Multiple Shooting: Newton's Method in MTH422

Multiple Shooting: No One Injured ! Multiple Shooting, MTH422

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: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 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 2 • Compute the Jacobian: 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 Part 2 • This means we need to solve the BVP: 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

Multiple Shooting Any questions? Multiple Shooting, MTH422

Multiple Shooting: Newton's Method in MTH422

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