DOCSLIB.ORG

On Positive Semidefinite Modification Schemes for Incomplete Cholesky Factorization

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Positive Semidefinite Modification Schemes for Incomplete Cholesky Factorization Preprint RAL-P-2013-005 On positive semidefinite modification schemes for incomplete Cholesky factorization J Scott, M Tuma April 2013 Submitted for publication in SIAM Journal on Scientific Computing RAL Library STFC Rutherford Appleton Laboratory R61 Harwell Oxford Didcot OX11 0QX Tel: +44(0)1235 445384 Fax: +44(0)1235 446403 email: [email protected] Science and Technology Facilities Council preprints are available online at: http://epubs.stfc.ac.uk ISSN 1361- 4762 Neither the Council nor the Laboratory accept any responsibility for loss or damage arising from the use of information contained in any of their reports or in any communication about their tests or investigations. On positive semidefinite modification schemes for incomplete Cholesky factorization Jennifer Scott1 and Miroslav T˚uma2 ABSTRACT Incomplete Cholesky factorizations have long been important as preconditioners for use in solving large- scale symmetric positive-definite linear systems. In this paper, we present a brief historical overview of the work that has been done over the last fifty years, highlighting key discoveries and rediscoveries. We focus in particular on the relationship between two important positive semidefinite modification schemes, namely that of Jennings and Malik and that of Tismenetsky. We present a novel view of their relationship and implement them in combination with a limited memory approach. We explore their effectiveness using extensive numerical experiments involving a large set of test problems arising from a wide range of practical applications. The experiments are used to isolate the effects of semidefinite modifications to enable their usefulness in the development of robust algebraic incomplete factorization preconditioners to be assessed. We show that we are able to compute sparse incomplete factors that provide robust, general-purpose preconditioners. Keywords: sparse matrices, sparse linear systems, positive-definite symmetric systems, iterative solvers, preconditioning, incomplete Cholesky factorization. AMS(MOS) subject classifications: 65F05, 65F50 1 Computational Science and Engineering Department, Rutherford Appleton Laboratory, Harwell Oxford, Oxfordshire, OX11 0QX, UK. Correspondence to: [email protected] Supported by EPSRC grant EP/I013067/1. 2 Institute of Computer Science, Academy of Sciences of the Czech Republic. Partially supported by the Grant Agency of the Czech Republic Project No. P201/13-06684 S. Travel support from the Academy of Sciences of the Czech Republic is also acknowledged. April 18, 2013 1 Introduction Iterative methods are widely used for the solution of large sparse symmetric linear systems of equations Ax = b. To increase their robustness, the system matrix A generally needs to be transformed by preconditioning. For positive-definite systems, an important class of preconditioners is represented by incomplete Cholesky (IC) factorizations, that is, factorizations of the form LLT in which some of the fill entries (entries that were zero in A) that would occur in a complete factorization are ignored. Over the last fifty years, many different algorithms for computing incomplete factorizations have been proposed and they have been used to solve problems from a wide range of application areas. In Section 2, we provide a brief historical overview and highlight important developments and significant contributions in the field. Some crucial ideas that relate to modifications and dropping are explained in greater detail in Section 3. Here we consider two important semidefinite modification schemes that were introduced to avoid factorization breakdown (that is, zero or negative pivots): that of Jennings and Malik [61, 62] from the mid 1970s and that of Tismenetsky [109] from the early 1990s. Variations of the Jennings-Malik scheme were adopted by engineering communities and it is recommended in, for example, [92] and used in experiments in [11] to solve hard problems, including the analysis of structures and shape optimization. The Tismenetsky approach has also been used to provide a robust preconditioner for some real-world problems, see, for example, [4, 9, 77, 78]. However, in his authoritative survey paper [10], Benzi remarks