Digital function generator utilizing cascade accumulation

• US 3,967,100 A
• Filed: 09/13/1974
• Issued: 06/29/1976
• Est. Priority Date: 11/12/1973
• Status: Expired due to Term

• Current Assignee: Naonobu Shimomura
• Original Assignee: Naonobu Shimomura
• Inventors: Shimomura, Naonobu
• Primary Examiner(s): Malzahn, David H.
• Application Number:
  US05/505,849
• Time in Patent Office: 655 Days
• Field of Search: 235/152, 235/197, 235/151.11
• US Class Current: 708/446
• CPC Class Codes: G06F 2207/5523 Calculates a power, e.g. the square, of a number or a function, e.g. polynomials
• CPC Class Codes: G06F 7/552 Powers or roots , e.g. Pythagorean sums

Abstract

This invention relates to apparatus for obtaining the numerical value of a function of a single or plural number of variables. The method of cascade accumulation is used in which memories M0, M1, M2, . . . , Mn are involved and under one cycle of manipulation the contents of memories M1, M2, . . . , Mn are added to the contents of M0, M1, M2, . . . , Mn-1 respectively. By loading a digital value determined by a₀, a₁, a₂, . . . , a_n into each of the memories as its initial value and repeating x cycles of the above mentioned manipulations, the digital value of the function

w = a.sub.0 + a.sub.1 x + a.sub.2 x.sup.2 + . . . + a.sub.n x.sup.n 1.

is obtained at the memory M0.

By extending the above principle to two dimensional operations, a digital function of two variables expressed by

w = a.sub.00 + a.sub.10 x.sub.1 + a.sub.20 x.sub.1.sup.2 + a.sub.30

x₁³ + . . . + a₀₁ x₂ + a₁₁ x₁ x₂ + a₂₁ x₁² x₂ + a₃₁ x₁³ x₂ + . . . + a₀₂ x₂² + a₁₂ x₁ x₂² + a₂₂ x₁² x₂² + a₃₂ x₁³ x₂² + . . . + a₀₃ x₂³ + a₁₃ x₁ x₂³ + a₂₃ x₁² x₂³ + a₃₃ x₁³ x₂³ + . . . + . . . 2.

is generated.

Further generalizing the cascade accumulation to n dimensions, a digital function of n variables

w =Σ a.sub.i i . . . i . . . i x.sub.1.sup.1 x.sub.2.sup.i . . .

x_jⁱ . . . x_nⁱ 3.

is generated.

Since many of the relations of scientific or industrial parameters given by mathematical formulae or numerical tables of empirical data can be expressed with sufficient practical accuracy by polynomials of a power series as shown above, this invention covers a broad field of function generation for practical purposes.

Very simple and reliable electrical circuits to be used for the above operation are disclosed, in one example of which only a shift register, a delay circuit and an adder are involved as the key elements in the logic circuit to generate a multi-variable function, and no complicated programming is needed.

Another type of function generator where w is an independent variable and x is the function of w in equation (1) is also disclosed.

Claims

1. Apparatus for obtaining a digital value of a function expressed by:

space="preserve" listing-type="equation">w = Σ

a.sub.i.sbsb.1i.sbsb.2 . . . i.sub..sbsb.j. . . i.sbsb.nx.sub.1.sup.i.sbsp.1 x.sub.2 .sup.i.sbsp.2 . . . x.sub.j.sup.i.sbsp.j . . . x.sub.n.sup.i.sbsp.n

where w is the function the value of which is to be obtained,

a_i.sbsb.1i .sbsb.2i.sbsb.j. . . i.sbsb.nn '"'"'s are constants,

x₁, x₂, . . . , x_j, . . . , x_n are the independent variables,

i_j = 0, 1, 2, . . . , m_j, and at least one of m_j '"'"'s ≧

2,

where j = 1, 2, . . . , n and represents the dimension of a multidimensional function, wherein n ≧

2.sup., comprising;

a. memories designated by multi-dimensional addresses (i₁, i₂, . . . , i_j, . . . , i_n) where i_j = 0, 1, 2, . . . , m_j and at least one m_j ≧

2, each memory having a word length and means for loading said memories with respective, specific, initial numerical values determined by the constants a_i .sbsb.1i.sbsb.2. . . i.sbsb.j . . . i.sbsb.n '"'"'s,

3. Apparatus as claimed in claim 1 wherein said means for performing algebraic additions comprises:

means for arranging said memories of multi-dimensional addresses in a shift register means,

delay means, the amount of delay thereof being selectable,

said adder means having first and second inputs, and said control means selecting the amount of delay of said delay means in accordance with the dimension j of the variable x_j corresponding to which said cycles of algebraic additions are to be performed, and reading out said shift register means and controlling said selective interconnecting means to interconnect said shift register means, said adder means and said delay means for applying digits read out from said shift register means to one of the inputs of said adder means and for applying the output digits from said adder means to the other input of said adder means through said delay means in accordance with the selected delay thereof and also to said shift register means for rewriting therein.

