H. J. Stetter. Eigenproblems are at the heart of polynomial system solving. SIGSAM Bulletin, 30(4):2225, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....basis computations (see section 2.4.1 and its generalization in section 2.4.2) In section 3, we will describe another way to compute implicitly the multiplication maps, based on resultant matrix computations. Our matrix approach is based on the following fundamental theorem (see [5] 65] [84]) Theorem 2.6 Assume that Z K (I) f 1 ; d g. 1. The eigenvalues of the linear operator M a (resp. M a ) are fa( 1 ) a( d )g. 2. The common eigenvectors of (M a ) a2A are (up to a scalar) 1 1 ; 1 d . Notice that if (x ) 2E is a monomial basis of ....

H. J. Stetter, Eigenproblems are at the heart of polynomial system solving, SIGSAM Bulletin, 30 (1996), pp. 22-25.

....of structured matrices associated with multivariate polynomials as well as the advancements of the study of the dual space, Bezoutians and algebraic residues introduced brieAEy in the previous section. We will start with recalling some denitions and techniques used in [1] 8] 13] 25] 28] [40]. Then, in sections 3.8, 3.103.12, we will develop some new techniques to be used in section 4. 3.1 Polynomial ring The denitions of the previous section and appendix A can be immediately extended to the n variate case, for any natural n. In this case, R = C [x] is replaced by the ring C [x 1 ; ....

.... algorithms for computing the zeros in Z(I) The rst operator that comes naturally in this study is the operator of multiplication by an element of A, based on (10) For any element a 2 A, we dene the map b 7 a b: An important property of this operator is given in the next theorem (see [1] [40], 26] Theorem 3.2.1 The set of the eigenvalues of the linear operator M a is exactly fa(i 1 ) a(i d )g. Proof. Let p(x) i2Z(I) a(x) Gamma a(i) This polynomial vanishes on Z(I) so that (according to the Nullstellensatz, see [9] there exists d = d p 2 N such that p(x) 2 I ....

[Article contains additional citation context not shown here]

H. J. Stetter, Eigenproblems are at the heart of polynomial system solving, SIGSAM Bulletin, 30, 4 (1996), pp. 2225.

....2 A. M a : A A (6) q 7 q a and denote M a the matrix of M a in a fixed basis (m i ) of A. The transposed matrix M t a represents the transposed map from the dual b A to itself. Recall that b A is the set of linear forms from A to C . Our approach is based on the following property [12] 1] [14]: Lemma 1 The linear forms 1 i : p 7 p(i) where i is any solution of P, are eigenvectors of all matrices (M t a ) a2A . The corresponding eigenvalues are a(i) As an application, if one chooses a( h 1 , then the eigenvalues of M a will yield the d L solutions h 1 = m , 1 m d L . ....

H. J. STETTER, "Eigenproblems are at the Heart of Polynomial System Solving", SIGSAM Bulletin, vol. 30, no. 4, pp. 22--25, 1996.

....this approach, which we divide into two stages: 1. of dening appropriate reduction to the matrix eigenvector problem and 2. of devising its eoeective solution algorithm. Though the reduction of a polynomial system to the matrix eigenproblem is a well known and well studied topic [1] 16] [25], there are several variations of this reduction leading to dioeerent frameworks, which may facilitate or complicate the subsequent computation. Our contribution at this stage is a simple unifying approach, based on the study of the associated maps, operators and functionals, which enables more ....

....suOEciently fast to a specied root minimizing or maximizing the absolute value of a xed polynomial. 2 Reduction of the solution of a polynomial system to matrix eigenproblem In this section we formalize the reduction of the solution of a polynomial system to the matrix eigenproblem (cf. 1] [25], 5] 19] 22] We denote by R = C [x 1 ; xn ] the ring of polynomials in the variables x = x 1 ; xn ) with coeOEcients in the eld of complex numbers C . Many of our results are valid for any algebraically closed eld K . N will denote the set of nonnegative integers. Let f 1 ....

H. J. Stetter. Eigenproblems are at the Heart of Polynomial System Solving. SIGSAM Bulletin, 30(4):2225, 1996.

....became practical and should allow various further improvements, for instance, by using parallel processing. 2 Reduction of the solution of a polynomial system to matrix eigenproblem In this section we formalize the reduction of the solution of a polynomial system to matrix eigenproblem (cf. 1] [32], 9] 7] 25] 26] We denote by R = C [x1 ; xn ] the ring of polynomials in the variables x = x1 ; xn) with coeOEcients in the eld of complex numbers C . Many of our results are valid for any algebraically closed eld K . To motivate and illustrate the material of this ....

....of e i ) in this decomposition, is just the linear form which associate to a its value at the point i i . We will extend this approach to the case of multivariate polynomial systems, which of course will require substantial further elaboration and algebraic formalism. We refer to [24] 25] [32] . for further details. Let f1 ; fm be m polynomials of R, dening the polynomial system f1 (x) 0; fm (x) 0. Let I be the ideal generated by these polynomials. We consider the case, where the quotient A = R=I is of nite dimension D over C . This implies that the set of ....

Stetter, H. J. Eigenproblems are at the Heart of Polynomial System Solving. SIGSAM Bulletin 30, 4 (1996), 2225.

....this approach, which we divide into two stages: 1. of dening appropriate reduction to the matrix eigenvector problem and 2. of devising its eoeective solution algorithm. Though the reduction of a polynomial system to the matrix eigenproblem is a well known and well studied topic [1] 16] [25], there are several variations of this reduction requiring various computational cost and leading to dioeerent frameworks, which may facilitate or complicate the subsequent computation. In particular the reduction stage based on the computation of a Gr#bner basis has major deciencies already ....

....suOEciently fast to a specied root minimizing or maximizing the absolute value of a xed polynomial. 2 Reduction of the solution of a polynomial system to matrix eigenproblem In this section we formalize the reduction of the solution of a polynomial system to the matrix eigenproblem (cf. 1] [25], 5] 19] 22] We denote by R = C [x 1 ; xn ] the ring of polynomials in the variables x = x 1 ; xn ) with coeOEcients in the eld of complex numbers C . Many of our results are valid for any algebraically closed eld K . N will denote the set of nonnegative integers. Let f 1 ....

H. J. Stetter. Eigenproblems are at the Heart of Polynomial System Solving. ACM SIGSAM Bulletin, 30(4):2225, ACM Press, New York, 1996.

....and for any polynomial f 1 2 R, the matrix of B f1 ;f 2 : f n 1 in these bases is of the form M f1 0 0 N f1 ; 2) where M f1 is the matrix of multiplication by f 1 in the basis (a 1 ; a d ) of A. The next results on elimination theory rely on this fundamental proposition (see [AS88, Ste96, Mou97]) Proposition 3.10 The eigenvalues of M f1 are f 1 (i) for i 2 V(f 1 = 0; fn 1 = 0) This proposition can be applied for instance to the computation of so called the Chow Form, dened as follows: Y i2V(f2=0; fn=0) u 0 u 1 i 1 : un i n ) i where i is the multiplicity ....

H. J. Stetter. Eigenproblems are at the Heart of Polynomial System Solving. SIGSAM Bulletin, 30(4):22 25, 1996.

....L f1 ; 2) where M f1 is the matrix of multiplication by f 1 in the basis (a 1 ; a d ) of A. Let us describe rst a direct application of this proposition, related to the Chow Form and based on the fact that the eigenvalues of M f1 are f 1 (i) for i 2 Z(f 2 = 0; fn 1 = 0) see [AS88, Ste96, Mou98]) Let us recall that the Chow form of (f 2 ; fn 1 ) is Y i2Z(f2=0; fn=0) u 0 u 1 i 1 : un i n ) i where i is the multiplicity of i 2 Z(f 2 = 0; fn 1 = 0) see [HP52] It can be shown that it is also the determinant det(u 0 I d u 1 M x1 Delta Delta Delta ....

H. J. Stetter. Eigenproblems are at the heart of polynomial system solving. SIGSAM Bulletin, 30(4):22 25, 1996.

....; 2) where M f1 is the matrix of multiplication by f 1 in the basis (a 1 ; a d ) of A. Let us describe rst a direct application of this proposition, related to the Chow Form and based on the fact that the eigenvalues of M f1 are f 1 (i) for i 2 Z(f 2 = 0; fn 1 = 0) see [AS88, Ste96, Mou97] Let us recall that the Chow form of (f 2 ; fn 1 ) is Y i2Z(f2=0; fn=0) u 0 u 1 i 1 : un i n ) i where i is the multiplicity of i 2 Z(f 2 = 0; fn 1 = 0) see [HP52] It can be shown that it is also the determinant det(u 0 I d u 1 M x1 Delta Delta ....

H. J. Stetter. Eigenproblems are at the Heart of Polynomial System Solving. SIGSAM Bulletin, 30(4):22 25, 1996.

....the inverses of Toeplitz matrices (cf. 3] 2 Generalization to the multivariate case Our next goal, is the extension of the approach and the results of the previous section to the multivariate case. We will start with recalling some definitions and techniques used in [1] 8] 11] 21] 24] [32]. Then, in sections 2.8, 2.10 2.12, we will develop some new techniques to be used in section 3. 2.1 Polynomial ring The definitions of the previous section and appendix A can be immediately extended to the n variate case, for any natural n. In this case, R = C [x] is replaced by the ring C ....

.... computing the zeros in Z(I) The first operator that comes naturally in this study is the operator of multiplication by an element of A, based on (10) For any element a 2 A, we define the map M a : A A b 7 a b: An important property of this operator is given in the next theorem (see [1] [32], 22] Theorem 2.2.1 The set of the eigenvalues of the linear operator M a is exactly fa(i 1 ) a(i d )g. Proof. Let p(x) Q i2Z(I) a(x) Gamma a(i) This polynomial vanishes on Z(I) so that (according to the Nullstellensatz, see [9] there exists d = d p 2 N such that p(x) d ....

[Article contains additional citation context not shown here]

H. J. Stetter, Eigenproblems are at the heart of polynomial system solving, SIGSAM Bulletin, 30, 4, ACM Press, New York (1996), pp. 22--25.

....became practical and should allow various further improvements, for instance, by using parallel processing. 2 Reduction of the solution of a polynomial system to matrix eigenproblem In this section we formalize the reduction of the solution of a polynomial system to matrix eigenproblem (cf. 1] [29], 8] 21] 22] We denote by R = C [x1 ; xn ] the ring of polynomials in the variables x = x1 ; xn) with coeOEcients in the eld of complex numbers C . Many of our results are valid for any algebraically closed eld K . To motivate and illustrate the material of this section, ....

....e i in this decomposition) consists of the linear forms associating to a its values at the points i i . We will extend this approach to the case of multivariate polynomial systems, which of course will require substantial further elaboration and algebraic formalism. We refer to [20] 21] 22] [29] for further details. Let f1 ; fm be m polynomials of R, de ning the polynomial system f1 (x) 0; fm (x) 0. Let I be the ideal generated by these polynomials. We consider the case, where the quotient algegra A = R=I is of nite dimension D over C . This implies that the set of ....

Stetter, H. J. Eigenproblems are at the Heart of Polynomial System Solving. SIGSAM Bulletin 30, 4 (1996) ACM Press, New York, 2225.

....to the first half of the 19th century, when, motivated by applications to celestial mechanics, Cauchy discovered his celebrated Interlacing Theorem for the eigenvalues. Recently the eigenproblem turned out to be highly important also for the solution of polynomial systems of equations [AuSt88] [Ste96], MP98] BMP98] The subject has enormous bibliography (see e.g. GL96] and references therein) and causes never stopping stream of research articles in the leading journals on numerical analysis and applied linear algebra. The importance of this topic is highly recognized by numerical analysts ....

....on the solution of the important open problem of proving the bound KA (D) O(D h ) for h 3, for the matrices A of this class. Solution of this problem would have immediately given us fast solution of the polynomial system because its common roots are exactly the eigenvectors of A [AuSt88] [Ste96], BMP98] ....

H. J. Stetter, Eigenproblems Are at the Heart of Polynomial System Solving, ACM Bulletin, 30, 4, ACM Press, New York, 22-25, 1996.

....However the condition number of the eigenvector matrix is a fairly reliable indicator. 2 i.e. R y 1 R1 = I N 1 ThetaN 1 . Such R y 1 are easily calculated from most numerical decompositions of R1 . This multiplication matrix approach to numerical rootfinding is quite recent [17, 14, 4], although its roots go back a century. So far as I know, the observation that it continues to work when A 0 and U span more than the null space of R is new. This is numerically useful, as it allows eigensystem size to be traded against elimination stability. This approach can be used to find all ....

H. J. Stetter. Eigenproblems are at the heart of polynomial system solving. SIGSAM Bulletin, 30:22--5, 1996.

....spectacularly unstable for ill conditioned B 1 or ill chosen q. This is not immediately obvious from the recovered eigenvalues or eigenvectors, but the condition number of the eigenvector matrix is a reliable indicator. This multiplication matrix approach to numerical rootfinding is quite recent [16, 13, 4], although its roots go back 2 i.e. B y 1 B1 = I N 1 ThetaN 1 . Such B y 1 are easily calculated from most numerical decompositions of B 1 . a long way. So far as I know, the observation that it continues to work when A 0 and U span more than the null space of B is new. This is ....

H. J. Stetter. Eigenproblems are at the heart of polynomial system solving. SIGSAM Bulletin, 30:22--5, 1996.

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H. J. Stetter. Eigenproblems are at the heart of polynomial system solving. SIGSAM Bulletin, 30(4):2225, 1996.

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H. J. Stetter. Eigenproblems are at the heart of polynomial system solving. SIGSAM Bulletin, 30(4):22--25, 1996.

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H. J. Stetter. Eigenproblems are at the Heart of Polynomial System Solving. SIGSAM Bulletin, 30(4):2225, 1996.

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H. J. Stetter, Eigenproblems are at the heart of polynomial system solving, SIGSAM Bulletin, 30, 4 (1996), pp. 2225.

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H. J. Stetter. Eigenproblems are at the heart of polynomial system solving. SIGSAM Bulletin, 30(4):2225, 1996.

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