Unitary Integrators and Applications to Continuous Orthonormalization Techniques

Dieci, Luca; Russell, Robert D.; Vleck, Erik S. Van

doi:10.1137/0731014

Cited by 118 publications

(86 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Lemma 48, it suffices to consider the Sacker-Sell spectrum associated with the underlying ODEs (15) and (22). Then, using similar arguments as in [23, Lemma 1.2], the proof is straightforward.…”

Section: Lemma 55mentioning

confidence: 99%

“…Since we have started with a strangeness-free system, it follows that the corresponding transformed matrixÂ partitioned as in (13) has a nonsingular blockÂ 22 .…”

Section: Theorem 12mentioning

confidence: 99%

“…Similarly, the equivalence transformation associated with V 1 , U 1 transforms the adjoint equation (22) to…”

Section: Lemma 55mentioning

confidence: 99%

“…The following results are obtained with Version 7.0 on an IBM computer with Intel CPU T2300 1.66 GHz. For the orthogonal integration, we have used the projected integration technique, see [22].…”

Section: Numerical Examplesmentioning

confidence: 99%

See 3 more Smart Citations

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Linh

Mehrmann

2009

J Dyn Diff Equat

View full text Add to dashboard Cite

Lyapunov and exponential dichotomy spectral theory is extended from ordinary differential equations (ODEs) to nonautonomous differential-algebraic equations (DAEs). By using orthogonal changes of variables, the original DAE system is transformed into appropriate condensed forms, for which concepts such as Lyapunov exponents, Bohl exponents, exponential dichotomy and spectral intervals of various kinds can be analyzed via the resulting underlying ODE. Some essential differences between the spectral theory for ODEs and that for DAEs are pointed out. It is also discussed how numerical methods for computing the spectral intervals associated with Lyapunov and Sacker-Sell (exponential dichotomy) can be extended from those methods proposed for ODEs. Some numerical examples are presented to illustrate the theoretical results.

show abstract

Section: Lemma 55mentioning

confidence: 99%

“…Since we have started with a strangeness-free system, it follows that the corresponding transformed matrixÂ partitioned as in (13) has a nonsingular blockÂ 22 .…”

Section: Theorem 12mentioning

confidence: 99%

“…Similarly, the equivalence transformation associated with V 1 , U 1 transforms the adjoint equation (22) to…”

Section: Lemma 55mentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Linh

Mehrmann

2009

J Dyn Diff Equat

View full text Add to dashboard Cite

show abstract

“…Because of its central role in applications, the unitary case has received attention early [6]. For an inexpensive structure preserving numerical integration method,…”

Section: Examplementioning

confidence: 99%

Matrix intersection problems for conditioning

Huhtanen

Seiskari

2017

Banach Center Publ.

View full text Add to dashboard Cite

Abstract. Conditioning of a nonsingular matrix subspace is addressed in terms of its best conditioned elements. Computationally the problem is seemingly challenging. By associating with the task an intersection problem with unitary matrices leads to a more accessible approach. A resulting matrix nearness problem can be viewed to generalize the so-called Löwdin problem in quantum chemistry. For critical points in the Frobenius norm, a differential equation on the manifold of unitary matrices is derived. Another resulting matrix nearness problem allows locating points of optimality more directly, once formulated as a problem in computational algebraic geometry.Key words. conditioning, matrix intersection problem, matrix nearness problem, Löwdin's problem, generalized eigenvalue problem AMS subject classifications. 15A12, 65F351. Introduction. This paper is concerned with the problem of conditioning of a nonsingular matrix subspace V of C n×n over C (or R). Matrix subspaces typically appear in large scale numerical linear algebra problems where assuming additional structure is quite unrealistic. Nonsingularity means that there exists invertible elements in V. The conditioning of V is then defined in terms of its best conditioned elements. In the applications that we have in mind, typically dim V ≪ n 2 . For example, in the generalized eigenvalue problem dim V = 2 only. In this paper the task of assessing conditioning is formulated as a matrix intersection problem for V and the set of unitary matrices.1 Since this can be done in many ways, the interpretation is amenable to computations through matrix nearness problems and versatile enough in view of addressing operator theoretic problems more generally.Denote by U (n) the set of unitary matrices in C n×n . The intersection problem for V and U (n), which are both smooth submanifolds of C n×n , can be formulated as a matrix nearness problem

show abstract

A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents

Gadella

Lara

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one‐dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.

show abstract

Unitary Integrators and Applications to Continuous Orthonormalization Techniques

Cited by 118 publications

References 23 publications

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Matrix intersection problems for conditioning

A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents

Contact Info

Product

Resources

About