Computer for operations with functions

Computer for operations with mathematical functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

(unlike the usual computer

Computer

A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

) operates with functions at the hardware

Hardware

Hardware is a general term for equipment such as keys, locks, hinges, latches, handles, wire, chains, plumbing supplies, tools, utensils, cutlery and machine parts. Household hardware is typically sold in hardware stores....

level (i.e. without programming these operations).

History

A thingy machine for operations with functions has been presented and developed and by Kartsev in the year 1967 . Among the operations of this computing machine were the functions addition, subtraction and multiplying, functions comparison, the same operations between a function and a number, finding the function maximum, computing indefinite integral, computing definite integral of derivative of two functions, derivative of two functions, shift of a function along the along X-axis etc. By its architecture this computing machine was (using the modern terminology) a vector processor

Vector processor

A vector processor, or array processor, is a central processing unit that implements an instruction set containing instructions that operate on one-dimensional arrays of data called vectors. This is in contrast to a scalar processor, whose instructions operate on single data items...

. In itthere has been used the fact that many of these operations may be interpreted as the known operation on vectors: addition and subtraction of functions - as addition and subtraction of vectors, computing a definite integral of two functions derivative— as computing the vector product of two vectors, function shift along the X-axis – as vector rotation about axes, etc. In the year 1966 Khmelnik had proposed a functions coding method , i.e. the functions representation by a "uniform" (for a function as a whole) positional code. And so the mentioned operations with functions are performed as unique computer operations with such codes on a "single" arithmetic unit.

The main idea

Positional code of an integer number

is a numeral notation of this digit

in a certain positional number system

Positional notation

Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...

of the following form:

Such code may be called "linear". Unlike it a positional code of one-variable

function

has the form:

and so it is flat and "triangular", as the digits in it comprise a triangle.

Such positional code of an integer number corresponds to a sum of the following form:

where

— is the radix of the said number system. Positional code of one-variable function correspond to a "double' code f the form:

where

— is an integer positive number, the number of a digit's values

— is a certain function of argument

.

Addition of positional codes of numbers is associated with the carry transfer to a highest digit according to the scheme:

Addition of positional codes of one-variable functions is also associated with the carry transfer to a highest digits according to the scheme:

Here the same transfer is carried simultaneously to "two" highest digits.

R-nary triangular code

A triangular code is called R-nary (and is denoted as

), if the numbers

take their values from the set

где

.
For example, a triangular code is a ternary code

, if

, and quaternary

, if

.

For R-nary triangular codes the following equalities are valid:

,
where a is an arbitrary number. There exists

of an arbitrary integer real number. In particular,

. Also there exists

of any function of the form

. For instance,

Single-digit addition

in R-nary triangular codes consists in the following:

in the given -digit there is determined the sum of the digits that are being added and two carries , transferred into this digit from the left, i.e.

this sum is presented in the form , where ,
is written in the -digit of summary code, and the carry from the given digit is carried into -digit and --digit.

This procedure is described (as also for one-digit addition of the numbers) by a table of one-digit addition, where all the values of the terms

and

must be present and all the values of carries appearing at decomposition of the sum

. Such a table may be synthesized for

Below we have written the table of one-digit addition for

: !Smk>

TK(Smk)

>-
|.

|style="background:#FFCC00"|0

0
|-
|.

|style="background:#FFCC00"|1

0
|-
|.

|style="background:#FFCC00"|(-1)

(-1)

0
|-
|.

|style="background:#FFCC00"|2

(-1)

1
|-
|.

|style="background:#FFCC00"|3

1
|-
|.

|style="background:#FFCC00"|4

1
|-
|.

(-1)

|style="background:#FFCC00"|(-2)

(-1)

(-1)
|-
|.

(-1)

|style="background:#FFCC00"|(-3)

(-1)

(-1)
|-
|.

(-1)

|style="background:#FFCC00"|(-4)

(-1)

One-digit subtraction

in R-nary triangular codes differs from the one-digit addition only by the fact that in the given

-digit the value

is determined by the formula

One-digit division by the parameter R

in R-nary triangular codes is based on using the correlation:

from this it follows that the division of each digit causes carries into two lowest digits. Hence, the digits result in this operation is a sum of the quotient from the division of this digit by R and two carries from two highest digits. Thus, when divided by parameter R

in the given -digit the following sum is determined

this sum is presented as , where ,
is written into --digit of the resulting code, and carry from the given digit is transferred into the -digit and -digit.