that Tismenetsky's idea \has unfortunately attracted surprisingly little attention". Benzi also highlights a serious drawback of the scheme which is that its memory requirements can be prohibitively large (in some cases, more than 70 per cent of the storage required for a complete Cholesky factorization is needed, see also [12]). We seek to gain a better understanding of the Jennings-Malik and Tismenetsky semidefinite modifications schemes and to explore the relationship between them. Our main contribution in Section 3 is new theoretical results that compare the 2-norms of the modifications to A that each approach makes. Then, in Section 4, we propose a memory-efficient variant of the Tismenetsky approach, optionally combined with the use of drop tolerances and the Jennings-Malik modifications to reduce factorization breakdowns. In Section 5, we report on extensive numerical experiments in which we aim to isolate the effects of the modifications so as to assess their usefulness in the development of robust algebraic incomplete factorization preconditioners. Finally, we draw some conclusions in Section 6. 2 Historical overview The rapid development of computational tools in the second half of the 20th century significantly pushed forward progress in solving systems of linear algebraic equations. Both software and hardware tools helped to solve still larger and more challenging problems from a wide range of engineering and scientific applications. This further motivated research into the development of improved algorithms and better understanding of their theoretical properties and practical limitations. In particular, within a very short period, important seminal papers were published by Hestenes and Stiefel [51] and Lanczos [74], on which most contemporary iterative solvers are based. The work of Householder and others in the 1950's and early 1960's on factorization approaches to matrix computations substantially changed the field (see [53]). Around the same time (and not long after Turing coined the term preconditioning in [111]), another set of important contributions led to the rapid development of sparse direct methods [93] and preconditioning of iterative methods by incomplete factorizations. In this paper, we consider preconditioning techniques based on incomplete factorizations that belong to the group of algebraically constructed techniques. To better understand contemporary forms of these methods and the challenges they face, it is instructive to summarize key milestones in their development. Here we follow basic algorithms without mentioning additional enhancements, such as block strategies, algorithms developed for parallel processing or for use within domain factorization or a multilevel framework. We also omit the pivoting techniques based on computing some auxiliary criteria for the factorization (like minimum discard strategy), mentioning them only when needed for modifying dropping 1 strategies. We concentrate on incomplete factorizations of symmetric positive-definite systems. Since some of the related ideas have been developed in the more general framework of incomplete LU factorizations for preconditioning non symmetric systems, we will refer to them as appropriate. 2.1 Early days motivated by partial differential equations The development of incomplete factorizations closely accompanies progress in the derivation of new computational schemes, especially for partial differential equations (PDEs) that still often motivate simple theoretical ideas as well as attempts to achieve fast and robust implementations, even when the incomplete factorizations are computed completely algebraically. Whilst there is a general acceptance that the solution of some discretized PDEs requires the use of physics-based or functional space-based preconditioners, the need for reliable incomplete factorizations is widespread. They can be used not only as stand-alone procedures but they are also useful in solving augmented systems, smoothing intergrid transformations and solving coarse grid systems. The earliest incomplete factorizations were described independently by several researchers. In the late 1950s, in a series of papers [20, 21] (see also [22]), Buleev proposed a novel way to solve systems of linear equations arising from two- and three-dimensional elliptic equations that had been discretized by the method of finite differences using five- and seven-point stencils, respectively. Buleev's