Verified error b oun d s f or multipl e roots o f syst e ms o f - PowerPoint PPT Presentation

Verified error b oun d s f or multipl e roots o f syst e ms o f nonlin ea r e qu a tions St ef Gr a ill a t LIP6/P E QU A N , Sor b onn e Univ e rsit é s , UPM C Univ P a ris 06 , C NRS Joint work with Si e g f ri ed M. Rump Journ ée s m é tho de s de su bd ivisions pour l e s syst è m e s singuli e rs IR CC yN , N a nt e s , Dece m be r 1 5 -16 , 2014 S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 1 / 42

Gener a l motiv a tions : s e l f -v a li da ting m e tho d s V e ri f y a ssumptions o f m a th e m a ti ca l th e or e ms on th e c omput e r M a king m a th e m a ti ca l proo f s with c omput e rs G e tting v e ri fied r e sults : → a n int e rv a l e n c losur e o f th e tru e r e sult → a n a pproxim a t e r e sult with a rigorous e rror b oun d Possi b ly with proo f o f uniqu e n e ss Be ing fa st a n d acc ur a t e Dea ling with “ill-pos ed pro b l e ms” S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 2 / 42

Gener a l motiv a tions ( c ont ’d ) P r oo fs wi t h c omp uters: how t o d o t h at ? wi t h c omp uter a lg ebra s y ste m s: e x act resu l ts but s om et im es no t effic i e n t wi t h flo at ing-n u m bers: fast but of e n w r ong resu l ts due t o r o u n d ing err o rs Po ss i b l e s ol ut ion : c omp ut ing wi t h flo at ing-poin t but ta king in t o acc o u n t a ll t h e r o u n d ing err o rs ! S . G ra ill at ( U niv. P ar i s 6 ) V er i f i ed err o r b o u n ds f o r m u l t ipl e r oo ts 3 / 4 2

Outline of the t a lk Prin c ipl e o f s e l f -v a li da ting m e tho d s 1 Multipl e roots o f polynomi a l syst e ms 2 Num e ri ca l e xp e rim e nts 3 S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 4 / 42

Outline of the t a lk Prin c ipl e o f s e l f -v a li da ting m e tho d s 1 Multipl e roots o f polynomi a l syst e ms 2 Num e ri ca l e xp e rim e nts 3 S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 5 / 42

Proving th a t a m a trix is nonsingul a r T e or e m 1 Let A be a matr ix a n d R a no t h er m atr ix suc h t h at ∥ I − RA ∥ < 1 . Te n A i s non s ing u l ar Proo f . B y c ontr a positiv e, i f A is singul a r , th e r e e xists x ≠ 0 su c h th a t A x = 0. T e n ( I − RA ) x = x a n d so ∥ I − RA ∥ ≥ 1. On a c omput e r , c hoos e f or R ≈ A − 1 a n d th e n c omput e ∥ I − RA ∥ with int e rv a l a rithm e ti c . S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 6 / 42

Proving th a t a m a trix is nonsingul a r with INTL AB L e t A be a m a trix o f d im e nsion n R = inv(A) C = eye(n) - R*intval(A) nonsingular = ( norm(C,1) < 1 ) I f nonsingul a r = 1 , th e n A is nonsingul a r. I f nonsingul a r = 0 , th e n w e ca n s a y nothing S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 7 / 42

A simpl e a ppro ac h L e t f ∶ R n → R n a n d ̂ x ∈ R n unknown su c h th a t f (̂ x ) = 0 L e t ̃ x ≈ ̂ x su c h th a t f (̃ x ) ≈ 0 F in d a b oun d f or ̃ x : a n int e rv a l X su c h th a t ̂ x ∈ X W e h a v e f ( x ) = 0 ⇔ g ( x ) = x w ith g ( x ) ∶ = x − R f ( x ) w ith de t ( R ) ≠ 0 . T e or e m 2 ( B rou w e r , 1912) Every continuous function from a closed ba ll o f a E u c li dea n sp ace to its e l f h a s a fi x ed point. S. Gr a ill a t (Uni v . P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 8 / 42

