noexact

if provided, exact factorization of F will not be attempted

optimize

if given then a post-processing step is done on the output, using Optimization:-NLPSolve to return an approximate factorization with smaller backward error. Optionally, it can be given as optimize=list with a list of extra options to be passed to optimization.

After a series of initial preprocessing steps designed to handle exact and degenerate cases, numerical factors of F are found from the a low rank approximation of its RuppertMatrix.

This command works for univariate polynomials by calling factor which finds the real linear and quadratic factors from the roots.

$\mathrm{with}\left(\mathrm{PolynomialTools}:-\mathrm{Approximate}\right)\:$

$F≔\mathrm{sort}\left(\mathrm{expand}\left(\left(x^2\+y^2-1\right)\left(x^3-y^3\+1\right)\right)\,\left[x\,y\right]\right)$

$F≔x^5+x^3{y}^2-x^2{y}^3-y^5-x^3+y^3+x^2+y^2-1$

$\mathrm{aF\_8}≔\mathrm{Factor}\left(\mathrm{expand}\left(F\+{10}^{-8}xy\right)\,\left[x\,y\right]\right)$

$\mathrm{aF\_8}≔-3.44406483254625\left(-0.552477515342034+9.46731687426374\times {10}^{\mathrm{-11}}x+1.34863064328525\times {10}^{\mathrm{-9}}y+0.552477517547611x^2+2.03989803334921\times {10}^{\mathrm{-9}}xy+0.552477516001717y^2\right)\left(-0.525550037745132-6.82222843397422\times {10}^{\mathrm{-10}}x-3.43514285807329\times {10}^{\mathrm{-9}}y-7.62609217128201\times {10}^{\mathrm{-10}}{x}^2+6.10950366484575\times {10}^{\mathrm{-10}}xy+6.99438205675422\times {10}^{\mathrm{-10}}{y}^2-0.525550038461344x^3-3.70256663288842\times {10}^{\mathrm{-10}}y{x}^2-8.19581626434466\times {10}^{\mathrm{-10}}{y}^{2}x+0.525550036601363y^3\right)$

$\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF\_8}\right)\right)\,\left[x\,y\right]\right)$

$1.x^5+0.9999999988x^3{y}^2-0.9999999958x^2{y}^3-0.9999999937y^5-0.9999999947x^3+0.9999999990y^3+0.9999999972x^2+0.9999999972y^2-0.9999999946$

$\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF\_8}\right)\,2\right)}{\mathrm{norm}\left(F\,2\right)}\right)$

$\mathrm{-9}$

$\mathrm{aF\_4}≔\mathrm{Factor}\left(\mathrm{expand}\left(F\+{10}^{-4}xy\right)\,\left[x\,y\right]\right)$

$\mathrm{aF\_4}≔3.47266293000014\left(-0.552885827441133-9.97108941071252\times {10}^{\mathrm{-6}}x+0.0000179791567479040y+0.552898569944407x^2+8.98886932463720\times {10}^{\mathrm{-6}}xy+0.552899121071376y^2\right)\left(0.520834024267643-5.33436675036546\times {10}^{\mathrm{-6}}x+0.0000175929390281592y+4.50859262549109\times {10}^{\mathrm{-6}}{x}^2-0.0000210937198292715xy+9.51805868068816\times {10}^{\mathrm{-6}}{y}^2+0.520828698460220x^3+2.48682307880352\times {10}^{\mathrm{-6}}y{x}^2-2.34026244957104\times {10}^{\mathrm{-6}}{y}^{2}x-0.520822689868282y^3\right)$

$\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF\_4}\right)\,6\right)\,\left[x\,y\right]\right)$

$1.00001x^5+1.00000x^3{y}^2-0.999991x^2{y}^3-0.999996y^5-0.999994x^3+1.00001y^3+1.00001x^2+1.00000y^2-0.999994$

$\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF\_4}\right)\,2\right)}{\mathrm{norm}\left(F\,2\right)}\right)$

$\mathrm{-5}$

$\mathrm{aF\_4I}≔\mathrm{Factor}\left(\mathrm{expand}\left(F\+{10}^{-4}Ixy\right)\,\left[x\,y\right]\right)$

$\mathrm{aF\_4I}≔\left(3.33619010359970+0.00138346582082882\mathrm{I}\right)\left(-0.563243378347060+0.\mathrm{I}+\left(3.16408238493893\times {10}^{\mathrm{-10}}-0.0000272293643435069\mathrm{I}\right)x+\left(\mathrm{-1.01117917827161}\times {10}^{\mathrm{-10}}+0.0000398337911251992\mathrm{I}\right)y+\left(0.563243374465927+0.0000246224780583257\mathrm{I}\right)x^2-\left(8.64543563279926\times {10}^{\mathrm{-9}}+0.0000388463617204220\mathrm{I}\right)xy+\left(0.563243379991283+0.0000319121660331265\mathrm{I}\right)y^2\right)\left(0.532173243975585-0.000251363881658109\mathrm{I}+\left(\mathrm{-3.36046554181828}\times {10}^{\mathrm{-11}}+7.37229024894790\times {10}^{\mathrm{-6}}\mathrm{I}\right)x-\left(1.32661446390877\times {10}^{\mathrm{-8}}+0.0000153645990045513\mathrm{I}\right)y-\left(3.07282917970803\times {10}^{\mathrm{-9}}+4.12354640896998\times {10}^{\mathrm{-6}}\mathrm{I}\right)x^2+\left(7.04839785801415\times {10}^{\mathrm{-9}}+0.0000239416896984294\mathrm{I}\right)xy-\left(8.60637220725151\times {10}^{\mathrm{-10}}+0.0000123372148927410\mathrm{I}\right)y^2+\left(0.532173243175538-0.000243948150310564\mathrm{I}\right)x^3-\left(2.53017116814042\times {10}^{\mathrm{-9}}+2.35423346961460\times {10}^{\mathrm{-6}}\mathrm{I}\right)y{x}^2+\left(\mathrm{-3.18762218104142}\times {10}^{\mathrm{-10}}+4.05319056283696\times {10}^{\mathrm{-6}}\mathrm{I}\right)y^{2}x+\left(\mathrm{-0.532173249337958}+0.000239782043422948\mathrm{I}\right)y^3\right)$

$\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF\_4I}\right)\,6\right)\,\left[x\,y\right]\right)$

$1.00000x^5+1.00000x^3{y}^2-1.00000x^2{y}^3-1.00000y^5+0.000115711\mathrm{I}{x}^{3}y-1.00000x^3+1.00000y^3+1.00000x^2-0.000113958\mathrm{I}xy+1.00000y^2-1.00000+0.\mathrm{I}$

$\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF\_4I}\right)\,2\right)}{\mathrm{norm}\left(F\,2\right)}\right)$

$\mathrm{-5}$

Gao, S.; Kaltofen, E.; May, J.; Yang, Z.; and Zhi, L. "Approximate factorization of multivariate polynomials via differential equations." Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC 2004),  pp. 167-174. Ed. J. Guitierrez. ACM Press, 2004.

Kaltofen, E.; May, J.; Yang, Z.; and Zhi, L. "Approximate factorization of multivariate polynomials using singular value decomposition." Journal of Symbolic Computation Vol. 43(5), (2008): 359-376.

The PolynomialTools:-Approximate:-Factor command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.