SOME STANDARD PROGRAMS FOR THE MINSK-1 COMPUTER

N. S. GOLOSOV, O. K. GORDEEV

(Presented by the scientific seminar of the computing laboratory of TPI)

It is often necessary to solve engineering problems in a complex domain of variation of the variables. This causes certain difficulties when programming such problems on the Minsk-1 computer. The SP developed for performing arithmetic operations with complex numbers in the system of the universal IS [1] (UIS) makes it possible, on the one hand, to simplify considerably the process of programming problems and, on the other, to write the program more compactly and logically. An example of this is the SP for finding the roots of a polynomial of degree \(n\) with complex or real coefficients, presented in the second part of the article.

Program for arithmetic operations with complex numbers (\(Z\)-program)

The program is arranged as a standard program in the UIS system. Its essential feature is that it is closely connected with the operation of the UIS itself, since after a call to the \(Z\)-program all arithmetic pseudocodes of the UIS are executed as operations with complex numbers in floating-point form, written in four cells:

\[ \begin{array}{rcll} A+0 & & p & \left.\begin{array}{l} \end{array}\right\} Rez \\[-2pt] A+1 & & q & \\[2pt] A+2 & & p & \left.\begin{array}{l} \end{array}\right\} Imz \\[-2pt] A+3 & & q & \end{array} \left\} z .\right. \]

The executive address of the arithmetic-operation pseudocodes is \(A+0\), i.e., the order of the real part of \(z\).

The call to the \(Z\)-program is common to all SPs in the UIS system, with the feature that it does not require sending numbers into working cells. Exit from the \(Z\)-program is carried out by executing the special command: 05 0306 0137, after which the UIS executes the following arithmetic pseudocodes with real numbers.

It should be noted that after the \(Z\)-program is enabled, logical pseudocodes are executed in the same way as before it was enabled, with the exception of 02, 04, which are executed according to the sign and according to the number of the complex part of the pseudo-accumulator.

Appendix: Program

Continuation

Program for computing the roots of a polynomial

Continuation

Continuation

1120–1141—working cells.

The program occupies 77 cells and makes it possible, without disturbing the generality of the programming process in the UIS system, to perform the necessary operations with complex numbers. Working cells: 0020–0037.

Finding the roots of a polynomial of the $n$-th degree

The roots of a polynomial of the form

$$ P_n(z)=z^n+a_1z^{n-1}+\ldots\ldots\ldots+a_{n-1}z+a_n, $$

where $a_1, a_2, \ldots, a_n$ are always written in complex form (if $a_n$ are real, then the imaginary part is zero), are found by iteration-

by Newton’s process in the complex domain of variation of \(Z\), by the formula [2]:

\[ z_{n+1}=z_n-\frac{P_n(z_n)}{P'_n(z_n)} . \]

\(P'_n(z_n)\) is the derivative computed at the point \(z_n\). The call to the SP is common to all subprograms in the UIS system:

where \(\langle A_1\rangle\): \(p\) is the order; \(\langle A_1+1\rangle\): \(q\) is the mantissa; \(n\) is the degree of the polynomial,

\(A a_1,\ A a_{n-1}\) are the addresses of the first and the \(n-1\)-st coefficients. The roots of the polynomial are placed in the positions of the coefficients. The accuracy of root finding \((10^{-6})\) is specified in cells 1112, 1114 and may be changed if desired.

In the computations this program uses the \(Z\)-program, which must have pseudocode 16, i.e. the first line is TX. The program is completely restored after operation and can be used for repeated calls.

The program length is \((n-1)=141\); the length of the computing part is 0106 cells. As can be seen, reducing arithmetic operations with complex numbers to ordinary pseudocodes made it possible, despite the relative complexity of the problem, to write a compact program.

References

1. N. S. Golosov, O. K. Gordeev. Universal Interpreting System (UIS). In the present collection.

2. B. P. Demidovich, I. A. Maron. Foundations of Computational Mathematics. Moscow, 1960.