A simpl e a ppro ac h ( c ont ’ d ) B y B r o u w e r fi x ed poin t t h e o r e m , x ) = ̂ x ) = 0 X ∈ IR n , g ( X ) ⊆ X ⇒ ∃̂ x ∈ X , g (̂ ⇒ f (̂ x W e j ust h a v e t o c h ec k g ( X ) ⊆ X a n d p r o v e de t ( R ) ≠ 0 . B ut n a i v e a pp r o ac h fa il s : g ( X ) ⊆ X − R f ( X ) ⊈ X S . G r a ill a t ( U ni v . P a r i s 6 ) V e r i f i ed e rr o r b o u n d s f o r m u l t ipl e r oo ts 9 / 4 2

B oun d s f or th e solution o f nonlin ea r syst e ms M ea n V a lu e T e or e m : i f f ∈ C 1 th e n f ( x ) = f (̃ x ) + M ( x − ̃ x ) with M = ( ∂ f ∂x ( ξ i )) i L e t Y ∶= X − ̃ x a n d g ( x ) − ̃ x − ̃ x ∈ X ⇒ = x − R f ( x ) x − R f (̃ x ) + ( I − RM )( x − ̃ = x ) ∈ − R f (̃ x ) + ( I − RM ) Y A s a c ons e qu e n ce − R f (̃ x ) + ( I − RM ) Y ⊆ Y ⇒ g ( X ) − ̃ x ⊆ Y ⇒ g ( X ) ⊆ X S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 10 / 42

B oun d s f or th e solution o f nonlin ea r syst e ms ( c ont ’ d ) T e o r e m 3 ( Ru mp , 19 8 3 ) Let f ∶ R n → R n with f = ( f 1 , . . . , f n ) ∈ C 1 , ˜ x ∈ R n , X ∈ IR n with 0 ∈ X a n d R ∈ R n × n be giv e n. L e t M ∈ IR n × n be giv e n su c h th a t {∇ f i ( ζ ) ∶ ζ ∈ ̃ x + X } ⊆ M i , ∶ . De not e b y I th e n × n i de ntit y m a tri x a n d a ssum e − R f (̃ x ) + ( I − RM ) X ⊆ in t ( X ) . T e n th e r e is a uni qu e ̂ x ∈ ̃ x + X w i t h f (̂ x ) = 0 . Mo r e o v e r , e v e ry m a tr i x ̃ M ∈ M i s non s ing u l a r . In p a rt i c u l a r , t h e J ac o b i a n J f (̂ ∂ x (̂ x ) = ∂ f x ) i s non s ing u l a r . S . G r a ill a t ( U niv. P a r i s 6 ) V e r i f i ed e rr o r b o u n d s f o r m u l t ipl e r oo ts 11 / 4 2

Rem a rk Not e th a t a n in c lusion o f th e r a ng e o f th e gr ad i e nts ∇ f i ov e r th e s e t ̃ x + X n eed s to be c omput ed . A c onv e ni e nt w a y to d o this in INTL AB is b y int e rv a l a rithm e ti c a n d th e gr ad i e nt tool b ox. F or a giv e n (M a tl ab ) f un c tion f , f or xs = ̃ x a n d a n int e rv a l v ec tor X , th e ca ll M = f ( gradientinit ( xs + X )) c omput e s a n in c lusion M . S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 12 / 42

Outline of the t a lk Prin c ipl e o f s e l f -v a li da ting m e tho d s 1 Multipl e roots o f polynomi a l syst e ms 2 Num e ri ca l e xp e rim e nts 3 S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 13 / 42

