0af58fb1-0af5-4c8b-aa75-491b1e918895 14 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

References:

http://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm

http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3215/pdf/imm3215.pdf

http://www.brnt.eu/phd/node10.html

http://www.it-weise.de/projects/book.pdf

Levenberg–Marquardt Method. 1 f x.0 ε maxIters LevenbergMarquardt.ref ν 2 : m x.0 f rows eval : x# F 12 / x# f i el (2^(i 1 m sum * : M Diag out m identity : i 1 m range out i i el M i i el : for out 31 line : k 0 : x x.0 : J X# f x Jacobian eval : λ J transpose J * Diag max eval : k maxIters < grad.F x# F x Gradient : grad.F 2^sqrt eval ε ≤ break J X# f x Jacobian eval : δ J transpose J * λ J transpose J * Diag * + (1 -^J transpose x f * - (* eval : x' x δ + eval : x' F x F < x x' : λ λ ν / eval : 21 line λ λ ν * eval : if k k 1 + : 51 line if 21 line while x k 11 mat 21 mat 101 line :

α 10 :

x Rosenbrock 1 x 1 el - α sqrt x 2 el x 1 el (2^- (* 21 mat :

1121 mat Rosenbrock 0021 mat

x Rosenbrock 0.5 - 1.5 21 mat 0.01 50 LevenbergMarquardt.ref 0.99765 0.99513 21 mat 2311 mat 21 mat

0.99765 0.99513 21 mat Rosenbrock 0.0023 0.0006 - 21 mat

sanity check

x y Rosenbrock 1 x - α sqrt y x 2^- (* 21 sys :

11 Rosenbrock 0021 sys

x y Rosenbrock out 0.5 - 1.5 21 sys ≡ 0.01 50100 LevenbergMarquardt 0.99779 0.9953 21 sys 811 mat 21 mat

out 1 el out 2 el Rosenbrock 0.0022 0.0009 - 21 sys

sanity check

x y Rosenbrock out 0.5 - 1.5 21 sys ≡ 105 -^0.01 50100 LevenbergMarquardt.CD 1121 sys 5011 mat 21 mat

out 1 el out 2 el Rosenbrock 0021 sys

sanity check

x y Rosenbrock out 0.5 - 1.5 21 sys ≡ 102 -^0.01 50100 LevenbergMarquardt.CD 0.99779 0.9953 21 sys 811 mat 21 mat

out 1 el out 2 el Rosenbrock 0.0022 0.0009 - 21 sys

sanity check

x y Rosenbrock 1 x - (2^10 y x 2^- (2^* + :

11 Rosenbrock 0

<(--- f:Re[m]->Re[n] with n≥m not satisfied

x y Rosenbrock out 0.5 - 1.5 21 sys ≡ 0.001 300100 LevenbergMarquardt 0.931 0.862 21 sys 30011 mat 21 mat

sanity check

out 1 el out 2 el Rosenbrock 0.005

x f 3 x 1 el * x 2 el x 3 el * cos - 32 / - 4 x 1 el (2^* 625 x 2 el (2^* - 2 x 2 el * + 1 - x 1 el x 2 el * - exp 20 x 3 el * + 10 π * 3 - 3 / + 31 mat :

cost function 12 / x# f transpose * x# f *

grad.F 12 / x# f transpose * x# f * 111 mat * 00031 mat Jacobian transpose :

grad.F 2^sqrt 209.5833

x f out 00031 mat ≡ 0.01 100100 LevenbergMarquardt 0.833121166762849 0.054936176061187 0.521361951077468 - 31 mat 1011 mat 21 mat

out f 0.0002 - 3.784 106 -^* - 4.2108 107 -^* - 31 mat

sanity check

x f out 00031 mat ≡ 105 -^0.01 100100 LevenbergMarquardt.CD 0.833196581863439 0.054943658308979 0.521361434378165 - 31 mat 10011 mat 21 mat

out f 1.25 1015 -^* 1.6093 1014 -^* - 6.9636 1015 -^* - 31 mat

sanity check

grad.F 12 / x# f transpose * x# f * 111 mat * 0.833172881822216 0.054941762440043 0.521346726307088 - 31 mat Jacobian transpose : 0.0004 - 0.0019 0.0059 31 mat

grad.F 2^sqrt 0.0062

x f x 1 + (2^1 + :

1 - f 1

x f out 2 ≡ 0.001 500000 LevenbergMarquardt 1.01815414026727 -

sanity check

out f 1.0003

x f out 2 ≡ 105 -^0.001 100000 LevenbergMarquardt.CD 1.0399579933443 -

sanity check

out f 1.0016

grad.F 12 / x f (2^* 1.00044773309445 - Jacobian :

x f

grad.F 2^sqrt 0.0009