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Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the inclusion of missing integrability conditions and the statement of of existence and uniqueness theorems for local analytic solutions of such systems. Further, they are useful in obtaining systems in a form more amenable to exact and approximate solution methods. Maple's \maple{dsolve} and \maple{pdsolve} algorithms for solving PDE and ODE often automatically call such routines during applications. Indeed, even casual users of Maple's dsolve and pdsolve commands have probably unknowingly used Maple's differential-elimination algorithms. Suppose that a system of PDE has been reduced by differential-elimination to a system whose automatic existence and uniqueness algorithm has been determined to be finite-dimensional. We present an algorithm for rewriting the output as a system of parameterized ODE. Exact methods and numerical methods for solving ODE and DAE can be applied to this form.

The advent of increasingly powerful language models has raised expectations for language-based interactions. However, controlling these models is a challenge, emphasizing the need to be able to investigate the feasibility and value of their application. We present PROMISE, a framework that facilitates the development of complex language-based interactions with information systems. Its use of state machine modeling concepts enables model-driven, dynamic prompt orchestration across hierarchically nested states and transitions. This improves the control of the behavior of language models and thus enables their effective and efficient use. In this technical report we show the benefits of PROMISE in the context of application scenarios within health information systems and demonstrate its ability to handle complex interactions. We also include code examples and present default user interfaces available as part of PROMISE., Comment: Title and abstract changed

Integro-differential-algebraic equations (IDAE)s are widely used in applications of engineering and analysis. When there are hidden constraints in an IDAE, structural analysis is necessary. But if derivatives of dependent variables appear in their integrals, the existing definition of the signature matrix for an IDAE cannot be satisfied. Moreover, if an IDAE has a singular Jacobian matrix after structural analysis by the Sigma-method, improved structural analysis methods are proposed to regularize it. However, the optimal value of an IDAE may be negative which can not ensure the termination of the regularization. Furthermore, overestimation of the signature matrix may also lead to failure of its structural analysis. In this paper, firstly, we redefine the signature matrix and introduce a definition of the degree of freedom for IDAEs. Thus, the termination of improved structural analysis methods can be guaranteed. Secondly, the detection method by points is proposed to deal with the problem of overestimation of signature matrix. Thirdly, the embedding method has proved to suitable for structural unamenable IDAEs, including those types that arise from symbolic cancellation and numerical degeneration. Finally, the global numerical method is applied to an example of two-stage drive system which can help to find all solutions for IDAEs by witness points. Hopefully, through the example of pendulum curtain, the approach for IDAEs proposed in this paper can be applied to integro-partial-differential-algebraic equations (IPDAE)s., Comment: 33 pages, 4 figures, conference

To find consistent initial data points for a system of differential-algebraic equations, requires the identification of its missing constraints. An efficient class of structural methods exploiting a dependency graph for this task was initiated by Pantiledes. More complete methods rely on differential-algebraic geometry but suffer from other issues (e.g. high complexity). In this paper we give a new class of efficient structural methods combined with new tools from numerical real algebraic geometry that has much improved completeness properties. Existing structural methods may fail for a system of differential-algebraic equations if its Jacobian matrix after differentiation is still singular due to symbolic cancellation or numerical degeneration. Existing structural methods can only handle degenerated cases caused by symbolic cancellation. However, if a system has parameters, then its parametric Jacobian matrix may be still singular after application of the structural method for certain values of the parameters. This case is called numerical degeneration. For polynomially nonlinear systems of differential-algebraic equations, numerical methods are given to solve both degenerated cases using numerical real algebraic geometry. First, we introduce a witness point method, which produces at least one witness point on every constraint component. This can help to ensure constant rank and detection of degeneration on all components of such systems. Secondly, we present a Constant Rank Embedding Lemma, and based on it propose an Index Reduction by Embedding (IRE) method which can construct an equivalent system with a full rank Jacobian matrix. Thirdly, IRE leads to a global structural differentiation method, to solve degenerated differential-algebraic equations on all components numerically. Application examples from circuits, mechanics, are used to demonstrate our method., Comment: arXiv admin note: substantial text overlap with arXiv:2111.08160

Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for polynomial systems of differential-algebraic equations, numerical methods are given to solve such cases using numerical real algebraic geometry. First, we propose an embedding method that for a given real analytic system constructs an equivalent system with a full-rank Jacobian matrix. Secondly, we introduce a witness point method, which can help to detect degeneration on all components of constraints of such systems. Thirdly, the two methods above lead to a numerical global structural analysis method for structurally unamenable differential-algebraic equations on all components of constraints., Comment: 27 pages, 5 figures,3 tables