approach was to extend the sweep method for solving the difference equations from the one-dimensional case. To do this efficiently, he modified the system matrix so that it could be expressed in the form of a scaled product of two triangular matrices. In contemporary terminology, these two matrices represent an incomplete factorization. A slightly different and more algebraic approach to deriving an incomplete factorization can be found in the work of Oliphant [89] in the early 1960s. Starting from a PDE model of steady-state diffusion, he obtained the incomplete factorization of a matrix A corresponding to the five-point stencil discretization by neglecting the fill-in during the factorization that occurs in sub-diagonal positions. An independent algebraic derivation of the methods of Buleev and Oliphant and their interpretation as instances of incomplete factorizations was given by Varga [115]. Varga also discussed
Load more

Recommended publications
  • Conjugate Gradient Method (Part 4)  Pre-Conditioning  Nonlinear Conjugate Gradient Method
    18-660: Numerical Methods for Engineering Design and Optimization Xin Li Department of ECE Carnegie Mellon University Pittsburgh, PA 15213 Slide 1 Overview Conjugate Gradient Method (Part 4) Pre-conditioning Nonlinear conjugate gradient method Slide 2 Conjugate Gradient Method Step 1: start from an initial guess X(0), and set k = 0 Step 2: calculate ( ) ( ) ( ) D 0 = R 0 = B − AX 0 Step 3: update solution D(k )T R(k ) X (k +1) = X (k ) + µ (k )D(k ) where µ (k ) = D(k )T AD(k ) Step 4: calculate residual ( + ) ( ) ( ) ( ) R k 1 = R k − µ k AD k Step 5: determine search direction R(k +1)T R(k +1) (k +1) = (k +1) + β (k ) β = D R k +1,k D where k +1,k (k )T (k ) D R Step 6: set k = k + 1 and go to Step 3 Slide 3 Convergence Rate k ( + ) κ(A) −1 ( ) X k 1 − X ≤ ⋅ X 0 − X κ(A) +1 Conjugate gradient method has slow convergence if κ(A) is large I.e., AX = B is ill-conditioned In this case, we want to improve convergence rate by pre- conditioning Slide 4 Pre-Conditioning Key idea Convert AX = B to another equivalent equation ÃX̃ = B̃ Solve ÃX̃ = B̃ by conjugate gradient method Important constraints to construct ÃX̃ = B̃ à is symmetric and positive definite – so that we can solve it by conjugate gradient method à has a small condition number – so that we can achieve fast convergence Slide 5 Pre-Conditioning AX = B −1 −1 L A⋅ X = L B L−1 AL−T ⋅ LT X = L−1B A X B ̃ ̃ ̃ L−1AL−T is symmetric and positive definite, if A is symmetric and positive definite T (L−1 AL−T ) = L−1 AL−T T X T L−1 AL−T X = (L−T X ) ⋅ A⋅(L−T X )> 0 Slide
    [Show full text]
  • Hybrid Algorithms for Efficient Cholesky Decomposition And
    Hybrid Algorithms for Efficient Cholesky Decomposition and Matrix Inverse using Multicore CPUs with GPU Accelerators Gary Macindoe A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of UCL. 2013 ii I, Gary Macindoe, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Signature : Abstract The use of linear algebra routines is fundamental to many areas of computational science, yet their implementation in software still forms the main computational bottleneck in many widely used algorithms. In machine learning and computational statistics, for example, the use of Gaussian distributions is ubiquitous, and routines for calculating the Cholesky decomposition, matrix inverse and matrix determinant must often be called many thousands of times for com- mon algorithms, such as Markov chain Monte Carlo. These linear algebra routines consume most of the total computational time of a wide range of statistical methods, and any improve- ments in this area will therefore greatly increase the overall efficiency of algorithms used in many scientific application areas. The importance of linear algebra algorithms is clear from the substantial effort that has been invested over the last 25 years in producing low-level software libraries such as LAPACK, which generally optimise these linear algebra routines by breaking up a large problem into smaller problems that may be computed independently. The performance of such libraries is however strongly dependent on the specific hardware available. LAPACK was originally de- veloped for single core processors with a memory hierarchy, whereas modern day computers often consist of mixed architectures, with large numbers of parallel cores and graphics process- ing units (GPU) being used alongside traditional CPUs.
