Abstract

A simple method to linearize formally a nonlinear automatic lens design problem is presented with two different solvers.

© 1990 Optical Society of America

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References

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  1. D. R. Buchele, “Damping Factor for the Least-Squares Method of Optical Design,” Appl. Opt. 7, 2433–2435 (1968).
    [CrossRef] [PubMed]
  2. G. H. Golub, C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403–420 (1970).
    [CrossRef]
  3. For example, C. R. Rao, S. K. Mitra, Generalized Inverse Matrices and its Application (Wiley, New York, 1971).
  4. Å. Björk, “Solving Linear Least-Squares Problems by Gram-Schmidt Orthogonalization,” B.I.T. 7, 1–21 (1967).
  5. P. Businger, G. H. Golub, “Linear Least-Squares Solutions by Householder Transformation,” Numer. Math. 7, 269–276 (1965).
    [CrossRef]
  6. A. Girard, “Calcul automatique en optique géométrique,” Rev. Opt. 37, 225–241 (1958).
  7. C. G. Wynne, “Lens Designing by Electronic Digital Computer: I,” Proc. Phys. Soc. London 73, 777–787 (1959).
    [CrossRef]
  8. J. Meiron, “Damped Least-Squares Method for Automatic Lens Design,” J. Opt. Soc. Am. 55, 1105–1109 (1965).
    [CrossRef]
  9. K. A. Levenberg, “A Method for the Solutions of Certain Nonlinear Problems in Least-Squares,” Q. Appl. Math. 2, 164–168 (1944).

1970 (1)

G. H. Golub, C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403–420 (1970).
[CrossRef]

1968 (1)

1967 (1)

Å. Björk, “Solving Linear Least-Squares Problems by Gram-Schmidt Orthogonalization,” B.I.T. 7, 1–21 (1967).

1965 (2)

P. Businger, G. H. Golub, “Linear Least-Squares Solutions by Householder Transformation,” Numer. Math. 7, 269–276 (1965).
[CrossRef]

J. Meiron, “Damped Least-Squares Method for Automatic Lens Design,” J. Opt. Soc. Am. 55, 1105–1109 (1965).
[CrossRef]

1959 (1)

C. G. Wynne, “Lens Designing by Electronic Digital Computer: I,” Proc. Phys. Soc. London 73, 777–787 (1959).
[CrossRef]

1958 (1)

A. Girard, “Calcul automatique en optique géométrique,” Rev. Opt. 37, 225–241 (1958).

1944 (1)

K. A. Levenberg, “A Method for the Solutions of Certain Nonlinear Problems in Least-Squares,” Q. Appl. Math. 2, 164–168 (1944).

Björk, Å.

Å. Björk, “Solving Linear Least-Squares Problems by Gram-Schmidt Orthogonalization,” B.I.T. 7, 1–21 (1967).

Buchele, D. R.

Businger, P.

P. Businger, G. H. Golub, “Linear Least-Squares Solutions by Householder Transformation,” Numer. Math. 7, 269–276 (1965).
[CrossRef]

Girard, A.

A. Girard, “Calcul automatique en optique géométrique,” Rev. Opt. 37, 225–241 (1958).

Golub, G. H.

G. H. Golub, C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403–420 (1970).
[CrossRef]

P. Businger, G. H. Golub, “Linear Least-Squares Solutions by Householder Transformation,” Numer. Math. 7, 269–276 (1965).
[CrossRef]

Levenberg, K. A.

K. A. Levenberg, “A Method for the Solutions of Certain Nonlinear Problems in Least-Squares,” Q. Appl. Math. 2, 164–168 (1944).

Meiron, J.

Mitra, S. K.

For example, C. R. Rao, S. K. Mitra, Generalized Inverse Matrices and its Application (Wiley, New York, 1971).

Rao, C. R.

For example, C. R. Rao, S. K. Mitra, Generalized Inverse Matrices and its Application (Wiley, New York, 1971).

Reinsch, C.

G. H. Golub, C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403–420 (1970).
[CrossRef]

Wynne, C. G.

C. G. Wynne, “Lens Designing by Electronic Digital Computer: I,” Proc. Phys. Soc. London 73, 777–787 (1959).
[CrossRef]

Appl. Opt. (1)

B.I.T. (1)

Å. Björk, “Solving Linear Least-Squares Problems by Gram-Schmidt Orthogonalization,” B.I.T. 7, 1–21 (1967).

J. Opt. Soc. Am. (1)

Numer. Math. (2)

P. Businger, G. H. Golub, “Linear Least-Squares Solutions by Householder Transformation,” Numer. Math. 7, 269–276 (1965).
[CrossRef]

G. H. Golub, C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403–420 (1970).
[CrossRef]

Proc. Phys. Soc. London (1)

C. G. Wynne, “Lens Designing by Electronic Digital Computer: I,” Proc. Phys. Soc. London 73, 777–787 (1959).
[CrossRef]

Q. Appl. Math. (1)

K. A. Levenberg, “A Method for the Solutions of Certain Nonlinear Problems in Least-Squares,” Q. Appl. Math. 2, 164–168 (1944).

Rev. Opt. (1)

A. Girard, “Calcul automatique en optique géométrique,” Rev. Opt. 37, 225–241 (1958).

Other (1)

For example, C. R. Rao, S. K. Mitra, Generalized Inverse Matrices and its Application (Wiley, New York, 1971).

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Equations (12)

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minimize Φ ( X ) = F T ( X ) F ( X ) .
Φ + 2 Φ Δ X = 0 ,
A T F 0 + [ A T A + k = 1 m ( F k 2 F k ) ] Δ X = 0 ,
( A T A + D ) Δ X = - A T F 0 ,
d i i = k = 1 m [ F k ( 2 F k / X i 2 ) ] and d i j = 0.
A = [ A D 1 / 2 ] ,
F = [ F O ] .
( A T A ) Δ X = - A T F 0 .
A = UTV T ,
Δ X = - ( VT + U T ) F 0 ,
A = QR ,
R Δ X = - Q T F 0 .

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