SU1265810A1 - Device for solving sets of algebraic equations - Google Patents

Abstract

The invention relates to analogue technology and is intended to solve systems of linear algebraic equations. The device contains three matrices of current-leading resistors, a block for setting initial conditions, a switch, a block of subtractors, two blocks of drivers of the signal module, two blocks of summation, a block of integrators, a control block. The device allows one to solve systems of linear algebraic equations with approximately given source data in the presence of apri (for a complete information on the smoothness of the solution. 2 ill. S.

Description

Yu

O) ate

cx The invention relates to analog computing, is designed to solve systems of algebraic equations, and can serve as the basis for creating specialized analog processors of hybrid computing systems. The aim of the invention is to improve the accuracy and expand the functional capabilities of the device by taking into account additional a priori information about solution smoothness. Figure 1 shows the block diagrams of the device; Fig. 2 is a block diagram of a control unit. The device contains three matrices 1-3 of current-setting resistors, switch 4, integrator block 5, block 6, readout, first block 7 of signal module drivers, second block of 8 modulator drivers, first and second block 9 and 10 summation, control block 11 and block 12 set the initial conditions. Block 11 contains a multiplier 13, an adder 14, and a comparator 15. The device operates as follows. Three matrices 1–3 of current-generating resistors are a model-extended matrix of coefficients of system A, a matrix of coefficients A transposed to A, and a matrix of coefficients C, respectively. The inputs of the control unit 11 receive signals corresponding to the numerical parameters oi and / 3, which characterize the errors in the coefficients of the matrix A and the right-hand sides L, respectively. An approximate, stable solution of the solved system of linear algebraic equations is formed at the output of the block 5 of integrators. The proposed device is described by a system of obedient differential equations with constant coefficients and some initial conditions:

1 / h

ABOUT

ABOUT

ABOUT

ABOUT

-1 / h o

ABOUT

ABOUT

1 / h -1 / h OO

О1 / h -1 / hО

001 / h-l / h

0001 / h +. x (o) Let us describe in detail the functions of each of the blocks included in the proposed device. Matrix I is intended to multiply the vector x (t) by the matrix A and subtract the vector of the right side b, i.e. matrix 1 is a resistive matrix, the input of which receives signals X | (t), i 1,2, ..., n, which are components of the dimensional vector x (t) x, (t) x (t), ..., x (t), and the output is formed by the signals ri (t), i, 2, ..., n, which are the components of the skew vector r (t) Ax (t) -b, r (t) r, (t) r (t), ..., i, (t). The matrix 1 of resistors. This multiplies the mismatch r (t) by the matrix A, i.e. the components of the vector r (t) are received at the input of the indicated block, and the components of the vector g (t) (t) Ax (t) - Ab are formed at the output. At the outputs of matrix 3 of resistors djrltl, the signals f (t) dt are formed by solving the equation ". (T)), the signals g (t) from the output of matrix 2 are fed to its first group of inputs, and the signals from the outputs of matrix 3 to the second group of inputs resistors. The matrix C is formed by multiplying the matrix D by D, where the matrix D converts the vector U (v) on the grid with a step h to the derivative J ---. dv

1 / h -1 / h

С i 0 r 0

0 0

The elements C of the matrix C take the values

if 1i-j, 5

WITH;; ABOUT,

with;; -l / h, if 1 i-j 1 1,

  2 / h, if i 2, ...,

P - ,

“2

Cj; 1 / h, if i I, p.

Note that the matrix C has a ribbon structure with a tape width of three.

. Control Unit II serves to check the condition

// r (t) // (A, (t) -b // fo // x (t) - x, // - t- / J, (2)

where // // is the norm of the vector; ot and / 3 are numeric parameters defining

If llrll Up (y // x - yij / + p, then and - O,

if llrll

Thus, during the execution of the condition // g // 4 "// x - XQ // + / 3, the control unit generates a signal U 1, which opens the keys of the switch 4.

In general, the device is defined by a system of differential equations (1). The solution of this system of steady-state differential equations is the normal solution of the original system of linear algebraic equations for the vector Xd. With XD O, the model allows one to obtain a solution with a minimum norm in the space Wj of functions having a generalized derivative integrable with a square.

In the initial state, at the outputs of the integrators of the block 5, the integrators are set to the voltages corresponding to the a priori well-known vector x defined by the problem statement.

