JP2009020619A - Inter-polynomial distance calculation apparatus, method, program, and recording medium, one variable nearest real polynomial calculatin apparatus, method, program, and recording medium - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To calculate a distance defined by l<SP>∞</SP>-norm between one variable real polynomial f= which is the nearest to one variable real polynomial f which does not have a zero point in a complex region D among one variable real polynomials having a zero point in the region D and one variable real polynomial f. <P>SOLUTION: In an inter-polynomial distance calculation apparatus, in the case of the polynomial f(x)=e<SB>0</SB>(x)+Σ<SB>j=1</SB><SP>n</SP>a<SB>j</SB>e<SB>j</SB>(x)(a<SB>j</SB>belongs to a set R), the minimum value is calculated from among absolute values ¾t¾ to be acquired correspondingly to the candidate of the zero point of f(x)+tq(x)=0 to be acquired for each combination of each division boundary C<SB>k</SB>of the region D and a polynomial q(x) belonging to a set äΣ<SB>j=1</SB><SP>n</SP>±e<SB>j</SB>(x)} (processing A). Also, the minimum value is calculated from among absolute values ¾τ¾ to be acquired correspondingly to the candidate of the zero point of f(x)+t<SB>μ</SB>e<SB>μ</SB>(x)+τp<SB>μ</SB>(x)=0 to be acquired for each combination of each division boundary C<SB>k</SB>and an integer μ belonging to ä1, 2, ..., n} and a polynomial p<SB>μ</SB>(x) belonging to a set äΣ<SB>1≤j≤n</SB>,<SB>j≠μ</SB>±e<SB>j</SB>(x)} (processing B). In this case, the minimum value between the values acquired by the processing A and the processing B is calculated as the minimum distance. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a univariate real polynomial f˜ closest to a certain univariate real polynomial f having no zero in the above-mentioned region among univariate real polynomials having a zero in a specified region in the set of all complex numbers. The present invention relates to a technique for obtaining a distance between a univariate real polynomial f and a univariate real polynomial f ~.

It is assumed that e ₀ (x), e ₁ (x), e ₂ (x),..., e _n (x) are univariate real polynomials (coefficients are real univariate polynomials) given in advance. However, if a set of all real numbers is expressed as R, e ₁ (x), e ₂ (x),..., E _n (x) are linearly independent on R. In other words, real c _1, c _2, ..., with respect to c _n, a _{c 1 e 1 (x) +} c 2 e 2 (x) + ... + c n e n (x) is identically 0 Suppose that c ₁ = c ₂ = ... = c _n = 0 only.

Let a ₁ , a ₂ ,..., a _{n be} real numbers given in advance. A set of all complex numbers is denoted as _Cg . At this time, consider the following problem. In this specification, the symbol ~ is added to the head of the immediately preceding character.
[Problem ☆]
Equation (1) univariate real polynomial f (x) which is expressed by a no zeros in advance given region D in the C _g. At this time, among the set of univariate real polynomials represented by Expression (2), the univariate real polynomials f to (x) having zeros in the region D and closest to the univariate real polynomials f (x) Find the distance between the univariate real polynomial f (x) and the univariate real polynomial f ~ (x). Find one such univariate real polynomial f ~ (x).

However, the distance d (f, g) between the two univariate real polynomials f and g is defined by the l ^∞ -norm [maximum norm] of the coefficient of fg. That is, the distance d (f, g) between the two univariate real polynomials f and g represented by the equation (3) is expressed as max {| a ₁ -b ₁ |, | a ₂ -b ₂ |, ..., | a _n -b _n |}. max {| a ₁ -b ₁ |, | a ₂ -b ₂ |,…, | a _n -b _n |} is a set | a ₁ -b ₁ |, | a ₂ -b ₂ |,…, | It represents the maximum of the original out of | a _n -b _n.

  In the notation of the polynomial, when it is desired to specify the variable x, it is expressed as f (x), g (x), and in other cases, it is expressed as f, g. Unless otherwise specified, “one-variable real polynomial” is abbreviated as “polynomial”.

There are practical benefits to thinking about the above problem. For example, in a control by a transfer function such as digital signal processing, there is a case where it is desired to set a coefficient of a polynomial so that a region D does not have a zero point in order to obtain a desired characteristic. However, since the coefficient can be specified only with finite accuracy, its setting generally involves an error. In order to determine whether or not a polynomial set with an error can surely have no zero in the region D on the assumption that such an error is involved, the above problem ☆ needs to be solved. When the viewpoint is changed, the above determination is performed when the coefficient of the polynomial f is moved while maintaining the property that the domain D has no zero in the domain D, which is known to have no zero in the area D. This limit is obtained as a distance [l ^∞ -norm] between the polynomial f and the nearest polynomial f˜.

Non-patent document 1 is cited in relation to the above problem ☆. This Non-Patent Document 1 deals with a problem similar to the above problem ☆ for a complex coefficient polynomial having a given complex number γ as a zero. Although Non-Patent Document 1 deals with complex coefficient polynomials, as suggested in Non-Patent Document 1, it can be systematized so that it can also be applied to the case of real coefficient polynomials (real polynomials).
The difference between the problem ☆ and the problem addressed by Non-Patent Document 1 is that the problem ☆ is addressed by a polynomial having a zero in the region where the problem ☆ is given. The point is to find the one with the smallest distance from the polynomial with one given point as the zero.

Non-patent document 2 can be cited as a study dealing with the problem that the location of the zero point is not a single point but an area.
In the non-patent document 2, the distance as the above l ^∞ - when solving the problem as a norm, is limited to the case with a zero point to a point rather than a region. When the problem is solved when the zero is located in a region, the distance between the two polynomials f (x) and g (x) is set to l ² -norm, that is, f (x) -g ( It is solved only when the vector defined by the coefficient of x) is defined as l ² -norm ((a ₁ -b ₁ ) ² +... + (a _n -b _n ) ² ) ^1/2 .

Further, when ε is a positive real number with respect to the polynomial f having no zero in the region D, the polynomial g satisfies d (f, g) ≦ ε, and the condition “region” within the specified coefficient error range Non-patent document 3 and patent document 1 can be cited as a method for determining whether or not there is an object that satisfies “having zero in D”.
If the determination methods shown in Non-Patent Document 3 and Patent Document 1 are used, is there a polynomial g satisfying d (f, g) ≦ r that satisfies the above condition for a certain real number r> 0? If the bisection method is applied to r using this presence / absence as a decision basis, an approximate value of the distance m between the polynomial that satisfies the above condition and is closest to the polynomial f Can be requested. That is, r is an initial value of H and 0 is an initial value of L so that a polynomial g satisfying d (f, g) ≦ r exists, and d (f, g) ≦ (H + L) / 2 is set. If there is a satisfying polynomial, H is replaced with (H + L) / 2, and if not, L is replaced with (H + L) / 2. By repeating such processing, H approaches the upper limit m from above. Accordingly, an approximate value of the distance m between the polynomial that satisfies the above conditions and that is closest to the polynomial f can be obtained.
HJ Stetter, "The Nearest Polynomial with a Given Zero, and Similar Problems," ACM SIGSAM Bulletin, Vol.33, No.4, pp.2-4, 1999. MA Hitz and E. Kaltofen, "Efficient algorithms for computing the nearest polynomial with constrained roots," Proc. 1998 International Symposium on Symbolic and Algebraic Computation (ISSAC98), pp.236-243, 1998. Hiroshi Sekikawa, Kiyoshi Shirayagi, “About the Location of Zeros of Interval Polynomials”, IEICE Transactions Volume A, Vol. J89-A, No. 3, pp.199-216, 2006. JP-A-2005-235121

  According to the method of Non-Patent Document 1, it is possible to obtain a polynomial that is closest to a given polynomial and has a given point as a zero. However, the method of Non-Patent Document 1 cannot be applied to the above problem ☆ in which a given “one point” is “region”.

Further, in the method of Non-Patent Document 2, a method for obtaining an approximate solution of an overdetermined simultaneous linear equation is used when measuring the distance with l ^∞ -norm. As described in Non-Patent Document 2, it is difficult to extend this when the location of the zero point is a region. In addition, when measuring the distance with l ² -norm, it can be handled even when the location of the zero point is a region, but since it is a method specialized for l ² -norm, when measuring the distance with l ^∞ -norm The technique of the said nonpatent literature 2 cannot be used. That is, the method of Non-Patent Document 2 cannot give an answer to the problem ☆.

  In the method of applying the bisection method to the determination method of Non-Patent Document 3 or Patent Document 1, if there is a polynomial g that satisfies the condition “having a zero in region D”, Although an approximate value of the distance between the polynomial f closest to the polynomial f and the polynomial f can be obtained, the exact distance or the nearest polynomial itself cannot be obtained. In the first place, since the bisection method is used, it is necessary to know the existence of a polynomial that satisfies the above condition, that is, r where there exists a polynomial g that satisfies d (f, g) ≦ r. Unless the existence of r is confirmed by some method, the determination method of Non-Patent Document 3 or Patent Document 1 cannot be used. Therefore, the method of applying the bisection method to the determination method of Non-Patent Document 3 or Patent Document 1 cannot give an answer to the problem ☆.

Therefore, the present invention provides an answer to the above problem ☆, that is, one univariate real polynomial having no zero in the region D among the one variable real polynomials having the zero in the designated region D in the set of all complex numbers. Inter-polynomial distance calculation apparatus, method, program, and recording medium for obtaining a distance defined by l ^∞ -norm between a univariate real polynomial f˜ closest to the polynomial f and the univariate real polynomial f, and a univariate real An object of the present invention is to provide a one-variable recent real polynomial calculation device, method, program, and recording medium for obtaining the polynomial f ~.

In order to solve the above problems, the present invention, real a _1, a _2, ..., and a _n, univariate real polynomial _{e 0 (x), e 1} (x), ..., and e _n (x), A finite number of partitioned boundaries C ₁ , C ₂ ,..., C _K represented by rational expressions constituting the bounds of the bounded closed complex region D in the set of all complex numbers are stored in the storage means, division boundary C _k [k = 1,2, ..., K] and the set ^{_{{Σ j = 1 n ± e}} j (x)} belonging polynomial q (x) and each combination in the resulting first equation f of (x ) + tq (x) = 0 candidates for the zero point x (first candidate search), and the absolute value of t | t | of the first equation obtained corresponding to each candidate obtained in the first candidate search The smallest one is determined (first distance candidate determination). Further, each partition boundary C _k , an integer μ belonging to the set {1, 2,..., N} and a polynomial p _μ (x) belonging to the set {Σ _{1 ≦ j ≦ n, j ≠ μ} ± e _j (x)}. The candidate of the zero point x of the second equation f (x) + t _μ e _μ (x) + τp _μ (x) = 0 obtained for each combination with (second candidate search) is obtained and obtained by the second candidate search A minimum one of absolute values | τ | of the second equation obtained corresponding to each candidate is obtained (second distance candidate determination). Then, the smallest value obtained by the first distance candidate determination and the second distance candidate determination is output (minimum distance output), and this is set as the minimum distance.

が恒等的に０となる実数c₁,c₂,…,c_nが存在するか否かを判定し、存在する場合には一変数実多項式f(x)と恒等的に０の多項式との距離を求め（恒等式調査）、上記の最小距離出力に代えて、第一距離候補決定と第二距離候補決定によって得られた値および恒等式調査によって得られた距離のうち最小のものを求めるとしてもよい。
恒等的に０の多項式との距離についても考慮を要求する場合の最小距離を求めることができる。 Also polynomial

Determines whether there is a real number c ₁ , c ₂ ,..., C _n that is uniquely 0, and if so, a univariate real polynomial f (x) and a polynomial that is 0 uniquely Is obtained (identity survey), and instead of the above minimum distance output, the smallest one of the values obtained by the first distance candidate determination and the second distance candidate determination and the distance obtained by the identity survey is obtained. It is good.
It is possible to obtain the minimum distance in the case where consideration is also required for the distance from the polynomial of 0 in identity.

Moreover, a subset X _Omega set ^{_{{Σ j = 1 n ± e}} j (x)} , per the set ^{_{{Σ j = 1 n ± e}} j (x)} any two polynomials M _1, M ₂ belonging to as a set of the original either when there are M ₁ = -M ₂ the relationship, first candidate search means, a combination of the above, each segment boundary C _k and the subset X _Omega belonging polynomial q (x) It is good also as a combination. Also, it sets any belonging to the subset X _mu of _{{Σ 1 ≦ j ≦ n,} j ≠ μ ± e j (x)} , the set _{{Σ 1 ≦ j ≦ n,} j ≠ μ ± e j (x)} When there is a relationship of P ₁ = −P _{2 for} the two polynomials P ₁ and P ₂ , the second candidate search means uses the above combinations as the partition boundaries C _k , A combination of an integer μ belonging to the set {1, 2,..., N} and a polynomial p _μ (x) belonging to the subset X _μ may be used.
In this case, the calculation cost for each process can be halved.

