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Newton’s Method in Review (1-D)

- Approximates xn given f and initial guess x0

Multiple Shooting, MTH422

Newton’s Method Expanded (n-D)

- To Solve the System F(x)=0, F:RnRn
- 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 xk

Multiple Shooting, MTH422

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

- 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 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

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

- 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

- 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

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

Multiple Shooting, MTH422

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

- If we take ω to be an arbitrary function we can produce:

Multiple Shooting, MTH422

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

- 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, Initial Guess

Multiple Shooting, MTH422

Multiple Shooting:An Example, First Iteration

Multiple Shooting, MTH422

Multiple Shooting:An Example, plots

Multiple Shooting, MTH422

Multiple Shooting: An Example, Initial Guess and Final Iteration

Multiple Shooting, MTH422

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