    [Show full text]
  • Matrix Inversion Using Cholesky Decomposition
    Matrix Inversion Using Cholesky Decomposition Aravindh Krishnamoorthy, Deepak Menon ST-Ericsson India Private Limited, Bangalore [email protected], [email protected] Abstract—In this paper we present a method for matrix inversion Upper triangular elements, i.e. : based on Cholesky decomposition with reduced number of operations by avoiding computation of intermediate results; further, we use fixed point simulations to compare the numerical ( ∑ ) accuracy of the method. … (4) Keywords-matrix, inversion, Cholesky, LDL. Note that since older values of aii aren’t required for computing newer elements, they may be overwritten by the I. INTRODUCTION value of rii, hence, the algorithm may be performed in-place Matrix inversion techniques based on Cholesky using the same memory for matrices A and R. decomposition and the related LDL decomposition are efficient Cholesky decomposition is of order and requires techniques widely used for inversion of positive- operations. Matrix inversion based on Cholesky definite/symmetric matrices across multiple fields. decomposition is numerically stable for well conditioned Existing matrix inversion algorithms based on Cholesky matrices. decomposition use either equation solving [3] or triangular matrix operations [4] with most efficient implementation If , with is the linear system with requiring operations. variables, and satisfies the requirement for Cholesky decomposition, we can rewrite the linear system as In this paper we propose an inversion algorithm which … (5) reduces the number of operations by 16-17% compared to the existing algorithms by avoiding computation of some known By letting , we have intermediate results. … (6) In section 2 of this paper we review the Cholesky and LDL decomposition techniques, and discuss solutions to linear and systems based on them.
    [Show full text]
  • AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 3: Positive-Definite Systems; Cholesky Factorization
    AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 3: Positive-Definite Systems; Cholesky Factorization Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 11 Symmetric Positive-Definite Matrices n×n Symmetric matrix A 2 R is symmetric positive definite (SPD) if T n x Ax > 0 for x 2 R nf0g n×n Hermitian matrix A 2 C is Hermitian positive definite (HPD) if ∗ n x Ax > 0 for x 2 C nf0g SPD matrices have positive real eigenvalues and orthogonal eigenvectors Note: Most textbooks only talk about SPD or HPD matrices, but a positive-definite matrix does not need to be symmetric or Hermitian! A real matrix A is positive definite iff A + AT is SPD T n If x Ax ≥ 0 for x 2 R nf0g, then A is said to be positive semidefinite Xiangmin Jiao Numerical Analysis I 2 / 11 Properties of Symmetric Positive-Definite Matrices SPD matrix often arises as Hessian matrix of some convex functional I E.g., least squares problems; partial differential equations If A is SPD, then A is nonsingular Let X be any n × m matrix with full rank and n ≥ m. Then T I X X is symmetric positive definite, and T I XX is symmetric positive semidefinite n×m If A is n × n SPD and X 2 R has full rank and n ≥ m, then X T AX is SPD Any principal submatrix (picking some rows and corresponding columns) of A is SPD and aii > 0 Xiangmin Jiao Numerical Analysis I 3 / 11 Cholesky Factorization If A is symmetric positive definite, then there is factorization of A A = RT R where R is upper triangular, and all its diagonal entries are positive Note: Textbook calls is “Cholesky
    [Show full text]
  • 1. Positive Definite Matrices a Matrix a Is Positive Definite If X>Ax > 0 for All Nonzero X
    CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8 GENE H. GOLUB 1. Positive Definite Matrices A matrix A is positive definite if x>Ax > 0 for all nonzero x. A positive definite matrix has real and positive eigenvalues, and its leading principal submatrices all have positive determinants. From the definition, it is easy to see that all diagonal elements are positive. To solve the system Ax = b where A is positive definite, we can compute the Cholesky decom- position A = F >F where F is upper triangular. This decomposition exists if and only if A is symmetric and positive definite. In fact, attempting to compute the Cholesky decomposition of A is an efficient method for checking whether A is symmetric positive definite. It is important to distinguish the Cholesky decomposition from the square root factorization.A square root of a matrix A is defined as a matrix S such that S2 = SS = A. Note that the matrix F in A = F >F is not the square root of A, since it does not hold that F 2 = A unless A is a diagonal matrix. The square root of a symmetric positive definite A can be computed by using the fact that A has an eigendecomposition A = UΛU > where Λ is a diagonal matrix whose diagonal elements are the positive eigenvalues of A and U is an orthogonal matrix whose columns are the eigenvectors of A. It follows that A = UΛU > = (UΛ1/2U >)(UΛ1/2U >) = SS and so S = UΛ1/2U > is a square root of A. 2. The Cholesky Decomposition The Cholesky decomposition can be computed directly from the matrix equation A = F >F .
