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DeterminantalRepresentations :: bivariateDiagEntries

bivariateDiagEntries -- computes diagonal entries and eigenvalues for a determinantal representation of a bivariate polynomial

Synopsis

Description

This method computes the eigenvalues and diagonal entries of a monic symmetric determinantal representation of a real bivariate polynomial f, or gives an error if certain necessary conditions for existence of such a representation are not met. For a symmetric determinantal representation f = det(I + x1A1 + x2A2), this method computes diagonal entries and eigenvalues of A1 and A2. The output is a 4-tuple of column vectors: (eigenvalues of A1, eigenvalues of A2, diagonal entries of A1, diagonal entries of A2).

When working over an InexactFieldFamily like RR or CC, the option Tolerance can be used to specify the internal threshold for checking equality (any floating point number below the tolerance is treated as numerically zero).

i1 : R = RR[x1, x2]

o1 = R

o1 : PolynomialRing
i2 : f = 15*x1^2 + 20*x1*x2 - 36*x2^2 + 20*x1 + 16*x2 + 1

         2                 2
o2 = 15x1  + 20x1*x2 - 36x2  + 20x1 + 16x2 + 1

o2 : R
i3 : bivariateDiagEntries f

o3 = (| 19.2195 |, | 18 |, | 17 |, | 15.5926 |)
      | .780456 |  | -2 |  | 3  |  | .407434 |

o3 : Sequence

See also

Ways to use bivariateDiagEntries :