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Multiple Shooting:. No One Injured !. Newton’s Method in Review (1-D). Approximates x n given f and initial guess x 0. Newton’s Method Expanded (n-D). To Solve the System F(x)=0, F:R n  R n We use X k+1 =x k -(F’(x k )) -1 F(x k ) Where F’(x k ) := J(x k )

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multiple shooting

Multiple Shooting:

No One Injured !

Multiple Shooting, MTH422

newton s method in review 1 d
Newton’s Method in Review (1-D)
  • Approximates xn given f and initial guess x0

Multiple Shooting, MTH422

newton s method expanded n d
Newton’s Method Expanded (n-D)
  • To Solve the System F(x)=0, F:RnRn
  • We use Xk+1=xk-(F’(xk))-1F(xk)
  • Where F’(xk) := J(xk)
  • J(xk)=Jacobian matrix of F at xk

Multiple Shooting, MTH422

jacobian matrix of x k
Jacobian matrix of xk

Multiple Shooting, MTH422

newton s method expanded part2
Newton’s Method Expanded Part2
  • In practice xk+1=xk-F’(xk))-1F(xk)

is never computed.

  • Use J(xk)(xk+1-xk)=-F(xk) instead,

which is of the form Ax=b.

  • Can be written:

J(xk)h=-F(xk),xk+1=xk+h

  • Which is a linear system.

Multiple Shooting, MTH422

newton s method an example
Newton’s Method: An Example
  • Solve the nonlinear system using Newton’s method:
  • f1: x+y+z=3
  • f2: x2+y2+z2=5
  • f3: ex+xy-xz=1
  • Where

F(x,y,z)=(x+y+z-3, x2+y2+z2-5, ex+xy-xz-1)

Multiple Shooting, MTH422

newton s method an example part 2
Newton’s Method: An Example Part 2
  • Compute the Jacobian:

Multiple Shooting, MTH422

newton s method an example part 3
Newton’s Method: An Example Part 3

Newton’s Method becomes:

(xk+1,yk+1,zk+1)=(xk,yk,zk)+(h1,h2,h3)

Multiple Shooting, MTH422

newton s method an example part 4
Newton’s Method: An Example Part 4

If (x0, y0, z0) = (0.2, 1.4, 2.6)

  • This method converges Quadratically

to the unique point p, such that F(p) = 0

  • ||xk+1-x*|| <= C||xk-x*||2

where x* is the exact solution, so

||errork+1|| <= C||errork||2

  • Reaches (0, 1, 2) in 5 iterations!

Multiple Shooting, MTH422

convergence of newton s method
Convergence of Newton’s Method
  • The error at each iteration is as follows:

Error ( ||h|| )

  • 6.372324 * 10-1
  • 3.079968 * 10-2
  • 6.701403 * 10-4
  • 3.175531 * 10-8
  • 1.136453 * 10-15

Multiple Shooting, MTH422

multiple shooting setup part 1
Multiple Shooting Setup Part 1
  • x’ = f(t,x) and g(x(a), x(b)) = 0
  • Which is a Boundary Value Problem (BVP) and can be rewritten as:
  • x’- f(ty, x) = 0 and g(x(a), x(b)) = 0

Multiple Shooting, MTH422

multiple shooting setup part 2
Multiple Shooting Setup Part 2
  • or F(x) = 0
  • This is a nonlinear system of equations.

Multiple Shooting, MTH422

multiple shooting newton s method part 1
Multiple Shooting:Newton’s Method Part 1
  • F’(xk+1(t))(xk+1(t)-xk(t)) = -F(xk(t)),

which is again of the form Ax=b

  • F’(xk+1(t))

is a very general version of the derivative,

called a Frechét Derivative.

Multiple Shooting, MTH422

multiple shooting newton s method part 2
Multiple Shooting:Newton’s Method Part 2
  • If we take ω to be an arbitrary function we can produce:

Multiple Shooting, MTH422

multiple shooting newton s method part 3
Multiple Shooting:Newton’s Method Part 3
  • We can make a similar case for H(x(t))
  • Next: G’(xk)(xk+1 - xk)=-G(xk),

and similar for H.

  • ω’ = fx(t, x)ω – (x’ – f – f(t, ω))
  • Baω(a) + Bbω(b) = -g(x(a), x(b))
  • ω’ is of the form ω’ = Aω + q
    • Quasilinearization

Multiple Shooting, MTH422

multiple shooting an example
Multiple Shooting: An Example
  • Compute a periodic solution

(with period τ) of the system:

  • x’ = f(x, λ)
  • x1’ = 10(x2-x1)
  • x2’ = λx1 – x2 –x1x3
  • x3’ = x1x2 – (8/3)x3
  • For λ = 24.05

Multiple Shooting, MTH422

multiple shooting an example part 2
Multiple Shooting:An Example Part 2
  • This means we need to solve the BVP:

Multiple Shooting, MTH422

multiple shooting an example plots
Multiple Shooting:An Example, plots

Multiple Shooting, MTH422

multiple shooting1
Multiple Shooting

Any questions?

Multiple Shooting, MTH422

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