Computer for operations with functions

The positional code of an integer number ${\displaystyle A}$ is a numeral notation of digits ${\displaystyle \alpha }$ in a certain positional number system of the form

${\displaystyle A=\alpha _{0}\alpha _{1}\dots \alpha _{k}\dots \alpha _{n}}$.

Such code may be called "linear". Unlike it a positional code of one-variable ${\displaystyle x}$ function ${\displaystyle F(x)}$ has the form:

${\displaystyle F(x)={\begin{pmatrix}\ &\ &\cdots \\\ &\ &\alpha _{22}\cdots \alpha _{2k}\cdots \\\ &\alpha _{11}&\alpha _{12}\cdots \alpha _{1k}\cdots \\\alpha _{00}&\alpha _{01}&\alpha _{02}\cdots \alpha _{0k}\cdots \end{pmatrix}}}$

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

The value of the positional number ${\displaystyle A}$ above is that of the sum

${\displaystyle A=\sum _{k=0}^{n}\alpha _{k}\rho ^{k}}$,

where ${\displaystyle \rho }$ is the radix of the said number system. The positional code of a one-variable function correspond to a 'double' code of the form

${\displaystyle F(x)=\sum _{k=0}^{n}\sum _{m=0}^{k}\alpha _{mk}R^{k}y^{k-m}(1-y)^{m}}$,

where ${\displaystyle R}$ is an integer positive number, quantity of values that taken ${\displaystyle \alpha }$, and ${\displaystyle y}$ is a certain function of argument ${\displaystyle x}$.

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

${\displaystyle \alpha _{k}\longrightarrow \alpha _{k+1}}$.

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

${\displaystyle {\begin{pmatrix}\ \ &\alpha _{k+1,m+1}\\\ \nearrow &\ \\\alpha _{k,m}\longrightarrow &\alpha _{k+1,m}\end{pmatrix}}}$.

Here the same transfer is carried simultaneously to two higher digits.

A triangular code is called R-nary (and is denoted as ${\displaystyle TK_{R}}$), if the numbers ${\displaystyle \alpha _{mk}}$ take their values from the set

${\displaystyle D_{R}=\{-r_{1},-r_{1}+1,\dots ,-1,0,1,\dots ,r_{2}-1,r_{2}\}}$, where ${\displaystyle r_{1},\;r_{2}\geq 0}$ and ${\displaystyle R_{}^{}=r_{1}+r_{2}+1}$.

For example, a triangular code is a ternary code ${\displaystyle TK_{3}}$, if ${\displaystyle \alpha _{mk}\in (-1,0,1)}$, and quaternary ${\displaystyle TK_{4}}$, if ${\displaystyle \alpha _{mk}\in (-2,-1,0,1)}$.
For R-nary triangular codes the following equalities are valid:

${\displaystyle {\begin{pmatrix}\ \ &0\\\ \nearrow &\ \\aR\longrightarrow &0\end{pmatrix}}={\begin{pmatrix}\ \ &a\\\ \nearrow &\ \\0\longrightarrow &a\end{pmatrix}},\quad {\begin{pmatrix}\ \ &a\\\ \nearrow &\ \\0\longrightarrow &0\end{pmatrix}}={\begin{pmatrix}\ \ &0\\\ \nearrow &\ \\aR\longrightarrow &-a\end{pmatrix}},\quad {\begin{pmatrix}\ \ &0\\\ \nearrow &\ \\0\longrightarrow &a\end{pmatrix}}={\begin{pmatrix}\ \ &-a\\\ \nearrow &\ \\aR\longrightarrow &0\end{pmatrix}}}$,

where ${\displaystyle a}$ is an arbitrary number. There exists ${\displaystyle TK_{R}}$ of an arbitrary integer real number. In particular, ${\displaystyle TK_{R}(\alpha )=\alpha }$. Also there exists ${\displaystyle TK_{R}}$ of any function of the form ${\displaystyle y^{k}}$. For instance, ${\displaystyle TK_{R}(y^{2})=(0\ 0\ 1)}$.

in R-nary triangular codes consists in the following:

• in the given ${\displaystyle (mk)}$-digit there is determined the sum ${\displaystyle S_{mk}^{}}$ of the digits that are being added ${\displaystyle \alpha _{mk},\ \beta _{mk}}$ and two carries ${\displaystyle p_{m,k-1},\ p_{m-1,k-1}}$, transferred into this digit from the left, i.e.

${\displaystyle S_{mk}^{}=\alpha _{mk}+\beta _{mk}+p_{m,k-1}+p_{m-1,k-1}}$,

• this sum is presented in the form ${\displaystyle S_{mk}^{}=\sigma _{mk}+Rp_{mk}}$, where ${\displaystyle \sigma _{mk}\in D_{R}}$,
• ${\displaystyle \sigma _{mk}}$ is written in the ${\displaystyle (mk)}$-digit of summary code, and the carry ${\displaystyle p_{mk}}$ from the given digit is carried into ${\displaystyle (m,k+1)}$-digit and ${\displaystyle (m+1,k+1)}$—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 ${\displaystyle \alpha _{mk}\in D_{R}}$ and ${\displaystyle \beta _{mk}\in D_{R}}$ must be present and all the values of carries appearing at decomposition of the sum ${\displaystyle S_{mk}^{}=\sigma _{mk}+Rp_{mk}}$. Such a table may be synthesized for ${\displaystyle R>2.}$
Below we have written the table of one-digit addition for ${\displaystyle R=3}$:

Smk	TK(Smk)		${\displaystyle \sigma _{mk}^{}}$	${\displaystyle p_{mk}^{}}$
.	.	0	.	.
0	0	0	0	0
.	.	0	.	.
1	1	0	1	0
.	.	0	.	.
(-1)	(-1)	0	(-1)	0
.	.	1	.	.
2	(-1)	1	(-1)	1
.	.	1	.	.
3	0	1	0	1
.	.	1	.	.
4	1	1	1	1
.	.	(-1)	.	.
(-2)	1	(-1)	1	(-1)
.	.	(-1)	.	.
(-3)	0	(-1)	0	(-1)
.	.	(-1)	.	.
(-4)	(-1)	(-1)	(-1)	(-1)

in R-nary triangular codes differs from the one-digit addition only by the fact that in the given ${\displaystyle (mk)}$-digit the value ${\displaystyle S_{mk}^{}}$ is determined by the formula

${\displaystyle S_{mk}^{}=\alpha _{mk}-\beta _{mk}+p_{m,k-1}+p_{m-1,k-1}}$.

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

${\displaystyle {\begin{pmatrix}\ \ &a\\\ \nearrow &\ \\0\longrightarrow &0\end{pmatrix}}={\begin{pmatrix}\ \ &0\\\ \nearrow &\ \\aR\longrightarrow &-a\end{pmatrix}}}$,

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 ${\displaystyle (mk)}$-digit the following sum is determined

${\displaystyle S_{mk}^{}=\alpha _{mk}/R-p_{m+1,k}/R+p_{m+1,k+1}}$,

• this sum is presented as ${\displaystyle S_{mk}^{}=\sigma _{mk}+p_{mk}/R}$, where ${\displaystyle \sigma _{mk}\in D_{R}}$,
• ${\displaystyle \sigma _{mk}}$ is written into ${\displaystyle (mk)}$—digit of the resulting code, and carry ${\displaystyle p_{mk}}$ from the given digit is transferred into the ${\displaystyle (m-1,k-1)}$-digit and ${\displaystyle (m-1,k)}$-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 ${\displaystyle S_{mk}^{}=\sigma _{mk}+p_{mk}/R}$, must be present. Such table may be synthesized for ${\displaystyle R>2.}$
Below the table will be given for the one-digit division by the parameter R for ${\displaystyle R=3}$:

Smk	TK(Smk)		${\displaystyle \sigma _{mk}^{}}$	${\displaystyle p_{mk}^{}}$
.	.	0	.	.
0	0	0	0	0
.	.	1	.	.
1	0	0	1	0
.	.	(-1)	.	.
(-1)	0	0	(-1)	0
.	.	0	.	.
1/3	1	(-1/3)	0	1
.	.	1	.	.
2/3	(-1)	1/3	1	(-1)
.	.	1	.	.
4/3	1	(-1/3)	1	1
.	.	2	.	.
5/3	(-1)	1/3	2	(-1)
.	.	0	.	.
(-1/3)	(-1)	1/3	0	(-1)
.	.	(-1)	.	.
(-2/3)	1	(-1/3)	(-1)	1
.	.	(-1)	.	.
(-4/3)	(-1)	1/3	(-1)	(-1)
.	.	(-2)	.	.
(-5/3)	1	(-1/3)	(-2)	1

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.

of R-nary triangular codes. Multiplication of a code ${\displaystyle TK_{R}'^{}}$ by ${\displaystyle (mk)}$-digit of another code ${\displaystyle TK_{R}''^{}}$ consists in ${\displaystyle (mk)}$-shift of the code ${\displaystyle TK_{R}'^{}}$, i.e. its shift k columns left and m rows up. Multiplication of codes ${\displaystyle TK_{R}'^{}}$ and ${\displaystyle TK_{R}''^{}}$ consists in subsequent ${\displaystyle (mk)}$-shifts of the code ${\displaystyle TK_{R}'^{}}$ and addition of the shifted code ${\displaystyle TK_{R}'^{}}$ with the part-product (as in the positional codes of numbers).

of R-nary triangular codes. The derivative of function ${\displaystyle F(x)}$, defined above, is

${\displaystyle {\frac {\partial F(x)}{\partial x}}={\frac {\partial y}{\partial x}}{\frac {\partial F(x)}{\partial y}}}$.

So the derivation of triangular codes of a function ${\displaystyle F(x)}$ consists in determining the triangular code of the partial derivative ${\displaystyle {\frac {\partial F(x)}{\partial y}}}$ and its multiplication by the known triangular code of the derivative ${\displaystyle {\frac {\partial y}{\partial x}}}$. The determination of the triangular code of the partial derivative ${\displaystyle {\frac {\partial F(x)}{\partial y}}}$ is based on the correlation

${\displaystyle {\frac {\partial }{\partial x}}{\begin{pmatrix}\ &\ &0\\\ &0&\alpha _{mk}\\0&0&0\end{pmatrix}}={\begin{pmatrix}\ &\ &(k-m)\alpha _{mk}\\\ &0&(k-2m)\alpha _{mk}\\0&0&(-m)\alpha _{mk}\end{pmatrix}}}$.

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.

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

${\displaystyle F(x)=\sum _{k=0}^{n}A_{k}y^{k}}$,

with integer coefficients ${\displaystyle A_{k}}$, may be represented by R-nary triangular codes, for these coefficients and functions ${\displaystyle y^{k}}$ 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 ${\displaystyle \alpha _{mk}R^{k}y^{k}(1-y)^{m}}$ in the positional expansion of the function (corresponding to this code) may be represented by a similar series.

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.

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.