5. Apparatus as claimed in claim 1 wherein said means for performing algebraic additions comprises:

means for arranging said memories of multi-dimensional addresses in a shift register means,

delay means, the amount of delay thereof being selectable, said adder means having first and second inputs, and

said control means selecting the amount of delay of said delay means in accordance with the dimension j of the variable x_j corresponding to which said cycles of algebraic additions are to be performed, and reading out said shift register means and controlling said selective interconnecting means to interconnect said shift register means, said delay means, and said adder means for applying digits read out from said shift register means to one of the inputs of said adder means and also through said delay means in accordance with the selected delay thereof to the other input of said adder means, and for applying the output digits from said adder means to said shift register means for rewriting therein.

7. Apparatus as claimed in claim 1 wherein there are included m_j +1 memories of said multi-dimensional addresses (i₁, i₂, . . . , i_j, . . . , i_n) where i_j = 0, 1, 2, . . . , m_j, and m_j adder means designated by numbers 1, 2, ..., m_j, and wherein said selective interconnecting means connects said memories and said adder means for applying the digits read out from the memories of addresses (i₁, i₂, . . . , i_j, . . . i_n) where i_j = 0, 1, 2, . . . , m_j -1, to one of the inputs of said adder means of number i_j +1, where i_j = 0, 1, 2, . . . , m_j -1, respectively, for applying the output digits from said adder means of number i_j + 1, where i_j = 1, 2, . . . , m_j -1, to the other input of said adder means of number i_j, where i_j = 1, 2, . . . , m_j -1, respectively, and for applying the output digits from said adder means of number i_j +1, where i_j = 0, 1, 2, . . . , m_j -1, for rewriting into the memories of addresses (i₁, i₂, . . . , i_j, . . . , i_n), where i_j = 0, 1, 2, . . . , m_j -1, respectively, and for applying the digits non-destructively read out from the memory of address (i₁, i₂, . . . , i_j, . . . , i_n) where i_j = m_j to the other input of said adder means of number m_j.

9. Apparatus as claimed in claim 1 wherein there are included m_j + 1 memories of said multi-dimensional addresses (i₁, i₂, . . . , i_j, . . . , i_n), where i_j = 0, 1, 2, . . . , m_j, and m_j adder means designated by numbers 1, 2, . . . , m_j, and wherein said selective interconnecting means connects said memories and said adder means for applying the digits read out from the memories of addresses (i₁, i₂, . . . , ij, . . . ,i_n), where i_j = 0, 1, 2, . . . , m_j- 1, to one of the inputs of said adder means of number i_j+ 1, where i_j = 0, 1, 2, . . . , m_j -1, respectively, for applying the digits read out from the memories of addresses (i₁, i₂, . . . , i.sub. j, . . . , i_n), where i_j = 1, 2, . . . , m_j -1, and the non-destructively read out digits from the memory of the address (i₁, i₂, . . . , i_j, . . . , i_n), where i_j = m_j, to the other input of said adder means of number i_j, where i_j = 1, 2, . . . , m_j, respectively, and applies the output of the adder means of number i_j +1, where i_j = 0, 1, 2, . . . , m_j -1 to the memories of addresses (i₁, i₂, . . . , i_j, . . . , i_n)where i_j = 0, 1, 2, . . . , m_j -1, respectively, for rewriting therein.

2. b. means for performing algebraic additions of the contents of the memories designated by the addresses (i₁, i₂, . . . , i_j-1, i_j + 1, i_j+1, . . . , i_n) to the contents of the memories designated by the addresses (i₁, i₂, . . . , i_j-1, i.sub. j, i_j+1, . . . , i_n) for i_j = 0, 1, 2, . . . , m_j -1 in a predetermined sequence, said algebraic additions being performed with respect to every combination of i₁, i₂, . . . , i_j-1 and at least i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0 in corresponding cycles, said means for performing algebraic additions including:

b-1. adder means,

b-2. means for selectively interconnecting the outputs and inputs of said memories and said adder means for each said cycle of algebraic additions and for maintaining said interconnections during each said cycle, and

b-3. means to control said interconnecting means and said adder means to perform a number of said cycles of algebraic additions in total corresponding to the value of each variable x_j for all of the variables x_j, where j = 1, 2, . . . , n whereby the value of the function w is obtained in the memory of address (0,0,0, . . . ,

     0).