の解ｓとして得るとすることができる。
この理論的な詳細は後述する。 Also, each partition boundary C _k is given by the injection γ _k (s) [where s∈S _k ] for the set S _k on the real number, and the real part and imaginary part of the complex number are represented by Re and Im, respectively. said first candidate search, the candidate of the first equation f (x) + tq (x ) = 0 segment boundaries C _k zeros on x,

It can be obtained as a solution s.
The theoretical details will be described later.

が最小となる条件を満足するｓとして得るとすることができる。
この理論的な詳細は後述する。 In addition, the second candidate search described above searches for a candidate of the zero point x on the partition boundary C _{k of} the second equation f (x) + t _μ e _μ (x) + τp _μ (x) = 0.

Can be obtained as s satisfying the condition that minimizes.
The theoretical details will be described later.

  Although a detailed theory will be described later, a univariate real polynomial f ~ corresponding to the minimum distance is obtained because a univariate real polynomial f ~ is obtained when the minimum distance is obtained based on the proposition, Theorem 1 and Theorem 2 described later. Just output.

Further, the computer can be operated as an interpolynomial distance calculation device by the interpolynomial distance calculation program that causes the computer to function as the interpolynomial distance calculation device of the present invention. Similarly, the computer can be operated as a single variable nearest real polynomial calculator by the one variable nearest real polynomial calculator that causes the computer to function as the single variable nearest real polynomial calculator.
And by using a computer-readable program recording medium that records such a program, other computers can function as an inter-polynomial distance calculation device, a one-variable recent real polynomial calculation device, or a program can be distributed. It becomes possible.

According to the present invention, the minimum interpolynomial distance is calculated by deriving an algebraic equation satisfying the minimum distance, using the interpolynomial distance as a variable, based on the proposition, theorem 1 and theorem 2 described later. As a result, among the one-variable real polynomials having zeros in the designated region D in the set of all complex numbers, the one-variable real polynomials f˜ closest to a certain one-variable real polynomial f having no zeros in the region D are obtained. The distance defined by the l ^∞ -norm with the univariate real polynomial f can be obtained. When the minimum distance is obtained, the univariate real polynomial f ~ can be obtained by the above theorem.

＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝ [theory]
Prior to the description of the embodiments of the present invention, the theory of the present invention will be described.
First, in the present invention, a predetermined region D in a set C _{g of} all complex numbers is a bounded and closed complex region, and its boundary C is a simple closed curve of a finite length under certain conditions. .
That is, the problems to be solved by the present invention are as follows.
=======================================
≪Problem≫
[conditions]
<1> Let e ₀ (x), e ₁ (x), e ₂ (x),..., E _n (x) be univariate real polynomials given in advance. However, for the set R of all real numbers, e ₁ (x), e ₂ (x),..., E _n (x) are linearly independent on R.
<2> Let a ₁ , a ₂ ,..., A _{n be} real numbers given in advance.
<3> formula (4) represented by univariate real polynomial f (x) is to have no zeros in advance given region D in the set C _g of the whole complex. However, the region D is a bounded and closed complex region, and the boundary C of the region D is a simple closed curve having a finite length.
<4> The boundary C is expressed as a union of one or more finite number of partitioned boundaries C _k . That is, the boundary C ^{_{is, C = ∪ k = 1 K}} C k = C 1 ∪C 2 ∪ ··· ∪C K ( integer K satisfies 1 ≦ K <∞.) And, C _j ∩C _k = φ It is expressed as (j ≠ k). However, φ represents an empty set. In addition, each segment boundary C _k is expressed by the injection γ _k ,
γ _k : S _k → C _k
Is displayed. Here, γ _k is a rational expression, and the denominator / numerator polynomial coefficients are limited to those expressed as a + bi using real algebraic numbers a and b and an imaginary unit i. S _k ⊂R is assumed to be one of [α, β], [α, ∞), (−∞, β], (−∞, ∞), where α and β are real algebraic numbers.
<5> The distance d (f, g) between two univariate real polynomials f and g is defined by the l ^∞ -norm of the coefficient of fg.
Under the above conditions, among the set of univariate real polynomials represented by equation (5), the univariate real polynomials f˜ (x having the zeros in the region D and closest to the univariate real polynomials f (x) ), Find the distance between the univariate real polynomial f (x) and the univariate real polynomial f ~ (x). Find one such univariate real polynomial f ~ (x).

=======================================

  As in the past, when it is desired to specify the variable x in the notation of the polynomial, it is expressed as f (x), g (x), and in other cases, it is simply expressed as f, g. . Unless otherwise specified, “one-variable real polynomial” is abbreviated as “polynomial”.

  The coefficient of the polynomial may be a real number in theory, but it is necessary and sufficient that the arithmetic operation and the equality judgment can be executed by a finite step algorithm in the calculation by a computer or the like. (Integer coefficient univariate algebraic real number root). In the following explanation, for convenience of explanation, it is assumed that the coefficient of the polynomial is a rational number.

Condition <3> From the text, the polynomial f (x) represented by the equation (4) does not have a zero in the region D.
If, a _1, a ₂ optionally ..., when the polynomial f which is obtained by selecting a _n (x) it is necessary to ensure that no zeros in the region D is the following process Just do it.

[Confirmation process that polynomial f (x) has no zero in region D]
The determination whether or not the zero of the polynomial f (x) exists in the region D (on the boundary C and inside thereof) is achieved by a known method.
First, it is determined whether or not the zero of the polynomial f (x) exists on the boundary C of the region D. For example, after the indication of the boundary C of the region D is substituted into the polynomial f (x), the presence or absence of a zero is determined according to an algorithm based on Sturm's theorem.
Here, as an example for convenience of explanation, each partition boundary C _k is expressed as φ _k (s) + i · ψ _k by the rational expressions φ _k (s) and ψ _k (s) with the parameter s and the parameter s as one variable. Assume that (s) is displayed [k = 1, 2,..., K]. i represents an imaginary unit. Here, the coefficients in the rational expressions φ _k (s) and ψ _k (s) may be real numbers in theory, as in the above, but in arithmetic operations such as a computer, the four arithmetic operations and the equality judgment are performed by a finite step algorithm. It is necessary and sufficient that it can be executed by the above. For this purpose, for example, a real algebraic number (a real number that becomes the root of an integer coefficient univariate algebraic equation) is assumed, and a specific example is a rational number. Hereinafter, the rational expressions φ _k (s) and ψ _k (s) will be described as being rational expressions of rational number coefficients. Range S _k of motion of the parameters s, corresponding to the condition <4> [α, β] , [α, ∞), (- ∞, β], (- ∞, ∞) and is either. [•, •] indicates a closed interval, (•, •] indicates a left-open / right-closed interval, [•, •] indicates a left-closed / right-open interval, and (•, •) indicates an open interval. It is injective between the boundary C _k .

The application of the algorithm based on Strum's theorem will be described. Since the same processing is performed for each section boundary C _k , the description will be given by omitting the horizontal letter k.
The polynomial f (φ (s) + i · ψ (s)) into which the representation φ (s) + i · ψ (s) of the partition boundary C of the region D is substituted is a rational coefficient rational expression ω (s) and τ (s) can be expressed as f (φ (s) + i · ψ (s)) = ω (s) + i · τ (s). Here, “polynomial f (x) is ₀ in φ (s ₀ ) + i · ψ (s ₀ )” and “ω (s) and τ (s) are ₀ at the same time when s = s _0”. "Is" is equivalent. In addition, “ω (s) and τ (s) become ₀ at the same time when s = s ₀ ” and “when ω (s) and τ (s) are expressed as irreducible rational coefficient rational expressions respectively. The fact that the numerator GCD (the greatest common polynomial) is _{0 at} s ₀ is equivalent. Therefore, an algorithm based on Strum's theorem is applied to the numerator GCD when ω (s) and τ (s) are expressed as irreducible rational coefficient rational expressions. For convenience of explanation, let G (s) be the GCD of the numerator when ω (s) and τ (s) are expressed as irreducible rational coefficient rational expressions. For this H (s), a sequence of polynomials H ₀ (s), H ₁ (s), H ₂ (s),..., H _r (s) is generated. However, these polynomial sequences are generated according to the following rules. H ₀ (s) = H (s), H ₁ (s) = dH (s) / ds [first derivative with respect to s. Leibniz notation], the remainder polynomial obtained by dividing H _j-1 (s) by H _j (s) is changed to -H _{j + 1} (s). That is, if the quotient polynomial is expressed as Q _j (s), the relationship H _j−1 (s) = H _j (s) Q _j (s) −H _{j + 1} (s) is established. Also, H _r-1 (s) is divisible by H _r (s). That is, when representing the quotient polynomial and Q _r (s), the relationship of _{H r-1 (s) =} H r (s) Q r (s) is established. In this case, the interval S = [a, b] _{in, H 0 (a), H} 1 (a), H 2 (a), ..., the number of positive and negative sign change in _{H r (a) w (a} ) If the number of positive and negative sign changes in H ₀ (b), H ₁ (b), H ₂ (b),..., H _r (b) is represented by w (b), the interval S = [a , B], the number of zeros of H (s) is w (a) -w (b). Here, it is assumed that a and b in the section S = [a, b] are not the zeros of H (s) (if they are zeros, the expression for the number changes slightly, but in the present invention, it is essential. I ’ll omit it because it ’s not a big part.) If w (a) -w (b) = 0, it is found that there is no zero, and if w (a) -w (b) ≠ 0, it is found that there is a zero.

Refer to Reference 1 for Sturm's theorem.
(Reference 1) Sadaharu Takagi, “Algebra Lecture (Revised New Edition)”, Kyoritsu Shuppan, 1965.

In the above, the determination as to whether or not the zero of the polynomial f (x) exists on the boundary C of the region D is based on the Sturm theorem after substituting the representation of the boundary C of the region D into the polynomial f (x). The description is based on an algorithm. However, it is not intended to be limited to algorithms based on Sturm's theorem. In short, it is only necessary to determine whether or not an algebraic equation of a univariate real coefficient has a root in a certain real section. For example, this may be determined by a sign variation method shown in Reference 2.
(Reference 2) GE Collins and R. Loos, "Real zeros of polynomials", Computer Algebra, Symbolic and Algebraic Computation, (B. Buchberger, GE Collins and R. Loos (ed)), pp.83-94, Springer -Verlag, 1983.

  If the zero of the polynomial f (x) does not exist on the boundary C, it is determined whether or not the zero of the polynomial f (x) exists within the region D. This determination is achieved by using the principle of declination. The determination based on the principle of declination is based on the premise that there is no zero of the polynomial f (x) on the boundary C of the region D. Therefore, first, the presence / absence of the zero on the boundary C is determined. It was.

Application of an algorithm based on the principle of declination will be described.
The indication φ _k (s) + i · ψ _k (s) of the segment boundary C _k is substituted into the polynomial f (x). At this time, f (x) = f (φ _k (s) + i · φ _k (s)) = u (φ _k (s), φ _k (s)) + i · v (φ _k (s), φ _k (S)). Therefore, the real roots of the real part u (φ (s), ψ (s)) and the imaginary part v (φ (s), ψ (s)) are obtained at s∈S. Similarly, roots are obtained for all partition boundaries, and a list S1 is created in which these roots are arranged in the order in which they appear. In addition, a list S2 for counting roots in the region surrounded by the boundary C is prepared (in the initial state, the list S2 is empty). The roots are sequentially read from the head of the list S1. First, the head element (root) A is extracted, and if the list S2 is an empty list, it is written in S2. When the list S2 is not empty, it is determined whether the first element (root) A and the first element (root) of the list S2 are both real part solutions or imaginary part solutions, and this determination is true. For example, the top element (root) is removed from the list S2. When the determination is false, the head element (root) A is added to the head of the list S2. This processing operation is performed by reading the root sequentially from the top of the list S1. As a result, 1/4 of the number of elements (roots) in the list S2 becomes the number of roots of the polynomial f (x) included in the boundary C. If this number is 0, it turns out that the zero of the polynomial f (x) does not exist in the region D. If this number is not 0, the zero of the polynomial f (x) exists in the region D. It turns out to be.

See Reference 3 for the counting of the zeros of the polynomial using the principle of declination.
(Reference 3) Teruo Hirano, “Counting the roots of polynomials using the principle of declination”, Formula processing, Vol.9, No.4, pp.34-35, 2003.

In the above description, it has been described that the determination as to whether or not the zero of the polynomial f (x) exists within the region D is performed by an algorithm based on the principle of declination. However, in short, it is only necessary to be able to determine whether or not the polynomial f (x) has a root inside the region D, and is not intended to be limited to an algorithm based on the principle of declination.
This is the end of the description of [confirmation process for polynomial f (x) having no zero in region D].

Next, all the zeros in the region D of the polynomial f˜ that are nonzeroly identical and are closest to the polynomial f exist on the boundary C, and among the points that can be zeros of the polynomial f˜ It shows that it is enough to examine only a finite number.
As a result, there are a finite number of candidates for the polynomial f˜, so that the minimum value of d (f, f˜) and the polynomial f˜ at that time can be obtained. If there is no such candidate, if there is a polynomial that is uniformly 0 in the set of polynomials represented by Equation (5), this polynomial is the nearest polynomial f ~. If there is no polynomial that is uniformly 0 in the set of polynomials represented by Equation (5), it is a polynomial having a zero in region D and belongs to the set of polynomials represented by Equation (5) It can be concluded that there is no polynomial.

