924], [-0.15581144445167106,
-1.128948696332049,0.02254152681632878] ])
e = MrcEigenValues(m)
print "e = ",e
for i in xrange(3):
print "e[",i,"] = ",e[i].n()
}}}
I used the symbolic ring, and I have no approximate values :
{{{
e = [0.665664016441795, -0.30389127563424534*sqrt(3) -
0.2595988182291682, 0.30389127563424534*sqrt(3) - 0.2595988182291682]
e[ 0 ] = 0.665664016442000
e[ 1 ] = -0.785953947604006
e[ 2 ] = 0.266756311146006
}}}
I coded it yesterday and not tested so much.
I posted long time ago a similar post on googlegroups sage and the answer
was same than yours : coerce to the solutions field.
Kind of maxima behaviour with the SR : try to keep all symbolic, rational
and with sqrt ... I am a math newbie in Galois's theory and expressing
roots with radicals.
--
Ticket URL: <http://trac.sagemath.org/ticket/20019#comment:2>
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