On the reduction of a symmetric matrix to tridiagonal form

Aasen, Jan Ole

doi:10.1007/bf01931804

Cited by 82 publications

(72 citation statements)

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“…However, this method in its basic form can only decompose a positive definite matrix due to the need to calculate diagonal square roots. To factorise an indefinite matrix efficiently, methods such as the Bunch and Kaufman method [22] or Aasen's method [23] need to be used. These are generally based on an LDL T decomposition [24] and can be used without significant efficiency reduction as compared to a Cholesky decomposition.…”

Section: Factorizing Matricesmentioning

confidence: 99%

Use of negative stiffness in failure analysis of concrete beams

Al‐Sabah

Laefer

2016

Engineering Structures

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A new concrete analysis method is presented entitled continuous, incremental-only, tangential analysis (CITA). CITA employs piece-wise linear stress-strain curve and a tangent elasticity modulus to calculate stiffness including parts with negative values. Since indefinite structure stiffness matrices generally indicate instability, traditionally they have been avoided. However, since CITA analysis involves introducing damage in steps, the full range of concrete behaviour including the softening portion under tensile cracking can be addressed. Herein CITA is verified against numerical and experimental results for concrete beams, thereby showing faster solutions for non-linear problems than sequentially-linear analysis, while reducing self-imposed restrictions against negative stiffness.

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Section: Factorizing Matricesmentioning

confidence: 99%

Use of negative stiffness in failure analysis of concrete beams

Al‐Sabah

Laefer

2016

Engineering Structures

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“…Let r be the least integer such that i4*>i = *(fc)- (2) If {A[k^\ > aX(k) where 0 < a < 1, perform a 1 x 1 pivot to obtain A^k~1', decrease k by 1 and return to (1). We will show that a good value for a is (1 +VÏ7)/8.…”

Section: «;mentioning

confidence: 99%

“…(4) If aX(fc)2 < L4(nVfc), then perform a 1 x 1 pivot to obtain A(k~l), decrease k by 1, and return to (1). (We need this test to guarantee stability.)…”

Section: Kmmentioning

confidence: 99%

Some Stable Methods for Calculating Inertia and Solving Symmetric Linear Systems

Bunch¹,

Kaufman²

1977

Mathematics of Computation

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“…The resulting L factors are used to transform (1) to a standard Hermitian eigenproblemÃz = λz, whereÃ = L −1 AL −H . After solving the standard Hermitian eigenproblem, the eigenvectors X of the generalized problem (1) are then computed by backsolving with the Cholesky factor, X = L −H Z. To solve the standard Hermitian (symmetric) eigenproblem of the formÃz = λz, finding its eigenvalues Λ and eigenvectors Z such thatÃ = ZΛZ H , the standard strategy follows three steps [1,12,24].…”

Section: Introductionmentioning

confidence: 99%

“…After solving the standard Hermitian eigenproblem, the eigenvectors X of the generalized problem (1) are then computed by backsolving with the Cholesky factor, X = L −H Z. To solve the standard Hermitian (symmetric) eigenproblem of the formÃz = λz, finding its eigenvalues Λ and eigenvectors Z such thatÃ = ZΛZ H , the standard strategy follows three steps [1,12,24]. First, reduce the matrix to a tridiagonal matrix T using an orthogonal transformation Q such thatÃ = QT Q H (called the "reduction phase").…”

Section: Introductionmentioning

confidence: 99%

Leading Edge Hybrid Multi-GPU Algorithms for Generalized Eigenproblems in Electronic Structure Calculations

Haidar

Solcà

Gates

et al. 2013

Lecture Notes in Computer Science

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Abstract. Today's high computational demands from engineering fields and complex hardware development make it necessary to develop and optimize new algorithms toward achieving high performance and good scalability on the next generation of computers. The enormous gap between the high-performance capabilities of GPUs and the slow interconnect between them has made the development of numerical software that is scalable across multiple GPUs extremely challenging. We describe and analyze a successful methodology to address the challenges-starting from our algorithm design, kernel optimization and tuning, to our programming model-in the development of a scalable high-performance generalized eigenvalue solver in the context of electronic structure calculations in materials science applications. We developed a set of leading edge dense linear algebra algorithms, as part of a generalized eigensolver, featuring fine grained memory aware kernels, a task based approach and hybrid execution/scheduling. The goal of the new design is to increase the computational intensity of the major compute kernels and to reduce synchronization and data transfers between GPUs. We report the performance impact on the generalized eigensolver when different fractions of eigenvectors are needed. The algorithm described provides an enormous performance boost compared to current GPU-based solutions, and performance comparable to state-of-the-art distributed solutions, using a single node with multiple GPUs.

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On the reduction of a symmetric matrix to tridiagonal form

Abstract: A stable algorithm for reducing a symmetric, non-definite matrix of order n to tridiagonal form, involving about n3/6 additions and multiplications is presented. L m 3 J

Cited by 82 publications

References 0 publications

Use of negative stiffness in failure analysis of concrete beams

Use of negative stiffness in failure analysis of concrete beams

Some Stable Methods for Calculating Inertia and Solving Symmetric Linear Systems

Leading Edge Hybrid Multi-GPU Algorithms for Generalized Eigenproblems in Electronic Structure Calculations

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