    [Show full text]
  • Cholesky Decomposition 1 Cholesky Decomposition
    Cholesky decomposition 1 Cholesky decomposition In linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a symmetric, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices and is an example of a square root of a matrix. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.[1] Statement If A has real entries and is symmetric (or more generally, is Hermitian) and positive definite, then A can be decomposed as where L is a lower triangular matrix with strictly positive diagonal entries, and L* denotes the conjugate transpose of L. This is the Cholesky decomposition. The Cholesky decomposition is unique: given a Hermitian, positive-definite matrix A, there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL*. The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite. The requirement that L have strictly positive diagonal entries can be dropped to extend the factorization to the positive-semidefinite case. The statement then reads: a square matrix A has a Cholesky decomposition if and only if A is Hermitian and positive semi-definite. Cholesky factorizations for positive-semidefinite matrices are not unique in general. In the special case that A is a symmetric positive-definite matrix with real entries, L can be assumed to have real entries as well.
    [Show full text]
  • The CMA Evolution Strategy: a Tutorial
    The CMA Evolution Strategy: A Tutorial Nikolaus Hansen June 28, 2011 Contents Nomenclature3 0 Preliminaries4 0.1 Eigendecomposition of a Positive Definite Matrix...............5 0.2 The Multivariate Normal Distribution.....................6 0.3 Randomized Black Box Optimization.....................7 0.4 Hessian and Covariance Matrices........................8 1 Basic Equation: Sampling8 2 Selection and Recombination: Moving the Mean9 3 Adapting the Covariance Matrix 10 3.1 Estimating the Covariance Matrix From Scratch................ 10 3.2 Rank-µ-Update................................. 12 3.3 Rank-One-Update................................ 14 3.3.1 A Different Viewpoint......................... 14 3.3.2 Cumulation: Utilizing the Evolution Path............... 16 3.4 Combining Rank-µ-Update and Cumulation.................. 17 4 Step-Size Control 17 5 Discussion 21 A Algorithm Summary: The CMA-ES 25 B Implementational Concerns 27 B.1 Multivariate normal distribution........................ 28 B.2 Strategy internal numerical effort........................ 28 B.3 Termination criteria............................... 28 B.4 Flat fitness.................................... 29 B.5 Boundaries and Constraints........................... 29 C MATLAB Source Code 31 1 D Reformulation of Learning Parameter ccov 33 2 Nomenclature We adopt the usual vector notation, where bold letters, v, are column vectors, capital bold letters, A, are matrices, and a transpose is denoted by vT. A list of used abbreviations and symbols is given in alphabetical order. Abbreviations CMA Covariance Matrix Adaptation EMNA Estimation of Multivariate Normal Algorithm ES Evolution Strategy (µ/µfI;Wg; λ)-ES, Evolution Strategy with µ parents, with recombination of all µ parents, either Intermediate or Weighted, and λ offspring. RHS Right Hand Side. Greek symbols λ ≥ 2, population size, sample size, number of offspring, see (5).
