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.

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

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

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

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

$\mathrm{aF\_8}≔-3.44406483254625\left(-0.552477515342034+9.46732507922508\times {10}^{\mathrm{-11}}x+1.34863064253940\times {10}^{\mathrm{-9}}y+0.552477517547611x^2+2.03989800976321\times {10}^{\mathrm{-9}}xy+0.552477516001717y^2\right)\left(-0.525550037745132-6.82222634889839\times {10}^{\mathrm{-10}}x-3.43514283815260\times {10}^{\mathrm{-9}}y-7.62609279982288\times {10}^{\mathrm{-10}}{x}^2+6.10950431663182\times {10}^{\mathrm{-10}}xy+6.99438216295762\times {10}^{\mathrm{-10}}{y}^2-0.525550038461344x^3-3.70256764216089\times {10}^{\mathrm{-10}}y{x}^2-8.19581622671544\times {10}^{\mathrm{-10}}{y}^{2}x+0.525550036601363y^3\right)$

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

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

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

$\mathrm{-9}$

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

$\mathrm{aF\_4}≔3.47266293000014\left(-0.552885827441133-9.97108941066633\times {10}^{\mathrm{-6}}x+0.0000179791567478723y+0.552898569944407x^2+8.98886932458016\times {10}^{\mathrm{-6}}xy+0.552899121071376y^2\right)\left(0.520834024267642-5.33436675047966\times {10}^{\mathrm{-6}}x+0.0000175929390278785y+4.50859262542559\times {10}^{\mathrm{-6}}{x}^2-0.0000210937198294713xy+9.51805868081277\times {10}^{\mathrm{-6}}{y}^2+0.520828698460220x^3+2.48682307870096\times {10}^{\mathrm{-6}}y{x}^2-2.34026244956660\times {10}^{\mathrm{-6}}{y}^{2}x-0.520822689868282y^3\right)$

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

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

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

$\mathrm{-5}$

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

$\mathrm{aF\_4I}≔\left(3.33619010335823+0.00138405371522788\mathrm{I}\right)\left(-0.563243378346106+0.\mathrm{I}+\left(3.15399383486919\times {10}^{\mathrm{-10}}-0.0000272293890141111\mathrm{I}\right)x+\left(\mathrm{-9.99721735744224}\times {10}^{\mathrm{-11}}+0.0000398337975347528\mathrm{I}\right)y+\left(0.563243374464763+0.0000246224743767121\mathrm{I}\right)x^2-\left(8.65008148054685\times {10}^{\mathrm{-9}}+0.0000388463671328584\mathrm{I}\right)xy+\left(0.563243379992084+0.0000319121754359758\mathrm{I}\right)y^2\right)\left(0.532173243933621-0.000251457665365619\mathrm{I}+\left(\mathrm{-3.38789333275673}\times {10}^{\mathrm{-11}}+7.37229906536008\times {10}^{\mathrm{-6}}\mathrm{I}\right)x-\left(1.32709965941180\times {10}^{\mathrm{-8}}+0.0000153645870134051\mathrm{I}\right)y-\left(3.07395251093478\times {10}^{\mathrm{-9}}+4.12354418671067\times {10}^{\mathrm{-6}}\mathrm{I}\right)x^2+\left(7.05065898735183\times {10}^{\mathrm{-9}}+0.0000239417006586643\mathrm{I}\right)xy-\left(8.60670859921420\times {10}^{\mathrm{-10}}+0.0000123372268822169\mathrm{I}\right)y^2+\left(0.532173243133265-0.000244041924962185\mathrm{I}\right)x^3-\left(2.53074133071583\times {10}^{\mathrm{-9}}+2.35423260814531\times {10}^{\mathrm{-6}}\mathrm{I}\right)y{x}^2+\left(\mathrm{-3.19409226541232}\times {10}^{\mathrm{-10}}+4.05319818908638\times {10}^{\mathrm{-6}}\mathrm{I}\right)y^{2}x+\left(\mathrm{-0.532173249298037}+0.000239875827130997\mathrm{I}\right)y^3\right)$

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

$1.x^5+1.00000x^3{y}^2-1.00000x^2{y}^3-1.00000y^5+0.000115711\mathrm{I}{x}^{3}y-1.00000x^3+1.00000y^3+1.00000x^2-0.000113958\mathrm{I}xy+1.00000y^2-1.00000+0.\mathrm{I}$

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

$\mathrm{-5}$

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.