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.

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

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

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

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

$\mathrm{aF\_8}≔-3.44406483319975\left(-0.552477515351686+9.46726890028623\times {10}^{\mathrm{-11}}x+1.34863083369451\times {10}^{\mathrm{-9}}y+0.552477517557262x^2+2.03989782080192\times {10}^{\mathrm{-9}}xy+0.552477516011368y^2\right)\left(-0.525550037636229-6.82222680826386\times {10}^{\mathrm{-10}}x-3.43514242374310\times {10}^{\mathrm{-9}}y-7.62609054545801\times {10}^{\mathrm{-10}}{x}^2+6.10950662565009\times {10}^{\mathrm{-10}}xy+6.99437842035687\times {10}^{\mathrm{-10}}{y}^2-0.525550038352441x^3-3.70256461964700\times {10}^{\mathrm{-10}}y{x}^2-8.19581361585493\times {10}^{\mathrm{-10}}{y}^{2}x+0.525550036492460y^3\right)$

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

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

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

$\mathrm{-9}$

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

$\mathrm{aF\_4}≔3.47266293000095\left(-0.552885827441145-9.97108941035032\times {10}^{\mathrm{-6}}x+0.0000179791567477263y+0.552898569944418x^2+8.98886932466931\times {10}^{\mathrm{-6}}xy+0.552899121071388y^2\right)\left(0.520834024267511-5.33436675048057\times {10}^{\mathrm{-6}}x+0.0000175929390281277y+4.50859262571895\times {10}^{\mathrm{-6}}{x}^2-0.0000210937198288496xy+9.51805868064093\times {10}^{\mathrm{-6}}{y}^2+0.520828698460088x^3+2.48682307894181\times {10}^{\mathrm{-6}}y{x}^2-2.34026244980658\times {10}^{\mathrm{-6}}{y}^{2}x-0.520822689868150y^3\right)$

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

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

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

$\mathrm{-5}$

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

$\mathrm{aF\_4I}≔\left(3.33619010286800+0.00138524690482115\mathrm{I}\right)\left(-0.563243378344182+0.\mathrm{I}+\left(3.13353783832099\times {10}^{\mathrm{-10}}-0.0000272294178703771\mathrm{I}\right)x+\left(\mathrm{-9.76499830597157}\times {10}^{\mathrm{-11}}+0.0000398338050312695\mathrm{I}\right)y+\left(0.563243374462419+0.0000246224700712112\mathrm{I}\right)x^2-\left(8.65949653567788\times {10}^{\mathrm{-9}}+0.0000388463734635401\mathrm{I}\right)xy+\left(0.563243379993721+0.0000319121864341227\mathrm{I}\right)y^2\right)\left(0.532173243848354-0.000251648003844275\mathrm{I}+\left(\mathrm{-3.44355626123618}\times {10}^{\mathrm{-11}}+7.37230937854983\times {10}^{\mathrm{-6}}\mathrm{I}\right)x-\left(1.32808396699356\times {10}^{\mathrm{-8}}+0.0000153645729853419\mathrm{I}\right)y-\left(3.07623209457749\times {10}^{\mathrm{-9}}+4.12354158710398\times {10}^{\mathrm{-6}}\mathrm{I}\right)x^2+\left(7.05525063405447\times {10}^{\mathrm{-9}}+0.0000239417134783695\mathrm{I}\right)xy-\left(8.60740959257295\times {10}^{\mathrm{-10}}+0.0000123372409064711\mathrm{I}\right)y^2+\left(0.532173243047371-0.000244232252846877\mathrm{I}\right)x^3-\left(2.53189723598475\times {10}^{\mathrm{-9}}+2.35423160039879\times {10}^{\mathrm{-6}}\mathrm{I}\right)y{x}^2+\left(\mathrm{-3.20721711648609}\times {10}^{\mathrm{-10}}+4.05320710982624\times {10}^{\mathrm{-6}}\mathrm{I}\right)y^{2}x+\left(\mathrm{-0.532173249216916}+0.000240066165610370\mathrm{I}\right)y^3\right)$

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

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

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

$\mathrm{-5}$

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.