4. Apparatus as claimed in claim 2 wherein said means for performing algebraic additions further includes complementer means connected in series with said delay means between the output of said adder means and said other input thereof for complementing the digits which pass through said delay means.

6. Apparatus as claimed in claim 4 wherein said means for performing algebraic additions further includes complementer means connected in series with said delay means between said shift register means and said other input of said adder means for complementing the digits which pass through said delay means.

8. Apparatus as claimed in claim 6, wherein each said adder means includes first and second inputs, and said means for performing algebraic additions further includes means to complement said output digits from said adder means of number i_j +1, where i_j = 1, 2, . . . , m_j -1, for supplying the complements of the said output digits to said other input of said adder means of number i_j, where i_j = 1, 2, . . . , m_j -1, respectively, and means to complement said non-destructively read out digits from said memory of address (i₁, i₂, . . . , i_j, . . . , i_n), where i_j = m_j, for supplying the complements of the said read out digits to said other input of said adder means of number m_j.

10. Apparatus as claimed in claim 8, wherein said means for performing algebraic additions further includes complementer means connected between said memories of addresses (i₁, i₂, . . . , i_j, . . . , i_n) where i_j = 1, 2, . . . , m_j and said other input of said adder means of the number i_j, where i_j = 1, 2, . . . , m_j, respectively, to complement the digits read from said memories for applying the complements of the said read out digits to the respective said adder means.

13. Apparatus as claimed in claim 10 wherein there is further included complementer means and means for connecting said complementer means in series with said delay means.

11. Apparatus for obtaining a digital value of a function expressed by:

space="preserve" listing-type="equation">w = a.sub.o + a.sub.1 x + a.sub.2 x.sup.2 + . . . + a.sub.i x.sup.i + . . . + a.sub.n x.sup.n

where w is the function the value of which is to be obtained,

a₀, a₁, a₂, . . . , a_i, . . . , a_n are constants,

x is the independent variable, and

n ≧

2 comprising;

plural memories designated by addresses 0, 1, 2, . . . , i, . . . , n, each having a word length and means for loading said memories with a specific, initial numerical value determined by the constants a₀, a₁, a₂, . . . , a_i, . . . a_n,

means for arranging said memories in a shift register means,

algebraic addition means having first and second inputs and an output,

delay means having an input and an output,

means connecting the output of said shift register to one of the inputs of said algebraic addition means,

means connecting the output of said algebraic addition means to the input of said shift register means, and also to the input of said delay means,

12. means connecting the output of said delay means to the other input of said algebraic addition means, and

means for repeatedly shifing said shift register a number of cycles corresponding to the value of the independent variable x.

15. Apparatus as claimed in claim 12 wherein there is further included complementer means and means for connecting said complementer means in series with said delay means.

14. Apparatus for obtaining a digital value of a function expressed by:

space="preserve" listing-type="equation">w = a.sub.0 + a.sub.1 x + a.sub.2 x.sup.2 + . . . + a.sub.i x.sup.i + . . . + a.sub.n x.sup.n

where w is the function the value of which is to be obtained,

a₀, a₁, a₂, . . . , a_i, . . . , a_n are constants,

x is the independent variable, and

n ≧

2 comprising;

plural memories designated by addresses 0, 1, 2, . . . , i, . . . , n, each having a word length and means for loading said memories with a specific, initial numerical value determined by the constants a₀, a₁, a₂, . . . , a_i, . . . , a_n,

means for arranging said memories in a shift register means,

algebraic addition means having first and second inputs and an output and a delay means having an input and an output,

means connecting the output of said shift register to one of the inputs of said algebraic addition means and also to the input of said delay means,

means connecting the output of said delay means to the other input of said algebraic addition means,

means connecting the output of said algebraic addition means to the input of said shift register, and

means for repeatedly shifting said shift register a number of cycles corresponding to the value of the independent variable x.

