Showing papers on "Triangular matrix published in 2011"

Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization

Abstract: SuiteSparseQR is a sparse QR factorization package based on the multifrontal method Within each frontal matrix, LAPACK and the multithreaded BLAS enable the method to obtain high performance on multicore architectures Parallelism across different frontal matrices is handled with Intel's Threading Building Blocks library The symbolic analysis and ordering phase pre-eliminates singletons by permuting the input matrix A into the form [R11R12; 0 A22] where R11 is upper triangular with diagonal entries above a given tolerance Next, the fill-reducing ordering, column elimination tree, and frontal matrix structures are found without requiring the formation of the pattern of ATA Approximate rank-detection is performed within each frontal matrix using Heath's method While Heath's method is not always exact, it has the advantage of not requiring column pivoting and thus does not interfere with the fill-reducing ordering For sufficiently large problems, the resulting sparse QR factorization obtains a substantial fraction of the theoretical peak performance of a multicore computer

Some Theorems on Leibniz Algebras

Abstract: If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L and M is a finite-dimensional irreducible L-bimodule, then all U-bimodule composition factors of M are isomorphic. If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L, then the nilpotent residual of U is an ideal of L. Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements.

SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix

Abstract: We describe an efficient implementation of an algorithm for computing selected elements of a general sparse symmetric matrix A that can be decomposed as A = LDLT, where L is lower triangular and D is diagonal. Our implementation, which is called SelInv, is built on top of an efficient supernodal left-looking LDLT factorization of A. We discuss how computational efficiency can be gained by making use of a relative index array to handle indirect addressing. We report the performance of SelInv on a collection of sparse matrices of various sizes and nonzero structures. We also demonstrate how SelInv can be used in electronic structure calculations.

A Note on the Schur Multiplier of a Nilpotent Lie Algebra

Abstract: For a nilpotent Lie algebra L of dimension n and dim (L 2) = m ≥ 1, we find the upper bound , where M(L) denotes the Schur multiplier of L. In case m = 1, the equality holds if and only if L ≅ H(1) ⊕ A, where A is an abelian Lie algebra of dimension n − 3 and H(1) is the Heisenberg algebra of dimension 3.

Matrices That Generate the Same Krylov Residual Spaces

Abstract: Given an n by n nonsingular matrix A and an n-vector v, we consider the spaces of the form AK k (A, v), k = 1,… n, where K k (A, v) is the k th Krylov space, equal to span v, Av,…, A k −1 v. We characterize the set of matrices B that, with the given vector v, generate the same spaces; i.e., those matrices B for which BK k (B,v) = AK k (A,v), for all k = 1, …, n. It is shown that any such sequence of spaces can be generated by a unitary matrix. If zero is outside the field of values of A, then there is a Hermitian positive definite matrix that generates the same spaces, and, moreover, if A is close to Hermitian then there is a nearby Hermitian matrix that generates the same spaces. It is also shown that any such sequence of spaces can be generated by a matrix having any desired eigenvalues.

The stable monomorphism category of a Frobenius category

Abstract: For a Frobenius abelian category A, we show that the category Mon(A) of monomorphisms in A is a Frobenius exact category; the associated stable category Mon(A) modulo projective objects is called the stable monomorphism category of A. We show that a tilting object in the stable category A of A modulo projective objects induces naturally a tilting object in Mon(A). We show that if A is the category of (graded) modules over a (graded) self-injective algebra A, then the stable monomorphism category is triangle equivalent to the (graded) singularity category of the (graded) 2× 2 upper triangular matrix algebra T2(A). As an application, we give two characterizations to the stable category of Ringel-Schmidmeier.

Enumerating (2+2)-free posets by indistinguishable elements

Abstract: A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two elements in a poset are indistinguishable if they have the same strict up-set and the same strict down-set. Being indistinguishable defines an equivalence relation on the elements of the poset. We introduce the statistic maxindist, the maximum size of a set of indistinguishable elements. We show that, under a bijection of Bousquet-Melou et al., indistinguishable elements correspond to letters that belong to the same run in the so-called ascent sequence corresponding to the poset. We derive the generating function for the number of (2+2)-free posets with respect to both maxindist and the number of different strict down-sets of elements in the poset. Moreover, we show that (2+2)-free posets P with maxindist(P) at most k are in bijection with upper triangular matrices of nonnegative integers not exceeding k, where each row and each column contains a nonzero entry. (Here we consider isomorphic posets to be equal.) In particular, (2+2)-free posets P on n elements with maxindist(P)=1 correspond to upper triangular binary matrices where each row and column contains a nonzero entry, and whose entries sum to n. We derive a generating function counting such matrices, which confirms a conjecture of Jovovic, and we refine the generating function to count upper triangular matrices consisting of nonnegative integers not exceeding k and having a nonzero entry in each row and column. That refined generating function also enumerates (2+2)-free posets according to maxindist. Finally, we link our enumerative results to certain restricted permutations and matrices.

