Abstract

New direct and implicit algorithms for optical matrix–vector and systolic array processors are considered. Direct rather than indirect algorithms to solve linear systems and implicit rather than explicit solutions to solve second-order partial differential equations are discussed. In many cases, such approaches more properly utilize the advantageous features of optical systolic array processors. The matrix-decomposition operation (rather than solution of the simplified matrix–vector equation that results) is recognized as the computationally burdensome aspect of such problems that should be computed on an optical system. The Householder QR matrix-decomposition algorithm is considered as a specific example of a direct solution. Extensions to eigenvalue computation and formation of matrices of special structure are also noted.

© 1983 Optical Society of America

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References

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  2. P. Mengert et al., U.S. Patent3,525,856 (6Oct.1966).
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  13. E. Issacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).
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1983

1982

D. Casasent, Appl. Opt. 21, 1859 (1982).
[CrossRef] [PubMed]

M. Carlotto, D. Casasent, Appl. Opt. 21, 147 (1982).
[CrossRef] [PubMed]

R. K. Montoye, D. H. Lawrie, IEEE Trans. Comput. C-31, 1076 (1982).
[CrossRef]

C. VanLoan, Math. Prog. Study 18, 102 (May1982).
[CrossRef]

1981

1979

1978

1977

M. A. Monahan et al., Proc. IEEE 65, 121 (Jan.1977).
[CrossRef]

1910

L. Richardson, Philos. Trans. R. Soc. London Ser. A, 210, 307 (1910).

Bulirsch, R.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer, New York, 1980).

Carlotto, M.

Casasent, D.

Caulfield, H. J.

Dvore, D.

Edison, A.

A. Edison, M. Noble, “Optical Analog Matrix Processors,” AD646060 (Nov.1966).

Goodman, J.

Goodman, J. W.

Gourlay, A. R.

A. R. Gourlay, G. A. Watson, Computational Methods for Matrix Eigenproblems (Wiley, London, 1973).

Issacson, E.

E. Issacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Jackson, J.

Keller, H. B.

E. Issacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Kumar, B. V. K. V.

Lawrie, D. H.

R. K. Montoye, D. H. Lawrie, IEEE Trans. Comput. C-31, 1076 (1982).
[CrossRef]

Meirovitch, L.

L. Meirovitch, Computational Methods in Structural Dynamics (Sijthoff and Noordhoff International Publishers B.V., Alpehn aan den Rijn, The Netherlands, 1980).

Mengert, P.

P. Mengert et al., U.S. Patent3,525,856 (6Oct.1966).

Monahan, M. A.

M. A. Monahan et al., Proc. IEEE 65, 121 (Jan.1977).
[CrossRef]

Montoye, R. K.

R. K. Montoye, D. H. Lawrie, IEEE Trans. Comput. C-31, 1076 (1982).
[CrossRef]

Newman, C.

Noble, M.

A. Edison, M. Noble, “Optical Analog Matrix Processors,” AD646060 (Nov.1966).

Psaltis, D.

Rhodes, W.

Richardson, L.

L. Richardson, Philos. Trans. R. Soc. London Ser. A, 210, 307 (1910).

Stoer, J.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer, New York, 1980).

VanLoan, C.

C. VanLoan, Math. Prog. Study 18, 102 (May1982).
[CrossRef]

Watson, G. A.

A. R. Gourlay, G. A. Watson, Computational Methods for Matrix Eigenproblems (Wiley, London, 1973).

Wilkinson, J. H.

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

Appl. Opt.

IEEE Trans. Comput.

R. K. Montoye, D. H. Lawrie, IEEE Trans. Comput. C-31, 1076 (1982).
[CrossRef]

Math. Prog. Study

C. VanLoan, Math. Prog. Study 18, 102 (May1982).
[CrossRef]

Opt. Commun.

H. J. Caulfield et al., Opt. Commun. 40, 86 (1981).
[CrossRef]

Opt. Lett.

Philos. Trans. R. Soc. London Ser. A

L. Richardson, Philos. Trans. R. Soc. London Ser. A, 210, 307 (1910).

Proc. IEEE

M. A. Monahan et al., Proc. IEEE 65, 121 (Jan.1977).
[CrossRef]

Other

A. Edison, M. Noble, “Optical Analog Matrix Processors,” AD646060 (Nov.1966).

P. Mengert et al., U.S. Patent3,525,856 (6Oct.1966).

L. Meirovitch, Computational Methods in Structural Dynamics (Sijthoff and Noordhoff International Publishers B.V., Alpehn aan den Rijn, The Netherlands, 1980).

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

E. Issacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

A. R. Gourlay, G. A. Watson, Computational Methods for Matrix Eigenproblems (Wiley, London, 1973).

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer, New York, 1980).

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Figures (1)

Fig. 1
Fig. 1

Schematic diagram of a frequency-multiplexed matrix–matrix systolic array processor.

Equations (26)

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u ( x , t ) t = c 2 2 u ( x , t ) x 2
u ( x , 0 ) = f ( x ) , for 0 x L ,
u ( 0 , t ) = u ( L , t ) = g ( t ) , for 0 t T .
u j n + 1 u j n Δ t = c 2 u j + 1 n 2 u j n + u j 1 n ( Δ x ) 2 ,
u j n + 1 = λ u j + 1 n + ( 1 2 λ ) u j n + λ u j 1 n for n 0 , 1 j J ,
u j n + 1 u j n Δ t = c 2 u j + 1 n + 1 2 u j n + 1 + u j 1 n + 1 + u j + 1 n 2 u j n + u j 1 n 2 ( Δ x ) 2 .
( 1 + 2 λ ) u j n + 1 λ u j + 1 n + 1 λ u j 1 n + 1 = λ u j + 1 n + ( 1 2 λ ) u j n + λ u j 1 n ,
A _ u _ n + 1 = b _ n ,
A _ x _ = b _ .
R _ x _ = Q _ T b _ = b _ .
A _ m x _ = b _ m ,
A _ m = [ R _ m V _ m 0 W _ N m ] ,
w _ m + 1 = ( w m + 1 , m + 1 , , W N , m + 1 ) T .
P _ m + 1 = [ I _ m 0 _ 0 _ I _ k m u _ m u _ m T ] ,
k m u _ m T u _ m = 2 .
u 1 = w m + 1 , m + 1 + l m sign ( w m + 1 , m + 1 ) = w m + 1 , m + 1 + p m + 1 ,
u i = w m + i , m + 1 , for 1 i N m .
l m 2 = i = 1 N m w m + i , m + 1 2 ,
k m = ( l m 2 + l m | w m + 1 , m + 1 | ) 1 .
P _ m + 1 A _ m = A _ m + 1
P _ m + 1 b _ m = b _ m + 1 .
P _ m + 1 = I _ k m u _ m u _ m T
A _ m = T _ m 1 A _ m 1 T _ m ,
B _ = A _ M = T _ 1 A _ T _ ,
A _ x _ = ( T _ B _ T _ 1 ) T _ ϕ _ = T _ B _ ϕ _ = T _ λ ϕ _ = λ T _ ϕ _ = λ x _ ,
A _ m = P _ m 1 A _ m 1 P _ m = P _ m A _ m 1 P _ m ,

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