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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 38, Iss. 4 — Apr. 1, 1948
  • pp: 308–309

An Electronic Method for Solving Simultaneous Equations1

ARTHUR C. HARDY and EDWARD C. DENCH  »View Author Affiliations

JOSA, Vol. 38, Issue 4, pp. 308-309 (1948)

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In connection with the requirements of color correction discussed in the preceding paper, we found that a set of n simultaneous equations can be solved by using each of the n error signals to control one of the independent variables. Stability of this equation-solving network can be achieved by the proper interconnection. In applying this procedure to the equations for color correction, it was necessary to find the sum of eight terms, each of which is the product of either three or four independent variables. It was found that these individual products could be computed by making use of the fact that, if n events have probabilities of occurrence a, b, . . .n, respectively, the probability of their simultaneous occurrence is the product of their individual probabilities abc . . . n. By generating rectangular waves of irrationally related frequencies whose positive pulse widths are proportional to the values of the unknown variables, and applying these signals to the grid of a vacuum tube in such a manner that its plate current flows only when all of the signals have positive values simultaneously, the average value of the plate current is proportional to the required product. When controlled in this way, eight vacuum tubes with their plates connected in parallel deliver an average current that is proportional to the required sum.

ARTHUR C. HARDY and EDWARD C. DENCH, "An Electronic Method for Solving Simultaneous Equations1," J. Opt. Soc. Am. 38, 308-309 (1948)

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  1. Extension of paper presented at the October, 1946 meeting of the O.S.A.
  2. Now with Raytheon Manufacturing Company.
  3. A. C. Hardy and F. L. Wurzburg, Jr., J. Opt. Soc. Am. 16, 300 (1948).
  4. It should be mentioned that this mode of interconnection does not lead to a solution of the equations for all possible values of the constants. However, it can be surmised from the nature of the procedure in color printing that R′ will generally depend to a greater extent on the value of c than on the values of m and y. Also, since G′ is chiefly dependent on m and B′ is chiefly dependent on y, the mode of interconnection described is, in fact, the preferred mode for the purpose at hand. Interest in the stability of multiple servo-networks was so stimulated by the war that it seems not unlikely that it may someday be possible to select the preferred mode of interconnection for any set of equations without reference to the underlying physical process or without trying the several possible modes of interconnection experimentally.
  5. It was shown in the preceding paper that c, m, and y represent dot ratios which always have fractional values lying between 0 and 1. The numerical values of (1-c) × (1-m) (1-y), c(1-m) (1-y), etc., are always positive fractions whose sum is unity.

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