deflate -- first-order deflation

Synopsis

• Usage:

  r = deflate(F,P); r = deflate(F,r); r = deflate(F,B), ...

• Inputs:
  • P, an instance of the type Point
  • F, a polynomial system
  • r, an integer
  • B, a matrix
• Optional inputs:
  • Variable => ..., default value null
• Outputs:
  • r, an integer, the rank used in the (last) deflation
• Consequences:

  • Attaches the keys Deflation and DeflationRandomMatrix which are MutableHashTables that (for rank r, a potential rank of the jacobian J of F) store the deflated system DF and a matrix B used to obtain it. Here B is a random matrix of size n x (r+1), where n is the number of variables and DF is obtained by appending to F the matrix equation J*B*[L_1,...,L_r,1]^T = 0. The polynomials of DF use the original variables and augmented variables L_1,...,L_r.

Description

The purpose of deflation is to restore quadratic convergence of Newton's method in a neighborhood of a singular isolated solution P. This is done by constructing an augmented polynomial system with a solution of strictly lower multiplicity projecting to P.

Apart from P, an instance of the type Point, one can pass various things as the second argument.

• an integer r specifies the rank of the Jacobian dF (that may be known to the user)
• a matrix B specifies a fixed (r+1)-by-n matrix to use in the deflation construction.
• a pair of matrices (B,M) specifies additionally a matrix that is used to squareUp.
• a list{(B1,M1),(B2,M2),...} prompts a chain of successive deflations using the provided pairs of matrices.

i1 : CC[x,y,z]

o1 = CC  [x..z]
       53

o1 : PolynomialRing

i2 : F = polySystem {x^3,y^3,x^2*y,z^2}

o2 = F

o2 : PolySystem

i3 : P0 = point matrix{{0.000001, 0.000001*ii,0.000001-0.000001*ii}}

o3 = P0

o3 : Point

i4 : isFullNumericalRank evaluate(jacobian F,P0)

o4 = false

i5 : r1 = deflate (F,P0)

o5 = 0

i6 : P1' = liftPointToDeflation(P0,F,r1) 

o6 = P0

o6 : Point

i7 : F1 = F.Deflation#r1

o7 = F1

o7 : PolySystem

i8 : P1 = newton(F1,P1')

o8 = P1

o8 : Point

i9 : isFullNumericalRank evaluate(jacobian F1,P1)

o9 = false

i10 : r2 = deflate (F1,P1)

o10 = 1

i11 : P2' = liftPointToDeflation(P1,F1,r2) 

o11 = P2'

o11 : Point

i12 : F2 = F1.Deflation#r2

o12 = F2

o12 : PolySystem

i13 : P2 = newton(F2,P2')

o13 = P2

o13 : Point

i14 : isFullNumericalRank evaluate(jacobian F2,P2)

o14 = true

i15 : P = point {take(coordinates P2, F.NumberOfVariables)}

o15 = P

o15 : Point

i16 : assert(residual(F,P) < 1e-50)	

Caveat

See also

• PolySystem -- a polynomial system
• newton -- Newton-Raphson method