First, by the non-patent document 3, the above-mentioned patent document 1, and a similar argument thereto,
=======================================
[Proposition] When the polynomial f ~ having the smallest distance is not equal to 0, its zero exists only on the boundary C of the region D.
=======================================
[Theorem 1] When the distance is minimum, the polynomial f ~ is equal to 0 or, except for at most one exception coefficient, | a _j -a ~ _j | = d (f, f ~) Satisfy.
=======================================
I understand that.

Prove [Proposition].
First, the following is shown.
“Polynomial f (x) = e ₀ (x) + Σ _{j = 1} ⁿ _aj e _j (x) (a _j ∈R) does not have a zero in the region D and is not equal to zero in the polynomial g ( Assume that x) = e ₀ (x) + Σ _{j = 1} ⁿ b _j e _j (x) (b _j ∈R) has a zero in the region D. At this time, the polynomial g˜ (x) = e ₀ (x) + Σ _{j = 1} ⁿ _b˜j e having a zero in the region D and satisfying d (f, g˜) <d (f, g) _{j (x) (b ~ j} ∈R) is present. (...)

An open disk (not including the boundary) having a radius r centered on the complex number z is represented as B (z; r). That is, B (z; r) = {w | w∈C _g , | zw | <r}. Let p (t, x) = f (x) + t (g (x) -f (x)). At this time, p (0, x) = f (x), p (1, x) = g (x), and for any t (0 ≦ t ≦ 1), p (t, x) is , E ₀ (x) + Σ _{j = 1} ⁿ c _j e _j (x) (c _j ∈ R).
Let ζ be the zero of the polynomial g present in the region D. Then, there exists ε> 0 that satisfies the condition “B (ζ; ε) ⊂D and arbitrary ξ∈B (ζ; ε), ξ ≠ ζ, g (ξ) ≠ 0”. If such ε does not exist, it contradicts that the polynomial g is not equal to zero.
Now, for any t (0 ≦ t ≦ 1), it is assumed that p (t, x) has no zero at the boundary of B (ζ; ε / 2). Then, according to Rouche's theorem, the number of zeros (0) of p (0, x) (= f (x)) in B (ζ; ε / 2) and p (1, x) (= g The number (zero or more) of zeros in (x)) is equal and contradictory. Therefore, there exists a certain 0 <τ <1, and p (τ, x) has a zero at the boundary of B (ζ; ε / 2), so p (τ, x) can be taken as g ~ (x) For example, d (f, g ~) = τd (f, g) <d (f, g) holds.
Therefore, the subject ★ was shown.

At this time, if the zeros of the polynomial f ~ at the minimum distance exist inside the region D, the polynomial has the zeros inside the region D and satisfies d (f, g ~) <d (f, f ~) g ~ (x) = e ₀ (x) + Σ _{j = 1} ⁿ b ~ _j e _j (x) (b ~ _j ∈ R) exists, and the polynomial f ~ is a polynomial when the distance is minimum. Contradicts something. Therefore, the zero of the polynomial f˜ when the distance is minimum exists only on the boundary C of the region D. This completes the proof of [Proposition].

Next, we prove [Theorem 1].
Polynomial f ~ (x) = e ₀ (x) + Σ _{j = 1} ⁿ a ~ _j e _j (x) (a ~ _j ∈ R) is not closest to polynomial f . Then, as described in [Proposition] above, the polynomial f˜ has a zero ζ only on the boundary C of the region D.

[1] d(f,g)=d(f,f~)
[2] |a_μ-b_μ|=d(f,f~)あるいは|a_η-b_η|=d(f,f~)
[3] 多項式ｇは、領域Ｄの境界Ｃ上に零点を持つ。 Suppose there are two or more j such that | a _j -a ~ _j | <d (f, f ~). Two of these j are selected, and these are set as μ and η. In other words, there are two subscripts μ and η, and | a _μ -a ~ _μ | and | a _η -a ~ _η | are smaller than d (f, f ~) (such μ, η If it doesn't exist, proof ends.)
It can be said that a contradiction arises from this assumption, or “there is a polynomial g represented by the equation (6) that satisfies the conditions [1], [2], [3]”.

[1] d (f, g) = d (f, f ~)
[2] | a _μ -b _μ | = d (f, f ~) or | a _η -b _η | = d (f, f ~)
[3] The polynomial g has a zero on the boundary C of the region D.

Hereinafter, b _μ and b _ν are written as t _μ + a _μ and t _ν + a _ν , respectively. Then, equation (7) can be said.

Therefore, the equation g (x) = 0 is equivalent to the simultaneous equations (8). _{However, u j (x) = Re} e j (x), v and _{j (x) = Im e j} (x). Re represents a real part, and Im represents an imaginary part.

The simultaneous equation (8) is regarded as a simultaneous equation of t _μ and t _ν with x as a parameter. Further, the zero ζ of the polynomial f ~ on the boundary C is taken from the above [proposition]. At this time, if the determinant (9) is 0 at x = ζ, | t _μ | = d (f, f˜) or | t _ν | = d (f, t _μ and t _ν can be moved until f ~) holds.

That is, the polynomial g expressed by Equation (6), [1] d (f, g) = d (f, f ~), [2] | a _μ -b _μ | = d (f, f ~) Or | a _η -b _η | = d (f, f ~), [3] There is a polynomial g that satisfies each condition of having a zero on the boundary C.

If the polynomial g is identically 0, then the proof ends. However, if the polynomial g is not identically 0, _j where | a _j -a ~ _j | <d (f, f ~) If there are still two or more, the above proof can be repeated.

If the determinant (9) is not 0 at x = ζ, the solutions t _μ and t _ν of the simultaneous equations (8) are uniquely determined and are continuous with respect to the parameter x unless the determinant (9) is 0. . Therefore, since t _μ and t _ν are continuous functions of x, they are written as t _μ (x) and t _ν (x). Then, | t _μ (x) | <d (f, f ~) and | t _ν (x) | <d (f, f ~) hold for all x∈B (ζ; ∈). A positive number ε exists.

Therefore, ξ is taken from the common part of the inside of the region D and the open disc B (ζ; ε). At this time, with respect to b _μ = t _μ (ξ) + a _μ and b _ν = t _ν (ξ) + a _ν , the polynomial g represented by the equation (10) is located inside the region D. Let ξ be the zero point.

Therefore, if the above [proposition] is applied to the polynomial (10), the polynomial g ~ represented by the expression (11), where d (f, g ~) <d (f, g) = d (f, There is something that holds f ~). This contradicts the assumption that the polynomial f ~ is closest to the polynomial f.
Therefore, [Theorem 1] was shown. This completes the proof of [Theorem 1].

Now, from Theorem 1, if there is a certain number mu, coefficients a ~ _mu about polynomial f ~ realizing the minimum value of the distance between the polynomial _{f e μ (x) | a} μ -a ~ μ | When <d (f, f ~), the number j different from μ is | a _j -a ~ _j | = d (f, f ~), so that σ _j = 1 or −1, The polynomial f ~ can be expressed as in 12).

Here, an equation (13) in which the right side of the equation (12) is generalized with t _μ and t as variables and x as a parameter is considered. Here, p _μ (x) is a polynomial belonging to the set {Σ _{1 ≦ j ≦ n, j ≠ μ} ± e _j (x)}. Note that when t = d (f, f˜) and t _μ = _a˜μ −a _μ , the left side of equation (13) is the same as the polynomial f˜ (x). In addition, t and _t μ Note also that can take the value of positive and negative.

Equation (13) is equivalent to simultaneous equations (14).

Here, the zero ζ on the boundary C of the polynomial f˜ (x) is taken. If determinant (15) is 0 at x = ζ, | t | can be made small while satisfying | t _μ | ≦ | t | and equation (13). This contradicts that the polynomial f˜ is a polynomial that realizes the minimum distance from the polynomial f.

Therefore, the determinant (15) is not 0 at x = ζ, and the determinant (15) is not 0 at the vicinity U around ζ [continuity of complex functions], and the simultaneous equations (14) The solution t can be expressed as Equation (16).

Since the right side of Expression (16) is continuous with respect to the parameter x, t (x) is written as a function of x. Then, | t (x) | takes the minimum value in U∩C at x = ζ. If not, g (x) = e ₀ (x) + Σ _{j = 1} ⁿ b _j e _j (x) (b _j ∈R), where ξ, which is a point belonging to the region D, is zero. In addition, there is a case in which d (f, g) <d (f, f˜) exists, which contradicts that the polynomial f˜ is a polynomial that realizes the minimum distance from the polynomial f.

From the above discussion, [Theorem 2] can be said.
=======================================
[Theorem 2] Polynomial f ~ (x) = e ₀ (x) + Σ _{j = 1} ⁿ a ~ _j e _j (x) (a ~ _j ∈ R) is a polynomial that is not equal to 0. Suppose that the region D has a zero and d (f, f˜) is minimum. In addition, there is a certain number _{μ, | a μ -a ~ μ} | <d (f, f ~) and. At this time, equation (13) is considered. It is assumed that ζ is a zero point on the boundary C of the polynomial f˜ (x), U is near ζ, and determinant (15) does not become zero at U. At this time, the absolute value of the right side of Expression (16) is the minimum value in U∩C at x = ζ.
=======================================

  From the above theorem, it is derived that it is sufficient to examine only a finite number of points that can be zeros of the polynomial f ~. This will be described by showing a specific calculation method.

[Calculation method based on the above theorem]
Hereinafter, the calculation method based on the above theorem will be described in two cases. In the first case, | a _j -a ~ _j | = d (f, f ~) holds for all j. In the second case, there is a certain number μ and | a _μ -a ~ _μ | <D (f, f ~).

≪First case≫
b _j = t _j + a _j At this time, polynomial _{g (x) = e 0 (} x) + Σ j = 1 n b j e j (x) (b j ∈R) can be expressed as equation (17).

Therefore, the simultaneous linear equation (18) is solved under the condition of x∈C and t ₁ ,..., T _n ∈R, and | t ₁ | =… = | t _n | Is the minimum distance. If there is no solution satisfying the condition, a polynomial belonging to the set of polynomials {e ₀ (x) + Σ _{j = 1} ⁿ b _j e _j (x) | b _j ∈R} is | a ₁ -b ₁ | = | There are no a ₂ −b ₂ | =... = | a _n −b _n |

The simultaneous linear equation (18) is equivalent to the equation (19).
Strictly speaking, the simultaneous linear equation (18) is equivalent to the equation f (x) + t (± e ₁ (x) ± e ₂ (x) ±... ± e _n (x)) = 0. However, since t is a variable that takes positive and negative values, the equation is substantially f (x) + t (e ₁ (x) ± e ₂ (x) ± ... ± e _n (x)) = 0, That is, equation (19) may be solved. However, in this sense, the simultaneous linear equation (18) is not equivalent only to the equation (19). The subset X _Omega set ^{_{{Σ j = 1 n ± e}} j (x)} = {± e 1 (x) ± e 2 (x) ± ... ± e n (x)}, the set {Σ _{j = 1} ^If any two polynomials M ₁ and M ₂ belonging to ⁿ ± e _j (x)} have a relationship of M ₁ = −M ₂ , select from the subset X _Ω when the set is based on either one The polynomial q (x) is equivalent to an equation with f (x) + tq (x) = 0. One of them is equation (19). Hereinafter, although discussion is advanced about an equation (19), generality is not lost.

Here, since ± takes all combinations of + and −, there are 2 ⁿ⁻¹ combinations in total, and each equation is expressed as shown in Expression (20). However, the polynomial q (x) here is one of e ₁ (x) ± e ₂ (x) ± ... ± e _n (x) [Generally, the polynomial q (x) is A subset X _Ω can be selected. ].
f (x) + tq (x) = 0 (20)

Hereinafter, each equation (20) is examined on the partition boundary C _k . Using the representation of the partition boundary C _k, the necessary and sufficient condition for the equation (20) to have a solution of x∈C and t∈R is that the equation (21) has a solution of s∈S _k and t∈R. is there.
_{f (γ k (s))} + tq (γ k (s)) = 0 (21)

Equation (21) is equivalent to simultaneous equations (22).

For a complex coefficient polynomial g (x) ∈C _g [x], a polynomial in which each coefficient is replaced with its complex conjugate is expressed as g ^ (x). For the complex coefficient rational expression h (x) = h ₁ (x) / h ₂ (x) (h ₁ (x), h ₂ (x) ∈C _g [x]), the complex coefficient rational expression h ^ Write ₁ (x) / h ^ ₂ (x) as h ^ (x). In this specification, the symbol ^ is added above the immediately preceding character.
At this time, Equation (23) holds for the real coefficient rational expression J [J (x) = f (x), q (x)].

  Therefore, when the simultaneous equations (22) are viewed as t equations, the coefficients are rational coefficient rational expressions.

When Re q (γ _k (s)) ≠ 0, t is expressed by Expression (24).

Substituting the right side of equation (24) into t in the second equation of simultaneous equations (22) yields equation (25).

That is, Expression (26) and Re q (γ _k (s)) ≠ 0.

When _{Re q (γ k (s)} ) = 0, from the simultaneous equations _{(22) Re f (γ k} (s)) = 0. At this time, if Im q (γ _k (s)) = 0, simultaneous equations (22) to Im f (γ _k (s)) = 0 and t is arbitrary. This contradicts the assumption that the polynomial f does not have a zero in region D. Therefore, when Re q (γ _k (s)) = 0, Im q (γ _k (s)) ≠ 0, and Equation (27) holds.

Substituting the right side of equation (27) into t in the first equation of simultaneous equations (22) yields equation (28).

That is, equation (29) and _{Im q (γ k (s)} ) ≠ 0.

When settled the above, the Re q (gamma _k (s)) and Im q (gamma _k (s)) are not 0 at the same time, it may simply solved equation (30).

When the left side of the equation (30) is not a constant, sεS _k that is a solution candidate is finite. When the left-hand side of equation (30) is identically 0, the candidate s solution has the formula (24), from equation (27), the absolute value of S _k of absolute values or of the formula (31) (32) Satisfy the minimum requirement above.

Therefore, when the absolute value of the equation (31) is not a constant, the zero point and the end point of the S _{k in} the equation (33) are candidate solutions, and when the absolute value of the equation (31) is a constant, any one point belonging to S _k May be a solution candidate.

The same applies to the formula (32), when the absolute value of formula (32) is not constant, a candidate end point of the zero point and S _k of formula (34) is a solution, when the absolute value of formula (32) is constant , S _k may be a candidate solution.

From the above, with respect to a finite number of s∈S _k , the absolute value of the equation (31) or the absolute value of the equation (32) is calculated, and the minimum value among them is adopted as the minimum distance in << first case >>. That's fine.
Further, when the value of Equation (31) or Equation (32) giving this minimum value is t ₀ , the polynomial f˜ that realizes the minimum distance | t ₀ | One of them is obtained by equation (35) using q (x) when the value t ₀ is given [see equation (20)].

≪Second case≫
Returning to Equation (13), the description will be made.
A certain μ is selected and one p _μ (x) is selected for a certain segment boundary C _k .
As described above, for the point s∈S _k corresponding to the zero x = ζ on the boundary C of the polynomial f˜, the indication γ _k (s) of the partitioned boundary C _k is substituted for x in the determinant (15). Things are not zero. With this in mind, Expression (36) is obtained by substituting the display γ _k (s) of the section boundary C _k into Expression (16).

When t (γ _k (s)) is a constant c, polynomials that are candidates for the polynomial _f˜ satisfy | t _μ (γ _k (s)) | <| c |. Therefore, one sεS _k satisfying | t _μ (γ _k (s)) | <| c | may be selected and used as a candidate for the parameter s in the selected μ and p _μ (x).
Specifically, when t (γ _k (s)) is a constant c, sεS _k is arbitrarily taken, and this s satisfies | t _μ (γ _k (s)) | <| c |. Then, this s may be a parameter s candidate. If arbitrarily selected s satisfies | t _μ (γ _k (s)) | ≧ | c |, the existence of candidates is examined for a combination of another partition boundary and p _μ (x). The reason is as follows. If all s′∈S _k satisfy | t _μ (γ _k (s ′)) | ≧ | c |, the selected p _μ (x) and the partition boundary C _k [ie, S _k ]. In the combination, there is no parameter s that should be a candidate for << problem >>. If s ₁ ∈ S _k satisfying | t _μ (γ _k (s)) | <| c | exists, s ₂ satisfying | t _μ (γ _k (s ₂ )) | = | c | ∈ S _k exists [intermediate value theorem]. Accordingly, in this case, since it is a target of << first case >>, it is not necessary to set s ₁ as a candidate in consideration of the fact that only one polynomial f ~ needs to be obtained.

t (gamma _k (s)) is not a constant _{| t (γ k (s)} ) | when takes the minimum value at s = [delta] is, [delta] is the end point if S _k is the zero of the formula (37) . Since the equation (37) is a rational expression, zero-point belonging to this of S _k is a finite number. That is, the number of candidates for parameter s is limited. In any case, it should be noted that the point where the denominator of Expression (36) is 0 is excluded.

By substituting γ _k (s) for x in the simultaneous equation (14) corresponding to the candidate s, the simultaneous equation (14) is determined as a binary simultaneous linear equation with respect to t and t _μ [∵ The parameter x is determined according to s. ]. Thus, the two yuan simultaneous by solving linear equations, t and t _mu solution t _<solution> and _t μ _<solution> is determined. At this time, the value of which absolute value is the smallest among the t _<solution> obtained corresponding to the s became candidate t ₀ Then, | t ₀ | adopted as the minimum distance in «second case» do it.
Also, one of the polynomials f˜ that realizes the minimum distance | t ₀ | from the polynomial f in the “second case” is t _{μ <solution>} [t _{μ <solution} when the value t ₀ is given. _{>, 0} . ] And p _μ (x) [p _{μ, 0} (x). Can be obtained by the equation (38) [see the equation (13)].

Since << the first case >> and << the second case >> do not have a theoretical prior relationship, either may be performed in advance. Therefore, the minimum distance obtained in << first case >> | t ₀ | and the minimum distance obtained in << second case >> | t ₀ | are compared, and the smaller one may be selected as the minimum distance. [If both are equal, it is sufficient to select one of them. ]. Furthermore, when not considering the case where the polynomial f˜ is equal to 0, the processing of << first case >> and << second case >> is performed on each partition boundary, and the minimum obtained for each partition boundary The smallest distance among the distances may be set as the minimum distance as an answer to << Problem >>. However, when considering the case where the polynomial f˜ is uniformly 0, the smallest distance among the minimum distances obtained for each section boundary, the polynomial f˜ = 0 which is identically 0, and the polynomial Compared with the distance d (f, 0) from f, the smaller one may be set as the minimum distance as an answer to << Problem >> [If both are equal, it is sufficient to select one of them. ].

If we take into account the fact that s and x are injective, and summarize the above explanation, the minimum distance as the answer to «Problem» is not taken into consideration when the case where the polynomial f ~ is identically 0 is not considered. Are candidates for zeros of equation (20) obtained for each combination of segmented boundaries C _k and q (x), and equation (13) obtained for each combination of segmented boundaries C _k , μ and p _μ (x) The zero point candidates are given as the minimum value of the absolute values of t obtained correspondingly. Further, when considering the case where the polynomial f˜ is equal to 0, in short, it is given as the smaller of the minimum value and d (f, 0). Therefore, the algorithm for obtaining the minimum distance when executed by a computer or the like is not particularly limited as long as it performs such an exhaustive search.

Further, the polynomial f˜ as the answer to the “question” may be a polynomial corresponding to the minimum distance as the answer to the “question” [if there are a plurality of minimum distances, it is sufficient to select one of them. . ].
This completes the explanation of [Theory].

[algorithm]
An example of an algorithm for obtaining the minimum distance d (f, f˜) will be described based on the above [Theory] [see FIGS. 1-1 and 1-2].
In the example algorithm:
=======================================
Input: Polynomial e ₀ (x), e ₁ (x), e ₂ (x), ..., e _n (x),
Rational numbers a ₁ , a ₂ ,…, a _n ,
Polynomial _{f (x) = e 0 (} x) + Σ j = 1 n a j e j boundary complex area D zeros of (x) is not present C = per ∪ _{k = 1 ^K} C _k, each segment boundary C _k However, each segment boundary C _k is expressed as injection γ _k : S _k → C _k .
Output: Minimum distance d (f, f ~)
However, when the polynomial f ~ does not exist, the output is -1.
=======================================
And Here, rational numbers are only specific examples of real algebraic numbers.

<Specific example of algorithm>
◎ Step S1
_{e 0 (x) + Σ j} = 1 n c j e j (x) = 0 real c _1, c ₂ satisfying, ..., if c _n is present, and T = d (f, 0) . When such real numbers c ₁ , c ₂ ,..., C _n do not exist, T = −1.
<Commentary>
Step S1 is a process for not excluding the case where the polynomial f˜ is equal to zero. The determination of the existence of real numbers c ₁ , c ₂ , ..., c _n is as follows: e ₀ (x) + Σ _{j = 1} ⁿ c _j e _j (x) as Σ _{i = 0} ^N w _i x ⁱ It is only necessary to solve simultaneous equations in which each coefficient w _i when expressed is 0 [however, it is not necessary to actually find a solution, and it is only necessary to determine the existence of a solution of simultaneous equations (known). ]. When the case where the polynomial f˜ is equal to 0 is not considered, only the initial value setting T = −1 is performed.

◎ Step S2
Details of the processing in step S2 are as follows.
It is determined whether or not there is an unselected division boundary (O step S20).
If there is an unselected segment boundary, one _Ck is selected (◯ step S21).
If there is no unselected segment boundary, the value of T is output as d (f, f ~) and the process ends (O step S22).
<Commentary>
Step S2 is a process for performing each of the subsequent steps S3 and S4 for all the division boundaries. When the processes in step S3 and step S4 have been executed for all the division boundaries, the process is completed.

◎ Step S3
The processing described in << First Case >> is performed. Details of the processing in step S3 are as follows.

○ Step S301
It is determined whether or not unselected q (x) exists.

○ Step S302
If unselected q (x) exists, select one of them.
After the process of step S304, equation (30) is solved for q (x) selected in the process of step S302.

○ Step S303
If there is no unselected q (x), the absolute value of Expression (31) or the absolute value of Expression (32) corresponding to all candidates s is calculated, and the minimum value | t ₀ | Ask for. T = -1 or T> | t ₀ | of the case, T = | t ₀ | to. That is, the value of T is rewritten.
After the process of step S303, the process of step S4 is performed.

○ Step S304
It is determined whether the left side of the equation (30) is equal to 0 or not. When the left side of equation (30) is not identically 0, the process proceeds to step S305, and when the left side of equation (30) is identically 0, the process proceeds to step S304c.

○ Step S304c
It is determined whether Re q (γ _k (s)) is equal to 0 or not. If Re q (γ _k (s)) is equal to 0, the process proceeds to step S309. If Re q (γ _k (s)) is not equal to 0, the process proceeds to step S306.

○ Step S305
When the left side of the equation (30) is not equal to 0, the solution s of the equation (30) is taken as a candidate.
After the process of step S305, the process of step S301 is performed.

○ Step S306
It is determined whether Equation (31) is a constant. If it is not a constant, the process proceeds to step S307. If it is a constant, the process proceeds to step S308.

○ Step S307
And candidate end points of zero and S _k of formula (33). Subsequently, the processing returns to step S301.

○ Step S308
Any one point belonging to the S _k as a candidate. Subsequently, the processing returns to step S301.

○ Step S309
It is determined whether or not Expression (32) is a constant. If it is not a constant, the process proceeds to step S310. If it is a constant, the process proceeds to step S311.

○ Step S310
And candidate end points of zero and S _k of formula (34).
After the process of step S310, the process returns to step S301.

○ Step S311
Any one point belonging to S _k is a candidate.
After the process of step S311, the process returns to step S301.

<Outline of Step S3>
In the process of step S304c, if Re q (γ _k (s)) is equal to 0, Im q (γ _k (s)) does not become 0 as already described with respect to equation (26). Therefore, it is not necessary to determine whether Im q (γ _k (s)) is equal to 0, and the process of step S309 is performed.

If Re q (γ _k (s)) is not equal to 0 in the process of step S304c, the process can return to the process of step S301 without performing the processes of steps S309 to S311. The reason will be described below.

If the processing result of step S306 is yes, the processing result of step S304 is yes and Re q (γ _k (s)) is not 0 as a polynomial, so Im q (γ _k (s)) is identically. Except for the case of 0, it can be expressed as a rational expression like Expression (39).

Therefore, the right side of Expression (39) is also a constant. By the way, when the point s where the denominators Re q (γ _k (s)) and Im q (γ _k (s)) on both sides of the equation (39) are 0 is returned to the simultaneous equations (22) which is the beginning of the discussion. , Re q (γ _k (s)) = 0, and Re f (γ _k (s)) = 0, the simultaneous equation (22) is only the second equation [the following equation], and at this time Im q (γ _{Since k} (s)) ≠ 0, any point belonging to S _k can be used as a candidate after all. Conversely, the same applies to the case of Im q (γ _k (s)) = 0. If Im q (γ _k (s)) is identically 0, the zero of Re q (γ _k (s)) is not in S _k , so in this case any point belonging to S _k No problem as a candidate.
What should be calculated in this paper is the absolute value of t, which is a constant as shown in equation (40). Therefore, it is sufficient to select one point belonging to S _k in the process of step S308.

If the result of the process in step S306 is no, the process in step S307 may be performed and the process may return to step S301.
If Re q (γ _k (s)) is not 0 as a polynomial, if there is a common factor of the denominator numerator on the left side of Equation (39), if it is divided and irreducible, the denominator in S _k There will be no zeros.
If there is a denominator zero s ₀ ∈S _k , then Re q (γ _k (s)) = 0, so Im q (γ _k (s)) is not equal to zero. Here, since the processing result of step S304 is yes, it is expressed as a rational expression as in Expression (39). Therefore, Im q (γ _k (s)) = 0, which contradicts that “Re q (γ _k (s)) and Im q (γ _k (s)) are not 0 simultaneously”.

In the process of step S304c, it is determined whether Re q (γ _k (s)) is equal to 0 or not, but as is clear from the above description, Im q (γ _k (s It may be determined whether or not)) is equal to zero. In this case, if this determination is no, the process of steps S309 to S311 is performed and the process returns to step S301. If the determination is yes, the process of steps S306 to S308 is performed and the process returns to step S301. Good.
This is the end of <Outline of Step S3>.

◎ Step S4
The processing described in << Second Case >> is performed. Details of the processing in step S4 are as follows.

○ Step S401
It is determined whether there is an unselected μ.

○ Step S402
If there is no unselected μ, the binary simultaneous equations obtained by substituting γ _k (s) for x in the simultaneous equations (14) corresponding to all candidates s are solved, and t and t solution t _<solution> and t _μ of _μ get the _<solution>. At this time, the value of which absolute value is the smallest among the t _<solution> obtained corresponding to the s became candidates t _0. When T = −1 or T> | t ₀ |, T = | t ₀ |. That is, the value of T is rewritten.
After step S402, step S20 is performed.

○ Step S403
If there is an unselected μ, one of them is selected.

○ Step S404
It is determined whether or not unselected p _μ (x) exists. If there is no unselected p _μ (x), the process proceeds to step S401, and if present, the process proceeds to step S405.

○ Step S405
One unselected p _μ (x) is selected with a leading σ _j of 1.
<Commentary>
“The leading σ _j is 1” means that σ ₂ = 1 when μ = 1, and σ ₁ = 1 when μ ≠ 1. Here, the reason why “the first σ _j is −1” is not selected is as follows. When μ = 1, p _μ (x) = σ ₂ e ₂ (x) +... + σ _n e _n (x) = ± e ₂ (x) ± ... ± e _n (x). The left side of 13) is f (x) + t _μ e _μ (x) + t · p _μ (x) = f (x) + t _μ e _μ (x) + t (± e ₂ (x) ± ... ± e _n (x)). Here, since t is a variable that takes a positive or negative value, f (x) + t _μ e _μ (x) + t (e ₂ (x) ±… ± e _n (x)) = Solve 0. The same applies to μ ≠ 1, and the equation is essentially f (x) + t _μ e _μ (x) + t (e ₁ (x) ±… ± e _n (x)) = 0. That's fine. Therefore, it is sufficient to select “the first σ _j is 1”. From this point of view, not limited to "one head of sigma _j is 1", may be selected to "what the head of the sigma _j is -1". As far Furthermore, the set a subset Y _mu of _{{Σ 1 ≦ j ≦ n,} j ≠ μ ± e j (x)} , set to _{{Σ 1 ≦ j ≦ n,} j ≠ μ ± e j (x)} When there is a relationship of P ₁ = −P _{2 for} any two polynomials P ₁ and P _{2 to which} they belong, the polynomial p _μ (x) is selected from the subset Y _μ as a set based on one of them. Can be. A specific example will be described. Two integers α and β determined corresponding to each μ, where α and β belong to the set {1, 2,..., N} and are different from each other. On the other hand, if μ ≠ α, the polynomial p _μ (x) is set (Σ _{1 ≦ j ≦ n, j ≠ μ} σ _j e _j (x) | σ _α = 1, σ _{j ≠ α} = ± 1}, and when μ = α, the polynomial p _μ (x) is a set {Σ _{1 ≦ j ≦ n, j ≠ μ} σ _j e _j (x) | σ _β = 1, σ Select from _{j ≠ β} = ± 1}. Of course, if μ ≠ α, the set of polynomials p _μ (x) {Σ _{1 ≦ j ≦ n, j ≠ μ} σ _j e _j (x) | σ _α = -1, σ _{j ≠ α} = ± 1} Or if μ = α, the polynomial p _μ (x) is a set {Σ _{1 ≦ j ≦ n, j ≠ μ} σ _j e _j (x) | σ _β = -1, You may make it select from (sigma) _{j <= beta} == ± 1}.

○ Step S406
It is determined whether Expression (36) is a constant c. If it is a constant c, the process proceeds to step S407. If it is not a constant, the process proceeds to step S410.

○ Step S407
Arbitrarily take one sεS _k . However, the point where the denominator of Expression (36) is 0 is excluded.

○ Step S408
It is determined whether or not s selected in the process of step S407 satisfies | t _μ (γ _k (s)) | <| c |. If s does not satisfy | t _μ (γ _k (s)) | <| c |, the process proceeds to step S404 without using s as a candidate.

○ Step S409
If s satisfies | t _μ (γ _k (s)) | <| c |, this s is a candidate. Subsequently, the process proceeds to step S404.

○ Step S410
And candidate end points of zero and S _k of formula (37). However, the point where the denominator of Expression (36) is 0 is excluded.
After the process of step S410, the process returns to step S404.

The above algorithm is only an example, and the order of processing can be changed or changed without departing from the logical predecessor relationship.
For example, the processing can be changed to an algorithm in which the processing in the 300s and the processing in the 400s are exchanged as a whole [see FIGS. Further, the processing content of step S1 is not limited to being performed prior to all processing. For example, it can be changed as follows. In step S1, only T = -1 is set. Then, after the process of step S22, the existence of the real numbers c ₁ , c ₂ ,..., C _n when the polynomial f˜ is uniformly 0 is determined without ending, and if it exists, T ₁ = Let d (f, 0). Further, T ₁ is compared with T at the stage of processing in step S22, and this minimum value is set as the minimum distance value. In the above algorithm, the 300th and 400th processes are performed for each segment boundary, but the 300th process may be performed for each selected segment for q (x) selected. , 400 processing may be performed at each segment boundary for the selected μ and p _μ (x).

Furthermore, in the above algorithm, rational [real algebraic number] a _1, a _2, ..., a polynomial f (x) in the complex domain D at _{a n = e 0 (x)} + Σ j = 1 n a j e It is assumed that the zero of _j (x) does not exist in advance, but the process of confirming that there is no zero in the complex domain D is performed in advance using the Sturm theorem and the principle of declination. It may be. In this case, indicators to show that not only display by rational expressions of each segment boundary C _k, bounded region surrounded by the boundary C is a simple closed curve consisting of the segment boundary C _k is a complex region D, For example, one point belonging to the inside of the complex region D is also input.

Further, although the above algorithm is for obtaining the minimum value of the distance between the polynomial f and the polynomial f˜, the polynomial f˜ may be output.
In this case, as an example, the above algorithm may be changed as follows (see FIGS. 3-1 and 3-2).
-Change step S1 as follows.
_{e 0 (x) + Σ j} = 1 n c j e j (x) = 0 real c _1, c ₂ satisfying, ..., if c _n is present, and T = d (f, 0) . When such real numbers c ₁ , c ₂ ,..., C _n do not exist, T = −1. The polynomial P is set to 0 when T = d (f, 0). When T = -1, the polynomial P is set to -1. The polynomial P = −1 means that “there is no polynomial that has a zero in the region D and belongs to the set of polynomials represented by the equation (5)”.
Step S22 is changed as follows.
If there is no unselected segment boundary, the value of T is output as d (f, f ~), the polynomial P is output as the polynomial f ~, and the process ends.
Step S303 is changed as follows.
If there is no unselected q (x), the absolute value of Expression (31) or the absolute value of Expression (32) corresponding to all candidates s is calculated, and the minimum value | t ₀ | Ask for. When T = −1 or T> | t ₀ |, T = | t ₀ |. That is, the value of T is rewritten. Further, when the value of T is rewritten, a polynomial obtained by Equation (35) using q (x) when the value t ₀ is given is defined as a polynomial P. That is, the polynomial P is rewritten. If the value of T is not rewritten, the polynomial P is not rewritten.
Step S402 is changed as follows.
If there is no unselected μ, the binary simultaneous equations obtained by substituting γ _k (s) for x in the simultaneous equations (14) corresponding to all candidates s are solved, and t and t solution t _<solution> and t _μ of _μ get the _<solution>. At this time, the value of which absolute value is the smallest among the t _<solution> obtained corresponding to the s became candidates t _0. When T = −1 or T> | t ₀ |, T = | t ₀ |. That is, the value of T is rewritten. Further, when the value of T is rewritten, a polynomial obtained by the equation (38) using t _{μ <solution>, 0} and p _{μ, 0} (x) when the value t ₀ is given is defined as a polynomial P. That is, the polynomial P is rewritten. If the value of T is not rewritten, the polynomial P is not rewritten.

  Note that the present invention is not limited to outputting the polynomial P together with the minimum distance T, and only the polynomial P may be output.

[First embodiment]
Below, 1st Embodiment of this invention is described according to the said algorithm example [refer FIG. 1-1, FIG. 1-2]. In the first embodiment, the polynomial f (x) = e ₀ (x) + Σj _{= 1} ⁿ _aj e _j (x) (a _j ∈R) and the polynomial f˜ (x) = e ₀ (x) + Σ _{j = 1} ⁿ a ~ _j e _j (x) (a ~ _j ∈ R)
FIG. 4 is a configuration block diagram illustrating the hardware configuration of the inter-polynomial distance calculation device (1) according to this embodiment.

  As illustrated in FIG. 4, the interpolynomial distance calculation device (1) includes an input unit (11) to which a keyboard or the like can be connected, an output unit (12) to which a liquid crystal display or the like can be connected, and an interpolynomial distance calculation device (1). ) A communication unit (13) to which a communication device (for example, a communication cable) that can communicate with the outside can be connected, and a CPU (Central Processing Unit) (14) [may include a cache memory and a register. ] RAM (15) as a memory, ROM (16), external storage device (17) as a hard disk, and these input unit (11), output unit (12), communication unit (13), CPU (14), The bus (18) is connected so that data can be exchanged between the RAM (15), the ROM (16), and the external storage device (17). If necessary, the inter-polynomial distance calculation device (1) may be provided with a device (drive) that can read and write a storage medium such as a CD-ROM. A physical entity having such hardware resources includes a general-purpose computer.

  The external storage device (17) of the inter-polynomial distance calculation device (1) stores a program necessary for calculating the minimum distance between the polynomial f and the polynomial f˜, data necessary for processing of the program, and the like. Yes. Data obtained by the processing of these programs is appropriately stored in a RAM or an external storage device. Hereinafter, a device such as a RAM (15) or a register for storing the calculation result, the address of the storage area, and the like will be simply referred to as “memory”.

More specifically, the external storage device (17) [or ROM, etc.] of the inter-polynomial distance calculation device (1) has an identity equation search program for investigating the case where the polynomial f˜ is equal to 0, As for the case where | a _j -a ~ _j | = d (f, f ~) holds for all j, the equation (20) obtained for each combination of the partition boundaries C _k and q (x) First candidate search program for obtaining zero candidate, absolute value | t of t [see equations (31) and (32)] obtained corresponding to each candidate obtained by the processing by the first candidate search program First distance candidate determination program for determining the smallest one in |, for polynomial f ~, | a _μ -a ~ _μ | <d (f, f ~) for some number μ and | a for j ≠ μ _j -a ~ _j | = zero d (f, f ~) segment boundary C _k and mu and p _mu equation (13) obtained for each combination of (x) as if holds Absolute value | t | of t [refer to Equation (13) and Equation (14)] obtained corresponding to each candidate obtained by the processing by the second candidate search program and the second candidate search program The second distance candidate determination program, the first candidate search program, the first distance candidate determination program, the second candidate search program, and the second distance candidate determination program for determining the smallest one are executed for each division boundary. A search control program for outputting the distance and data necessary for the processing of these programs are stored. In addition, a control program for controlling processing based on these programs is also stored and stored as appropriate.

  In the interpolynomial distance calculation device (1), each program stored in the external storage device (17) [or ROM, etc.] and data necessary for processing each program are read into the memory (20) as necessary. The CPU (14) interprets and processes it as appropriate. As a result, the CPU (14) has a predetermined function (identity search unit, first candidate search program unit, first distance candidate determination unit, second candidate search unit, second distance candidate determination unit, search control unit, control unit). Is realized.

  With reference to FIGS. 1-1 and 1-2, the inter-polynomial distance calculation processing in the present embodiment will be described.

The external storage device of the polynomial distance calculation device (1) (17), previously, the polynomial _{e 0 (x), e 1} (x), e 2 (x), ..., e n (x), rational [real Algebraic numbers] a ₁ , a ₂ ,..., A _n , polynomial f (x) = e ₀ (x) + Σ _{j = 1} ⁿ _aj e _j (x) = per ∪ _{k = 1} ^K C _k, each segment boundary C _k display [by rational expressions of each segment boundary C _k is injective gamma _k: is displayed in S _k → C _k. ] Is previously stored as data.

  These pieces of information may not be stored in advance in the external storage device (17) of the interpolynomial distance calculation device (1), but may be input from the input unit (11), for example. The drive may be read from a recording medium storing these information and the like, and can be changed as appropriate.

Further, information stored in advance as data is not limited to such information. For example, for the polynomials e ₀ (x), e ₁ (x), e ₂ (x), ..., e _n (x), rather than storing the polynomial itself, the coefficients are stored for the terms of x of each degree You may make it keep. Further, for example, if an expression representing the boundary C is input from the input unit (11), the CPU (14) that interprets and executes a program for appropriately dividing the boundary C into divided boundaries is stored in the memory (20). The stored boundary C may be read from the memory (20) and the segment boundary may be output.

  In addition, in the details of the present invention, not only numerical calculation processing but also mathematical expression processing such as rational expressions and polynomials are necessary, but numerical calculation processing and mathematical expression processing itself are achieved in the same manner as known techniques, A detailed description of the arithmetic processing method and the like is omitted (for example, Risa / Asir, Maple (registered trademark), MATHEMATICA (registered trademark No. 2312968), Gnuplot can be used as a software capable of mathematical expression processing indicating the technical level of this point. For Risa / Asir, refer to the Internet <URL: http: //hpc.cs.ehime-u.ac.jp/risa/> [searched July 2, 2007]. For Maple, refer to the Internet <URL: http://www.cybernet.co.jp/maple/> [Search July 2, 2007] For Gnuplot, for example, the Internet <URL: http: // See www.gnuplot.info/> [searched July 2, 2007].

First, the control unit (190) of the interpolynomial distance calculation device (1) receives the polynomials e ₀ (x), e ₁ (x), e ₂ (x),..., E _n (x) from the external storage device (17). ), rational [real algebraic number] _{a 1, a 2, ...,} a n, reads the display gamma _k by rational expressions of each segment boundary C _k, and stores each in a predetermined storage area of the memory (20) deep. Hereinafter, when the description “reads XX from memory” is used, it means “read XX from a predetermined storage area where XX is stored in the memory”.

The identity checking unit (110) of the inter-polynomial distance calculation device (1) reads the polynomials e ₀ (x), e ₁ (x), e ₂ (x),..., E _n (x) from the memory (20). , real c _1, c ₂ satisfying _{e 0 (x) + Σ j} = 1 n c j e j (x) = 0, ..., determine the presence of c _n. Identity check unit (110), such real c _1, c _2, ..., a T set to if there is _{c n T = d (f,} 0) in a predetermined storage area of the memory (20) When such real numbers c ₁ , c ₂ ,..., C _n do not exist, T = −1 is set and T is stored in a predetermined storage area of the memory (20) [FIG. Step S1].

Next, the search control unit (120) of the inter-polynomial distance calculation device (1) determines whether or not there is an unselected segment boundary [Step S20 in FIG. If there is no unselected section boundary, output processing for displaying T stored in the memory (20) on a liquid crystal display or the like is performed, and the inter-polynomial distance calculation processing is terminated [step S22 in FIG. 1-1]. If there is an unselected partition boundary, one is selected from the unselected partition boundaries, and the information is stored in a predetermined storage area of the memory (20) [step S21 in FIG. 1-1].
As a specific example, the search control unit (120) may perform each process for the partition boundary C _k (k = 1, 2,..., K) by incrementing from k = 1 to k = K.
Further, the search control unit (120) determines a first candidate search unit (130), a first distance candidate determination unit (140), and a first distance candidate until it is determined in step S20 that there is no unselected segment boundary. Each operation | movement of a 2 candidate search part (150) and a 2nd distance candidate determination part (160) is controlled.

Then, the first candidate search unit of the polynomial distance calculation device (1) (130) cooperates with the memory (20), the polynomial f ~ for _{| a j -a ~ j | =} d (f, f ~ ) Holds for all j, zero candidate candidates of the equation (20) obtained for each combination of the partition boundaries C _k and q (x) selected in step S21 are obtained, and these candidates are stored in the memory (20). [Steps S301 to S311 in FIG. 1-1. However, step S303 is excluded. ]. In the case of conforming to the above algorithm, the zero candidate stored in the memory (20) is given by s belonging to S _k corresponding to the segment boundary C _k selected in step S21.

Next, the first distance candidate determination unit (140) of the inter-polynomial distance calculation device (1) cooperates with the memory (20) to obtain each candidate obtained by a series of processes by the first candidate search unit (130). T that is determined and stored in the memory (20) at the present stage is | t obtained corresponding [formula (31), equation (32) refer to Fig absolute value of | t | smallest of | t ₀ When T = −1 or T> | t ₀ |, the value of T stored in the memory (20) is rewritten to | t ₀ | and stored. However, if T = −1 or T> | t ₀ | does not hold, the value of T stored in the memory (20) at this stage is maintained [step S303 in FIG. 1-1].

Next, the second candidate search unit (150) of the inter-polynomial distance calculation device (1) cooperates with the memory (20) to | a _μ −a˜ _μ | <d for a polynomial f˜ with a certain number μ. When (f, f ~) and j ≠ μ, | a _j -a ~ _j | = d (f, f ~) holds, and the segment boundaries C _k , μ, and p _μ (x) selected in step S21 The candidates of the zero point of the equation (13) obtained for each combination are obtained, and these candidates are stored in a predetermined storage area of the memory (20) [Steps S401 to S409 in FIG. 1-2. However, step S402 is excluded. ]. In the case of conforming to the above algorithm, the zero candidate stored in the memory (20) is given by s belonging to S _k corresponding to the segment boundary C _k selected in step S21.

Next, the second distance candidate determination unit (160) of the inter-polynomial distance calculation device (1) cooperates with the memory (20) to obtain each candidate obtained by a series of processes by the second candidate search unit (150). The minimum value | t ₀ | of the absolute value | t | of t [see Equation (13) and Equation (14)] obtained correspondingly is determined, and T stored in the memory (20) at this stage is determined. When T = −1 or T> | t ₀ |, the value of T stored in the memory (20) is rewritten to | t ₀ | and stored. However, if T = −1 or T> | t ₀ | does not hold, the value of T stored in the memory (20) at this stage is maintained [step S402 in FIG. 1-2].

[Second Embodiment]
Below, 2nd Embodiment of this invention is described according to the said algorithm example [refer FIG. 3-1, FIG. 3-2]. In the second embodiment, the polynomial f (x) = e ₀ (x) + Σj _{= 1} ⁿ _aj e _j (x) (a _j ∈R) and the polynomial f˜ (x) = e ₀ (x) + Σ _{j = 1} ⁿ a ~ _j e _j (x) This relates to the calculation of polynomial f ~ (x) that gives the minimum distance to (a ~ _j ∈ R) [one variable recent real polynomial calculation]. Here, for convenience of explanation, it is assumed that the polynomials f to (x) are calculated in addition to the calculation of the minimum distance of the first embodiment.

  The hardware configuration of the interpolynomial distance calculation and single variable nearest real polynomial calculation apparatus according to the second embodiment is the same as the hardware configuration of the interpolynomial distance calculation apparatus (1) shown in FIG. To do. Therefore, in the second embodiment, the inter-polynomial distance calculation device (1) of the first embodiment is renamed as the inter-polynomial distance calculation / univariate nearest real polynomial calculation device (1).

In the external storage device (17) [or ROM, etc.] of the inter-polynomial distance calculation and one-variable recent real polynomial calculation device (1), the identity survey program for investigating the case where the polynomial f ~ is equal to 0 The equation (20) obtained for each combination of the partition boundaries C _k and q (x), assuming that | a _j -a ~ _j | = d (f, f ~) holds for all j. Absolute value of t [see formula (31), formula (32)] obtained corresponding to each candidate obtained by the processing by the first candidate search program and the first candidate search program A first distance / polynomial candidate determination program for determining a minimum value of t | and calculating a polynomial corresponding to the minimum value. For a polynomial f˜, | a _μ −a˜ _μ | <d (f, f ~) is a j ≠ in _{μ | a j -a ~ j |} = d (f, f ~) classification border as if that is true C _k and mu and p _mu (x) Second candidate search program for determining a candidate of the zero point of the resulting equation (13) for each combination of, corresponding for each candidate obtained in the process according to the second candidate search program Second distance / polynomial candidate determination program for determining a minimum of absolute values | t | of t [see formulas (13) and (14)] obtained in this way and calculating a polynomial corresponding to the minimum value A search for outputting a minimum distance and a polynomial f˜ by executing the first candidate search program, the first distance / polynomial candidate determination program, the second candidate search program, and the second distance / polynomial candidate determination program for each partition boundary Control programs and data necessary for processing these programs are stored. In addition, a control program for controlling processing based on these programs is also stored and stored as appropriate.

  In the inter-polynomial distance calculation and one-variable recent real polynomial calculation device (1), each program stored in the external storage device (17) [or ROM, etc.] and data necessary for the processing of each program are stored in memory as necessary. (20) is read and interpreted and executed by the CPU (14) as appropriate. As a result, the CPU (14) has predetermined functions (identity search unit, first candidate search program unit, first distance / polynomial candidate determination unit, second candidate search unit, second distance / polynomial candidate determination unit, search control unit) The control unit).

  With reference to FIGS. 3A and 3B, the inter-polynomial distance calculation and univariate nearest real polynomial calculation processing in the present embodiment will be described.

In the external storage device (17) of the inter-polynomial distance calculation / univariate recent real polynomial calculation device (1), the polynomials e ₀ (x), e ₁ (x), e ₂ (x) _,. (x), rational numbers [real algebraic numbers] a ₁ , a ₂ , ..., a _n , polynomials f (x) = e ₀ (x) + Σ _{j = 1} ⁿ a _j e _j (x) exists , per boundary ^{_{C = ∪ k = 1 K C}} k of the complex domain D without the division boundary C _k display [by rational expressions of each segment boundary C _k is injective gamma _k: is displayed in S _k → C _k. ] Is previously stored as data. This point is the same as in the first embodiment. For example, these pieces of information may be input from the input unit (11), or the polynomials e ₀ (x), e ₁ (x), e ₂ (x ),..., E _n (x), instead of storing the polynomial itself, the coefficients of the terms of x of each order may be stored. Further, as described above, the numerical calculation process and the mathematical expression process are achieved in the same manner as in the known technique.

First, the controller (290) of the inter-polynomial distance calculation / univariate nearest real polynomial calculation device (1) receives the polynomials e ₀ (x), e ₁ (x), e ₂ (x) from the external storage device (17). _{, ..., e n (x)} , rational [real algebraic number] _{a 1, a 2, ...,} a n, a predetermined read the display gamma _k by rational expressions of each segment boundary C _k, each memory (20) Stored in the storage area.

The identity search unit (210) of the interpolynomial distance calculation and univariate recent real polynomial calculation device (1) reads the polynomials e ₀ (x), e ₁ (x), e ₂ (x),. e _n (x) is read, and the existence of real numbers c ₁ , c ₂ ,..., c _n satisfying e ₀ (x) + Σj _{= 1} ⁿ _cj e _j (x) = 0 is determined. The identity checking unit (210) sets T = d (f, 0) and P = 0 when such real numbers c ₁ , c ₂ ,..., C _n exist, and stores T and P in the memory (20). of stored in a predetermined storage area, such real c _1, c _2, ..., if c _n is not present, T = -1, the memory (20) the T and P is set to P = -1 [Step S1a in FIG. 3A].

Next, the search control unit (220) of the inter-polynomial distance calculation / univariate nearest real polynomial calculation device (1) determines whether or not there is an unselected section boundary [step S20 in FIG. If there is no unselected section boundary, output processing for displaying T and P stored in the memory (20) on a liquid crystal display or the like is performed, and the inter-polynomial distance calculation processing is terminated [step S22a in FIG. 3-1]. . If there is an unselected partition boundary, one is selected from the unselected partition boundaries and the information is stored in a predetermined storage area of the memory (20) [step S21 in FIG. 3-1].
As a specific example, as in the case of the first embodiment, the search control unit (220) increments the partition boundary C _k (k = 1, 2,..., K) from k = 1, and k = K It suffices to carry out each processing until.
Further, the search control unit (220) determines a first candidate search unit (230) and a first distance / polynomial candidate determination unit (240), which will be described later, until it is determined in step S20 that there is no unselected segment boundary. The operations of the second candidate search unit (250) and the second distance / polynomial candidate determination unit (260) are controlled.

Next, the first candidate search unit (230) of the inter-polynomial distance calculation / univariate recent real polynomial calculation device (1) cooperates with the memory (20) to obtain the polynomial f˜ | a _j −a˜ _j |. Assuming that = d (f, f ~) holds for all j, candidates for zeros of the equation (20) obtained for each combination of the segment boundary C _k and q (x) selected in step S21 are obtained. These candidates are stored in a predetermined storage area of the memory (20) [Steps S301 to S311 in FIG. However, step S303a is excluded. ]. In the case of conforming to the above algorithm, the zero candidate stored in the memory (20) is given by s belonging to S _k corresponding to the segment boundary C _k selected in step S21.

Next, the first distance / polynomial candidate determination unit (240) of the inter-polynomial distance calculation / univariate nearest real polynomial calculation device (1) operates in cooperation with the memory (20) by the first candidate search unit (230). The minimum value | t ₀ | of the absolute value | t | of t [refer to equations (31) and (32)] obtained corresponding to each candidate obtained in the series of processes is determined, and the memory at this stage When the value of T stored in (20) is T = −1 or T> | t ₀ |, the value of T stored in the memory (20) is rewritten to | t ₀ | and stored. However, if T = −1 or T> | t ₀ | does not hold, the value of T stored in the memory (20) at the present stage is maintained. Further, the first distance / polynomial candidate determination unit (240), when rewriting the value of T, uses the polynomial obtained by equation (35) using q (x) when the value t ₀ is given as the polynomial P. The polynomial stored in the memory (20) is rewritten to the polynomial P determined here and stored. However, if T = −1 or T> | t ₀ | does not hold, the polynomial P stored in the memory (20) at the present stage is maintained [step S303a in FIG. 3-1].

Next, the second candidate search unit (250) of the inter-polynomial distance calculation / univariate nearest real polynomial calculation device (1) cooperates with the memory (20) to | a _μ for a polynomial f˜ with a certain number _{μ. -a ~ μ | <d (f} , f ~) at is j ≠ mu a _{| a j -a ~ j | =} d (f, f ~) and classification boundary C _k selected in step S21 as if the holds The candidates of the zero point of the equation (13) obtained for each combination of μ and p _μ (x) are obtained, and these candidates are stored in a predetermined storage area of the memory (20) [Steps S401 to S409 in FIG. 3-2. However, step S402a is excluded. ]. In the case of conforming to the above algorithm, the zero candidate stored in the memory (20) is given by s belonging to S _k corresponding to the segment boundary C _k selected in step S21.

Next, the second distance candidate determination unit (260) of the inter-polynomial distance calculation / univariate nearest real polynomial calculation device (1) cooperates with the memory (20) to perform a series of operations by the second candidate search unit (150). The minimum value | t ₀ | of the absolute value | t | of t [refer to the equations (13) and (14)] obtained corresponding to each candidate obtained by the processing is determined, and the memory (20 When the value of T stored in () is T = −1 or T> | t ₀ |, the value of T stored in the memory (20) is rewritten to | t ₀ | and stored. However, if T = −1 or T> | t ₀ | does not hold, the value of T stored in the memory (20) at the present stage is maintained. Further, when the value of T is rewritten, the second distance candidate determination unit (260) uses the equation (38) using t _{μ <solution>, 0} and p _{μ, 0} (x) when the value t ₀ is given. ) Is obtained as a polynomial P, and the polynomial stored in the memory (20) is rewritten and stored in the polynomial P obtained here. However, if T = −1 or T> | t ₀ | does not hold, the polynomial P stored in the memory (20) at the present stage is maintained [step S402a in FIG. 3-2].

  In the second embodiment, the minimum distance and the polynomial f ~ are output. However, if only the polynomial f ~ is output, the inter-polynomial distance calculation and one-variable recent real polynomial calculation apparatus (1) is a single variable. Recently, it will operate as a real polynomial calculator.

  The inter-polynomial distance calculation device, the inter-polynomial distance calculation method, the one-variable nearest real polynomial calculation device, and the one-variable nearest real polynomial calculation method according to the present invention are not limited to the above-described embodiments, and depart from the spirit of the present invention. Changes can be made as appropriate without departing from the scope. In addition, the processing described in the above embodiment may be executed not only in time series according to the order of description but also in parallel or individually as required by the processing capability of the device that executes the processing. .

  Further, when the processing function in the interpolynomial distance calculation device and the one-variable recent real polynomial calculation device described in the above embodiment is realized by a computer, the inter-polynomial distance calculation device and the one-variable nearest real polynomial calculation device should have the functions The processing contents are described by a program. By executing this program on a computer, the processing functions of the inter-polynomial distance calculation device and the one-variable recent real polynomial calculation device are realized on the computer.

  The program describing the processing contents can be recorded on a computer-readable recording medium. As the computer-readable recording medium, for example, any recording medium such as a magnetic recording device, an optical disk, a magneto-optical recording medium, and a semiconductor memory may be used. Specifically, for example, as a magnetic recording device, a hard disk device, a flexible disk, a magnetic tape or the like, and as an optical disk, a DVD (Digital Versatile Disc), a DVD-RAM (Random Access Memory), a CD-ROM (Compact Disc Read Only). Memory), CD-R (Recordable) / RW (ReWritable), etc., magneto-optical recording medium, MO (Magneto-Optical disc), etc., semiconductor memory, EEP-ROM (Electronically Erasable and Programmable-Read Only Memory), etc. Can be used.

  The program is distributed by selling, transferring, or lending a portable recording medium such as a DVD or CD-ROM in which the program is recorded. Furthermore, the program may be distributed by storing the program in a storage device of the server computer and transferring the program from the server computer to another computer via a network.

  A computer that executes such a program first stores, for example, a program recorded on a portable recording medium or a program transferred from a server computer in its storage device. When executing the process, the computer reads a program stored in its own recording medium and executes a process according to the read program. As another execution form of the program, the computer may directly read the program from a portable recording medium and execute processing according to the program, and the program is transferred from the server computer to the computer. Each time, the processing according to the received program may be executed sequentially. Also, the program is not transferred from the server computer to the computer, and the above-described processing is executed by a so-called ASP (Application Service Provider) type service that realizes the processing function only by the execution instruction and result acquisition. It is good. Note that the program in this embodiment includes information that is used for processing by an electronic computer and that conforms to the program (data that is not a direct command to the computer but has a property that defines the processing of the computer).

  In this embodiment, the inter-polynomial distance calculation device and the one-variable recent real polynomial calculation device are configured by executing a predetermined program on the computer. However, at least a part of these processing contents is hardware. It may be realized as an example.

  A specific example of the minimum distance calculation according to the algorithm shown in FIGS. 1-1 and 1-2 will be described.

f (x) = x ² +2, and for j = 0,1,2, e _j (x) = x ^2−j . Region D is defined as a region composed of the circumference and the inside of a unit circle on the Gaussian plane. The boundary C of the region D, that is, the circumference C of the unit circle is a union of C ₁ = {1} and C ₂ . However, C ₁ and C ₂ are expressed as in Expression (41).

At this time, a polynomial f ~ (x) = e ₀ (x) + a ~ ₁ e ₁ (x) + a ~ ₂ e ₂ (x) (a ~ ₁ , a having a zero in the region D and closest to the polynomial f Let us show the behavior of the algorithm when finding ~ ₂ ∈R).

・ Step S1
Since real numbers c ₁ and c ₂ that satisfy e ₀ (x) + c ₁ e ₁ (x) + c ₂ e ₂ (x) = 0 do not exist, T = −1.

・ Step S20
An unselected partition boundary exists. Proceed to step S21.

Step S21
To select a segment boundary C _1.

Step S301 to step S311
q (x) is two ways of x + 1 and x-1.
When q (x) = x + 1, equation (30) is established equally. Further, the condition Re q (γ ₁ (s)) ≠ 0 always holds because 2 ≠ 0. On the other hand, Im q (γ ₁ (s)) ≠ 0 is not always established because 0 ≠ 0. Equation (31) is a constant 3/2. Equation (32) is not considered because the denominator is zero. Therefore, the candidate for s is only one point and one. At this time, | t | = 3/2.
When q (x) = x−1, equation (30) is established equally. Further, the condition Re q (γ ₁ (s)) ≠ 0 is not always satisfied because 0 ≠ 0. On the other hand, since Im q (γ ₁ (s)) ≠ 0 is also 0 ≠ 0, it is not always established. Therefore, in this case, there is no candidate s.
From the above, the minimum value of | t | for all candidates s is 3/2, and T = 3/2.

Step S401 to step S409
When μ = 1, only p _μ (x) = 1.
When μ = 2, only p _μ (x) = x.
In any case, there is no candidate because the denominator of equation (36) is zero, and no T substitution occurs.

・ Step S20
An unselected partition boundary exists. Proceed to step S21.

Step S21
To select a classification boundary C _2.

Step S301 to step S311
q (x) is two ways of x + 1 and x-1.
For q (x) = x + 1, equation (30) ^{(2s 3 -6s) / (s} 2 +1) is ² = 0, which condition "2s ² / (s ² +1) ≠ 0 Or solve under -2s / (s ² +1) ≠ 0 ". This is equivalent to solving s ² -3 = 0. Therefore, the candidate for s is ± √3. At this time, | t | = 1 for s = ± √3.
When q (x) = x-1, Expression (30) is (-2s ³ + 6s) / (s ² +1) ² = 0, and this is the condition “-2 / (s ² +1) ≠ Solve under 0 or -2 s / (s ² +1) ≠ 0 ". This is equivalent to solving s ² -3 = 0. Therefore, the candidate for s is ± √3. At this time, | t | = 1 for s = ± √3.
From the above, the minimum value of | t | for all candidates s is 1, and the value of T is changed from 3/2 to 1.

Step S401 to step S409
When μ = 1, p _μ (x) = 1 is examined. At this time, Expression (36) becomes 1 (constant), and t1 = (− 2s ² +2) / (s ² +1). The value of s satisfying | t ₁ | <1 is a candidate, but since s ² <3 or 3s ² > 1, it is not an empty set but | t | = 1 (constant).
When μ = 2, p _μ (x) = x is examined. At this time, Expression (36) is (2s ² −2) / (s ² +1). Since this is not a constant, t (γ ₂ (s)) is differentiated by s to obtain 8s / (s ⁴ + 2s ² +1). Therefore, the only candidate for s is 0. At this time, | t | = 2.
From the above, the minimum value of | t | in this case is 1. Since the value of T stored at this stage is 1, no replacement of T occurs.

・ Step S20
There is no unselected partition boundary. Proceed to step S22.

Step S22
Outputs the value of T, that is, 1 and ends.

  In the present invention, for example, when it is desired to set a coefficient of a polynomial so that the complex area D does not have a zero, the polynomial set with an error is definitely in the complex area D on the assumption that the coefficient setting involves an error. When not having a zero, it can be used to determine the error limit. As an example of such a case, it is useful for designing a digital filter (for example, an IIR (Infinite Impulse Response) filter) in digital signal processing.

The flowchart of the distance calculation process between polynomials of 1st Embodiment (the 2). The flowchart of the distance calculation process between polynomials of 1st Embodiment (the 2). The flowchart (the 2) of the modification of the distance calculation process between polynomials of 1st Embodiment. The flowchart (the 2) of the modification of the distance calculation process between polynomials of 1st Embodiment. The flowchart (the 2) of the distance calculation between polynomials of 2nd Embodiment and the one-variable recent real polynomial calculation process. The flowchart (the 2) of the distance calculation between polynomials of 2nd Embodiment and the one-variable recent real polynomial calculation process. The block diagram which illustrated the hardware constitutions of the distance calculation apparatus (1) between polynomials. The functional block diagram which shows the processing function of the distance calculation process between polynomials. The functional block diagram which shows the processing function of the distance calculation between polynomials, and the one-variable recent real polynomial calculation process.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Interpolynomial distance calculation apparatus 110 An equality type | formula search part 120 Search control part 130 1st candidate search part 140 1st distance candidate determination part 150 2nd candidate search part 160 2nd distance candidate determination part 210 An equality type | formula search part 220 Search Control unit 230 First candidate search unit 240 First distance / polynomial candidate determination unit 250 Second candidate search unit 260 Second distance / polynomial candidate determination unit

Claims (20)

The set of all real numbers and _{R, a 1, a 2,} ..., a real number given in advance respectively _{a n, e 0 (x)} , e 1 (x), ..., given in advance e _n a (x), respectively Univariate real polynomial

A set of univariate real polynomials, assuming that the bounded and closed region D in the set of complex numbers has no zeros

Among the one-variable real polynomials f to (x) having zeros in the region D and closest to the one-variable real polynomial f (x), the one-variable real polynomial f (x) and the one-variable real variable An inter-polynomial distance calculation device that calculates the distance from the polynomial f ~ (x) as l ^∞ -norm,
Said a _1, a _2, ..., finite and a _n, the e ₀ (x), e ₁ where (x), ..., and e _n (x), is represented by a rational expression constituting the boundary of the region D Storage means for storing the segment boundaries C ₁ , C ₂ ,..., C _K ;
Each said segment boundary C _k [where, k = 1,2, ..., K] and the set ^{_{{Σ j = 1 n ± e}} j (x)} polynomial q (x) and the first obtained for each combination of belonging to First candidate search means for finding candidates for zero point x of equation f (x) + tq (x) = 0;
First distance candidate determining means for obtaining the smallest one of the absolute values | t | of t of the first equation obtained corresponding to each of the candidates obtained by the first candidate searching means;
Each of the partition boundaries C _k , an integer μ belonging to the set {1, 2,..., N}, and a polynomial p _μ (x) belonging to the set {Σ _{1 ≦ j ≦ n, j ≠ μ} ± e _j (x)} Second candidate search means for obtaining candidates for a zero point x of the second equation f (x) + t _μ e _μ (x) + τp _μ (x) = 0 obtained for each combination of
Second distance candidate determining means for obtaining a minimum one of absolute values | τ | of τ of the second equation obtained corresponding to each of the candidates obtained by the second candidate searching means;
An inter-polynomial distance calculation device comprising: a first distance candidate determination unit; and a minimum distance output unit that outputs a minimum value among the values obtained by the second distance candidate determination unit. Polynomial

It is determined whether or not there exist real numbers c ₁ , c ₂ ,..., C _n that are _equal to 0, and if they exist, they are equal to 0 with the above-described univariate real polynomial f (x). Equipped with an identity survey means to find the distance to the polynomial,
Instead of the minimum distance output means,
A second minimum distance output means for obtaining and outputting the minimum value among the values obtained by the first distance candidate determination means and the second distance candidate determination means and the distance obtained by the identity search means; The inter-polynomial distance calculation device according to claim 1, wherein: The subset X _Omega above the set ^{_{{Σ j = 1 n ± e}} j (x)}, per the set ^{_{{Σ j = 1 n ± e}} j (x)} any two polynomials M ₁ belonging to, M ₂ When there is a relationship M ₁ = -M ₂ ,
The first candidate search means includes:
The above combination
Each said segment boundary C _k polynomial distance calculation device according to claim 1, claim 2, characterized in that the combination of the polynomial q (x) belonging to the subset X _Omega. Any belonging to the set a subset Y _mu of _{{Σ 1 ≦ j ≦ n,} j ≠ μ ± e j (x)}, the set _{{Σ 1 ≦ j ≦ n,} j ≠ μ ± e j (x)} When there is a relationship P ₁ = −P _{2 for} the two polynomials P ₁ and P ₂ of
The second candidate search means includes:
The above combination
2. A combination of each of the partition boundaries C _k , an integer μ belonging to the set {1, 2,..., N} and a polynomial p _μ (x) belonging to the subset Y _μ. The interpolynomial distance calculation apparatus according to claim 3. Each of the partition boundaries C _k is given by a bijection γ _k (s) [where s∈S _k ] for a set S _k on a real number, and the real part and imaginary part of a complex number are represented by Re and Im, respectively.
The first candidate search means includes:
The candidate of the zero point x on the partition boundary C _{k of} the first equation f (x) + tq (x) = 0 is

The inter-polynomial distance calculation apparatus according to claim 1, wherein the polynomial distance is obtained as a solution s. Each of the partition boundaries C _k is given by a bijection γ _k (s) [where s∈S _k ] for a set S _k on a real number, and the real part and imaginary part of a complex number are represented by Re and Im, respectively.
The second candidate search means includes:
The candidate of the zero point x on the partition boundary C _{k of} the second equation f (x) + t _μ e _μ (x) + τp _μ (x) = 0 is

The inter-polynomial distance calculation device according to claim 1, wherein s is obtained as s satisfying a condition that minimizes. The set of all real numbers and _{R, a 1, a 2,} ..., a real number given in advance respectively _{a n, e 0 (x)} , e 1 (x), ..., given in advance e _n a (x), respectively Univariate real polynomial

A set of univariate real polynomials, assuming that the bounded and closed region D in the set of complex numbers has no zeros

Among the one-variable real polynomials f to (x) having zeros in the region D and closest to the one-variable real polynomial f (x), the one-variable real polynomial f (x) and the one-variable real A distance calculation method between polynomials that calculates the distance from the polynomial f ~ (x) as l ^∞ -norm,
The storage unit, the a _1, a _2, ..., and a _n, the _{e 0 (x), e 1} (x), ..., e n a (x), rational expression constituting the boundary of the region D , A finite number of segment boundaries C ₁ , C ₂ ,..., C _K represented by
Is first candidate search means, each said segment boundary C _k [where, k = 1, 2, ..., K] and the set ^{_{{Σ j = 1 n ± e}} j (x)} belonging polynomial q (x) and the A first candidate search step for obtaining a candidate of a zero point x of the first equation f (x) + tq (x) = 0 obtained for each combination;
The first distance candidate determining means obtains the smallest one of the absolute values | t | of t of the first equation obtained corresponding to each of the candidates obtained in the first candidate searching step. A distance candidate determination step;
The second candidate searching means belongs to each of the above-described partition boundaries C _k and the integer μ belonging to the set {1, 2,..., N} and the set {Σ _{1 ≦ j ≦ n, j ≠ μ} ± e _j (x)}. A second candidate search step for obtaining candidates for a zero point x of the second equation f (x) + t _μ e _μ (x) + τp _μ (x) = 0 obtained for each combination with the polynomial p _μ (x);
The second distance candidate determining means obtains the smallest one of the absolute values | τ | of the second equation obtained corresponding to the candidates obtained in the second candidate searching step. A distance candidate determination step;
A distance calculation method between polynomials, wherein the minimum distance output means includes a minimum distance output step for outputting a minimum value among the values obtained in the first distance candidate determination step and the second distance candidate determination step. The identity survey means is the identity

It is determined whether or not there exist real numbers c ₁ , c ₂ ,..., C _n that are _equal to 0, and if they exist, they are equal to 0 with the above-described univariate real polynomial f (x). An identity search step to find the distance to the polynomial,
Instead of the minimum distance output step,
A second minimum distance output means obtains and outputs the smallest one of the values obtained in the first distance candidate determination step and the second distance candidate determination step and the distance obtained in the identity search step. 8. The interpolynomial distance calculation method according to claim 7, further comprising a minimum distance output step. The set of all real numbers and _{R, a 1, a 2,} ..., a real number given in advance respectively _{a n, e 0 (x)} , e 1 (x), ..., given in advance e _n a (x), respectively Univariate real polynomial

A set of univariate real polynomials, assuming that the bounded and closed region D in the set of complex numbers has no zeros

Is a one-variable recent real polynomial calculation device that calculates a one-variable real polynomial f to (x) having a zero in the region D and closest to the one-variable real polynomial f (x),
Said a _1, a _2, ..., finite and a _n, the e ₀ (x), e ₁ where (x), ..., and e _n (x), is represented by a rational expression constituting the boundary of the region D Storage means for storing the segment boundaries C ₁ , C ₂ ,..., C _K ;
The first obtained for each combination of each of the above-described partitioned boundaries C _k [k = 1, 2,..., K] and the polynomial q (x) belonging to the set {Σ _{j = 1} ⁿ ± e _j (x)}. First candidate search means for finding candidates for zero point x of equation f (x) + tq (x) = 0;
Among the absolute values | t | of the first equation obtained corresponding to each of the candidates obtained by the first candidate search means, the smallest one is obtained and the polynomial corresponding to the minimum value is calculated. First polynomial candidate determination means for
Each of the partition boundaries C _k , an integer μ belonging to the set {1, 2,..., N}, and a polynomial p _μ (x) belonging to the set {Σ _{1 ≦ j ≦ n, j ≠ μ} ± e _j (x)} Second candidate search means for obtaining candidates for a zero point x of the second equation f (x) + t _μ e _μ (x) + τp _μ (x) = 0 obtained for each combination of
Of the absolute values | τ | of the second equation obtained corresponding to the candidates obtained by the second candidate search means, the smallest one is obtained and the polynomial corresponding to the minimum value is calculated. Second polynomial candidate determination means for
A polynomial output means for obtaining a minimum one of the values obtained by the first polynomial candidate determination means and the second polynomial candidate determination means and outputting a polynomial corresponding to the minimum value; Univariate nearest real polynomial calculator. Polynomial

It is determined whether or not there exist real numbers c ₁ , c ₂ ,..., C _n that are _equal to 0, and if they exist, they are equal to 0 with the above-described univariate real polynomial f (x). Equipped with an identity survey means to find the distance to the polynomial,
Instead of the polynomial output means,
A polynomial corresponding to the minimum value is obtained by obtaining a minimum one of the values obtained by the first polynomial candidate determining means and the second polynomial candidate determining means and the distance obtained by the identity examining means. 10. The one-variable nearest real polynomial calculation device according to claim 9, further comprising second polynomial output means for outputting. The subset X _Omega above the set ^{_{{Σ j = 1 n ± e}} j (x)}, per the set ^{_{{Σ j = 1 n ± e}} j (x)} any two polynomials M ₁ belonging to, M ₂ When there is a relationship M ₁ = -M ₂ ,
The first candidate search means includes:
The above combination
The univariate nearest real polynomial calculation device according to claim 9 or 10, wherein each of the partition boundaries C _k is a combination of a polynomial q (x) belonging to the subset _XΩ . Any belonging to the set a subset X _mu of _{{Σ 1 ≦ j ≦ n,} j ≠ μ ± e j (x)}, the set _{{Σ 1 ≦ j ≦ n,} j ≠ μ ± e j (x)} When there is a relationship P ₁ = −P _{2 for} the two polynomials P ₁ and P ₂ of
The second candidate search means includes:
The above combination
10. A combination of each of the partition boundaries C _k , an integer μ belonging to the set {1, 2,..., N} and a polynomial p _μ (x) belonging to the subset X _μ. The univariate nearest real polynomial calculation device according to claim 11. Each of the partition boundaries C _k is given by a bijection γ _k (s) [where s∈S _k ] for a set S _k on a real number, and the real part and imaginary part of a complex number are represented by Re and Im, respectively.
The first candidate search means includes:
The candidate of the first equation f (x) + tq (x ) = 0 Segment boundaries C _k zeros on x,

The single-variable nearest real polynomial calculation device according to claim 9, which is obtained as a solution s of Each of the partition boundaries C _k is given by a bijection γ _k (s) [where s∈S _k ] for a set S _k on a real number, and the real part and imaginary part of a complex number are represented by Re and Im, respectively.
The second candidate search means includes:
The candidate of the zero point x on the partition boundary C _{k of} the second equation f (x) + t _μ e _μ (x) + τp _μ (x) = 0 is

14. The univariate nearest real polynomial calculation device according to claim 9, wherein s satisfies the condition that satisfies the minimum value. The set of all real numbers and _{R, a 1, a 2,} ..., a real number given in advance respectively _{a n, e 0 (x)} , e 1 (x), ..., given in advance e _n (x) is respectively Univariate real polynomial

A set of univariate real polynomials, assuming that the bounded and closed region D in the set of complex numbers has no zeros

Is a one-variable recent real polynomial calculation method for calculating a one-variable real polynomial f to (x) having a zero in the region D and closest to the one-variable real polynomial f (x),
The storage unit, the a _1, a _2, ..., and a _n, the _{e 0 (x), e 1} (x), ..., e n a (x), rational expression constituting the boundary of the region D , A finite number of segment boundaries C ₁ , C ₂ ,..., C _K represented by
The first candidate search means determines whether each of the partition boundaries C _k [k = 1, 2,..., K] and the polynomial q (x) belonging to the set {Σ _{j = 1} ⁿ ± e _j (x)} A first candidate search step for obtaining a candidate of a zero point x of the first equation f (x) + tq (x) = 0 obtained for each combination;
The first polynomial candidate determination means obtains the smallest one of the absolute values | t | of the first equation obtained corresponding to each of the candidates obtained in the first candidate search step. And a first polynomial candidate determination step for calculating a polynomial corresponding to the minimum value,
The second candidate searching means belongs to each of the above-described partition boundaries C _k and the integer μ belonging to the set {1, 2,..., N} and the set {Σ _{1 ≦ j ≦ n, j ≠ μ} ± e _j (x)}. A second candidate search step for obtaining candidates for a zero point x of the second equation f (x) + t _μ e _μ (x) + τp _μ (x) = 0 obtained for each combination with the polynomial p _μ (x);
The second polynomial candidate determination means obtains the minimum one of the absolute values | τ | of the second equation obtained corresponding to the candidates obtained in the second candidate search step. And a second polynomial candidate determination step for calculating a polynomial corresponding to the minimum value,
Polynomial output, wherein the polynomial output means obtains the minimum value among the values obtained in the first polynomial candidate determination step and the second polynomial candidate determination step, and outputs a polynomial corresponding to the minimum value And a univariate nearest real polynomial calculation method. The identity survey means is the identity

It is determined whether or not there exist real numbers c ₁ , c ₂ ,..., C _n that are _equal to 0, and if they exist, they are equal to 0 with the above-described univariate real polynomial f (x). An identity search step to find the distance to the polynomial,
Instead of the above polynomial output step,
The second polynomial output means obtains the smallest one of the values obtained in the first polynomial candidate determination step and the second polynomial candidate determination step and the distance obtained in the identity search step. The method according to claim 15, further comprising a second polynomial output step of outputting a polynomial corresponding to the minimum value.       An inter-polynomial distance calculation program for causing a computer to function as the inter-polynomial distance calculation device according to any one of claims 1 to 6.       A computer-readable recording medium on which the inter-polynomial distance calculation program according to claim 17 is recorded.       A univariate nearest real polynomial calculation program for causing a computer to function as the univariate nearest real polynomial calculation device according to any one of claims 9 to 14.       A computer-readable recording medium on which the one-variable recent real polynomial calculation program according to claim 19 is recorded.

Images