Sufficient conditions for positive definiteness of tridiagonal matrices revisited

Anđelić, Milica; Fonseca, Carlos M. da

doi:10.1007/s11117-010-0047-y

Cited by 27 publications

(13 citation statements)

References 12 publications

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“…Therefore, the derivative of the smoothing spline at the knots satisfies the matrix equation

T m = D f,

where m = ( m 0 , m 1 ,…, m N ) T is the vector of slopes, T is the tridiagonal matrix

T = {((), \begin{matrix} 2 & 1 \\ 1 & 4 & 1 \\ 1 & 4 & 1 \\ ⋱ & ⋱ & ⋱ \\ 1 & 4 & 1 \\ 1 & 2 \end{matrix})}_{(N + 1) \times (N + 1)},

and

D = \frac{3}{h} {((), \begin{matrix} - 1 & 1 \\ - 1 & 0 & 1 \\ - 1 & 0 & 1 \\ ⋱ & ⋱ & ⋱ \\ - 1 & 0 & 1 \\ - 1 & 1 \end{matrix})}_{(N + 1) \times (N + 1)} .

Note that the diagonal entries of T are all positive and T is strictly diagonally dominant, meaning that in each and every row the sum of the absolute values of the off‐diagonal elements is less than the absolute value of the diagonal element. Consequently, the symmetric tridiagonal matrix T is positive definite [see, e.g., Demmel , , Theorem 2.9; Andelić and da Fonseca , , Theorem 1.2]. The most efficient numerical method of solving the tridiagonal system T m = u , where u = D f , is based on the Cholesky decomposition T = R T R , where, because T is a symmetric positive definite tridiagonal matrix, R is an upper triangular matrix with all zeros above the first superdiagonal [ Schwarz , ; Golub and Van Loan , ].…”

Section: Differentiation Technique Based On Smoothing Splinesmentioning

confidence: 99%

The most intense current sheets in the high‐speed solar wind near 1 AU

Podesta

2017

JGR Space Physics

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Electric currents in the solar wind plasma are investigated using 92 ms fluxgate magnetometer data acquired in a high‐speed stream near 1 AU. The minimum resolvable scale is roughly 0.18 s in the spacecraft frame or, using Taylor's “frozen turbulence” approximation, one proton inertial length di in the plasma frame. A new way of identifying current sheets is developed that utilizes a proxy for the current density J obtained from the derivatives of the three orthogonal components of the observed magnetic field B. The most intense currents are identified as 5σ events, where σ is the standard deviation of the current density. The observed 5σ events are characterized by an average scale size of approximately 3di along the flow direction of the solar wind, a median separation of around 50di or 100di along the flow direction of the solar wind, and a peak current density on the order of 0.5 pA/cm2. The associated current‐carrying structures are consistent with current sheets; however, the planar geometry of these structures cannot be confirmed using single‐point, single‐spacecraft measurements. If Taylor's hypothesis continues to hold for the energetically dominant fluctuations at kinetic scales 1

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“…Therefore, the derivative of the smoothing spline at the knots satisfies the matrix equation

T m = D f,

where m = ( m 0 , m 1 ,…, m N ) T is the vector of slopes, T is the tridiagonal matrix

T = {((), \begin{matrix} 2 & 1 \\ 1 & 4 & 1 \\ 1 & 4 & 1 \\ ⋱ & ⋱ & ⋱ \\ 1 & 4 & 1 \\ 1 & 2 \end{matrix})}_{(N + 1) \times (N + 1)},

and

D = \frac{3}{h} {((), \begin{matrix} - 1 & 1 \\ - 1 & 0 & 1 \\ - 1 & 0 & 1 \\ ⋱ & ⋱ & ⋱ \\ - 1 & 0 & 1 \\ - 1 & 1 \end{matrix})}_{(N + 1) \times (N + 1)} .

Section: Differentiation Technique Based On Smoothing Splinesmentioning

confidence: 99%

The most intense current sheets in the high‐speed solar wind near 1 AU

Podesta

2017

JGR Space Physics

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“…Lemma 3.2. (See [5]) Let A m be the real symmetric tridiagonal matrix definied in (1.2), with diagonal entries positive. If…”

Section: Eigenvalues and Eigenvectorsmentioning

confidence: 99%

“…(See[5]) The real symmetric tridiagonal matrix A m , defined in (1.2), is positive definite if and only if its principal minors det A s , for s = 1, ..., m, are positive. The eigenvector of the matrix A m given in (1.2) associated with eigenvalue λ s , for s = 1, .…”

mentioning

confidence: 99%

Invariant regions and existence of global solutions to a generalized m-component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix

Abdelmalek¹,

Rebiai²,

Abdelmalek³

2021

APAM

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“…Therefore, we take the time derivative of Eq. (20) and require that _ U i be given by the sum of pairwise heat transfer laws of form Eq. (4).…”

Section: Problem Formulation Consider a Network Of N Single Integratmentioning

confidence: 99%

“…A tridiagonal symmetric matrix with main diagonal given by ða 1 ; a 2 ; …; a n Þ and first super-and subdiagonals given by [20]. Therefore, Eq.…”

Section: Extensions To Directed Networkmentioning

confidence: 99%

Thermodynamics-Based Control of Network Systems

Berg

Maithripala

Hui

et al. 2013

Journal of Dynamic Systems, Measurement, and Control

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The zeroth and first laws of thermodynamics define the concepts of thermal equilibrium and thermal energy. The second law of thermodynamics determines whether a particular transfer of thermal energy can occur. Collectively, these fundamental laws of nature imply that a closed collection of thermodynamic subsystems will tend to thermal equilibrium. This paper generalizes the concepts of energy, entropy, and temperature to undirected and directed networks of single integrators, and demonstrates how thermodynamic principles can be applied to the design of distributed consensus control algorithms for networked dynamical systems.

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Sufficient conditions for positive definiteness of tridiagonal matrices revisited

Abstract: We review several sufficient conditions for the positive definiteness of a tridiagonal matrix and propose a different approach to the problem, recalling and comprising little-known results on chain sequences.

Cited by 27 publications

References 12 publications

The most intense current sheets in the high‐speed solar wind near 1 AU

The most intense current sheets in the high‐speed solar wind near 1 AU

Invariant regions and existence of global solutions to a generalized m-component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix

Thermodynamics-Based Control of Network Systems

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