Computer for operations with functions - Revision history

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	Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 \| isbn=5-7707-6131-8 }} ([~~http~~://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 \|volume= 4 }}(see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|work=Mathematics in Computers \|location=Israel \|year=2004 \| isbn=978-0-557-07520-1 }} (see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref>		Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 \| isbn=5-7707-6131-8 }} ([https://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 \|volume= 4 }}(see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|work=Mathematics in Computers \|location=Israel \|year=2004 \| isbn=978-0-557-07520-1 }} (see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref>

	== History ==		== History ==

RJFJR: /* Positional code for functions of many variables {{cite journal |last=Khmelnik |first=S.I. | title=Several types of positional functions codes |publisher= Cybernetics, USSR Academy of Sciences |year=1970 |volume= 5 }} (see also here in Russian) */ move ref out of heading

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Positional code for functions of many variables {{cite journal |last=Khmelnik |first=S.I. | title=Several types of positional functions codes |publisher= Cybernetics, USSR Academy of Sciences |year=1970 |volume= 5 }} (see also here in Russian): move ref out of heading

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	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.		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 <ref name=" Khmelnik3">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Several types of positional functions codes \|publisher= Cybernetics, USSR Academy of Sciences \|year=1970 \|volume= 5 }} (see also [http://lib.izdatelstwo.com/Papers2/s17.pdf here in Russian])</ref>==		== Positional code for functions of many variables ==
			Source:<ref name=" Khmelnik3">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Several types of positional functions codes \|publisher= Cybernetics, USSR Academy of Sciences \|year=1970 \|volume= 5 }} (see also [http://lib.izdatelstwo.com/Papers2/s17.pdf here in Russian])</ref>

	Positional code for function of two variables is depicted on Figure 1. It corresponds to a "triple" sum of the form:: <math>F(x,v) = \sum_{k=0}^{n} \sum_{m1=0}^{k} \sum_{m2=0}^{k} \alpha_{m1,m2,k} R^k y^{k-m1}(1-y)^{m1} z^{k-m2}(1-z)^{m2}</math>,<br>		Positional code for function of two variables is depicted on Figure 1. It corresponds to a "triple" sum of the form:: <math>F(x,v) = \sum_{k=0}^{n} \sum_{m1=0}^{k} \sum_{m2=0}^{k} \alpha_{m1,m2,k} R^k y^{k-m1}(1-y)^{m1} z^{k-m2}(1-z)^{m2}</math>,<br>

RJFJR: /* Positional codes of one-variable functions */ move ref out of heading

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Positional codes of one-variable functions: move ref out of heading

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	A computing machine for operations with functions was presented and developed by Mikhail Kartsev in 1967.<ref name="Malinovsky"/> Among the operations of this computing machine were the functions addition, subtraction and multiplication, 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 X-axis etc. By its [[computer architecture\|architecture]] this computing machine was (using the modern terminology) a [[vector processor]] or ''array processor'', a [[central processing unit]] (CPU) that implements an instruction set containing instructions that operate on [[array data structure\|one-dimensional array]]s of data called ''vectors''. In it there 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.<ref name="Malinovsky"/> In 1966 Khmelnik had proposed a functions coding method,<ref name="Khmelnik1"/> 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 logic unit\|arithmetic unit]].<ref name="Khmelnik2"/>		A computing machine for operations with functions was presented and developed by Mikhail Kartsev in 1967.<ref name="Malinovsky"/> Among the operations of this computing machine were the functions addition, subtraction and multiplication, 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 X-axis etc. By its [[computer architecture\|architecture]] this computing machine was (using the modern terminology) a [[vector processor]] or ''array processor'', a [[central processing unit]] (CPU) that implements an instruction set containing instructions that operate on [[array data structure\|one-dimensional array]]s of data called ''vectors''. In it there 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.<ref name="Malinovsky"/> In 1966 Khmelnik had proposed a functions coding method,<ref name="Khmelnik1"/> 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 logic unit\|arithmetic unit]].<ref name="Khmelnik2"/>

	== Positional codes of one-variable functions <ref name="Khmelnik1"/><ref name="Khmelnik2"/>==		== Positional codes of one-variable functions ==
			Source:<ref name="Khmelnik1"/><ref name="Khmelnik2"/>