17. Apparatus as claimed in claim 14 wherein there are further included complementer means connected to the outputs of said memories of addresses 1, 2, . . . , n to complement the digits read out from said memories of addresses 1, 2, . . . , n thereby to supply the complement of the digits as read out to said other inputs of said algebraic addition means of number 1, 2, . . . , n, respectively.

16. Apparatus for obtaining a digital value of a function expressed by:

space="preserve" listing-type="equation">w = a.sub.0 + a.sub.1 x + a.sub.2 x.sup.2 + . . . + a.sub.i x.sup.i + + a.sub.n x.sup.n

where w is the function the value of which is to be obtained,

a₀, a₁, a₂, . . . , a_i, . . . , a_n are constants,

x is the independent variable, and

n ≧

2

comprising;

plural memories designated by addresses 0, 1, 2, . . . i, . . . , n, each having a word length and means for loading said memories with a specific, initial numerical value determined by the constants a₀, a₁, a₂, . . . , a_i, . . . , a_n,

algebraic addition means designated by numbers 1, 2, . . . , i, . . . , n each having first and second inputs and an output,

means for reading out said memories of addresses 0, 1, 2, . . . , n - 1 and for non-destructively reading out said memory of address n, and for supplying the read out digits from the memories of addresses 0, 1, 2, . . . , n - I to one of the inputs of said algebraic addition means designated by respective numbers 1, 2, . . . , n and for supplying the digits read out from the memories of addresses 1, 2, . . . , n - 1 and the digits non-destructively read out from the memory of address n to the other input of said algebraic addition means of number 1, 2, . . . , n, respectively, and means for connecting the outputs of said algebraic addition means of number 1, 2, . . . , n to the inputs of said memories of addresses 0, 1, 2, . . . , n-1, respectively for rewriting therein, and

means to repeatedly read out all said memories a number of times corresponding to the value of the independent variable x.

18. Apparatus for obtaining a digital value of a function expressed by:

space="preserve" listing-type="equation">x = b.sub.1 w + b.sub.2 w.sup.2 + . . . + b.sub.n w.sup.n

where w is the function the value of which is to be obtained,

b₁, b₂, . . . , b_n are constants,

x is the independent variable,

n is a positive integer and n ≧

2, comprising;

a. a cascade accumulation means comprising memory means designated by addresses 0, 1, 2, . . . , n each having a word length, means for loading each of said memory means with a respective, specific, initial numerical value, and means for arranging said memory means serially as a shift register means, an algebraic addition means having first and second inputs and one output and a delay means having an input and an output, means for reading out the digits stored in said shift register means, means leading the read out digits from said shift register means to one of the inputs of said algebraic addition means, means leading the output digits from said algebraic addition means to the input of said shift register means for rewriting therein and also to the input of said delay means, and means for leading the output digits from said delay means to the other input of said algebraic addition means, wherein one cascade accumulation is performed by shifting said shift register means one cycle,

b. means for generating a number of command signals corresponding to the value of said independent variable x,

c. a further non-destructively read memory means designated by an address d loaded with a specific numerical value, and a further algebraic addition means designated by number 0 having first and second inputs and one output.

d. means for reading out said memory means designated by address 0 in said cascade accumulation means and means for leading output digits therefrom to one of the inputs of said algebraic addition means designated by number 0, means for reading out said memory means designated by address d and means for leading the output digits therefrom to the other input of said algebraic addition means designated by number 0, means for leading the output digits from said algebraic addition means designated by number 0 to the input of said memory means designated by address 0 in said cascade accumulation means for rewriting therein, and means for reading out said memory means designated by address 0 in said cascade accumulation means and said memory means designated by address d once, corresponding to each of said command signals,

e. means to produce a number of secondary command signals corresponding to the increment of the content of said memory means designated by address 0 in said cascade accumulation means at each command signal, and

f. means for performing said one cascade accumulation of said cascade accumulation means in response to each of said secondary command signals,

whereby the value of said function w is obtained in said memory means designated by address 0 in said cascade accumulation means.

