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References

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  1. D. P. Feder, Appl. Opt. 2, 1209 (1963).
    [CrossRef]
  2. C. G. Wynne, P. M. J. H. Wormell, Appl. Opt. 2, 1233 (1963).
    [CrossRef]
  3. J. Merion, J. Opt. Soc. Amer. 55, 1105 (1965).
    [CrossRef]
  4. H. H. Rosenbrock, Computer J. 3, 175 (1960).
    [CrossRef]
  5. M. S. Shapiro, M. Goldstein, A Collection of Mathematical Computer Routines. AEC Computing and Applied Mathematics Center, Courant Institute of Mathematical Sciences, New York University, NYO–1480–14, 1Feb.1965.
  6. M. J. D. Powell, Computer J. 7, 155 (1965).
    [CrossRef]

1965 (2)

J. Merion, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

M. J. D. Powell, Computer J. 7, 155 (1965).
[CrossRef]

1963 (2)

1960 (1)

H. H. Rosenbrock, Computer J. 3, 175 (1960).
[CrossRef]

Feder, D. P.

Goldstein, M.

M. S. Shapiro, M. Goldstein, A Collection of Mathematical Computer Routines. AEC Computing and Applied Mathematics Center, Courant Institute of Mathematical Sciences, New York University, NYO–1480–14, 1Feb.1965.

Merion, J.

J. Merion, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

Powell, M. J. D.

M. J. D. Powell, Computer J. 7, 155 (1965).
[CrossRef]

Rosenbrock, H. H.

H. H. Rosenbrock, Computer J. 3, 175 (1960).
[CrossRef]

Shapiro, M. S.

M. S. Shapiro, M. Goldstein, A Collection of Mathematical Computer Routines. AEC Computing and Applied Mathematics Center, Courant Institute of Mathematical Sciences, New York University, NYO–1480–14, 1Feb.1965.

Wormell, P. M. J. H.

Wynne, C. G.

Appl. Opt. (2)

Computer J. (2)

H. H. Rosenbrock, Computer J. 3, 175 (1960).
[CrossRef]

M. J. D. Powell, Computer J. 7, 155 (1965).
[CrossRef]

J. Opt. Soc. Amer. (1)

J. Merion, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

Other (1)

M. S. Shapiro, M. Goldstein, A Collection of Mathematical Computer Routines. AEC Computing and Applied Mathematics Center, Courant Institute of Mathematical Sciences, New York University, NYO–1480–14, 1Feb.1965.

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Figures (3)

Fig. 1
Fig. 1

Parabolic test functions and the path of the iterative solution.

Fig. 2
Fig. 2

Dependence of total number of iterations on the size of the damping factor.

Fig. 3
Fig. 3

(a) Iterative convergence with full damping. (b) Iterative convergence with damping halved.

Equations (17)

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y i ( x 1 , x 2 , , x n ) = c i ,             i = 1 , m
f i = y i ( x 1 , x 2 , , x n ) - c i ,             i = 1 , m
ϕ = f 1 2 + f 2 2 + + f m 2 ,
j = 1 n ( 2 ϕ x k x j ) Δ x j = - ϕ x k ,             k = 1 , n .
- { 2 i = 1 m f i f i x k } = - 2 A T f ,
a i k = f i / x k .
{ 2 i = 1 m ( f i x k f i x j + f i ( 2 f i x k x j ) Δ x j ) } = 2 A T A Δ x + { 2 i = 1 m f i ( 2 f i x k x j ) Δ x j } .
A T A Δ x = - A T f .
{ 2 i = 1 m f i ( 2 f i x j 2 ) Δ x j } = 2 B T f Δ x ,
b i j = 2 f i / x j 2 .
( A T A + B T fI ) Δ x = - A T f
f 1 = 3 ( 2 x 1 - x 2 ) 2 - 7.5 ( x 1 + 2 x 2 ) , f 2 = ( 1.2 x 1 - 1 - c ) - 0.8 ( x 2 + 0.5 ) 2 .
[ 24 f 1 - 12 f 1 - 12 f 1 ( 6 f 1 - 1.6 f 2 ) ] .
p = [ ( 24 f 1 ) 2 + ( 6 f 1 - 1.6 f 2 ) 2 ] 1 2 / ( 2 ) 1 2 .
( A T A + p I ) Δ x = - A T f .
q = ( 1 n i = 1 n c i i 2 ) 1 2 .
A T A [ 1 + ( p / q ) I ] Δ x = - A T f .

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