	=== The main idea ===		=== The main idea ===

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	Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 \| isbn=5-7707-6131-8 }} ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 \|volume= 4 }}(see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|~~publisher~~="Mathematics in Computers" \|location=Israel \|year=2004 \| isbn=978-0-557-07520-1 }} (see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref>		Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 \| isbn=5-7707-6131-8 }} ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 \|volume= 4 }}(see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|work=Mathematics in Computers \|location=Israel \|year=2004 \| isbn=978-0-557-07520-1 }} (see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref>

	== History ==		== History ==

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	Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])~~\| isbn=5-7707-6131-8 }}~~</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf ~~see also~~ here in Russian) ~~\|volume= 4 }}~~</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf ~~see also~~ here in Russian)~~\| isbn=978-0-557-07520-1 }}~~</ref>		Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 \| isbn=5-7707-6131-8 }} ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 \|volume= 4 }}(see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 \| isbn=978-0-557-07520-1 }} (see also [http://lib.izdatelstwo.com/Papers2/s7.pdf here in Russian])</ref>

	== History ==		== History ==
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	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.		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 <ref name=" Khmelnik3">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Several types of positional functions codes \|publisher= Cybernetics, USSR Academy of Sciences \|year=1970 (http://lib.izdatelstwo.com/Papers2/s17.pdf ~~see also~~ here in Russian) ~~\|volume= 5 }}~~</ref>==		== Positional code for functions of many variables <ref name=" Khmelnik3">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Several types of positional functions codes \|publisher= Cybernetics, USSR Academy of Sciences \|year=1970 \|volume= 5 }} (see also [http://lib.izdatelstwo.com/Papers2/s17.pdf here in Russian])</ref>==

	Positional code for function of two variables is depicted on Figure 1. It corresponds to a "triple" sum of the form:: <math>F(x,v) = \sum_{k=0}^{n} \sum_{m1=0}^{k} \sum_{m2=0}^{k} \alpha_{m1,m2,k} R^k y^{k-m1}(1-y)^{m1} z^{k-m2}(1-z)^{m2}</math>,<br>		Positional code for function of two variables is depicted on Figure 1. It corresponds to a "triple" sum of the form:: <math>F(x,v) = \sum_{k=0}^{n} \sum_{m1=0}^{k} \sum_{m2=0}^{k} \alpha_{m1,m2,k} R^k y^{k-m1}(1-y)^{m1} z^{k-m2}(1-z)^{m2}</math>,<br>

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	Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])\| isbn=5-7707-6131-8 }}</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian) \|volume= 4 }}</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian)\| isbn=978-0-557-07520-1 }}</ref>		Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])\| isbn=5-7707-6131-8 }}</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian) \|volume= 4 }}</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian)\| isbn=978-0-557-07520-1 }}</ref>

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	{{null\|date=May 2011}}		{{null\|date=May 2011}}
	A '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])\| isbn=5-7707-6131-8 }}</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian) \|volume= 4 }}</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian)\| isbn=978-0-557-07520-1 }}</ref>		Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])\| isbn=5-7707-6131-8 }}</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian) \|volume= 4 }}</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian)\| isbn=978-0-557-07520-1 }}</ref>

	== History ==		== History ==

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	where <math>R</math> is an integer positive number, number of values of the digit <math>\alpha_{m_1, \ldots, m_a,k}</math>, and <math>y_i(x_i)</math> certain functions of arguments <math>x_i</math>. 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 <math>\alpha_{m1,m2,m3,k}</math>, and the circles contain the values of indexes <math>{m1,m2,m3,k}</math> of the corresponding digit. A positional hyperpyramidal code is called R-nary (and is denoted as <math>GPK_R</math>), if the numbers <math>\alpha_{m_1, \ldots, m_a,k}</math> assume the values from the set <math>D_R</math>. At the codes addition <math>GPK_R</math> the carry extends on ''a''-dimensional cube, containing <math>2^a</math> digits, and hence <math>R \geq (2^{a-1}-1)</math>.		where <math>R</math> is an integer positive number, number of values of the digit <math>\alpha_{m_1, \ldots, m_a,k}</math>, and <math>y_i(x_i)</math> certain functions of arguments <math>x_i</math>. 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 <math>\alpha_{m1,m2,m3,k}</math>, and the circles contain the values of indexes <math>{m1,m2,m3,k}</math> of the corresponding digit. A positional hyperpyramidal code is called R-nary (and is denoted as <math>GPK_R</math>), if the numbers <math>\alpha_{m_1, \ldots, m_a,k}</math> assume the values from the set <math>D_R</math>. At the codes addition <math>GPK_R</math> the carry extends on ''a''-dimensional cube, containing <math>2^a</math> digits, and hence <math>R \geq (2^{a-1}-1)</math>.

