Symbolic-Numeric Sparse Interpolation of Multivariate Polynomials

Transcription

1 Symbolic-Numeric Sparse Inerpolaion o Mulivariae Polynomials Mark Giesbrech George Labahn Wen-shin Lee School o Compuer Science Universiy o Waerloo

2 Approimae inerpolaion o polynomials Suppose we can sample he value o a mulivariae polynomial y 0 0 How do we recover he eplici represenaion? 5y 0 y3 0 5y 4y4 y Wha i n is sparse?

3 Approimae inerpolaion o polynomials (con) When n c d dn n is sparse: In eac arihmeic: Zippel s probabilisic inerpolaion (979) variable by variable Ben-Or/Tiwari deerminisic algorihm (988) all variables a once Wha abou loaing poin arihmeic?

4 Ben-Or/Tiwari algorihm: y 5y 0 y3 0 5y 4y4 Pick p Evaluae a 0 sequence a 0 0 p relaively prime: say p 30 a 3 a, p a a is linearly generaed Compue he minimal generaor G z algorihm or solving a Hankel sysem G z z z3 3 3 by Berlekamp/Massey z z Roos o G z are non-zero erms in a 0 p, yp: y 4 y 4 y y 3 Recover coeiciens by solving a Vandermonde sysem

5 Gaspard Clair Franois Marie Riche de Prony Essai epérimenal e analyique sur les lois de la dilaabilié e sur celles de la orce epansive de la vapeur de l eau e de la vapeur de l alkool, à di erenes empéraures J de l École Polyechnique :4 76, 795 For a uncion F :, and ind F, δ such ha F F e δ 0,

6 pi n Mehods: Prony (795) Ben-Or/Tiwari (988) A sum o eponenial uncions F c eδ n A polynomial c d dn n Pick p pn relaively prime Solve g, i0 0 g F i Evaluae F 0 F : Compue he minimal G ha F i generaes p i i 0 F Evaluae p 0 p p e δ are zeros o p d G z z g z g0 G z pdn n z g are zeros o z g0 3 Deermine c rom e δ i and 3 Deermine c rom p d evaluaions o F and evaluaions o pdn n

7 F F Numerical challenges in Prony s mehod When inding he generaing polynomial G z z g z g0 done via solving F 0 F F F F F F g 0 g g F F F H 0 Hankel sysem H 0 is oen ill-condiioned Roo-inding is no a good hing o do numerically: very sensiive o perurbaions in g To recover coeiciens c in F c, eδ need o solve a Vandermonde sysem ha migh be ill-condiioned

8 p p b Numerical challenges in Prony s mehod () Consider univariae d b pd H 0 ma : min 0? p p p p p p b b b b b b b b b H 0 V D V Tr Diicul o ind roos in G (and when hey re no) z i b b are clusered Moreover, in Ben-Or/Tiwari algorihm Recover numerical mulivariae erms?

9 On he uni circle: primiive roos o uniy d p deg, ω eπi p β ωd ω ω ω ω ω ω ω ω β β β β β β β β β β H 0 V D V Tr Beer or condiioning: H 0 V Tr and V Tr ma k k β βk Roo inding is easier: perurbing k-h coeicien o by ε k changes a roo β by ε β The eponens d can be recovered: powers o he primiive roo o uniy β

10 ω However, β may sill be clusered on he uniy circle poorly condiioned V H 0 poorly condiioned Randomly choose a roo o uniy: ω random s p esπi p d β ωd Wih high probabiliy, ω d disribued relaively uniormly around he uni circle ω ω ω ω ω ω ω β β β β β β β β β β H 0 V D V Tr Wih high probabiliy: V well-condiioned k β βk (no oo small) H 0 beer-condiioned

11 pi n ωi n Mulivariae case: primiive roos o uniy wih orders relaively prime m ω k p ep s k πi pn p k and p ω i pn relaively prime Each erm β is a power o ω m ep d Compue round log ωm β Recover eponens d remainder algorihm: d d s m p dn πi m by he (reverse) Chinese dn s n m p n Numerical mulivariae erms can be recovered Recall: In he eac arihmeic, he original Ben-Or/Tiwari algorihm evaluae or p p i pn relaively prime

12 Generalized eigenvalue reormulaion Golub, Milanar, and Varah 999 For F F e δ F F F F v β F F 0 F F v H H 0 has (eigenvalue) soluions e δ eδ or β avoid consrucion o he generaing polynomial G avoid roo inding on generaing polynomial G he generalized eigenvalue problem has numerically more sable algorihms (even when no a roos o uniy) z z

13 p Sparse inerpolaion via generalized eigenvalues d p 0 p p p p p p p v β p p p p 3 p p p p p v H 0 H Soluions or β : β β Obain candidaes or d β approimae p d round log p β d pd round pd log p β Compue by solving he associaed Vandermonde sysem

14 Sparsiy in he arge polynomial? Binary search guess T Cabay-Meleshko algorihm (993) deec irs badly condiioned leading minor o H 0 Knowledge o sparsiy may no necessary an upper bound T may sill lead o a good resul

15 Oher issues Dieren bases: can develop or Chebyshev, Pochhammer, and oher bases Sensiiviy analysis: ull sensiiviy analysis o obain sronger guaranees on oupu (especially degrees) Beer resuls or sample poins over : blocking and oher echniques may obain beer sabiliy