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### 65.2 Functions and Variables for hompack

Function: hompack_polsys (eqnlist, varlist [, iflg1, epsbig, epssml, numrr])

  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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