    [Show full text]
  • Reducing Dimensionality in Text Mining Using Conjugate Gradients and Hybrid Cholesky Decomposition
    (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 8, No. 7, 2017 Reducing Dimensionality in Text Mining using Conjugate Gradients and Hybrid Cholesky Decomposition Jasem M. Alostad The Public Authority for Applied Education and Training (PAAET), College of Basic Education P.O.Box 23167, Safat 13092, Kuwait Abstract—Generally, data mining in larger datasets consists The data reduction is the process of reducing the size or of certain limitations in identifying the relevant datasets for the dimensionality of the data, however, the representation of the given queries. The limitations include: lack of interaction in the data should be retained. Selection of instance is one better way required objective space, inability to handle the data sets or to reduce the data by reducing the total number of instances. In discrete variables in datasets, especially in the presence of spite of many efforts to deal with such instances, data mining missing variables and inability to classify the records as per the algorithm, however undergoes severe challenges due to non- given query, and finally poor generation of explicit knowledge for applicability of datasets with large instances. Hence, the a query increases the dimensionality of the data. Hence, this computational complexity of the system increases with larger paper aims at resolving the problems with increasing data instances [3], [4] and leads to problems in scaling, increased dimensionality in datasets using modified non-integer matrix storage requirements and clustering accuracy. The other factorization (NMF). Further, the increased dimensionality arising due to non-orthogonally of NMF is resolved with problems associated with larger data instances include, Cholesky decomposition (cdNMF).
    [Show full text]
  • Stable Computations of Generalized Inverses of Positive Semidefinite
    Stable computations of generalized inverses of positive semidefinite matrices A. Markopoulos, Z. Dostál, T. Kozubek, P. Kovár,ˇ T. Brzobohatý, and R. Kuceraˇ Centre of Excellence IT4I, VŠB-Technical University of Ostrava, Czech Republic [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] Summary. Effective implementation of some efficient FETI methods assumes application of a direct method for solving a system of linear equations with a symmetric positive semidefinite matrix A. The latter usually comprises a triangular decomposition of a nonsingular diagonal block AJJ of the stiffness matrix A of a subdomain and an effective evaluation of the action of a generalized inverse of the corresponding Schur complement. The other posibility consists in a regularization of A resulting in a non-singular matrix Ar whose (standard) inverse is the generalized inverse to A. It avoids the necessity to identify zero pivots with negligible fill-in. We review both these techniques. 1 Introduction Due to the rounding errors, effective elimination of the displacements of “floating” subdomains is a nontrivial ingredient of implementation of FETI methods, as it can be difficult to recognize the positions of zero pivots when the nonsingular diagonal block of A is ill-conditioned. Moreover, even if the zero pivots are recognized prop- erly, it turns out that the ill-conditioning of the nonsingular submatrix defined by the nonzero pivots can have a devastating effect on the precision of the solution. Most of the results are related to the first problem, i.e., to identify reliably the zero pivots.
    [Show full text]
  • Hierarchical Sparse Cholesky Decomposition with Applications to High-Dimensional Spatio-Temporal filtering
    Hierarchical sparse Cholesky decomposition with applications to high-dimensional spatio-temporal filtering Marcin Jurek∗ Matthias Katzfuss∗y Abstract Spatial statistics often involves Cholesky decomposition of covariance matrices. To ensure scalability to high dimensions, several recent approximations have assumed a sparse Cholesky factor of the precision matrix. We propose a hierarchical Vec- chia approximation, whose conditional-independence assumptions imply sparsity in the Cholesky factors of both the precision and the covariance matrix. This remarkable property is crucial for applications to high-dimensional spatio-temporal filtering. We present a fast and simple algorithm to compute our hierarchical Vecchia approxima- tion, and we provide extensions to non-linear data assimilation with non-Gaussian data based on the Laplace approximation. In several numerical comparisons, our methods strongly outperformed alternative approaches. Keywords: state-space model, spatio-temporal statistics, data assimilation, Vecchia approxima- tion, hierarchical matrix, incomplete Cholesky decomposition 1 Introduction Symmetric positive-definite matrices arise in spatial statistics, Gaussian-process inference, and spatio-temporal filtering, with a wealth of application areas, including geoscience (e.g., Cressie, 1993; Banerjee et al., 2004), machine learning (e.g., Rasmussen and Williams, 2006), data assimilation (e.g., Nychka and Anderson, 2010; Katzfuss et al., 2016), and the analysis arXiv:2006.16901v1 [stat.CO] 30 Jun 2020 of computer experiments (e.g., Sacks et al., 1989; Kennedy and O'Hagan, 2001). Inference in these areas typically relies on Cholesky decomposition of the positive-definite matrices. However, this operation scales cubically in the dimension of the matrix, and it is thus com- putationally infeasible for many modern problems and applications, which are increasingly high-dimensional.
