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PolynomialTools[Approximate]

  

Factor

  

compute approximate factorization

 

Calling Sequence

Parameters

Options

Description

Examples

References

Compatibility

Calling Sequence

Factor(F, vars)

Factor(F, vars, options)

Parameters

F

-

polynom({numeric,complex(numeric)})

vars

-

set or list of variables

Options

• 

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.

Description

• 

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.

Examples

withPolynomialTools:-Approximate:

Fsortexpandx2+y21x3y3+1,x,y

Fx5+x3y2x2y3y5x3+y3+x2+y21

(1)

aF_8FactorexpandF+108xy,x,y

aF_83.444064833199750.5524775153516869.46729745577350×10−11x1.34863065022021×10−9y0.552477517557262x22.03989792764670×10−9xy0.552477516011369y20.5255500376362306.82222545322290×10−10x3.43514241338219×10−9y7.62609004008427×10−10x2+6.10950533033342×10−10xy+6.99437727875974×10−10y20.525550038352441x33.70256462533906×10−10yx28.19581364011395×10−10y2x+0.525550036492460y3

(2)

sortfnormalexpandaF_8,x,y

1.x5+0.9999999988x3y20.9999999958x2y30.9999999937y50.9999999947x3+0.9999999990y3+0.9999999972x2+0.9999999972y20.9999999946

(3)

ilog10normexpandFaF_8,2normF,2

−9

(4)

aF_4FactorexpandF+104xy,x,y

aF_43.472662930000940.5528858274411459.97108941000953×10−6x+0.0000179791567479051y+0.552898569944418x2+8.98886932467533×10−6xy+0.552899121071388y20.5208340242675115.33436675006424×10−6x+0.0000175929390281242y+4.50859262542757×10−6x20.0000210937198292382xy+9.51805868057149×10−6y2+0.520828698460089x3+2.48682307869211×10−6yx22.34026244961223×10−6y2x0.520822689868150y3

(5)

sortfnormalexpandaF_4,6,x,y

1.00001x5+1.00000x3y20.999991x2y30.999996y50.999994x3+1.00001y3+1.00001x2+1.00000y20.999994

(6)

ilog10normexpandFaF_4,2normF,2

−5

(7)

aF_4IFactorexpandF+104Ixy,x,y

aF_4I−3.336190102508570.00138612068811785I0.5632433783427660.I+−3.11854963158353×10−10+0.0000272294190559388Ix+9.59484023866126×10−110.0000398338053388040Iy0.563243374460697+0.0000246224698940000Ix2+8.66638205488023×10−9+0.0000388463737235990Ixy0.563243379994912+0.0000319121868867062Iy20.5321732437858970.000251787385887176I+−3.48434702385247×10−11+7.37230980399005×10−6Ix1.32880444269199×10−8+0.0000153645724041642Iy3.07790140400092×10−9+4.12354147889653×10−6Ix2+7.05861311843272×10−9+0.0000239417140052928Ixy8.60791491508778×10−10+0.0000123372414848047Iy2+0.5321732429844540.000244371634452508Ix32.53274232099494×10−9+2.35423155822770×10−6Iyx2+−3.21682849408933×10−10+4.05320747808123×10−6Iy2x+−0.532173249157492+0.000240205547653828Iy3

(8)

sortfnormalexpandaF_4I,6,x,y

1.x5+1.00000x3y21.00000x2y31.00000y5+0.000115711Ix3y1.00000x3+1.00000y3+1.00000x20.000113958Ixy+1.00000y21.00000+0.I

(9)

ilog10normexpandFaF_4I,2normF,2

−5

(10)

References

  

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.

Compatibility

• 

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

• 

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

See Also

factor

RuppertMatrix