Patent GB-656137-A (GB)
Title: Improvements in or relating to apparatus for solving simultaneous equations
IPC: G06G7/34
Filed: 1947-08-08
Abstract: 656,137. Calculating-apparatus: INTERCHEMICAL CORPORATION. Aug. 8, 1947, No. 21862. Convention date, Nov. 26, 1942. [Class 106 (i)] [Also in Group XL (c)] Apparatus for solving a set of simultaneous equations fi(x 1 , x 2  ... x n )=Ai, (i=1,2 ... n), comprises means for generating arbitary electric signals representing the unknown quantities x, means for computing the functions fi of the unknown quantities in the several equations, and means controlling the signal-generating means in response to the difference between the computed values and the quantities Ai. Fig. 1 illustrates the principle of the invention. Amplifiers X1 ... Xn, e.g. as described in Specification 258,315, [Class 40 (v)], generate signals representing the unknowns x 1  ... x n  respectively, which signals are sent to computers f1 ... fn which may be partly electrical and partly optical or mechanical and produce output signals which are opposed by signals from circuits A1 ... An. Each difference is fed back by a subtraction device S1, S2 ... or Sn to one of the amplifiers X1 ... Xn, thus modifying the x-signals. This process is repeated automatically until a stable state is attained when the required values of the unknowns are given by output signals O1, 02 ... On, and may be passed to an indicating, recording or other device. The right amplifier X1 ... Xn to be connected to each device S 1 ... Sn to attain this stability may be determined empirally or by calculation. In the latter case, the unknown is found which is dominant in each function, i.e. the one which causes the greatest change in the function for a given increment in the unknown, and the subtraction device associated with this function is connected to the amplifier corresponding to this unknown. The procedure is described in detail in the Specification. The circuits A1 ... An may represent quantities which vary with time when the apparatus will provide a continuous solution of the equations. In the form shown in Fig. 3, the invention is used to solve the equation : A1=a x 1  - b x 2 2 ; A2=c x 1 x 2 +dx 2 . All the quantities are represented by A.C. voltages, power being supplied by a generator 10. The values A1, A2 are entered by adjusting potentiometers 11, 12 and the coefficients a, b, c, d by adjusting potentiometers 17, 13, 14, 17<SP>1</SP> respectively. The amounts b x 2 <SP>2</SP>, c x 1  x 2  are obtained by multiplication in a known device 24, 25 in which the amplified x-signals are passed through the two coils of a moving coil dynamometer type of galvanometer, and the product is measured by a variable autotransformer the current through which is made proportional to the coefficient b or c. The auto-transformer is on the shaft of an electric motor actuated through a torque amplifier to counterbalance the torque produced by the coils. The amounts a x 1 , d x 2  are obtained by passing the signals x 1 , x 2  through the potentiometers 17, 17<SP>1</SP> respectively. Signals representing the functions f1, f2 are thus obtained at junction points 19, 19<SP>1</SP> and are amplified and passed to mixing circuits S1, S2 respectively (described in Specification 506,259, [Group XL ], in opposition to signals from circuits A1, A2, the difference between these signals being fed back to the amplifiers X1, X2. The results are read on voltmeters O1, 02. The apparatus shown in part in Figs. 6, 7 and 8 is used for solving " colour equations " in three unknowns c, m and y (less than unity), each of the computed functions Xl, Y<SP>1</SP> Zl, comprising the sum over one column of the products illustrated in the boxes such as 56 in Fig. 6, multiplied by coefficients such as Xw. The signals X<SP>1</SP>, Y<SP>1</SP>, Z<SP>1</SP> are opposed by signals X, Y, Z respectively from photocells in circuits used in scanning colour-separation photographs of a coloured original to be reproduced, and the difference (X - X<SP>1</SP>), for example, is sent out from a subtraction device, which may include a potentiometer, to a high-gain D.C. amplifier, stage 1, Fig. 7, which sends out a signal representing the unknown c. This is combined with a signal from a triangular-wave generator, stage 2, and fed to a trigger circuit, stage 3, which consequently sends out a high frequency wave for the fraction c of the total time. This wave is converted into pulses of square-wave form by a detector, stage 4, and amplified, stage 5. It may also be inverted stage 6, so that pulses representing c, and (l - c) are generated, and collected in the appropriate boxes, Fig. 6, which make up the computers. Each box or collector, Fig. 8, includes a valve which conducts only when pulses are being received by its grid circuit from all three squarewave generators ; as shown, these represent c, m, L - y, so that the output pulses represent the product cm.(l - - y). A potentiometer is adjusted to represent the coefficient Xcm by a grid bias, and the pulses produced, together with those from the other collectors belonging to one computer, are passed through a filter 80 which produces a D.C. signal X<SP>1</SP> proportion to the sum of the average anode currents. The computers may also comprise devices as described in U.S.A. Specification 2,286,730, the coefficients may be entered by adjusting inductances as described in Specification 389,524 and the subtraction devices may also comprise means for opposing magnetic flux as described in Specifications 279,909, [Class 106 (i)], and 331,809, [Class 40 (iv)].