19. Apparatus for obtaining a digital value of a function expressed by:

space="preserve" listing-type="equation">x = b.sub.1 w + b.sub.2 w.sup.2 + . . . + b.sub.n w.sup.n

where w is the function the value of which is to be obtained,

b₁, b₂, . . . , b_n are constants,

x is the independent variable,

n is a positive integer and n ≧

2, comprising;

a. a cascade accumulation means comprising memory means designated by addresses 0, 1, 2, . . . , n each having a word length, means for loading each of said memory means with a respective, specific, initial numerical value, and means for arranging said memory means serially as a shift register means, an algebraic addition means having first and second inputs and one output and a delay means having an input and an output, means for reading out the digits stored in said shift register means, means leading the read out digits from said shift register means to one of the inputs of said algebraic addition means and also to the input of said delay means, means leading the output digits from said algebraic addition means to the input of said shift register means for rewriting therein, and means for leading the output digits from said delay means to the other input of said algebraic addition means, wherein one cascade accumulation is performed by shifting said shift register means one cycle,

b. means for generating a number of command signals corresponding to the value of said independent variable x,

c. a further non-destructively read memory means designated by an address d loaded with a specific numerical value, and a further algebraic addition means designated by number 0 having first and second inputs and one output,

d. means for reading out said memory means designated by address 0 in said cascade accumulation means and means for leading output digits therefrom to one of the inputs of said algebraic addition means designated by number 0, means for reading out said memory means designated by address d and means for leading the output digits therefrom to the other input of said algebraic addition means designated by number 0, means for leading the output digits from said algebraic addition means designated by number 0 to the input of said memory means designated by address 0 in said cascade accumulation means for rewriting therein, and means for reading out said memory means designated by address 0 in said cascade accumulation means and said memory means designated by address d once, coresponding to each of said command signals,

e. means to produce a number of secondary command signals corresponding to the increment of the content of said memory means designated by address 0 in said cascade accumulation means at each command signal,

f. means for performing said one cascade accumulation of said cascade accumulation means in response to each of said secondary command signals, and

whereby the value of said function w is obtained in said memory means designated by address 0 in said cascade accumulation means.

23. Apparatus as recited in claim 19, wherein said set of operations in the j th dimension comprises performing said element of operation regarding the j th dimension with respect to every combination of the numerical values of i₁, i₂, . . . , i_j-1 and only for i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0, and said executing means executes said set of operations in the j th dimension with respect to every dimension according to the order of j = n-1, . . . , 2, 1 to obtain the value of the function w in the memory means of multi-dimensional address (0, 0, . . . , 0).

24. Apparatus as claimed in claim 19, wherein there is further included complementer means connected in series with said delay means between the output of said algebraic addition means and the other input of said algebraic addition means for any one of said set of operations regarding the j th dimension.

20. apparatus for obtaining a digital value of a function expressed by:

space="preserve" listing-type="equation">x = b.sub.1 w + b.sub.2 w.sup.2 + . . . + b.sub.n w.sup.n

where w is the function the value of which is to be obtaind,

b₁, b₂, . . . , b_n are constants,

x is the independent variable,

n is a positive integer and n ≧

2,

comprising;

a. a cascade accumulation means comprising memory menas designated by addresses 0, 1, 2, . . . , n each having a word length, means for loading each of said memory means with a respective, specific, initial numerical value, algebraic addition means designated by numbers 1, 2, . . . , n each having first and second inputs and one output, means for reading out said memory means designated by addresses 0, 1, 2, . . . , n-1 and for non-destructively reading out said memory means designated by address n, and for leading the digits thus read out from the memory means designated by addresses 0, 1, 2, . . . , n-1 to one of the inputs of said algebraic addition means designated by numbers 1, 2, . . . , n respectively, and for leading the digits read out from said memory means designated by addresses 1, 2, . . . , n-1 and the non-destructively read out digits from said memory means designated by address n to the other inputs of said algebraic addition means designated by numbers 1, 2, . . . , n respectively, and means for leading the output digits from said algebraic addition means designated by numbers 1, 2, . . . , n to said memory means designated by addresses 0, 1, 2, . . . , n-1 respectively for rewriting therein, wherein one cascade accumulation is performed by reading out said memory means designated by addresses 0, 1, 2, . . . , n, once,

b. means for generating a number of command signals corresponding to the value of said independent variable x,

c. a further non-destructively read memory means designated by an address d loaded with a specific numerical value, and a further algebraic addition means designated by number 0 having first and second inputs and one output,

d. means for reading out said memory means designated by address 0 in said cascade accumulation means and means for leading output digits therefrom to one of the inputs of said algebraic addition means designated by number 0, means for reading out said memory means designated by address d and means for leading the output digits therefrom to the other input of said algebraic addition means designated by number 0, means for leading the output digits from said algebraic addition means designated by numeer 0 to the input of said memory means designated by address 0 in said cascade accumulation means for rewriting therein, and means for reading out said memory means designated by address 0 in said cascade accumulation means and said memory means designated by address d once, corresponding to each of said command signals,

e. means to produce a number of secondary command signals corresponding to the increment of the content of said memory means designated by address 0 in said cascade accumulation means at each command signal,

f. means for performing said one cascade accumulation of said cascade accumulation means in response to each of said secondary command signals, and

the value of said function w is obtained in said memory means designated by address 0 in said cascade accumulation means.

