Abstract

Prior iterative approaches to optical eigenfunction solution have at least three major problems: slow convergence (sometimes); decreasing accuracy after the first solution; and imperfect parallel renormalization (leading to poor use of system dynamic range and hence poor accuracy). We introduce new approaches and algorithms to solve these problems. The new algorithms lead to a tight error bound on eigenvalues and an automatic handling of degenerate or near degenerate eigenvalues. Applications are discussed.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. A. Heinz, J. O. Artman, S. H. Lee, Appl. Opt. 9, 2161 (1970).
    [CrossRef] [PubMed]
  2. M. A. Monahan, R. P. Bocker, K. Bromley, A. Louie, “Incoherent Electro-Optical Processing with CCD’s,” at International Optical Computing Conference Digest (IEEE Catalog 75 CH0941-5C) (Apr. 1975).
  3. J. W. Goodman, A. R. Dias, L. M. Woody, Opt. Lett. 2, 1 (1978).
    [CrossRef] [PubMed]
  4. H. J. Caulfield, W. T. Rhodes, M. J. Foster, S. Horvitz, Opt. Commun. 40, 86 (1981).
    [CrossRef]
  5. H. J. Caulfield, D. Dvore, J. Goodman, W. T. Rhodes, Appl. Opt. 20, 2283 (1981).
  6. B. V. K. Vijaya Kumar, D. Casasent, Appl. Opt. 20, 3707 (1981).
    [CrossRef]
  7. D. Psaltis, D. Casasent, M. Carlotto, Opt. Lett. 4, 348 (1979).
    [CrossRef] [PubMed]
  8. J. W. Goodman, M. S. Song, Appl. Opt. 21, 502 (1982).
    [CrossRef] [PubMed]
  9. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).
  10. H. Hotelling, J. Educ. Psychol. 24, 417, 498 (1933).
    [CrossRef]
  11. A. Ralston, First Course in Numerical Analysis (McGraw-Hill, New York, 1965), p. 486.
  12. A. Gamba, Am. J. Phys. 49, 187 (1981).
    [CrossRef]
  13. A. S. Householder, Principles of Numerical Analysis (McGraw-Hill, New York, 1953), p. 156.

1982 (1)

1981 (4)

B. V. K. Vijaya Kumar, D. Casasent, Appl. Opt. 20, 3707 (1981).
[CrossRef]

H. J. Caulfield, W. T. Rhodes, M. J. Foster, S. Horvitz, Opt. Commun. 40, 86 (1981).
[CrossRef]

H. J. Caulfield, D. Dvore, J. Goodman, W. T. Rhodes, Appl. Opt. 20, 2283 (1981).

A. Gamba, Am. J. Phys. 49, 187 (1981).
[CrossRef]

1979 (1)

1978 (1)

1970 (1)

1933 (1)

H. Hotelling, J. Educ. Psychol. 24, 417, 498 (1933).
[CrossRef]

Artman, J. O.

Bocker, R. P.

M. A. Monahan, R. P. Bocker, K. Bromley, A. Louie, “Incoherent Electro-Optical Processing with CCD’s,” at International Optical Computing Conference Digest (IEEE Catalog 75 CH0941-5C) (Apr. 1975).

Bromley, K.

M. A. Monahan, R. P. Bocker, K. Bromley, A. Louie, “Incoherent Electro-Optical Processing with CCD’s,” at International Optical Computing Conference Digest (IEEE Catalog 75 CH0941-5C) (Apr. 1975).

Carlotto, M.

Casasent, D.

Caulfield, H. J.

H. J. Caulfield, D. Dvore, J. Goodman, W. T. Rhodes, Appl. Opt. 20, 2283 (1981).

H. J. Caulfield, W. T. Rhodes, M. J. Foster, S. Horvitz, Opt. Commun. 40, 86 (1981).
[CrossRef]

Dias, A. R.

Dvore, D.

H. J. Caulfield, D. Dvore, J. Goodman, W. T. Rhodes, Appl. Opt. 20, 2283 (1981).

Foster, M. J.

H. J. Caulfield, W. T. Rhodes, M. J. Foster, S. Horvitz, Opt. Commun. 40, 86 (1981).
[CrossRef]

Gamba, A.

A. Gamba, Am. J. Phys. 49, 187 (1981).
[CrossRef]

Goodman, J.

H. J. Caulfield, D. Dvore, J. Goodman, W. T. Rhodes, Appl. Opt. 20, 2283 (1981).

Goodman, J. W.

Heinz, R. A.

Horvitz, S.

H. J. Caulfield, W. T. Rhodes, M. J. Foster, S. Horvitz, Opt. Commun. 40, 86 (1981).
[CrossRef]

Hotelling, H.

H. Hotelling, J. Educ. Psychol. 24, 417, 498 (1933).
[CrossRef]

Householder, A. S.

A. S. Householder, Principles of Numerical Analysis (McGraw-Hill, New York, 1953), p. 156.

Lee, S. H.

Louie, A.

M. A. Monahan, R. P. Bocker, K. Bromley, A. Louie, “Incoherent Electro-Optical Processing with CCD’s,” at International Optical Computing Conference Digest (IEEE Catalog 75 CH0941-5C) (Apr. 1975).

Monahan, M. A.

