Previous: Introduction to hompack [Contents][Index]
Finds the roots of the system of polynomials in the variables varlist in the system of equations in eqnlist. The number of equations must match number of variables. Each equation must be a polynomial with variables in varlist. The coefficients must be real numbers.
The optional keyword arguments provide some control over the algorithm. epsbig is the local error tolerance allowed by the path tracker, defaulting to 1e-4. epssml is the accuracy desired for the final solution, defaulting to 1d-14.. iflg1, defaulting to 0, controls the algorithm as follows:
0
If the problem is to be solved without calling polsys
’ scaling
routine, sclgnp
, and without using the projective
transformation.
1
If scaling but no projective transformation is to be used.
10
If no scaling but projective transformation is to be used.
11
If both scaling and projective transformation are to be used.
hompack_polsys
returns a list. The elements of the list are:
retcode
Indicates whether the solution is valid or not.
0
Solution found without problems
1
Solution succeeded but iflg2
indicates some issues with a
root. (That is, iflg2
is not all ones.)
-1
NN
, the declared dimension of the number of terms in the
polynomials, is too small. (This should not happen.)
-2
MMAXT
, the declared dimension for the internal coefficient and
degree arrays, is too small. (This should not happen.)
-3
TTOTDG
, the total degree of the equations, is too small.
(This should not happen.)
-4
LENWK
, the length of the internal real work array, is too
small. (This should not happen.)
-5
LENIWK
, the length of the internal integer work array, is too
small. (This should not happen.)
-6
iflg1 is not 0 or 1, or 10 or 11. (This should not happen; an
error should be thrown before polsys
is called.)
roots
The roots of the system of equations. This is in the same format as
solve
would return.
iflg2
Information on how the path for the m’th root terminated:
1
Normal return
2
Specified error tolerance cannot be met. Increase epsbig and epssml and rerun.
3
Maximum number of steps exceeded. To track the path further, increase numrr and rerun the path. However, the path may be diverging, if the lambda value is near 1 and the roots values are large.
4
Jacobian matrix does not have full rank. The algorithm has failed (the zero curve of the homotopy map cannot be followed any further).
5
The tracking algorithm has lost the zero curve of the homotopy map and is not making progress. The error tolerances epsbig and epssml were too lenient. The problem should be restarted with smaller error tolerances.
6
The normal flow newton iteration in stepnf
or rootnf
failed to converge. The error tolerance epsbig may be too
stringent.
7
Illegal input parameters, a fatal error.
lambda
The final lambda value for the m-th root, where lambda is the continuation parameter.
arclen
The arc length of the m-th root.
nfe
The number of jacobian matrix evaluations required to track the m-th root.
(%i1) hompack_polsys([x1^2-1, x2^2-2],[x1,x2]); (%o1) [0, [[x1 = (-1.354505666901954e-16*%i)-0.9999999999999999, x2 = 3.52147935979316e-16*%i-1.414213562373095], [x1 = 1.0-5.536432658639868e-18*%i, x2 = (-4.213674137126362e-17*%i)-1.414213562373095], [x1 = (-9.475939894034927e-17*%i)-1.0, x2 = 2.669654624736742e-16*%i+1.414213562373095], [x1 = 9.921253413273088e-18*%i+1.0, x2 = 1.414213562373095-5.305667769855424e-17*%i]],[1,1,1,1], [1.0,1.0,0.9999999999999996,1.0], [4.612623769341193,4.612623010859902,4.612623872939383, 4.612623114484402],[40,40,40,40]]
The analytical solution can be obtained with solve:
(%i2) solve([x1^2-1, x2^2-2],[x1,x2]); (%o2) [[x1 = 1,x2 = -sqrt(2)],[x1 = 1,x2 = sqrt(2)],[x1 = -1,x2 = -sqrt(2)], [x1 = -1,x2 = sqrt(2)]]
We see that hompack_polsys
returned the correct answer except
that the roots are in a different order and there is a small imaginary
part.
Another example, with corresponding solution from solve:
(%i1) hompack_polsys([x1^2 + 2*x2^2 + x1*x2 - 5, 2*x1^2 + x2^2 + x2-4],[x1,x2]); (%o1) [0, [[x1 = 1.201557301700783-1.004786320788336e-15*%i, x2 = (-4.376615092392437e-16*%i)-1.667270363480143], [x1 = 1.871959754090949e-16*%i-1.428529189565313, x2 = (-6.301586314393093e-17*%i)-0.9106199083334113], [x1 = 0.5920619420732697-1.942890293094024e-16*%i, x2 = 6.938893903907228e-17*%i+1.383859154368197], [x1 = 7.363503717463654e-17*%i+0.08945540033671608, x2 = 1.557667481081721-4.109128293931921e-17*%i]],[1,1,1,1], [1.000000000000001,1.0,1.0,1.0], [6.205795654034752,7.722213259390295,7.228287079174351, 5.611474283583368],[35,41,48,40]] (%i2) solve([x1^2+2*x2^2+x1*x2 - 5, 2*x1^2+x2^2+x2-4],[x1,x2]); (%o2) [[x1 = 0.08945540336850383,x2 = 1.557667386609071], [x1 = 0.5920619554695062,x2 = 1.383859286083807], [x1 = 1.201557352500749,x2 = -1.66727025803531], [x1 = -1.428529150636283,x2 = -0.9106198942815954]]
Note that hompack_polsys
can sometimes be very slow. Perhaps
solve
can be used. Or perhaps eliminate
can be used to
convert the system of polynomials into one polynomial for which
allroots
can find all the roots.
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