21. Apparatus for obtaining a digital value of a function w expressed by:

space="preserve" listing-type="equation">w = Σ

a.sub.i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n .x.sub.1.sup.i.sbsp.1 x.sub.2.sup.i.sbsp.2. . . x.sub.j.sup.i.sbsp.j. . . x.sub.n.sup.i.sbsp.n

where a_i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n '"'"'s are constants,

x_j '"'"'s are the independent variables,

n is a positive integer and n ≧

2,

j = 1, 2, . . . , n,

i_j = 0, 1, 2, . . . , m_j, and

m_j '"'"'s are positive integers and at least one of m_j '"'"'s ≧

2, comprising;

a. memory means each having a word length and designated by a multi-dimensional address (i₁, i₂, . . . , i_j, . . . , i_n) where i_j can assume any one of the numerical values of 0, 1, 2, . . . , m_j, and represents a component of said multi-dimensional address in the j th dimension,

b. means for loading each of said memory means of multi-dimensional addresses with a respective specific initial numerical value determined by the constants a_i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n '"'"'s,

c. means for performing an element of operation in the j th dimension comprising;

1. means for connecting said memory means designated by multi-dimensional addresses (i₁, i₂, . . . , i_j-1, i_j, i_j+1, . . . , i_n) each having a serial number i_j = 0, 1, 2, . . . , m_j as a component in the j th dimension and having identical components in the other dimensions, serially to form a shift register means,

22. algebraic addition means having two inputs and one output,delay means having an input and an output,means for selectively supplying the output digits from said shift register means to one of the inputs of said algebraic addition means,means for selectively supplying the output digits from said algebraic addition means to the input of said shift register means for rewriting therein,means for selectively supplying the output digits from said algebraic addition means to the input of said delay means,means for selectively supplying the output digits from said delay means to the other input of said algebraic addition means, andmeans for shifting said shift register means a number of cycles corresponding to the value of the independent variable x_j, and wherein

a set of operations in the j th dimension comprises performing said element of operation regarding the j th dimension with respect to every combination of the numerical values of i₁, i₂, . . . , i_j-1 and at least i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0, and

d. means for executing said set of operations in the j th dimension with respect to every dimension to obtain the value of the function w in the memory means of multi-dimensional address (0, 0, . . . , 0).

27. Apparatus as recited in claim 22, wherein said set of operations in the j th dimension comprises performing said element of operation regarding the j th dimension with respect to every combination of the numerical values of i₁, i₂, . . . , i_j-1 and only for i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0, and said executing means executes said set of operations in the j th dimension with respect to every dimension according to the order j = n, n-1, . . . , 2, 1 to obtain the value of the function w in the memory means of multi-dimensional address (0, 0, . . . , 0).

28. Apparatus as claimed in claim 22, wherein there is further included complementer means connected in series with said delay means between the output of said register means and the other input of said algebraic addition means for any one of said set of operations regarding the j th dimension.

34. Apparatus as recited in claim 28, wherein said set of operations in the j th dimension comprises performing said element of operation regarding the j th dimension with respect to every combination of the numerical values of i₁, i.sub. 2, . . . , i_j-1 and only for i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0, and said executing means executes said set of operations in the j th dimension with respect to every dimension according to the order j = n, n-1, . . . , 2, 1 to obtain the value of the function w in the memory means of multi-dimensional address (0, 0, . . . , 0).

35. Apparatus as claimed in claim 28 wherein there are further included complementer means connected in series with each of said second inputs of said algebraic addition means designated by numbers 1, 2, . . . , m_j for any one of said set of operations regarding the j th dimension.

