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Multiple Shooting:

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Jacobian matrix of xk. Multiple Shooting, MTH422. 5. Newton's Method Expanded Part2 ... Compute the Jacobian: Multiple Shooting, MTH422. 8. Newton's Method: An ... – PowerPoint PPT presentation

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Title: Multiple Shooting:


1
Multiple Shooting
  • No One Injured !

2
Newtons Method in Review (1-D)
  • Approximates xn given f and initial guess x0

3
Newtons Method Expanded (n-D)
  • To Solve the System F(x)0, FRn?Rn
  • We use Xk1xk-(F(xk))-1F(xk)
  • Where F(xk) J(xk)
  • J(xk)Jacobian matrix of F at xk

4
Jacobian matrix of xk
5
Newtons Method Expanded Part2
  • In practice xk1xk-F(xk))-1F(xk)
  • is never computed.
  • Use J(xk)(xk1-xk)-F(xk) instead,
  • which is of the form Axb.
  • Can be written
  • J(xk)h-F(xk),xk1xkh
  • Which is a linear system.

6
Newtons Method An Example
  • Solve the nonlinear system using Newtons method
  • f1 xyz3
  • f2 x2y2z25
  • f3 exxy-xz1
  • Where
  • F(x,y,z)(xyz-3, x2y2z2-5, exxy-xz-1)

7
Newtons Method An Example Part 2
  • Compute the Jacobian

8
Newtons Method An Example Part 3
  • Newtons Method becomes
  • (xk1,yk1,zk1)(xk,yk,zk)(h1,h2,h3)

9
Newtons 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
  • xk1-x lt Cxk-x2
  • where x is the exact solution, so
  • errork1 lt Cerrork2
  • Reaches (0, 1, 2) in 5 iterations!

10
Convergence of Newtons 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

11
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

12
Multiple Shooting Setup Part 2
  • or F(x) 0
  • This is a nonlinear system of equations.

13
Multiple ShootingNewtons Method Part 1
  • F(xk1(t))(xk1(t)-xk(t)) -F(xk(t)),
  • which is again of the form Axb
  • F(xk1(t))
  • is a very general version of the derivative,
  • called a Frechét Derivative.

14
Multiple ShootingNewtons Method Part 2
  • If we take ? to be an arbitrary function we can
    produce

15
Multiple ShootingNewtons Method Part 3
  • We can make a similar case for H(x(t))
  • Next G(xk)(xk1 - xk)-G(xk),
  • and similar for H.
  • ? fx(t, x)? (x f f(t, ?))
  • Ba?(a) Bb?(b) -g(x(a), x(b))
  • ? is of the form ? A? q
  • Quasilinearization

16
Multiple Shooting An Example
  • Compute a periodic solution
  • (with period t) of the system
  • x f(x, ?)
  • x1 10(x2-x1)
  • x2 ?x1 x2 x1x3
  • x3 x1x2 (8/3)x3
  • For ? 24.05

17
Multiple ShootingAn Example Part 2
  • This means we need to solve the BVP

18
Multiple ShootingAn Example, Initial Guess
19
Multiple ShootingAn Example, First Iteration
20
Multiple ShootingAn Example, plots
21
Multiple Shooting An Example, Initial Guess and
Final Iteration
22
Multiple Shooting
  • Any questions?
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