			== See also ==

			* [[Hardware acceleration]]
			* [[Digital signal processor]]

	==References==		==References==

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History

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	== History ==		== History ==
	A computing machine for operations with functions was presented and developed by Mikhail Kartsev in 1967.<ref name="Malinovsky"/> Among the operations of this computing machine were the functions addition, subtraction and multiplication, 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 [[computer architecture\|architecture]] this computing machine was (using the modern terminology) a [[vector processor]] or ''array processor'', a [[central processing unit]] (CPU) that implements an instruction set containing instructions that operate on [[array data structure\|one-dimensional array]]s of data called ''vectors''. In it there 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.<ref name="Malinovsky"/> In 1966 Khmelnik had proposed a functions coding method,<ref name="Khmelnik1"/> 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 logic unit\|arithmetic unit]].<ref name="Khmelnik2"/>		A computing machine for operations with functions was presented and developed by Mikhail Kartsev in 1967.<ref name="Malinovsky"/> Among the operations of this computing machine were the functions addition, subtraction and multiplication, 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 X-axis etc. By its [[computer architecture\|architecture]] this computing machine was (using the modern terminology) a [[vector processor]] or ''array processor'', a [[central processing unit]] (CPU) that implements an instruction set containing instructions that operate on [[array data structure\|one-dimensional array]]s of data called ''vectors''. In it there 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.<ref name="Malinovsky"/> In 1966 Khmelnik had proposed a functions coding method,<ref name="Khmelnik1"/> 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 logic unit\|arithmetic unit]].<ref name="Khmelnik2"/>

	== Positional codes of one-variable functions <ref name="Khmelnik1"/><ref name="Khmelnik2"/>==		== Positional codes of one-variable functions <ref name="Khmelnik1"/><ref name="Khmelnik2"/>==

← Previous revision		Revision as of 00:23, 10 January 2022
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			{{Multiple issues\|
	{{lead too short\|date=May 2011}}		{{lead too short\|date=May 2011}}
			{{Technical\|date=January 2022}}
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	{{null\|date=May 2011}}		{{null\|date=May 2011}}
	Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])\| isbn=5-7707-6131-8 }}</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian) \|volume= 4 }}</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian)\| isbn=978-0-557-07520-1 }}</ref>		Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])\| isbn=5-7707-6131-8 }}</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian) \|volume= 4 }}</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian)\| isbn=978-0-557-07520-1 }}</ref>

← Previous revision		Revision as of 00:22, 10 January 2022
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	{{multiple issues\|
	{{context\|date=May 2011}}
	{{lead too short\|date=May 2011}}		{{lead too short\|date=May 2011}}
	}}

	{{null\|date=May 2011}}		{{null\|date=May 2011}}
	A '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])\| isbn=5-7707-6131-8 }}</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian) \|volume= 4 }}</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian)\| isbn=978-0-557-07520-1 }}</ref>		Within [[computer engineering]] and [[computer science]], a '''computer for operations with (mathematical) functions''' (unlike the usual [[computer]]) operates with [[Function (mathematics)\|functions]] at the [[computer hardware\|hardware]] level (i.e. without programming these operations).<ref name="Malinovsky">{{cite book \|last= Malinovsky \|first=B.N. \|title= The history of computer technology in their faces (in Russian) \|publisher=Firm "KIT" \|location=Kiew \|year=1995 ([http://lib.ru/MEMUARY/MALINOWSKIJ/0.txt see also here http://www.sigcis.org/files/SIGCISMC2010_001.pdf and english version here])\| isbn=5-7707-6131-8 }}</ref><ref name=" Khmelnik1">{{cite journal \|last=Khmelnik \|first=S.I. \| title=Coding of functions \|publisher= Cybernetics, USSR Academy of Sciences \|year=1966 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian) \|volume= 4 }}</ref><ref name=" Khmelnik2">{{cite book \|last=Khmelnik \|first=S.I. \|title= Computer Arithmetic of Functions. Algorithms and Hardware Design \|publisher="Mathematics in Computers" \|location=Israel \|year=2004 (http://lib.izdatelstwo.com/Papers2/s7.pdf see also here in Russian)\| isbn=978-0-557-07520-1 }}</ref>

	== History ==		== History ==