Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences

Greenbaum, Anne

doi:10.1016/0024-3795(89)90285-1

Cited by 144 publications

(194 citation statements)

References 6 publications

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“…Each iteration is O n 2 x so this is not in itself a large gain over explicit factorization. However convergence is significantly faster if the eigenvalues of H are tightly clustered away from zero: if the eigenvalues are covered by intervals [99,47,48] 11 . Preconditioning (see below) aims at achieving such clustering.…”

Section: Gradient Descentmentioning

confidence: 99%

Bundle Adjustment — A Modern Synthesis

Triggs

McLauchlan

Hartley

et al. 2000

Lecture Notes in Computer Science

3,004

2,203

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This paper is a survey of the theory and methods of photogrammetric bundle adjustment, aimed at potential implementors in the computer vision community. Bundle adjustment is the problem of refining a visual reconstruction to produce jointly optimal structure and viewing parameter estimates. Topics covered include: the choice of cost function and robustness; numerical optimization including sparse Newton methods, linearly convergent approximations, updating and recursive methods; gauge (datum) invariance; and quality control. The theory is developed for general robust cost functions rather than restricting attention to traditional nonlinear least squares.

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Section: Gradient Descentmentioning

confidence: 99%

Bundle Adjustment — A Modern Synthesis

Triggs

McLauchlan

Hartley

et al. 2000

Lecture Notes in Computer Science

3,004

2,203

View full text Add to dashboard Cite

show abstract

“…[12,Theorem 2]). In investigating the robustness of BiCG, we are only interested in the convergence of r n because it is the convergence of r n that drives the convergence of the true residual b − Ax n .…”

Section: Analysis Of the Finite Precision Bicgmentioning

confidence: 99%

“…Finite precision analyses of conjugate gradient-type and Lanczos-type algorithms have played an important role in understanding these algorithms. The pioneering work is due to C. Paige [19,20] and A. Greenbaum [12]. Paige showed in [19,20] that the loss of orthogonality comes with but does not prevent convergence of the Ritz values, i.e., useful results can still be obtained from the algorithm even when the iterates deviate significantly from what would have been produced in exact arithmetic.…”

Section: Introductionmentioning

confidence: 99%

“…A generalization to the nonsymmetric case was given by Bai [1], and the near-breakdowns and loss of biorthogonality were discussed by Day [5]. Greenbaum established backward stability results in a generalized sense [12], showing that the iterative residuals produced by the finite precision conjugate gradient algorithm are equivalent to what would have been produced by applying the exact CG to a larger matrix. Some estimates on the larger matrix were given, which may vary from step to step.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the finite precision bi-conjugate gradient algorithm for nonsymmetric linear systems

Tong

1999

Math. Comp.

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Abstract.In this paper we analyze the bi-conjugate gradient algorithm in finite precision arithmetic, and suggest reasons for its often observed robustness. By using a tridiagonal structure, which is preserved by the finite precision bi-conjugate gradient iteration, we are able to bound its residual norm by a minimum polynomial of a perturbed matrix (i.e. the residual norm of the exact GMRES applied to a perturbed matrix) multiplied by an amplification factor. This shows that occurrence of near-breakdowns or loss of biorthogonality does not necessarily deter convergence of the residuals provided that the amplification factor remains bounded. Numerical examples are given to gain insights into these bounds.

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“…A growing body of work suggests that non-trivial clusters of Ritz values are only found very close to eigenvalues of A (see Wülling [32,33], Knizhnerman [14, Theorem 2], Strakoš and Greenbaum [30], and Greenbaum [12]). That is, if we find two or more eigenvalues of T (m) that are very close to each other, they normally indicate the location of an eigenvalue of A; we call such Ritz values doubly-converged.…”

Section: Background and Methodologymentioning

confidence: 99%

Experiences with a Lanczos Eigensolver in High-Precision Arithmetic

Alperovich

Druinsky

Toledo

2014

Parallel Processing and Applied Mathematics

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Abstract. We investigate the behavior of the Lanczos process when it is used to find all the eigenvalues of large sparse symmetric matrices. We study the convergence of classical Lanczos (i.e., without reorthogonalization) to the point where there is a cluster of Ritz values around each eigenvalue of the input matrix A. At that point, convergence to all the eigenvalues can be ascertained if A has no multiple eigenvalues. To eliminate multiple eigenvalues, we disperse them by adding to A a random matrix with a small norm; using high-precision arithmetic, we can perturb the eigenvalues and still produce accurate double-precision results. Our experiments indicate that the speed with which Ritz clusters form depends on the local density of eigenvalues and on the unit roundoff, which implies that we can accelerate convergence by using high-precision arithmetic in computations involving the Lanczos iterates.

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Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences

Cited by 144 publications

References 6 publications

Bundle Adjustment — A Modern Synthesis

Bundle Adjustment — A Modern Synthesis

Analysis of the finite precision bi-conjugate gradient algorithm for nonsymmetric linear systems

Experiences with a Lanczos Eigensolver in High-Precision Arithmetic

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