M. A. Monahan, R. P. Bocker, K. Bromley, A. Louie, “Incoherent Electro-Optical Processing with CCD’s,” at International Optical Computing Conference Digest (IEEE Catalog 75 CH0941-5C) (Apr. 1975).

Psaltis, D.

Ralston, A.

A. Ralston, First Course in Numerical Analysis (McGraw-Hill, New York, 1965), p. 486.

Rhodes, W. T.

H. J. Caulfield, D. Dvore, J. Goodman, W. T. Rhodes, Appl. Opt. 20, 2283 (1981).

H. J. Caulfield, W. T. Rhodes, M. J. Foster, S. Horvitz, Opt. Commun. 40, 86 (1981).
[CrossRef]

Song, M. S.

Vijaya Kumar, B. V. K.

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).

Woody, L. M.

Am. J. Phys. (1)

A. Gamba, Am. J. Phys. 49, 187 (1981).
[CrossRef]

Appl. Opt. (4)

J. Educ. Psychol. (1)

H. Hotelling, J. Educ. Psychol. 24, 417, 498 (1933).
[CrossRef]

Opt. Commun. (1)

H. J. Caulfield, W. T. Rhodes, M. J. Foster, S. Horvitz, Opt. Commun. 40, 86 (1981).
[CrossRef]

Opt. Lett. (2)

Other (4)

M. A. Monahan, R. P. Bocker, K. Bromley, A. Louie, “Incoherent Electro-Optical Processing with CCD’s,” at International Optical Computing Conference Digest (IEEE Catalog 75 CH0941-5C) (Apr. 1975).

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).

A. Ralston, First Course in Numerical Analysis (McGraw-Hill, New York, 1965), p. 486.

A. S. Householder, Principles of Numerical Analysis (McGraw-Hill, New York, 1953), p. 156.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (51)

Equations on this page are rendered with MathJax. Learn more.

e m · e n = δ m n .
V 0 = a 1 e 1 + a 2 e 2 + , , + a N e N ,
V 1 = A V 0 .
V 2 = A V 1 = A 2 V 0 , V p = A V p - 1 = A p V 0 .
A p a k e k = a k A p e k = a k λ k p e k ,
V p = a 1 λ 1 p e 1 + a 2 λ 2 p e 2 + , , + a N λ N p e N .
λ l > λ m
V p a l λ l p e l .
V p V p - 1 .
V p λ l V p - 1 .
A = E Λ E T ,
EE T = E T E = I .
A n = E Λ n E T .
A n = i = 1 N λ i n e i e i T ,
λ 1 λ 2 λ N .
A n λ 1 n e 1 e 1 T .
A n λ 1 n ( e 1 e 1 T + e 2 e 2 T ) ,
T r A = i N A i i n = i N λ i n .
λ 1 n < T r A n < N λ 1 n .
( 1 N ) 1 / n ( T r A n ) 1 / n < λ 1 < ( T r A n ) 1 / n .
λ 1 = [ 1 + ( 1 N ) 1 / n ] ( T r A n ) 1 / n / 2 ,
± δ = [ 1 - ( 1 N ) 1 / n ] ( T r A n ) 1 / n / 2.
G k l = V k T A V l .
2 M > > > 1 ,
V 1 = ( A - λ ˜ 1 I ) V .
( λ 1 λ 2 ) m ( δ λ 1 - λ 2 + δ ) k ,
Q ( μ , B ) = ( A - μ I ) 2 - B 2 I
q i = b i 2 - B 2 ,
b i = λ i - μ .
( λ i - μ ) < ( λ j - μ ) < ( λ k - μ ) for all k i k j ,
( q j / q i ) m = [ ( λ j - μ ) 2 - B 2 ( λ i - μ ) 2 - B 2 ] m .
λ s m λ 1 10 - s .
1 - 10 - s > ( λ j - μ ) 2 - B 2 ( λ i - μ ) 2 - B 2 .
1 - 10 - s > 1 - ( λ j - λ i ) 2 λ j 2 ,
δ = [ 1 - ( 1 / N ) 1 / P 2 ] ( T r A P ) 1 / P ( T r A P ) 1 / P 10 - s .
( a 2 ) i j = k = 1 N a i k a k j .
S ( t ) = S 0 t / τ ,
d i j ( t ) = δ i j - S ( t )
max ( δ i j ) = S 0 t 0 / τ .
P ( x ) = a 0 X n = a 1 X n - 1 + + a n = 0.
P ( X ) = a 0 ( X n + b 1 X n - 1 + + b n ) ,
b k = a k / a 0 .
C = [ 0 1 0 0 0 0 1 0 0 0 0 1 - b n - b n - 1 - b n - 2 - b 1 ]
det ( C - λ I ) = 0 ,
det ( C - λ I ) = ( - 1 ) n P ( λ ) / a 0 .
C = [ 0 1 0 0 0 0 1 0 0 0 0 1 - b 4 - b 3 - b 2 - b 1 ] .
det ( C - λ I ) = det [ X 1 0 0 0 λ 1 0 0 0 λ 1 - b 4 - b 3 - b 2 - b 1 ] = P ( λ ) / a 0 .
P 1 ( X ) = P 2 ( X ) ,
P N ( X ) = P 1 ( X ) - P 2 ( X )
P 1 ( X ) = P 2 ( X ) = = P N ( X ) = 0 ,
Q ( X ) = i = 1 N [ P i ( X ) ] 2 .

Metrics