25. Apparatus for obtaining a digital value of a function w expressed by:

space="preserve" listing-type="equation">w = Σ

a.sub.i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n x.sub.1.sup.i.sbsp.1 x.sub.2.sup.i.sbsp.2. . . x.sub.j.sup.i.sbsp.j. . . x.sub.n.sup.i.sbsp.n

where

a_i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n '"'"'s are constants.

x_j '"'"'s are the independent variables,

n is a positive integer and n ≧

2,

j = 1, 2, . . . , n,

i_j = 0, 1, 2, . . . , m_j, and

m_j '"'"'s are positive integers and at least one of m_j '"'"'s ≧

2, comprising;

a. memory the each having a word length and designated by a multi-dimensional address (i₁, i₂, . . . , i_j, . . . , i_n) where i_j can assume any one of th numerical values of 0, 1, 2, . . . , m_j, and represents a component of said multi-dimensional address in the j th dimension,

b. means for loading each of said memory means of multidimensional addresses with a respective specific initial numerical value determined by the constants a_i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n '"'"'s,

c. means for performing an element of operation in the j th dimension comprising;

31. Apparatus as recited in claim 25, wherein said set of operations in the j th dimension comprises performing said element of operation regarding the j th dimension with respect to every combination of the numerical values of i₁, i₂, . . . , i_j-1 and only for i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0, and said executing means executes said set of operations in the j th dimension with respect to every dimension according to the order j = n, n-1, . . . , 2, 1 to obtain the value of the function w in the memory means of multi-dimensional address (0, 0, . . . , 0).

32. Apparatus as claimed in claim 25 wherein there are further included complementer means connected in series with each of said second inputs of said algebraic addition means designated by numbers 1, 2, . . . , m_j for any one of said set of operations regarding the j th dimension.

26. means for connecting said memory means, designated by multi-dimensional addresses (i₁, i₂, . . . , i_j-1, i_j, i_j+1, . . . , i_n) each having a serial number i_j = 0, 1, 2, . . . , m_j as a component in the j th dimension and having identical components in the other dimensions, serially to form a shift register means,algebraic addition means having two inputs and one output.delay means having an input and an output,means for selectively supplying the output digits from said shift register means to one of the inputs of said algebraic addition means,means for selectively supplying the output digits from said algebraic addition means to the input of said shift register means for rewriting therein,means for selectively supplying the output digits from said shift register means to the input of said delay means,means for selectively supplying the output digits from said delay means to the other input of said algebraic addition means, andmeans for shifting said shift register means a number of cycles corresponding to the value of the independent variable x_j, and wherein

a set of operations in the j th dimension comprises performing said element of operation regarding the j th dimension with respect to every combination of the numerical values of i₁, i₂, . . . , i_j-1 and at least i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0 and

d. means for executing said set of operations in the j th dimension with respect to every dimension to obtain the value of the function w in the memory means of multi-dimensional address (0, 0, . . . , 0).

29. Apparatus for obtaining a digital value of a function w expressed by:

space="preserve" listing-type="equation">w = Σ

a.sub.i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n x.sub.1.sup.i.sbsp.1 x.sub.2.sup.i.sbsp.2. . . x.sub.j.sup.i.sbsp.j. . . x.sub.n.sup.i.sbsp.n

where a_i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n '"'"'s are constants,

x_j '"'"'s are the independent variables,

n is a positive integer and n ≧

2,

j = 1, 2, . . . , n,

i_j = 0, 1, 2, . . . , m_j, and

m_j '"'"'s are positive integers and at least one of m_j '"'"'s ≧

2, comprising;

a. memory means each having a word length and designated by a multi-dimensional address (i₁, i₂, . . . , i_j, . . . , i_n) where i_j can assume any one of the numerical values of 0, 1, 2, . . . , m_j, and represents a component of said multi-dimensional address in the j th dimension,

b. means for loading each of said memory means of multidimensional addresses with a respective specific initial numerical value determined by the constants a_i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n '"'"'s,

c. means for performing an element of operation in the j th dimension comprising;

30. memory means designated by multi-dimensional addresses (i₁, i₂, . . . , i_j-1, i_j, i_j+1, . . . , i_n) each having a serial number i_j = 0, 1, 2, . . . , m_j as the component in the j th dimension ahd having identical components in the othe dimensions,algebraic addition means designated by numbers 1, 2, . . . , m_j each having first and second inputs and one output,means for reading out said memory means designated by multi-dimensional addresses each having a serial number i_j = 0, 1, 2, . . . , m_j -1 as a component in the j th dimension and non-destructively reading out said memory means designated by multi-dimensional address having number i_j = m_j as the component in the j th dimension, and for selectively supplying the read out digits from the memory means designated by multi-dimensional addresses each having a serial number i_j = 0, 1, 2, . . . , m_j -1 as a component in the j th dimension to said first inputs of said algebraic addition means designated by numbers 1, 2, . . . , m_j respectively, and for selectively supplying the output digits from said algebraic addition means designated by numbers 2, 3, . . . , m_j and the nondestructively read out digits from said memory means designated by multi-dimensional address having number i_j = m_j as the component in the j th dimension to said second inputs of said algebraic addition means of number 1, 2, . . . , m_j respectively, and means for selectively supplying the outputs of said algebraic addition means designated by numbers 1, 2, . . . , m_j to the inputs of said memory means designated by the multi-dimensional addresses each having a serial number 0, 1, 2, . . . , m_j -1 as the component in the j th dimension respectively for rewriting therein,means for reading out said memory means designated by multi-dimensional addresses each having a serial number i_j = 0, 1, 2, . . . , m_j as the component in the j th dimension a number of times corresponding to the value of the independent variable x_j, and wherein