High performance matrix inversion based on LU factorization for multicore architectures

Abstract: The goal of this paper is to present an efficient implementation of an explicit matrix inversion of general square matrices on multicore computer architecture. The inversion procedure is split into four steps: 1) computing the LU factorization, 2) inverting the upper triangular U factor, 3) solving a linear system, whose solution yields inverse of the original matrix and 4) applying backward column pivoting on the inverted matrix. Using a tile data layout, which represents the matrix in the system memory with an optimized cache-aware format, the computation of the four steps is decomposed into computational tasks. A directed acyclic graph is generated on the fly which represents the program data flow. Its nodes represent tasks and edges the data dependencies between them. Previous implementations of matrix inversions, available in the state-of-the-art numerical libraries, are suffer from unnecessary synchronization points, which are non-existent in our implementation in order to fully exploit the parallelism of the underlying hardware. Our algorithmic approach allows to remove these bottlenecks and to execute the tasks with loose synchronization. A runtime environment system called QUARK is necessary to dynamically schedule our numerical kernels on the available processing units. The reported results from our LU-based matrix inversion implementation significantly outperform the state-of-the-art numerical libraries such as LAPACK (5x), MKL (5x) and ScaLAPACK (2.5x) on a contemporary AMD platform with four sockets and the total of 48 cores for a matrix of size 24000. A power consumption analysis shows that our high performance implementation is also energy efficient and substantially consumes less power than its competitors.

Joint spectral radius theory for automated complexity analysis of rewrite systems

Abstract: Matrix interpretations can be used to bound the derivational complexity of term rewrite systems. In particular, triangular matrix interpretations over the natural numbers are known to induce polynomial upper bounds on the derivational complexity of (compatible) rewrite systems. Recently two different improvements were proposed, based on the theory of weighted automata and linear algebra. In this paper we strengthen and unify these improvements by using joint spectral radius theory.

Reorthogonalized Block Classical Gram--Schmidt

Abstract: A new reorthogonalized block classical Gram--Schmidt algorithm is proposed that factorizes a full column rank matrix $A$ into $A=QR$ where $Q$ is left orthogonal (has orthonormal columns) and $R$ is upper triangular and nonsingular. With appropriate assumptions on the diagonal blocks of $R$, the algorithm, when implemented in floating point arithmetic with machine unit $\macheps$, produces $Q$ and $R$ such that $\| I- Q^{T} Q \|_2 =O(\macheps)$ and $\| A-QR \|_2 =O(\macheps \| A \|_2)$. The resulting bounds also improve a previous bound by Giraud et al. [Num. Math., 101(1):87-100,\ 2005] on the CGS2 algorithm originally developed by Abdelmalek [BIT, 11(4):354--367,\ 1971]. \medskip Keywords: Block matrices, Q--R factorization, Gram-Schmidt process, Condition numbers, Rounding error analysis.

The number of conjugacy classes in pattern groups is not a polynomial function

Abstract: A famous open problem due to Graham Higman asks if the number of conjugacy classes in the group of n x n unipotent upper triangular matrices over the q-element field can be expressed as a polynomial function of q for every fixed n. We consider the generalization of the problem for pattern groups and prove that for some pattern groups of nilpotency class two the number of conjugacy classes is not a polynomial function of q.

A new practical approach to GNSS high-dimensional ambiguity decorrelation

Abstract: Based on both the lower and the upper triangular Cholesky decomposition algorithms, the (inverse) lower triangular Cholesky integer transformation and the (inverse) upper triangular Cholesky integer transformation are defined, and the (inverse) paired Cholesky integer transformation is proposed. Then, for the case of high-correlation ambiguity, a multi-time (inverse) paired Cholesky integer transformation is given. In addition, a simple and practical criterion is presented to solve the uniqueness problem of the integer transformation. It is verified by an example that (1) the (inverse) paired Cholesky integer transformation is very convenient and very efficient in practical computation; (2) the (inverse) paired Cholesky integer transformation is better than both the (inverse) lower triangular Cholesky integer transformation and the (inverse) upper triangular Cholesky integer transformation; and that (3) the inverse paired Cholesky integer transformation outperforms the paired Cholesky integer transformation slightly in the most cases.

