, Norm_L2:=proc(f,x0,r) local i
, N:=0; c:=2/Pi^n/r^(2*n)
, ); end: Procédure S de sélection S:=proc(f,x0,r) local eta,i; global Sf,Sf1,alpha0; global x,y
, Sf1}; fi; od; end: Procédure S de sélection détaillée S_print:=proc(f,x0,r) local eta, else Sf:={op(Sf)
, Détail des calculs de F0 :=S(f ) Sf:={}: g:=[seq
, , vol.1
, =.97783e-8+.20061e-3*x+.19940e-3*y+.99950*x^2+2.0012*x*y+.9995*y^2+.33333*x^3+y^2*x évaluation en (0,0) = 9, pp.7783-7793
, = .10330e-7+.20085e-3*y+.19904e-3*x+1.9978*x*y+1.0006*x^2+1.0005*y^2+x^2*y-1.*y^2*x évaluation en (0,0) = 1, pp.330-340
, Déflation et itérations successives deflation:=proc(X0) local f, X, x0, y0, J, F0,F1,i; global
, 4,1)*J(1,2)/J(1,1): F1:=map(mtaylor, vol.4
, op(solve(F1
, alpha_theorem:=proc(f,x0,rho) local
,
, J:=MatrixInverse
,
Semilocal convergence of Newton's method for singular systems with constant rank derivatives, J. Korean Society of Mathematical Education Ser. B : Pure and Applied Mathematics, vol.18, pp.97-111, 2011. ,
DOI : 10.7468/jksmeb.2011.18.2.097
Complexity and Real Computation, 1998. ,
DOI : 10.1007/978-1-4612-0701-6
Using algebraic geometry, vol.185, 2005. ,
Multiple zeros of nonlinear systems, Mathematics of Computation, vol.80, pp.2143-2168, 2011. ,
Computing the multiplicity structure in solving polynomial systems, Proceedings of the 2005 international symposium on Symbolic and algebraic computation, pp.116-123, 2005. ,
DOI : 10.1145/1073884.1073902
Newton's method at singular points. i, SIAM Journ.al on Numerical Analysis, vol.17, pp.66-70, 1980. ,
Newton's method at singular points. ii, SIAM Journal on Numerical Analysis, vol.17, pp.465-471, 1980. ,
Convergence rates for Newton's method at singular points, SIAM Journal on Numerical Analysis, vol.20, issue.2, pp.296-314, 1983. ,
Points fixes, zéros et la méthode de Newton, 2006. ,
On simple double zeros and badly conditioned zeros of analytic functions of n variables, Mathematics of computation, pp.319-327, 2001. ,
The approximation of one matrix by another of lower rank, Psychometrika, vol.1, pp.211-218, 1936. ,
Géométrie des points épais, Bull. Soc. math. France, vol.106, pp.399-416, 1978. ,
UTV expansion pack : Special-purpose rank-revealing algorithms, Numerical Algorithms, vol.40, pp.47-66, 2005. ,
Multiplicity hunting and approximating multiple roots of polynomial systems, Recent Advances in Real Complexity and Computation, vol.604, pp.105-128, 2014. ,
On location and approximation of clusters of zeros : Case of embedding dimension one, Foundations of Computational Mathematics, vol.7, pp.1-58, 2007. ,
URL : https://hal.archives-ouvertes.fr/hal-00186739
On location and approximation of clusters of zeros of analytic functions, Foundations of Computational Mathematics, vol.5, pp.257-311, 2005. ,
URL : https://hal.archives-ouvertes.fr/hal-00186736
On solving nonlinear equations with simple singularities or nearly singular solutions, SIAM review, vol.27, pp.537-563, 1985. ,
Newton's method for singular problems when the dimension of the null space is > 1, SIAM Journal on Numerical Analysis, vol.18, pp.145-149, 1981. ,
Analysis of Newton's method at irregular singularities, SIAM Journal on Numerical Analysis, vol.20, pp.747-773, 1983. ,
On deflation and multiplicity structure, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01250388
Perturbation theory for linear operators, vol.132, 2013. ,
A new acceleration method for Newton's method at singular points, SIAM journal on numerical analysis, vol.20, pp.1001-1009, 1983. ,
Geometric analysis of the Bergman kernel and metric, 2013. ,
Quadratic Newton iteration for systems with multiplicity, Foundations of Computational Mathematics, vol.2, pp.247-293, 2002. ,
URL : https://hal.archives-ouvertes.fr/hal-00186730
Newton's method with deflation for isolated singularities of polynomial systems, Theoretical Computer Science, vol.359, issue.1, pp.111-122, 2006. ,
Computing isolated singular solutions of polynomial systems : case of breadth one, SIAM Journal on Numerical Analysis, vol.50, pp.354-372, 2012. ,
Computing the multiplicity structure of an isolated singular solution : Case of breadth one, Journal of Symbolic Computation, vol.47, pp.700-710, 2012. ,
Verified error bounds for isolated singular solutions of polynomial systems, SIAM Journal on Numerical Analysis, vol.52, pp.1623-1640, 2014. ,
Deflation and certified isolation of singular zeros of polynomial systems, Proceedings of the 36th international symposium on Symbolic and algebraic computation, pp.249-256, 2011. ,
URL : https://hal.archives-ouvertes.fr/inria-00556021
Low rank approximation : algorithms, implementation, applications, 2011. ,
Symmetric gauge functions and unitarily invariant norms, The quarterly journal of mathematics, vol.11, pp.50-59, 1960. ,
Isolated points, duality and residues, Journal of Pure and Applied Algebra, vol.117, pp.469-493, 1997. ,
URL : https://hal.archives-ouvertes.fr/inria-00125278
Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations, Journal of mathematical analysis and applications, vol.123, pp.199-221, 1987. ,
Deflation algorithm for the multiple roots of a system of nonlinear equations, Journal of mathematical analysis and applications, vol.96, issue.2, pp.463-479, 1983. ,
Convergence of the Newton process to multiple solutions, Numerische Mathematik, vol.9, pp.23-37, 1966. ,
On Newton's method for singular problems, SIAM Journal on Numerical Analysis, vol.15, pp.993-996, 1978. ,
Newton's method and high order singularities, Computers & Mathematics with Applications, vol.5, pp.79-86, 1979. ,
Function theory in the unit ball of C n, 2008. ,
Ueber unendliche viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen, vol.2, pp.317-365, 1870. ,
On the application of Newton's method in a singular case, USSR Computational Mathematics and Mathematical Physics, vol.7, pp.72-85, 1967. ,
Newton's method for singular nonlinear equations using approximate left and right nullspaces of the jacobian, Applied numerical mathematics, vol.54, pp.256-265, 2005. ,
Matrix Perturbation Theory, 1990. ,
Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Mathematische Annalen, vol.71, pp.441-479, 1912. ,
Convergence criterion of Newton's method for singular systems with constant rank derivatives, Journal of Mathematical Analysis and Applications, vol.345, pp.689-701, 2008. ,
On Newton's rule and Sylvester's theorems, Journal of Pure and Applied Algebra, vol.65, pp.293-309, 1990. ,
Newton's method for singular problems and its application to boundary value problems, Journal of mathematics, vol.17, pp.27-88, 1983. ,
Campus de l'École Polytechnique, 1 rue Honoré d'Estienne d'Orves, Bâtiment Alan Turing, CS35003, p.91120 ,
Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9 ,