Verific a tion o f multipl e roots V e ri fica tion m e tho d f or c omputing gu a r a nt eed (r ea l or c ompl e x) e rror b oun d s f or d ou b l e roots o f syst e ms o f nonlin ea r e qu a tions. To c ir c umv e nt th e prin c ipl e pro b l e m o f ill-pos ed n e ss w e prov e th a t a slightly p e rtur bed syst e m o f nonlin ea r e qu a tions h a s a d ou b l e root. F or e x a mpl e, f or a giv e n univ a ri a t e f un c tion f ∶ R → R w e c omput e two int e rv a ls X , E ⊆ R with th e prop e rty th a t th e r e e xists ̂ x ∈ X a n d ̂ e ∈ E su c h th a t ̂ x is a d ou b l e root o f f ( x ) ∶= f ( x ) − ̂ e . I f th e f un c tion f h a s a d ou b l e root , typi ca lly th e int e rv a l E is a v e ry n a rrow int e rv a l a roun d z e ro. S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 14 / 42

Verific a tion o f multipl e roots T e typi ca l s ce n a rio in th e univ a ri a t e ca s e is a f un c tion f ∶ R → R with a d ou b l e root ̂ x , i. e . f (̂ x ) = f ′ (̂ x ) = 0 a n d f ′′ (̂ x ) ≠ 0. C onsi de r , f or e x a mpl e, 1 8 x 7 − 1 8 3 x 6 + 764 x 5 − 167 5 x 4 + 2040 x 3 − 1336 x 2 + 416 x − 4 8 f ( x ) = = ( 3 x − 1 ) 2 ( 2 x − 3 )( x − 2 ) 4 S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 1 5 / 42

Verific a tion o f multipl e roots V e ri fica tion m e tho d s f or multipl e roots o f polynomi a ls a lr ead y e xist (Rump , 2003). A s e t c ont a ining k roots o f a polynomi a l is c omput ed, b ut no in f orm a tion on th e tru e multipli c ity ca n be giv e n. A hy b ri d a lgorithm ba s ed on th e m e tho d s o f (Rump , 2003) is impl e m e nt ed in a lgorithm verifypoly in INTL AB . C omputing in c lusions X1 , X2 a n d X3 o f th e simpl e root x 1 = 1 . 5, th e d ou b l e root x 2 = 1 / 3 a n d th e qu ad rupl e root x 3 = 2 o f f b y a lgorithm verifypoly in INTL AB w e o b t a in th e f ollowing. >> X1 = verifypoly(f,1.3), X2 = verifypoly(f,.3), X3 = verifypoly(f,2.1) intval X1 = [ 1.49999999999904, 1.50000000000078] intval X2 = [ 0.33333316656015, 0.33333343640539] intval X3 = [ 1.99741678159164, 2.00363593397305] S. Gr a ill a t (Univ. P a ris 6) V e ri f i ed e rror b oun d s f or multipl e roots 16 / 42

Verific a tion o f multipl e roots ( c ont ’d ) Te accurac y o f t h e in c l us ion o f t h e d o ub l e r oo t x 2 = 1 / 3 i s m uc h l ess t h a n t h at o f t h e s impl e r oo t x 1 = 1. 5, a n d t hi s i s t ypi ca l. I f w e p erturb f in t o ̃ f ( x ) ∶= f ( x ) − ε f o r s om e s m a ll rea l c on sta n t ε a n d look at a p erturbed r oo t ̃ x + h ) o f ̃ f (̂ f , t h e n x ) h 2 + O( h 3 ) 0 = ̃ f (̂ 2 f ′′ (̂ x + h ) = − ε + 1 √ impli es 2 ε / f ′′ (̂ x ) . h ∼ In g e n era l flo at ing-poin t c omp utat ion s are affl i cted wi t h a re l at iv e err o r o f s iz e ε ≈ 10 − 1 6 . T i s h as t h e sa m e effect as a p erturbat ion o f t h e giv e n fu n ct ion f in t o ̃ t hi s in c l us ion t o be o f better re l at iv e accurac y t h a n √ ε ≈ 10 − 8 . f . But f o r d o ub l e r oo ts, w e ca nno t e xp ect S . G ra ill at ( U niv. P ar i s 6 ) V er i f i ed err o r b o u n ds f o r m u l t ipl e r oo ts 1 7 / 4 2