Almost settling the hardness of noncommutative determinant

Abstract: In this paper, we study the complexity of computing the determinant of a matrix over a noncommutative algebra. In particular, we ask the question: "Over which algebras is the determinant easier to compute than the permanent?" Towards resolving this question, we show the following results for noncommutative determinant computation: [Hardness] Computing the determinant of an n x n matrix whose entries are themselves 2 x 2 matrices over any field of zero or odd characteristic is as hard as computing the permanent over the field. This extends the recent result of Arvind and Srinivasan, which required the entries to be matrices of dimension linear in n. [Easiness] The determinant of an n x n matrix whose entries are themselves d x d upper triangular matrices can be computed in poly(nd) time. Combining the above with the decomposition theorem for finite dimensional algebras (and in particular exploiting the simple structure of 2 x 2 matrix algebras), we can extend the above hardness and easiness statements to more general algebras as follows. Let A be a finite dimensional algebra over a finite field of odd characteristic with radical R(A). [Hardness] If the quotient A/R(A) is noncommutative, then computing the determinant over the algebra A is as hard as computing the permanent. [Easiness] If the quotient A/R(A) is commutative, and furthermore R(A) has nilpotency index d (i.e., d is the smallest integer such that R(A)d =0), then there exists a poly(nd)-time algorithm that computes determinants over the algebra A. In particular, for any constant dimensional algebra A over a finite field of odd characteristic, since the nilpotency index of R(A) is at most a constant, we have the following dichotomy theorem: if A/R(A) is commutative then efficient determinant computation is possible, and otherwise determinant is as hard as permanent.

Partitioned Triangular Tridiagonalization

Abstract: We present a partitioned algorithm for reducing a symmetric matrix to a tridiagonal form, with partial pivoting. That is, the algorithm computes a factorization PAPT = LTLT, where, P is a permutation matrix, L is lower triangular with a unit diagonal and entries’ magnitudes bounded by 1, and T is symmetric and tridiagonal. The algorithm is based on the basic (nonpartitioned) methods of Parlett and Reid and of Aasen. We show that our factorization algorithm is componentwise backward stable (provided that the growth factor is not too large), with a similar behavior to that of Aasen’s basic algorithm. Our implementation also computes the QR factorization of T and solves linear systems of equations using the computed factorization. The partitioning allows our algorithm to exploit modern computer architectures (in particular, cache memories and high-performance blas libraries). Experimental results demonstrate that our algorithms achieve approximately the same level of performance as the partitioned Bunch-Kaufman factor and solve routines in lapack.

Lie Triple Derivations on Triangular Matrices

Abstract: Let be the algebra of all n × n upper triangular matrices over a commutative unital ring , and let be a 2-torsion free unital -bimodule. We show that every Lie triple derivation is a sum of a standard Lie derivation and an antiderivation.

The Development and Use of Methods of Lr Type

Abstract: Given a non-singular matrix, factorize it into the product of two matrices, and then form a new matrix by multiplying the two factors together in reverse order. Repeat the process and so produce a sequence of matrices, all similar to the given matrix. This is the essence of ani LR method. For several factorization techniques the sequence converges, quite generally, to a triangular matrix from which the eigenvalues may be read off. The LR transformations discussed here preserve the form of a Hessenberg matrix (ai, = 0 if i > j+ 1) and for such matrices the work involved in the transformation from one N X N nmatrix to the next is proportional to N2 as against N3 for full nmatrices. Using this and other devices the latest versions of the transformation are among the best available methods for finding the eigenvalues of general matrices. The evolution of these algorithms can be traced back to classical function theory, each method being numerically more feasible than its predecessor.

A note on graded gelfand–kirillov dimension of graded algebras

Abstract: In this paper, we consider associative P.I. algebras over a field F of characteristic 0, graded by a finite group G. More precisely, we define the G-graded Gelfand–Kirillov dimension of a G-graded P.I. algebra. We find a basis of the relatively free graded algebras of the upper triangular matrices UTn(F) and UTn(E), with entries in F and in the infinite-dimensional Grassmann algebra, respectively. As a consequence, we compute their graded Gelfand–Kirillov dimension with respect to the natural gradings defined over these algebras. We obtain similar results for the upper triangular matrix algebra UTa, b(E) = UTa+b(E)∩Ma, b(E) with respect to its natural ℤa+b × ℤ2-grading. Finally, we compute the ℤn-graded Gelfand–Kirillov dimension of Mn(F) in some particular cases and with different methods.

Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules

Abstract: A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and RMS is an S-R-bimodule. In the main theorem of this paper we show that if TS is a tilting S-module, then under certain homological conditions on the S-module MS, one can extend TS to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and TS, and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and MS is finitely generated. In this case, (S′,R,M′) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, MS has a finite projective resolution and ExtSn(MS, S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, HomS(M, S)).

A supercharacter table decomposition via power-sum symmetric functions

Abstract: We give an $AB$-factorization of the supercharacter table of the group of $n\times n$ unipotent upper triangular matrices over $\FF_q$, where $A$ is a lower-triangular matrix with entries in $\ZZ[q]$ and $B$ is a unipotent upper-triangular matrix with entries in $\ZZ[q^{-1}]$. To this end we introduce a $q$ deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommutative variables. The factorization is obtain from the transition matrices between the supercharacter basis, the $q$-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group $S_n$ given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular we compute the determinant of the supercharacter table.