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SumsOfSquares :: coefficient field

coefficient field -- the role of the coefficient field

The SumsOfSquares package works with two coefficient rings: the rational numbers $\QQ$ and the real numbers $\RR$. Almost all operations in this package rely on a numerical semidefinite programming Solver. When calling such a solver, even if the input was a polynomial with rational coefficients, the result is numerical. The package makes some effort to round and return a rational result, but this can fail, independent of whether a rational sum-of-squares decomposition exists or not. In case of failure, a real result is returned. The following example of Scheiderer is a sum of squares, but does not admit any rational decomposition. Consequently the package must return a real solution.

i1 : f = library("Scheiderer", QQ[x,y,z])

      4      3    4     2          2      2 2      3      3    4
o1 = x  + x*y  + y  - 3x y*z - 4x*y z + 2x z  + x*z  + y*z  + z

o1 : QQ[x..z]
i2 : sol = solveSOS (f);
i3 : sosPoly sol

                         2                         2                                     2 2                       2                          2                                       2 2                      2                        2                                     2 2                      2                       2                                     2 2                    2                        2                                    2 2                     2                       2                                     2 2
o3 = 3.30451e-9*(.347571x  - .0338169x*y + .768917y  + .336255x*z - .203872y*z + .363573z )  + 5.71263e-9*(.318116x  + .679504x*y - .00674275y  + .148906x*z + .644087y*z - .00320062z )  + .612609*(- .489507x  + .515354x*y - .174264y  + .126103x*z - .330005y*z + .582769z )  + .889193*(- .389141x  + .18471x*y + .191399y  + .590218x*z - .140298y*z - .640138z )  + 1.85689*(.622038x  + .195424x*y - .441075y  + .199982x*z - .56497y*z - .145418z )  + 3.9924*(.00407971x  - .446355x*y - .38363y  + .678607x*z + .309532y*z + .311869z )

o3 : SOSPoly

Given a sum-of-squares decomposition with real coefficients, it is often useful to ignore the squares with very small coefficients. The function clean(RR,SOSPoly) removes the squares whose coefficients are smaller than a given tolerance.

i4 : clean (0.001, sosPoly sol)

                        2                        2                                     2 2                      2                       2                                     2 2                    2                        2                                    2 2                     2                       2                                     2 2
o4 = .612609*(- .489507x  + .515354x*y - .174264y  + .126103x*z - .330005y*z + .582769z )  + .889193*(- .389141x  + .18471x*y + .191399y  + .590218x*z - .140298y*z - .640138z )  + 1.85689*(.622038x  + .195424x*y - .441075y  + .199982x*z - .56497y*z - .145418z )  + 3.9924*(.00407971x  - .446355x*y - .38363y  + .678607x*z + .309532y*z + .311869z )

o4 : SOSPoly

See also