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1 B EHAVIOUR OF THE N EWTON PROCESS IN PRESENCE OF A MULTIPLE ISOLATED ROOT , CONSEQUENCES AND APPLICATIONS Jean-Luc Laurent Volery Thesis under the direction of Jean-Claude Yakoubsohn Laboratory MIP - University of Toulouse III A, Algorithms

  1. 1 B EHAVIOUR OF THE N EWTON PROCESS IN PRESENCE OF A MULTIPLE ISOLATED ROOT , CONSEQUENCES AND APPLICATIONS Jean-Luc Laurent Volery Thesis under the direction of Jean-Claude Yakoubsohn Laboratory MIP - University of Toulouse III A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

  2. 2 Position of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Schr¨ oder and Rall’s contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – Schr¨ oder’s operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – Multivariable case : Rall’s flag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Corank 1 zeroes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 – The simple-double zeros case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 – The (generalized) Whitney’s singularities case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 – Principal results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Corank at least 1 singularities of generic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 – First order singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 – Thom-Boardman’s varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 – Thom-Boardman’s flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 The main construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – Intrinsic derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 – Intrinsic flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 – Genericity conditions simplified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Geometric-Numeric computation of the Boardman symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 – One variable case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 – n -variables case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Application to bifurcation problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 – Reduction step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 – Geometrical aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 – Numerical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

  3. 3 P OSITION OF THE PROBLEM 1 – Position of the problem Let f = ( f 1 , . . . , f n ) = 0 be a system of • polynomial functions • analytic functions defined on a connected open subset U ⊂ C n in n complex variables ; Let ζ a zero of this system of finite multiplicity, and thus isolated in f − 1 ( { 0 } ) . Goal : approximate numerically ζ with the classical Newton’s operator N f : C n s → C n s z �→ z − Df ( z ) − 1 f ( z ) If ζ is a regular root of the system, let us mention Smale’s γ -theorem : Theorem 1 ( γ -Theorem) Let ψ ( u ) = 1 − 4 u + u 2 „ � Df ( ζ ) − 1 D k f ( ζ ) � « 1 k − 1 γ ( f, ζ ) := sup k ! k � 2 A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

  4. 4 P OSITION OF THE PROBLEM if a given z 0 ∈ C n satisfies √ u := γ ( f, ζ ) � z 0 − ζ � < 5 − 17 4 then the Newton sequence, initialized at z 0 , is well-defined and converge quadratically to ζ with „ « 2 k − 1 u � z k − ζ � ≤ � z 0 − ζ � , k ≥ 0 ψ ( u ) Reference : . Cucker, M. Shub, and S. Smale, Complexity and real computation , Springer-Verlag, 1998 L. Blum, F However, in the singular case, we can observe experimentally that, if Newton’s algorithm converge to ζ , then the convergence is linear due to a geometric grow in one direction of space . What we propose here : � Geometric caracterisation of directions of linear convergence � Quantitative analysis of the behaviour of Newton’s method with a γ -theorem in the spirit of the preceeding result. A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

  5. 5 S CHR ¨ ODER AND R ALL ’ S CONTRIBUTION 2 – Schr¨ oder and Rall’s contribution 2.1 – Schr¨ oder’s operator f a complex polynomial (or holomorphic function) ζ a zero of f of multiplicity µ < + ∞ ,that is : f ( ζ ) = f ′ ( ζ ) = . . . = f ( µ − 1) ( ζ ) = 0 f ( µ ) ( ζ ) � = 0 and suppose the Newton’s iterates ( z k ) k � 0 converge to ζ Rate of convergence : lim k → + ∞ η k where η k := ε k +1 − ε k and ε k := z k − ζ „ µ − 1 « ε k + O ( ε 2 ε k +1 = k ) µ the convergence of the z k ’s is geometric with a rate µ − 1 µ f ( z ) oder : If the Corrected Newton’s method defined by N µ,f ( z ) := z − µ Schr¨ f ( µ ) ( z ) converge, then the convergence is quadratic. A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