-1 / h OO

2 / h -l / h OO

-1 / h 2 / h -1 / h About

O-1 / h 2 / h -1 / h

OO-1 / h 1 / h

error of setting the initial data // 4 D // 4 ci; // lb // s; l A and fib are the errors of setting the matrix A and the vector of the right parts L, respectively. Signal

and, // X (t) - X // i: / X; (t) - i «l

multiplied by oi on block 13 and added to / 3 on adder 14. At the output of the adder, a signal is generated; // X - + ft, which is compared with a signal using comparator I5

and, // r (t) // «i / r; (t) / ...

Depending on the ratio of the signals AND I0 // xx, y / + A, control block II generates a control signal U, which is fed to the control input of switch 4.

xj / + p, then “1.

In the process of integrating the system of ordinary differential equations (1) using blocks 1-5, the outputs of block 6 continuously form a vector of difference x (t) - x, the outputs of block 7 are a vector whose components are modules of the vector X (t) - X i.e. / X; (t) - X, /,

at the output of block 9, the voltage corresponds to the norm // x (t) –Hd // of the vector x (t) –x. At the outputs of block 8, a vector is continuously formed, the components of which are modules (t) / of the mismatch vector v (t), at the output of block 10 there is a voltage corresponding to the norm // r (t) //, the vector r (t) . Hereinafter

Claims (1)

U, (t) // -G / g; a) /,. those. the second vector norm is used. In control block II, the consistency check of the obtained solution with the initial data error is carried out in accordance with the conditions (2 When the specified inequality is fulfilled, control block II generates a signal at the output and opening switches of switch 4. At the same time, the output of block 5 fixes the voltage corresponding to a stable approximation solution of the original system of linear algebraic equations. Thus, the proposed device for solving systems of linear algebraic equations allows one to automatically obtain to give an approximate stable solution of the problem with greater accuracy by taking into account the available a priori information about the smoothness of the solution. Note that provided that the coefficient of the matrix A and the right-hand part vector are exactly specified, the proposed device implements a method of establishing it in a pure form and allows obtaining a normal algebraic system , the same as the known device. In the process of integrating the system of differential equations (1) in the proposed device, the vector rate (tb // (I + s) V / monotonically decreases from some .nachalnogo values defined by the original system Neuve viscous algebraic, equations for X XQ, to zero at a t- (if to OJ. At the same time, the norm of the vector z (t) // (I + c) x (t) // as the integral of Zj (t) increases monotonically from a certain initial value of 2r (o), // (I + c) Xjj // to a final value at. Thus, if the source data is known approximately, i.e. x 0 ff Of stopping the process of integrating under some tt, according to conditions (2), at the output of block 5 of the integrators an approximate stable solution of the problem is formed, whose minimum function is 2j; (t) // (I + c) x (t ) //. Thus, the approximate solution of a system of linear algebraic equations is obtained, which has a minimum length and at the same time a minimum norm of the first derivative characterizing the smoothness of the solution. Consequently, the proposed device allows, in contrast to the known, to form an approximate stable solution that has the minimum length and is the smoothest at the same time, thereby achieving the goal of expanding the functionality of the device by taking into account additional a priori information about the solution smoothness, while improving the accuracy of the solution. The proposed device allows solving not only ill-conditioned, but also well-conditioned algebraic systems. An apparatus for solving systems of linear algebraic equations, comprising a block for setting initial conditions, first and second matrices of current setting resistors, an integrator block, first and second blocks of drivers of a signal module, groups of outputs of which are connected to groups of inputs of the first and second summation blocks, switch, a control unit and a subtractor unit, the output group of which is connected to the input group of the first block of drivers of the signal module, the output unit of the task unit is the start These conditions are connected to the inputs of the initial setup of the integrator unit and to the first group of inputs of the subtractors unit, the second group of inputs of which is connected to the group of outputs of the integrator unit and to the terminals of the first group of current supply resistors of the first matrix, the view ports of the second group of modules signal and to the outputs of the first group of tokadojush 1x resistors of the second matrix, the group of outputs is commuted. A torus is connected to a group of information inputs of an integrator block, characterized in that, in order to improve accuracy, a third matrix of current supply resistors is entered into it, and the control unit consists of a comparator, adder and flea multiplication, the output of which is connected to the first adder input, the output of which connected to the first input of the comparator, the output of which is connected to the control input of the switch, the information input of which is combined with the terminals of the first and second groups of current-conducting resistors of the third matrix, the conclusions of the third the group of current-setting resistors of the third matrix is connected to the terminals of the second 1 Qpiaiabi of the current resistors of the second matrix, the input of the error matrix of the constant coefficients of the system of equations of the device is connected to the first input of the multiplication unit, the second input is connected to the output of the first summation unit, the input of the error of the right vector The parts of the system of equations of the device are connected to the sum of the torus, the output of the second summation block is connected to the second input of the comparator. .2