a set of operations in the j th dimension comprises performing said element of operation regarding the j th dimension with respect to every combination of the numerical values of i₁, i₂, . . . , i_j-1 and at least i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0, and

d. means for executing said set of operations in the j th dimension with respect to every dimension to obtain the value of the function w in the memory means of multi-dimensional address (0, 0, . . . , 0).

33. Apparatus for obtaining a digital value of a function w expressed by:

space="preserve" listing-type="equation">w = Σ

a.sub.i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n x.sub.1.sup.i.sbsp.1 x.sub.2.sup.i.sbsp.2. . . x.sub.j.sup.i.sbsp.j. . . x.sub.n.sup.i.sbsp.n

where

a_i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n '"'"'s are constants,

x_j '"'"'s are the independent variables,

n is a positive integer and n ≧

2,

j = 1, 2, . . . , n,

i_j = 2, 1, 2, . . . , m_j, and

m_j '"'"'s are positive integers and at least one of m_j '"'"'s ≧

2, comprising;

a. memory means each having a word length and designated by a multi-dimensional address (i₁, i₂, . . . , i_j, . . . , i_n) where i_j can assume any one of the numerical values of 0, 1, 2, . . . , m_j, and represents a component of said multi-dimensional address in the j th dimension.

b. means for loading each of said memory means of multidimensional addresses with a respective initial numerical value determined by the constants a_i.sbsb.1i.sbsb.2. . . i.sbsb.j. . . i.sbsb.n '"'"'s,

c. means for performing an element of operation in the j th dimension comprising;

1. memory means designated by multi-dimensional addresses (i₁, i₂, . . . , i_j-1, i_j, i_j+1, . . . , i_n) each having a serial number i_j = 0, 1, 2, . . . , m_j as the component in the j th dimension and having identical components in the other dimensions,

2. algebraic addition means designated by numbers 1, 2 . . . , m_j each having first and second inputs and one output,

3. means for reading out said memory means designated by multi-dimensional addresses each having a serial number i_j = 0, 1, 2, . . . , m.sub. j -1 as a component in the j th dimension, and for non-destructively reading out said memory means designated by the multi-dimensional address having number i_j = m_j as the component =80 j th and for selectively supplying the read j92 digits from the memory means designated by multi-dimensional addresses each having a serial number i_j = 0, 1, 2, . . . , m.sub. -1 as the component in the j th dimension to said first inputs of said algebraic addition means designated by numbers 1, 2, . . . , m_j respectively, and for selectively supplying the read out digits from the memory means designated by multi-dimensional addresses each having a serial number i_j = 1, 2, . . . , m_j as the component in the j th dimension to said second inputs of said algebraic addition means designated by numbers 1, 2, . . . , m_j respectively, and for selectively supplying the output digits from said algebraic addition means designated by numers 1, 2, . . . , m_j to the inputs of said memory means designated by multi-dimensional addresses each having a serial number i_j = 0, 1, 2, . . . , m_j- 1 as a component in the j th dimension respectively for rewriting therein,

4. means for reading out said memory means designated by multi-dimensional addresses each having a serial number i_j = 0, 1, 2, . . . , m_j as the component in the j th dimension a number of times corresponding to the value of the independent variable x_j, and wherein

a set of operations in the j th dimension comprises performing said element of operation regarding the j th dimension with respect to every combination of the numerial values of i_j, i₂, . . . , i_j-1 and at least i_j+1 = 0, i_j+2 = 0, . . . , i_n = 0, and

d. means for executing said set of operations in the j th dimension with respect to every dimension to obtain the value of the function w in the memory means of multi-dimensional address (0, 0, . . . ,

     0).