SU551665A1 - Device for solving systems of algebraic equations - Google Patents

Description

one

The invention relates to computing and can be used in computing devices designed to solve a wide class of systems of algebraic equations.

A device for solving systems of algebraic equations is known, which contains an oezistor matrix and equilibration blocks, made in the form of DC operational amplifiers. This device stably solves such systems of linear algebraic equations whose square matrix of coefficients has all eigenvalues with a positive integer part.

A two-unit device for solving systems of algebraic equations is also known, which uses two matrices of resistors and balancing units in the form of operational amplifiers. However, the use of such devices is limited by the fact that the operating voltage levels are significantly lower than the voltage levels allowed by the datasheet using operational amplifiers.

The closest technical solution to the invention is a device for solving systems of algebraic equations, containing equilibration blocks and a matrix of resistors, the horizontal buses of which are connected to the zero potential bus through appropriate power sources, and the vertical tires are connected to the inputs of the corresponding repeaters. zi with solving systems of linear algebraic equations with only square matrices.

The purpose of the invention is an extension of the class of solved equations.

This is achieved by the fact that in the device for solving systems of algebraic equations, each equilibration block contains resistors and operational amplifiers, in the feedback of the first of which two chains of serially connected large-scale resistors are connected in parallel, the common point of the large-scale resistors of these chains is connected through the first matching resistor with the output of the corresponding repeater and through the input scale resistor - with the input of the second operational amplifier, in the feedback of which A chain of series-connected scale resistors, the common point of which is connected to the common point of the scale resistors of the second chain of the first operational amplifier, is connected; the output of the second operational amplifier is connected to the input of the corresponding repeater through the second matching resistor. The drawing shows the scheme of the proposed device for solving systems of algebraic equations. The proposed device contains a matrix of resistors 1, repeaters 2, equilibration units 3 with positive transmission coefficients, matching resistors 4-4 and current sources 5 for specifying the right-hand vector. Each equilibration unit 3 consists of two operational amplifiers 6, large-scale resistors 7 with single conductivities, an input large-scale resistor 8, and two large-scale resistors 9, whose conductivity L-Balance unit 3 has a positive transfer coefficient of equal. The matrix of resistors 1 models the matrix of coefficients A of the solved system of linear algebraic equations. In the steady state for the model, the following equations are valid.

| ".G, /."

.., 0,

(one)

.-t-f 0

one

t

AND",

f 2 a

,, i ° -; r - .. o.

t

tp

"

 Sq,

, -2v

1 ° i-n "r, -n

Oh ii-f

voltage across horizontal buses of the resistor 1 matrix; X, are the voltages on the vertical 50 buses of the matrix of resistors 1, where 77 and m, respectively, are the number of spinning and horizontal tires (in general, ti 55 may not be equal to tp); currents of sources 5; the voltage at the output i of the balancing unit 3 (i 1.2 ... n); 60

y q ax-q f

or

Substituted equations (5) and (6) i

equal (4), we have

(G-t-cx --bo () 0

(7)

Matrices b and a are selected to satisfy the condition

G- (t7-E)

OR

(-b-Ef G,

but

where E is the identity matrix.

Claims (1)

Then equation (7) takes on the form of resistors 4 2 and 43. The coefficients of the matrix of the original matrix equation Ax fCZ)) are conveniently represented in mats -bf 0, 0. x, are diagonal matrices whose elements are the sum of the elements of the rows and columns respectively matrices A; transposed matrix A; diagonal conductivity matrix of resistors; 4 4 2; diagonal matrix of transfer coefficients of balancing units 3; vectors, respectively, with the components f. f (3) we obtain. Thus, the proposed scheme is described by equation (9). Here, the matrix Q is the matrix of the intrinsic conductivities of the nodes Vi and therefore is a diagonal matrix with positive elements, and the matrix A Q A will be positively defined, hence the design of the device will be stable and give a solution. In the case of a square non-singular matrix A, this solution is unique, and it will be found by the scheme even of the crit of poor conditionality of the matrix A. Dp of the proof, multiply equation (9) on the left by the matrix (A Q A) and get K (A rAQ f АЧ, i.e. we obtain the solution of the system of equations (1). If the system of equations (1) is redefined (mn n, matrix A is right angular), the proposed scheme gives a unique solution, and this solution is the best, minimizing the sum of squares of the modules of the equations (1 ). The best solution for redefined systems is are used left Gaussian transformation:. Equation A can be interpreted as follows: The initial equation (1) is multiplied by a diagonal matrix Q QAx Qf This identity transformation that does not change the set of solutions of equations {1)... Therefore, if (11) is multiplied by A, then the solution of equation (9) thus obtained will coincide with the solution of equation (10), which is the best. We show that due to the special inclusion of resistors 7, 8 and 9 and operational amplifiers 6, the balancing unit 3 is stable. In the dynamic mode, taking into account the parasitic capacitances of the amplifiers C, the equilibration circuitry is described by the system of differential equations d -IT-- I - m -1. + C B N B C B N f voltage at the inputs of the amplifiers; the diagonal matrix of the intrinsic conductivities of the nodes of the block — the gain of the operational amplifier with open-loop feedback. For the stability of the equilibration unit, and therefore the proposed device, it is necessary that the matrix C B N B has eigenvalues with positive real parts. Since the matrix is diagonal with positive entries, it is sufficient for this that the matrix B N B is positive definite. This condition is satisfied, since the matrix N is diagonal with positive elements. Therefore, the balancing units 3 are stable. The proposed device, in contrast to the known devices of similar purpose, allows solving systems of linear algebraic equations with both square and rectangular (with overdetermined, i.e., mn) matrixes of coefficients. This feature allows you to apply the proposed device instead of the digital computer when designing such technical systems, the relationship between the parameters in which has the form of an overdetermined system of linear algebraic equations. The technical and economic effect of the application of the proposed device is determined by its low cost, significant computer time saving and the non-exclusion of the digital computer, due to the not very high demands on the accuracy of determining the parameters of technical systems at the stage of design calculations. An apparatus for solving systems of algebraic equations, comprising equilibration blocks and a resistor matrix, the horizontal buses of which are connected to the zero potential bus through appropriate current sources, and the vertical buses are connected to the inputs of corresponding repeaters, in order to expand the class of equations to be solved, in it, each equilibration block contains resistors and operational amplifiers, in the feedback of the first of which two chains are connected in parallel flax-connected scale resistors, the common point of the scale resistors of the first of these chains is connected via the first terminating resistor to the output of the corresponding repeater and via the input scale resistor to the input of the second operational amplifier, in feedback of which a chain of series-connected resistors is connected, the common point of which connected to the common point of the scale resistors of the second chain of the first operational amplifier, the output of the second operational amplifier through the second co The read resistor is connected to the input of the corresponding repeater. R n Lj I). eleven tH