This procedure is described by the table of one-digit division by parameter R, where all the values of terms and all values of carries, appearing at the decomposition of the sum

, must be present. Such table may be synthesized for

Below the table will be given for the one-digit division by the parameter R for

: !Smk> |. >

TK(Smk)			>- \|.	.	0	.	\|style="background:#FFCC00"\|0	0	0	0	0 \|- \|.	.	1	.	\|style="background:#FFCC00"\|1	0	0	1	0 \|- \|.	.	(-1)	.	\|style="background:#FFCC00"\|(-1)	0	0	(-1)	0 \|- \|.	.	0	.	\|style="background:#FFCC00"\|1/3	1	(-1/3)	0	1 \|- \|.	.	1	.	\|style="background:#FFCC00"\|2/3	(-1)	1/3	1	(-1) \|- \|.	.	1	.	\|style="background:#FFCC00"\|4/3	1	(-1/3)	1	1 \|- \|.	.	2	.	\|style="background:#FFCC00"\|5/3	(-1)	1/3	2	(-1) \|- \|.	.	0	.	\|style="background:#FFCC00"\|(-1/3)	(-1)	1/3	0	(-1) \|- \|.	.	(-1)	.	\|style="background:#FFCC00"\|(-2/3)	1	(-1/3)	(-1)	1 \|- \|.	.	(-1)	.	\|style="background:#FFCC00"\|(-4/3)	(-1)	1/3	(-1)	(-1)
.	(-2)	.	\|style="background:#FFCC00"\|(-5/3)	1	(-1/3)	(-2)	1

Addition and subtraction

of R-nary triangular codes consists (as in positional codes of numbers) in subsequently performed one-digit operations. Mind that the one-digit operations in all digits of each column are performed simultaneously.

Multiplication

of R-nary triangular codes. Multiplication of a code

-digit of another code

consists in

-shift of the code

, i.e. its shift k columns left and m rows up. Multiplication of codes

and

consists in subsequent

-shifts of the code

and addition of the shifted code

with the part-product (as in the positional codes of numbers).

Derivation

of R-nary triangular codes. The derivative of function

, defined above, is

.
So the derivation of triangular codes of a function

consists in determining the triangular code of the partial derivative

and its multiplication by the known triangular code of the derivative

. The determination of the triangular code of the partial derivative

is based on the correlation

.
The derivation method consists of organizing carries from mk-digit into (m+1,k)-digit and into (m-1,k)-digit, and their summing in the given digit is performed in the same way as in one-digit addition.

Coding and decoding

of R-nary triangular codes. A function represented by series of the form

with integer coefficients

, may be represented by R-nary triangular codes, for these coefficients and functions

have R-nary triangular codes (which was mentioned in the beginning of the section). On the other hand, R-nary triangular code may be represented by the said series, as any term

in the positional expansion of the function (corresponding to this code) may be represented by a similar series.

Truncation

of R-nary triangular codes. This is the name of an operation of reducing the number of "non"-zero columns. The necessity of truncation appears at the emergence of carries beyond the digit net. The truncation consists in division by parameter R. All coefficients of the series represented by the code are reduced R times, and the fractional parts of these coefficients are discarded. The first term of the series is also discarded. Such reduction is acceptable if it is known that the series of functions converge. Truncation consists in subsequently performed one-digit operations of division by parameter R. The one-digit operations in all the digits of a row are performed simultaneously, and the carries from lower row are discarded.

Scale factor

R-nary triangular code is accompanied by a scale factor M, similar to exponent for floating-point number. Factor M permits to display all coefficients of the coded series as integer numbers. Factor M is multiplied by R at the code truncation. For addition factors M are aligned, to do so one of added codes must be truncated. For multiplication the factors M are also multiplied.

Positional code for functions of many variables

Positional code for function of two variables is depicted on Figure 1. It corresponds to a "triple" sum of the form::

,

where

is an integer positive number, number of values of the figure

, and

— certain functions of arguments

correspondingly. On Figure 1 the nodes correspond to digits

, and in the circles the values of indexes

of the corresponding digit are shown. The positional code of the function of two variables is called "pyramidal". Positional code is called R-nary (and is denoted as

), if the numbers

assume the values from the set

. At the addition of the codes

the carry extends to four digits and hence

.

A positional code for the function from several variables corresponds to a sum of the form

where

is an integer positive number, number of values of the digit

, and

certain functions of arguments

. A positional code of a function of several variables is called "hyperpyramidal". Of Figure 2 is depicted for example a positional hyperpyramidal code of a function of three variables. On it the nodes correspond to the digits

, and the circles contain the values of indexes

of the corresponding digit. A positional hyperpyramidal code is called R-nary (and is denoted as

), if the numbers

assume the values from the set

. At the codes addition

the carry extends on a-dimensional cube, containing

digits, and hence

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