Multiple Shooting:

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Multiple Shooting

• No One Injured !

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Newtons Method in Review (1-D)

• Approximates xn given f and initial guess x0

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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

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Jacobian matrix of xk
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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.

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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)

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Newtons Method An Example Part 2

• Compute the Jacobian

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Newtons Method An Example Part 3

• Newtons Method becomes
• (xk1,yk1,zk1)(xk,yk,zk)(h1,h2,h3)

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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!

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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

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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

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Multiple Shooting Setup Part 2

• or F(x) 0
• This is a nonlinear system of equations.

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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.

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Multiple ShootingNewtons Method Part 2

• If we take ? to be an arbitrary function we can
  produce

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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

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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

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Multiple ShootingAn Example Part 2

• This means we need to solve the BVP

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Multiple ShootingAn Example, Initial Guess
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Multiple ShootingAn Example, First Iteration
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Multiple ShootingAn Example, plots
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Multiple Shooting An Example, Initial Guess and
Final Iteration
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Multiple Shooting

• Any questions?