  6. 6 S CHR ¨ ODER AND R ALL ’ S CONTRIBUTION 2.2 – Multivariable case : Rall’s flag f = ( f 1 , . . . , f n ) = 0 a system of n polynomial (or analytic) functions of n complex variables z 1 , . . . , z n ; ζ = ( ζ 1 , . . . , ζ n ) a zero of this system of multiplicity 1 < µ < + ∞ ; µ is the dimension of the local algebra C [ x 1: n ] ζ / ( f 1: n ) in the polynomial case and C { x 1: n } ζ / ( f 1: n ) in the analytic case. Rall defined the flag of vector spaces at the root : N 1 = ker Df ( ζ ) ⊃ N 2 := N 1 ∩ ker D 2 f ( ζ ) ⊃ . . . ⊃ N µ = { 0 } where the D k f ( ζ ) , 1 � k � µ are view has linear operators. Thus, the kernel of D 2 f ( ζ ) is the vector space { X ∈ ζ C n ; D ( Df )( ζ )( X, . ) = 0 } He got a unique decomposition of the source space : C n = N ⊥ 1 ⊕ N 1 = N ⊥ 1 ⊕ ( N ⊥ 2 ⊕ N 2 ) = N ⊥ 1 ⊕ . . . ⊕ N ⊥ µ − 1 ⊕ N µ − 1 If we denote by p k and p ⊥ k the orthogonal projections onto N k and N ⊥ k respectively, then Rall’s conjecture can A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

  7. 7 S CHR ¨ ODER AND R ALL ’ S CONTRIBUTION be expressed has follow : k ( ε 1 − k − 1 � p ⊥ ε 0 ) � = O ( � ε 0 � 2 ) , 1 � k � µ k where ε 0 = z 0 − ζ and ε 1 = N f ( z 0 ) − ζ . Thus, if Rall’s conjecture was correct, we could define the sequence ( y k ) k � 1 : y k = ( p ⊥ 1 ( z k ) , p ⊥ 2 (2 z k − z k − 1 ) , . . . , p µ − 1 ( µz k − ( µ − 1) z k − 1 )) and state � y k − ζ � = O ( � z k − 1 − ζ � 2 ) . Unfortunately, this construction works only for the case simple-double zeroes and the proof he gave is wrong in general. References : oder, ¨ osung der gleichungen , Math. Annalen 2 , 317 − 365 ( 1870 ) E. Schr¨ Uber unendlich viele algorithmen zur auflˆ L. B. Rall, Convergence of the Newton process to multiple solutions , Numerische Mathematik 9 , 23 − 27 ( 1966 ) A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

  8. 8 S CHR ¨ ODER AND R ALL ’ S CONTRIBUTION all’s example : f 1 = x 2 1 − x 1 x 2 + x 2 2 + x 1 − 2 f 2 = 3 x 2 1 + 2 x 1 x 2 + 2 x 2 − 7 = (1 , 1) is a root of multiplicity 2 ! 2 1 Df (1 , 1) = 8 4 1.001 ! 2 − 1 − 1 2 1.0008 D 2 f (1 , 1) = 1.0006 6 2 2 0 1.0004 ker Df (1 , 1) = { 2 x 1 + x 2 = 0 } 1.0002 Rad = { (0 , 0) } 1 0.9998 re � p 1 (2 ε 1 − ε 0 ) � = O ( � ε 0 � 2 ) , the Newton ite- 0.9996 s converge quadratically to the tangent line (1 , 1) + 0.9994 Df (1 , 1) and the rate of convergence over this line is 0.9992 . 0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1.0004 1.0006 1.0008 1.001 A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

  9. 9 S CHR ¨ ODER AND R ALL ’ S CONTRIBUTION ounter-example to Rall’s conjecture : Whitney’s pleat f 1 = x 3 0.4 1 + x 1 x 2 , f 2 = x 2 Σ( f ) = Σ 1 ( f ) = { 3 x 2 1 + x 2 = 0 } 0.2 T (0 , 0) Σ 1 ( f ) = { x 2 = 0 } = ker Df (0 , 0) 0 e singular locus of f is the set of points of corank 1 . –0.2 e can show that the rate of convergence given by ’s result is not the right one : the points in blue cor- –0.4 pond to the rate 1 / 2 while the red ones correspond / 3 . –0.4 –0.2 0 0.2 0.4 A, Algorithms Project’s Seminar Jean-Luc Laurent Volery May 31, 2002

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