Providing a single point spectrum for Runge-Kutta schemes of high stage order based on perturbed collocation

2017 ◽  
Author(s):  
Tim Steinhoff
Keyword(s):  
Point Spectrum ◽  
Single Point ◽  
High Stage ◽  
Stage Order ◽  
Runge Kutta

On implicit Runge-Kutta methods with high stage order

1997 ◽  
Vol 37 (1) ◽  
pp. 221-226 ◽  
Author(s):  
C. Bendtsen
Keyword(s):  
High Stage ◽  
Stage Order ◽  
Runge Kutta

Multivariate Visualiztion and Analysis Tools for Segmentation and Classification of Vibrational Spectroscopic Images

1997 ◽  
Vol 3 (S2) ◽  
pp. 865-866
Author(s):  
F. Delaglio ◽  
L.H. Kidder ◽  
I.W. Levin ◽  
E.N. Lewis
Keyword(s):  
High Resolution ◽  
Vibrational Spectrum ◽  
Point Spectrum ◽  
Dimensional Space ◽  
Single Point ◽  
Full Potential ◽  
Multivariate Approach ◽  
Macroscopic Level ◽  
Chemical And Biological Systems

Vibrational spectroscopic imaging has already demonstrated enormous potential for studying a variety of chemical and biological systems at both the microscopic and macroscopic level. However, these spectral images are large and complicated, typically consisting of tens of thousands of pixels, each with an associated high-resolution vibrational spectrum, leading to data sizes upwards of 64 megabytes. In order to realize the full potential of these spectral images, we must find ways to query the data so that specific questions can be answered.We illustrate a multivariate approach to this challenge, where each pixel is considered to be a single point in a multivariate (N-dimensional) space. The variables (coordinates) of the point in N dimensions are simply the intensities of the N-point spectrum associated with the pixel. In this representation, pixels with similar spectra will tend to cluster together in the multivariate space, since they will have similar coordinates.

scholarly journals On order 5 symplectic explicit Runge-Kutta Nyström methods

2000 ◽  
Vol 4 (2) ◽  
pp. 143-150 ◽  
Author(s):  
Lin-Yi Chou ◽  
P. W. Sharp
Keyword(s):  
Extra Cost ◽  
Optimal Method ◽  
New Methods ◽  
Nyström Methods ◽  
Stage Order ◽  
Principal Error ◽  
Numerical Performance ◽  
Free Parameters ◽  
Runge Kutta ◽  
Error Coefficients

Order five symplectic explicit Runge-Kutta Nyström methods of five stages are known to exist. However, these methods do not have free parameters with which to minimise the principal error coefficients. By adding one derivative evaluation per step, to give either a six-stage non-FSAL family or a seven-stage FSAL family of methods, two free parameters become available for the minimisation. This raises the possibility of improving the efficiency of order five methods despite the extra cost of taking a step.We perform a minimisation of the two families to obtain an optimal method and then compare its numerical performance with published methods of orders four to seven. These comparisons along with those based on the principal error coefficients show the new method is significantly more efficient than the five-stage, order five methods. The numerical comparisons also suggest the new methods can be more efficient than published methods of other orders.

SOLITON SECTOR OF THE XY MODEL

1996 ◽  
Vol 10 (13n14) ◽  
pp. 1685-1693
Author(s):  
HUZIHIRO ARAKI
Keyword(s):  
Ground State ◽  
Point Spectrum ◽  
Single Point ◽  
Ground States ◽  
Finite Energy ◽  
Particle Energy ◽  
Xy Model ◽  
Analytic Dependence ◽  
Sudden Appearance ◽  
Infinite Multiplicity

We study soliton sectors of the XY model by using known results and methods about its ground states. In the regions of parameters for which ground states are not unique, we show that (1) there are two soliton sectors depending on parameters of the model analytically in a well-defined sense, (2) the only sectors with “finite energy” are ground state and soliton sectors, and (3) the sudden appearance of additional ground states at a pair of specific values of parameters (despite analytic dependence of other ground states on parameters at those specific values), which were found in earlier study of ground states, can be understood as the degeneracy of one particle energy in the soliton sector (which has a continuous spectrum at other values of parameters) to a single point spectrum with infinite multiplicity at the specific values of parameters.

scholarly journals Rational functions with maximal radius of absolute monotonicity

2014 ◽  
Vol 17 (1) ◽  
pp. 159-205 ◽  
Author(s):  
Lajos Lóczi ◽  
David I. Ketcheson
Keyword(s):  
Initial Value Problems ◽  
Rational Functions ◽  
Strong Stability ◽  
Strong Stability Preserving ◽  
Algebraic Numbers ◽  
Initial Value ◽  
Stage Order ◽  
Absolute Monotonicity ◽  
Runge Kutta ◽  
Radius Of Absolute Monotonicity

AbstractWe study the radius of absolute monotonicity $R$ of rational functions with numerator and denominator of degree $s$ that approximate the exponential function to order $p$. Such functions arise in the application of implicit $s$-stage, order $p$ Runge–Kutta methods for initial value problems, and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with $p=2$ and $R>2s$, disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with two or three parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge–Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.

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