Question

1. Let x = (*) and 1. Let x = and r1(x) = xí + až – 3, r2(x) = x]e"? – 4, T3(z) = c+T2 – 2. Approximate the minimizer of }(r1(x)2 + r2(x)2 + r3(x)2) by: (a) Applying one step of the Gauss-Newton method with initial guess x(0) = (.). (b) Applying one step of the Levenberg-Marquardt method with initial guess 2(0) = () and 1 = 2. (Use AI rather than I diag(AT A).)

Answer 1

The given problem is

8, (x) = x2 + x2 – 3: Sr (2) = x, e de 4 83 (X) = x² + 22 - 2.

with variables x​​​​​1, x​​​​​2​​​.

The iteration formula for Gauss-Newton method is

X (RH) = x (*) _ (Just gek) ) y (x) y (k) where g (k) = 88. (x(K)) ) 1. (*): [r3(x(K)] J(K) - Jacobian matrix at x

The iteration formula for Levenber-Marquadt method is

x(x+1)= (Just Jok) + A 2) - (- Just :) & (K) where y(K) = 18; (***)) ? 182(x (K) , [r3(x(K) Jek) - Jacobian matrix at x CK)

Now we find the Jacobian matrix and then proceed to the solutions.

Now, Doy = 2*, ; 27. 212 ; ; dHz da, ax Bxis DEI ; son 3x2 :. Then, the Jacobian matrix is : J = (2x, 2x2 7 Tetr x, en (3x The initial guess is a® (0) Then, at xo), Jo 1 ,J = TO 2 102 you"" - [ ?]] ]

Then, ra 1/3 x Co Ther, x" ) - ( Jos gros) gosto +6)-(2)233) (0)-(2)(0) 2. 2. So, one step of the Gauss-Newton methed gives us X 12 x, = 4/3, K2=4/2, to mine mise 17.(x)? + V2(X)* + V3CX)3), and therefore Ź (ricostrz(X)*+ Y3 (292) At x, = 4/3 , x 2 = 4/3. X, (x) = 5/ 9 8, (x)² = 25/81 V2 (x) = 1.119837737 S (X) = 216

nr Wom So, the minimum volue is 2.16554.3011. b) The initial quess is x) = (0)2=2. & Then, at xro), J")1201 gro) glo) = (144 ) 14 2 NA a googwr ap 6 1)*356 ) Then. (56973421) = ( ) 3 Then, zodat (7673971334_5°) y"? ( 2 ) ::))) ( O)(O)(O) So, one step of the Lavenberg - Marquadt Method gives us x (2) El, x, = X 2 = 1 / 2

N to minimise ² (2, (a)? 12 (17783(x²) Dt x, = x 2 = 12 (7,1203 (6)+()-3)* = 2 (7z (x)) = (" - 4) = 10.08468537 (13/)* = ((1) + 4 =2) 2 So, the minimum value is 9.112655187. ) 2 2