Given a $k\times n$ integer primitive matrix $\bf{A}$ (i.e., a matrix can be extended to an $n\times n$ unimodular matrix over the integers) with the maximal absolute value of entries $\|\bf{A}\|$ bounded by {an integer} $\lambda$ from above, we study the probability that the $m\times n$ matrix extended from $\bf{A}$ by appending other $m-k$ row vectors of dimension $n$ with entries chosen randomly and independently from the uniform distribution over $\{0, 1,\ldots, \lambda-1\}$ is still primitive. We present a complete and rigorous proof of a lower bound on the probability, which is at least a constant for fixed $m$ in the range $[k+1, n-4]$. As an application, we prove that there exists a fast Las Vegas algorithm that completes a $k\times n$ primitive matrix $\bf{A}$ to an $n\times n$ unimodular matrix within expected $\tilde{O}(n^{\omega}\log \|\bf{A}\|)$ bit operations, where $\tilde{O}$ is big-$O$ but without log factors, $\omega$ is the exponent on the arithmetic operations of matrix multiplication.

Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper formulates the notion of numerical factorization based on the geometry of polynomial spaces and the stratification of factorization manifolds. Furthermore, this paper establishes the existence, uniqueness, Lipschitz continuity, condition number, and convergence of the numerical factorization to the underlying exact factorization, leading to a robust and efficient algorithm with a MATLAB implementation capable of accurate polynomial factorizations using floating point arithmetic even if the coefficients are perturbed.

We propose a method for tracing implicit real algebraic curves defined by polynomials with rank-deficient Jacobians. For a given curve $f^{-1}(0)$, it first utilizes a regularization technique to compute at least one witness point per connected component of the curve. We improve this step by establishing a sufficient condition for testing the emptiness of $f^{-1}(0)$. We also analyze the convergence rate and carry out an error analysis for refining the witness points. The witness points are obtained by computing the minimum distance of a random point to a smooth manifold embedding the curve while at the same time penalizing the residual of $f$ at the local minima. To trace the curve starting from these witness points, we prove that if one drags the random point along a trajectory inside a tubular neighborhood of the embedded manifold of the curve, the projection of the trajectory on the manifold is unique and can be computed by numerical continuation. We then show how to choose such a trajectory to approximate the curve by computing eigenvectors of certain matrices. Effectiveness of the method is illustrated by examples.

We present a new method for visualizing implicit real algebraic curves inside a bounding box in the $2$-D or $3$-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing space curve in higher-dimensional space. Since the topology of a curve near a singular point of it is not numerically stable, we trace only the curve outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is $\epsilon$-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the projection of the curve in $\mathbb{R}^2$, which is important for applications such as solving bi-parametric polynomial systems. The algorithm starts by computing three types of key points of the curve, namely the intersection of the curve with small spheres centered at singular points, regular critical points of every connected components of the curve, as well as intersection points of the curve with the given bounding box. It then traces the curve starting with and in the order of the above three types of points. This basic scheme is further enhanced by several optimizations, such as grouping singular points in natural clusters, tracing the curve by a try-and-resume strategy and handling "pseudo singular points". The effectiveness of the algorithm is illustrated by numerous examples. This manuscript extends our preliminary results that appeared in CASC 2018.

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 figures

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 figures

The celebrated integer relation finding algorithm PSLQ has been successfully used in many applications. PSLQ was only analyzed theoretically for exact input data, however, when the input data are irrational numbers, they must be approximate ones due to the finite precision of the computer. When the algorithm takes empirical data (inexact data with error bounded) instead of exact real numbers as its input, how do we theoretically ensure the output of the algorithm to be an exact integer relation? In this paper, we investigate the PSLQ algorithm for empirical data as its input. Firstly, we give a termination condition for this case. Secondly, we analyze a perturbation on the hyperplane matrix constructed from the input data and hence disclose a relationship between the accuracy of the input data and the output quality (an upper bound on the absolute value of the inner product of the exact data and the computed integer relation), which naturally leads to an error control strategy for PSLQ. Further, we analyze the complexity bound of the PSLQ algorithm for empirical data. Examples on transcendental numbers and algebraic numbers show the meaningfulness of our error control strategy., Comment: 23 pages; Math. Comp

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety.