W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theor. Comp. Sci., 36:309--317, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are a matrix inversion, a matrix multiplication, calculation of a determinant and computation of the coefficients of a characteristic polynomial. Each takes O(MM(t) time, except from the last step which takes O(MM (t) log t) time, for arbitrary matrices, due to an algorithm by Keller Gehrig [27]. To establish the bit complexity bound, the transformation of Theorem 13 is used. The bit size of the coefficients of the ffl polynomial representing the determinant is O(ts) since the original matrix entries have size s and its order is t. Hence modular arithmetic may be used over k = O(ts) ....

W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theor. Comp. Sci., 36:309--317, 1985.

....discussions on this subject and survey of current solutions. We denote by M(n) the number of operations in F required for n n matrix multiplications. The characteristic polynomial of a general matrix A can be computed at cost of O(n 3 ) or O(M(n) log n) operations by the method of Keller Gehrig [10]. The Frobenius normal form can be computed in O(n 3 ) as achieved by Storjohann [19, 20] and Storjohann and the author [21] while the randomized Las Vegas algorithms of Giesbrecht [7] and of Eberly [3, x4.3] give the best known asymptotic complexity O(M(n) log n) The problem we address is to ....

W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theoretical Computer Science, 36:309-317, 1985.

.... along the first coordinates by a factor of 2 and the asymptotic worst case bit complexity of Sidedness by a factor of O(log d) The O(log d) factor follows from a bound of MM(d)O(log d) on the complexity of computing all coefficients of the characteristic polynomial of an arbitrary square matrix [KG85] However, this algorithm is nontrivial and we have, instead, implemented a simpler O(d 3 ) method. In fact, each Sidedness test starts with computing det d 1 . If this is nonzero then there is no need for the perturbed primitive, which is more expensive. The computation of this determinant ....

W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theor. Comp. Sci., 36:309--317, 1985.

....(ffl) and Delta d (ffl) obtained from perturbation (2) has complexity O(MM(d) under the algebraic model. Proof Both predicates perform certain matrix manipulation operations, each taking O(MM(d) steps, including the characteristic polynomial computation, due to an algorithm by Keller Gehrig [KG]. 2 Proposition 5.4 Perturbation (2) is valid and does not change the running time complexity of the Orientation and Transversality predicates under the algebraic or the bit model. These results hold regardless of whether long or modular arithmetic is used. Proof At branching nodes, the sign of ....

Keller-Gehrig W., Fast Algorithms for the Characteristic Polynomial, Theor. Comp. Sci., Vol. 36, pp. 309-317, 1985.

.... ) wjA 2 i 1 wj : jA 2 i 1 1 w i ; it is easily seen that the vectors Aw;A 2 w; A n w can be computed from the above powers of A and from a given vector w using O(MM(n) operations, if MM(n) 2 n 2 ) for 0 see Borodin and Munro [5] page 128, or Keller Gehrig [16]. Furthermore, if 0 h log n and w 1 ; w 2 ; w 2 h 2 F n 1 , then it is also possible to compute the matrix vector products A j w i for 0 j dn=2 h e and 1 i 2 h at this cost. Clearly, since (A T ) 2 i ) A (2 i ) T , one could also compute ....

W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theoretical Computer Science, 36:309-317, 1985.

....The sequential arithmetic complexity of this problem over any field F (to be denoted by CP (n) is closely related to the complexity, M(n) of n Theta n matrix multiplication, M(n) Omega Gamma n 2 ) M(n) O(n ) 2:376. Namely, we have quite tight bounds, CP (n) O(M(n) log n) [K G85], CP (n) Omega Gamma M(n) BP94] exercise 2.6, p. 213) Furthermore, for 1 generic matrices A we have CP (n) O(M(n) K G85] To state the parallel complexity estimates, we will assume the customary arithmetic EREW PRAM model of parallel computing [KR90] under which every processor is ....

....M(n) of n Theta n matrix multiplication, M(n) Omega Gamma n 2 ) M(n) O(n ) 2:376. Namely, we have quite tight bounds, CP (n) O(M(n) log n) K G85] CP (n) Omega Gamma M(n) BP94] exercise 2.6, p. 213) Furthermore, for 1 generic matrices A we have CP (n) O(M(n) [K G85]. To state the parallel complexity estimates, we will assume the customary arithmetic EREW PRAM model of parallel computing [KR90] under which every processor is capable of performing in unit time a single operation in F. We will write O(t; p) to denote the simultaneous bounds O(t) on time and ....

W. Keller-Gehrig, Fast Algorithms for Characteristic Polynomial, Theoretical Computer Science, 36, 309-317, 1985.

....Problems like matrix inversion, computation of the determinant or of all coefficients of the characteristic polynomial, LR decomposition and over the complex numbers also QR decomposition and unitary transformation to Hessenberg form are all known to be as hard as matrix multiplication. See [3, 4, 6, 7, 11, 12, 15]. In this paper we study some computational problems in linear algebra that are not specified by a function but by a relation. Let F denote a field of characteristic zero that may be endowed with an ordering . The reader may keep in mind the two important examples F = C or F = R. A problem is ....

W. Keller--Gehrig, Fast algorithms for the characteristic polynomial, Theor. Computer Science 36(1985), 309--317.

....9 that an unsymmetric modification of Lanczos algorithm reduces it to the tridiagonal form. Generic n Theta n matrices cover all n Theta n matrices but ones forming an algebraic variety of a lower dimension. Furthermore, by combining and extending some known techniques, Keller Gehrig in [K G85] devised a solution algorithm that reduced any n Theta n input matrix A to the triangular Frobenius form. All these algorithms are performed at nearly optimal arithmetic cost of O(M(n) log n) ops, whereas Omega Gamma M(n) is a lower bound on the arithmetic complexity of the computation of c A ....

....are performed very effectively on modern supercomputers [GL96] Q94] Theorem 3.1 Given an n Theta n matrix A, it suffices to use O(M(n) log n) ops in order to compute a nonsingular matrix W , its inverse W Gamma1 , and the matrix WAW Gamma1 in the triangular Frobenius form. Proof. See [K G85]. 2 Keller Gehrig s algorithm of [K G85] supporting Theorem 3.1, involves several steps of modified block Gaussian elimination, which makes it generally ineffective for sparse and or structured matrices A. We also recall the following results. Theorem 3.2 (see [GL96] p.314) For a pair of n ....

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W. Keller-Gehrig, Fast Algorithms for Characteristic Polynomial, Theoretical Computer Science, 36, 309-317, 1985.

....are a matrix inversion, a matrix multiplication, calculation of a determinant and computation of the coefficients of a characteristic polynomial. Each takes O(MM(t) time, except from the last step which takes O(MM(t) log t) time, for arbitrary matrices, due to an algorithm by Keller Gehrig [24]. To establish the bit complexity bound, the transformation of Theorem 13 is used. The bit size of the coefficients of the ffl polynomial representing the determinant is O(ts) since the original matrix entries have size s and its order is t. Hence modular arithmetic may be used over k = O(ts) ....

W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theor. Comp. Sci., 36:309--317, 1985.

.... of M(A;B; We may write a linear system which solutions gives the entries of LH (x) This linear system is computed by first isolating the lexicographically first independent columns of M(A;B; Since the matrix is block Krylov, we can restrict ourselves to Theta 2 matrices as done in [25]. The cost is that of O(log( solutions of L( The fact that nd terminates the proof. 2 4.2 Modules over a P.I.D These results can be expressed using the language of modules. Well known from a theoretical point of view [19] this approach has not been often use to derive practical algorithms ....

....[31,42] given A, this diagram allows to develop algorithms to compute A 0 if T (x) and LH (x) are known. Conversely, given T (x) it is possible to compute its Hermite form if A and A 0 are known. Results for the computation of a shift Hessenberg form of a matrix, such that the algorithm in [25], could thus be used for another proof of proposition 4.1. This would lead to the same complexity bounds. 5 Cost analysis By using the results of previous sections we establish new cost bounds for the computation of the Hermite normal form of a matrix polynomial of dimension n and of degree d. ....

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W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theoretical Computer Science, 36:309--317, 1985.

....Toeplitz perturbation B 2 R n Thetan of A. This is done for sparse matrices using the linear recurrence methods of Wiedemann (1986) and Kaltofen Saunders (1991) modulo sufficiently many randomly chosen primes. An algorithm is also presented for dense matrices along the lines of the method of Keller Gehrig (1985) . Care is taken to avoid bad primes, modulo which the problem changes locally. Finally, in Section 4 we tie the techniques together into an algorithm for the Smith form. The cost of the sparse algorithm will be measured in terms of 4 M. Giesbrecht the number of modular matrix vector products ....

W. Keller-Gehrig, Fast algorithms for the characteristic polynomial. Theor. Computer Science 36 (1985), 309--317.

....are a matrix inversion, a matrix multiplication, calculation of a determinant and computation of the coefficients of a characteristic polynomial. Each takes O(MM(t) time, except from the last step which takes O(MM(t) log t) time, for arbitrary matrices, due to an algorithm by Keller Gehrig [KG85] To establish the bit complexity bound, the transformation of theorem 6.4.3 is used. The bit size of the coefficients of the ffl polynomial representing the determinant is O(ts) since the original matrix entries have size s and its order is t. Hence modular arithmetic may be used over k = O(ts) ....

W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theor. Comp. Sci., 36:309--317, 1985.

....coefficients of P (z) and Q(z) not in mere approximations for them. If the matrix A is dense we can begin by computing the characteristic polynomial of A. This can be done using O(n 3 ) arithmetical operations using a classical method attributed to Danilevski in [Faddeev and Faddeeva 1963] Keller Gehrig [1985] has shown that it can also be done using only O(M(n) arithmetical operations, where M(n) is the number of arithmetical operations required for multiplying two n Thetan matrices. The best upper bound on M(n) is currently O(n 2:376 ) Coppersmith and Winograd 1990] We are especially ....

....complexity of computing Av is O(jEj) O(dn) so the overall complexity of this stage is O(jEj jV j) O(dn 2 ) This naive approach should be used when d is relatively small compared to n. If d is of the order of n then the complexity of this stage will be O(n 3 ) This can be reduced (see [Keller Gehrig 1985], for example) to O(M(n) log n) using fast matrix multiplications. Although stage 1 seems to be more straightforward than the following stage 2, it turns out to be the dominant stage in terms of the computational complexity of the problem. The problem we have to solve in stage 2 could be solved ....

W. Keller-Gehrig, "Fast algorithms for the characteristic polynomial", Theoret. Comput. Sci. 36 (1985), 309--317.

.... the perturbation is the linearity in ffl in de nition (1) This leads to expression (4) which demonstrates that there is no explicit symbolic computation in terms of ffl, since the computation of the coeOEcients of the characteristic polynomial given the matrix entries is an algebraic operation [KG85] This essentially allows for an eOEcient computation of det d 1 (ffl) Theorem 4.1 [ECS97, Th. 15, 17] Perturbation (1) is valid with respect to Ordering and Sidedness. Suppose that there exist n distinct input parameters and that determinant sign determination is implemented by determinant ....

.... along the rst coordinate by a factor of 2 and the asymptotic worst case bit complexity of Sidedness by a factor of O(log d) The O(log d) factor follows from a bound of MM(d)O(log d) on the complexity of computing all coeOEcients of the characteristic polynomial of an arbitrary square matrix [KG85] However, this algorithm is nontrivial and we have, instead, implemented a simpler O(d 3 ) method. In fact, each Sidedness primitive starts with computing det d 1 . If this is nonzero then there is no need for the perturbed primitive, which is more expensive. The computation of this ....

W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theor. Comp. Sci., 36:309317, 1985.

....are a matrix inversion, a matrix multiplication, calculation of a determinant and computation of the coefficients of a characteristic polynomial. Each takes O(MM(t) time, except from the last step which takes O(MM(t) log t) time, for arbitrary matrices, due to an algorithm by Keller Gehrig [30]. To establish the bit complexity bound, the transformation of Theorem 13 is used. The bit size of the coefficients of the ffl polynomial representing the determinant is O(ts) since the original matrix entries have size s and its order is t. Hence modular arithmetic may be used over k = O(ts) ....

W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theor. Comp. Sci., 36:309--317, 1985.

.... applications a la fois dans le cas o u l anneau des coefficients est Z [12] et dans le cas o u cet anneau est un anneau de polynomes a une variable [3, 6] D un point de vue algorithmique, on sait calculer la forme en temps s equentiel polynomial sur les entiers [10] et sur les polynomes [9, 7, 13]. En parall ele, trouver un algorithme rapide est toujours une question ouverte dans le cas entier, question qui repose sur la parall elisation du pgcd d entiers [4] Dans le cas polynomial, on ne connaissait jusqu alors que des algorithmes rapides mais probabilistes [7, 8, 5] le probl eme ....

.... B (3) 2 ( B (oe) 2 ( 0 0 C s3 (x) B (oe) 3 ( 0 0 : C s oe (x) 3 7 7 7 7 7 7 5 ; where C s i (x) is the companion matrix associated with s i (x) an example is given at appendix B) By construction, C is in polycyclic form (for this form, see [13,14]) i.e each matrix B (l) j is zero except its last column. From such a form, a transformation toward the Frobenius form could be computed [13,7] however, this is not currently our purpose since the form itself is already obtained. It remains to get convinced that the computations can be done in ....

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W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theoretical Computer Science, 36:309--317, 1985.

....Factors of T on V . Assume now that we are given vectors w 1 ; wn 2 V for V such that w 1 ; w k generate a modular cyclic basis for V , and let V i ; T i be as in Fact 1.3, for 0 i k. How do we find k and the invariant factors f 1 ; f k We adapt an algorithm of Keller Gehrig (1985) to accomplish this with O(MM(n) log n) operations in K. Fact 2.1 (Keller Gehrig 1985) Let T 2 K n Thetan and u 1 ; un 2 V . Matrices H i 2 K n Thetad i for 1 i n with the following properties can be computed with O(MM(n) log n) operations in K: i) P 1in d i = n, where d i 2 N ....

....for V such that w 1 ; w k generate a modular cyclic basis for V , and let V i ; T i be as in Fact 1.3, for 0 i k. How do we find k and the invariant factors f 1 ; f k We adapt an algorithm of Keller Gehrig (1985) to accomplish this with O(MM(n) log n) operations in K. Fact 2. 1 (Keller Gehrig 1985). Let T 2 K n Thetan and u 1 ; un 2 V . Matrices H i 2 K n Thetad i for 1 i n with the following properties can be computed with O(MM(n) log n) operations in K: i) P 1in d i = n, where d i 2 N may equal 0 for notational convenience; ii) for 1 i n, H i = u i jT u i j Delta ....

[Article contains additional citation context not shown here]

W. Keller-Gehrig, Fast algorithms for the characteristic polynomial, Theor. Computer Science, 36 (1985), pp. 309--317.

....d Vandermonde matrix takes at most O(d 2 ) arithmetic steps [23] while computing M or N as a matrix product takes O(MM(d) operations. Computing det(M Gamma ffl I d 1 ) or det(N Gamma ffl I d ) is a characteristic polynomial computation for which there exists an algorithm by Keller Gehrig [16] requiring O(MM (d) log d) operations. This algorithm is purely numeric as it transforms matrix M or N respectively to a new matrix that contains the coefficients of the characteristic polynomial in the last column. A brief discussion of modular arithmetic is in order here because, besides being ....

W. Keller-Gehrig, Fast algorithms for the characteristic polynomial, Theor. Comp. Sci., 36 (1985), pp. 309--317.

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K-G W. Keller-Gehrig, Fast Algorithms for Characteristic Polynomial, Theoretical Computer Science, 36, 309-317, 1985.

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W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theoretical Computer Science, 36:309-- 317, 1985. The Computer Journal, Vol. 36, No. 5, 496 Dinesh Manocha

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