368 research outputs found
Error Estimation of Numerical Solvers for Linear Ordinary Differential Equations
Solving Linear Ordinary Differential Equations (ODEs) plays an important role
in many applications. There are various numerical methods and solvers to obtain
approximate solutions. However, few work about global error estimation can be
found in the literature. In this paper, we first give a definition of the
residual, based on the piecewise Hermit interpolation, which is a kind of the
backward-error of ODE solvers. It indicates the reliability and quality of
numerical solution. Secondly, the global error between the exact solution and
an approximate solution is the forward error and a bound of it can be given by
using the backward-error. The examples in the paper show that our estimate
works well for a large class of ODE models.Comment: 13 pages,6 figure
Exact Bivariate Polynomial Factorization in Q by Approximation of Roots
Factorization of polynomials is one of the foundations of symbolic
computation. Its applications arise in numerous branches of mathematics and
other sciences. However, the present advanced programming languages such as C++
and J++, do not support symbolic computation directly. Hence, it leads to
difficulties in applying factorization in engineering fields. In this paper, we
present an algorithm which use numerical method to obtain exact factors of a
bivariate polynomial with rational coefficients. Our method can be directly
implemented in efficient programming language such C++ together with the GNU
Multiple-Precision Library. In addition, the numerical computation part often
only requires double precision and is easily parallelizable
A Short Note on Zero-error Computation for Algebraic Numbers by IPSLQ
The PSLQ algorithm is one of the most popular algorithm for finding
nontrivial integer relations for several real numbers. In the present work, we
present an incremental version of PSLQ. For some applications needing to call
PSLQ many times, such as finding the minimal polynomial of an algebraic number
without knowing the degree, the incremental PSLQ algorithm is more efficient
than PSLQ, both theoretically and practically.Comment: 4 page
Geometric involutive bases for positive dimensional polynomial ideals and SDP methods
Geometric involutive bases for polynomial systems of equations have their
origin in the prolongation and projection methods of the geometers Cartan and
Kuranishi for systems of PDE. They are useful for numerical ideal membership
testing and the solution of polynomial systems. In this paper we further
develop our symbolic-numeric methods for such bases. We give methods to
explicitly extract and decrease the degree of intermediate systems and the
output basis. Algorithms for the numerical computation of involutivity criteria
for positive dimensional ideals are also discussed.
We were also motivated by some remarkable recent work by Lasserre and
collaborators who employed our prolongation projection involutive criteria as a
part of their semi-definite based programming (SDP) method for identifying the
real radical of zero dimensional polynomial ideals. Consequently in this paper
we begin an exploration of the interaction between geometric involutive bases
and these methods particularly in the positive dimensional case. Motivated by
the extension of these methods to the positive dimensional case we explore the
interplay between geometric involutive bases and the new SDP methods
Design optimization of mode-matched bulk-mode piezoelectric micro-gyroscopes through modal analysis
Bulk piezoelectric micro-gyroscope is a miniaturized inertial sensor that
uses a differential thickness-shear bulk mode of a PZT block as the drive mode
of the gyroscope. In the paper, a second differential thickness-extensional
mode is identified for the sense mode and mode-matching is proposed for the
first time by proper design of the device geomtries. Through finite element
modal analysis, the frequencies of drive mode and sense mode are obtained when
the length of the PZT block varies from 4.8mm to 5.6mm and the width of the PZT
block varies from 3.0mm to 4.0mm. Using a fitting method, the empirical
formulae with an excellent fit are induced to predict the influence of the
length and the width of the PZT block on the drive and sense mode frequencies.
Based on these empirical formulae, the mode-matching equations are introduced.
The analysis results show that for a given thickness of the PZT block, the
effect of the width on the drive mode frequency is prominant. Conversly, the
effect of length on the sense mode frequency is dominant. The resonance
frequencies, kinetic energy ratios, scale factors of gyroscope are compared to
evaluate the mode quality. The results show that the kinetic energy in y-axis
direction of the drive mode and the kinetic energy in z-axis direction of the
sense mode increase with the thickness of the PZT block, and consequently the
scale factor of the gyroscope increases. For a constant thickness of the PZT
block the scale factor will decrease as the length increases. Through design
optimization we present a 20 times improvement in the scale factor of the
mode-matched gyroscope. Given the thickness of PZT block, the length and the
width will be determined by the mode-matching equations mentioned. Generally,
the analysis suggests that the resolution of the gyroscope improves by
increasing the thickness PZT block.Comment: 10 pages, 5 figure
Real Root Isolation of Polynomial Equations Based on Hybrid Computation
A new algorithm for real root isolation of polynomial equations based on
hybrid computation is presented in this paper. Firstly, the approximate
(complex) zeros of the given polynomial equations are obtained via homotopy
continuation method. Then, for each approximate zero, an initial box relying on
the Kantorovich theorem is constructed, which contains the corresponding
accurate zero. Finally, the Krawczyk interval iteration with interval
arithmetic is applied to the initial boxes so as to check whether or not the
corresponding approximate zeros are real and to obtain the real root isolation
boxes. Meanwhile, an empirical construction of initial box is provided for
higher performance. Our experiments on many benchmarks show that the new hybrid
method is more efficient, compared with the traditional symbolic approaches.Comment: 23 pages. Accepted by ASCM2012. Some typos have been correcte
Optimal Solution of Linear Ordinary Differential Equations by Conjugate Gradient Method
Solving initial value problems and boundary value problems of Linear Ordinary
Differential Equations (ODEs) plays an important role in many applications.
There are various numerical methods and solvers to obtain approximate solutions
represented by points. However, few work about optimal solution to minimize the
residual can be found in the literatures. In this paper, we first use Hermit
cubic spline interpolation at mesh points to represent the solution, then we
define the residual error as the square of the L2 norm of the residual obtained
by substituting the interpolation solution back to ODEs. Thus, solving ODEs is
reduced to an optimization problem in curtain solution space which can be
solved by conjugate gradient method with taking advantages of sparsity of the
corresponding matrix. The examples of IVP and BVP in the paper show that this
method can find a solution with smaller global error without additional mesh
points.Comment: 9 pages,6 figure
Facial Reduction and SDP Methods for Systems of Polynomial Equations
The real radical ideal of a system of polynomials with finitely many complex
roots is generated by a system of real polynomials having only real roots and
free of multiplicities. It is a central object in computational real algebraic
geometry and important as a preconditioner for numerical solvers. Lasserre and
co-workers have shown that the real radical ideal of real polynomial systems
with finitely many real solutions can be determined by a combination of
semi-definite programming (SDP) and geometric involution techniques. A
conjectured extension of such methods to positive dimensional polynomial
systems has been given recently by Ma, Wang and Zhi.
We show that regularity in the form of the Slater constraint qualification
(strict feasibility) fails for the resulting SDP feasibility problems. Facial
reduction is then a popular technique whereby SDP problems that fail strict
feasibility can be regularized by projecting onto a face of the convex cone of
semi-definite problems.
In this paper we introduce a framework for combining facial reduction with
such SDP methods for analyzing and positive dimensional real ideals of real
polynomial systems. The SDP methods are implemented in MATLAB and our geometric
involutive form is implemented in Maple. We use two approaches to find a
feasible moment matrix. We use an interior point method within the CVX package
for MATLAB and also the Douglas-Rachford (DR) projection-reflection method.
Illustrative examples show the advantages of the DR approach for some
problems over standard interior point methods. We also see the advantage of
facial reduction both in regularizing the problem and also in reducing the
dimension of the moment matrices. Problems requiring more than one facial
reduction are also presented
Structural index reduction algorithms for differential algebraic equations via fixed-point iteration
Motivated by Pryce's structural index reduction method for differential
algebraic equations (DAEs), we show the complexity of the fixed-point iteration
algorithm and propose a fixed-point iteration method with parameters. It leads
to a block fixed-point iteration method which can be applied to large-scale
DAEs with block upper triangular structure. Moreover, its complexity analysis
is also given in this paper.Comment: 19 page
Recursive Geman-McClure method for implementing second-order Volterra filter
The second-order Volterra (SOV) filter is a powerful tool for modeling the
nonlinear system. The Geman-McClure estimator, whose loss function is
non-convex and has been proven to be a robust and efficient optimization
criterion for learning system. In this paper, we present a SOV filter, named
SOV recursive Geman-McClure, which is an adaptive recursive Volterra algorithm
based on the Geman-McClure estimator. The mean stability and mean-square
stability (steady-state excess mean square error (EMSE)) of the proposed
algorithm is analyzed in detail. Simulation results support the analytical
findings and show the improved performance of the proposed new SOV filter as
compared with existing algorithms in both Gaussian and impulsive noise
environments
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