    [Show full text]
  • LAPACK Working Note
    ISSN 1360-1725 UMIST LAPACK-Style Codes for Level 2 and 3 Pivoted Cholesky Factorizations C. Lucas Numerical Analysis Report No. 442 February 2004 Manchester Centre for Computational Mathematics Numerical Analysis Reports DEPARTMENTS OF MATHEMATICS Reports available from: And over the World-Wide Web from URLs Department of Mathematics http://www.ma.man.ac.uk/nareports University of Manchester ftp://ftp.ma.man.ac.uk/pub/narep Manchester M13 9PL England LAPACK-Style Codes for Level 2 and 3 Pivoted Cholesky Factorizations Craig Lucas¤ February 5, 2004 Abstract Fortran 77 codes exist in LAPACK for computing the Cholesky factor- ization of a symmetric positive de¯nite matrix using Level 2 and 3 BLAS. In LINPACK there is a Level 1 BLAS routine for computing the Cholesky factorization with complete pivoting of a symmetric positive semi-de¯nite matrix. We present two new algorithms and Fortran 77 LAPACK-style codes for computing this pivoted factorization: one using Level 2 BLAS and one using Level 3 BLAS. We show that on modern machines the new codes can be many times faster than the LINPACK code. Also, with a new stopping criterion they provide more reliable rank detection and can have a smaller normwise backward error. 1 Introduction The Cholesky factorization of a symmetric positive de¯nite matrix A 2 Rn£n has the form A = LLT ; where L 2 Rn£n is a lower triangular matrix with positive diagonal elements. If A is positive semi-de¯nite, of rank r, there exists a Cholesky factorization with complete pivoting ([6, Thm. 10.9], for example).
    [Show full text]
  • A Scalable High Performant Cholesky Factorization for Multicore with GPU Accelerators LAPACK Working Note #223
    A Scalable High Performant Cholesky Factorization for Multicore with GPU Accelerators LAPACK Working Note #223 Hatem Ltaief1, Stanimire Tomov1, Rajib Nath1, Peng Du1, and Jack Dongarra1,2,3? 1 Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville 2 Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 3 School of Mathematics & School of Computer Science, University of Manchester {ltaief, tomov, rnath1, du, dongarra}@eecs.utk.edu Abstract. We present a Cholesky factorization for multicore with GPU accelerators. The challenges in developing scalable high performance al- gorithms for these emerging systems stem from their heterogeneity, mas- sive parallelism, and the huge gap between the GPUs’ compute power vs the CPU-GPU communication speed. We show an approach that is largely based on software infrastructures that have already been devel- oped for homogeneous multicores and hybrid GPU-based computing. The algorithm features two levels of nested parallelism. A coarse-grained parallelism is provided by splitting the computation into tiles for con- current execution between GPUs. A fine-grained parallelism is further provided by splitting the work-load within a tile for high efficiency com- puting on GPUs but also, in certain cases, to benefit from hybrid com- putations by using both GPUs and CPUs. Our resulting computational kernels are highly optimized. An efficient task scheduling mechanism ensures a load balanced execution over the entire multicore with GPU accelerators system. Furthermore, the communication overhead is min- imized by trading off the amount of memory allocated on GPUs. This results in a scalable hybrid Cholesky factorization of unprecedented per- formance.
    [Show full text]