Matrix Sign Research Articles

Alternating sign matrices and totally symmetric plane partitions

Alternating sign matrices and totally symmetric plane partitions

A Study of Algorithms for the p th Root of Matrix

Some results for Pth root of square matrix are revived. It shows that matrix sign function and Wiener- Hopf factorization plays important role in Pth root of matrix. Some new algorithms for computing P th root numerically can design by these results. We can analyze Stability properties of iterative methods for convergence.

Adaptive robust integrated guidance and control for thrust-vector-controlled aircraft by solving LQR online

In order to enhance the matching relationship between guidance subsystem and control subsystem of the thrust-vector-controlled aircraft, a kind of sufficient modeling and adaptive robust design facing to integrated guidance and control (IGC) is proposed. With respect to the researched aircraft, a linear state-dependent mathematical model facing to IGC design is first established. Based on the as-built model, a new kind of adaptive robust IGC law is proposed and it is composed by an adaptive optimal IGC law and a robustness-improved IGC law. In order to guarantee the global stability of time-varying control system, the adaptive optimal IGC law is designed by solving Riccati matrix equation on line and using matrix Sign function method. Furthermore, in order to enhance the robust ability against the unmatched system uncertainties, the robustness-improved IGC law is designed by using dynamic surface control approach and extended state observing strategy. Simulation results present that, the proposed IGC scheme presents more performance advantages compared with traditional IGC schemes, including the improvement of guidance precision and attitude stabilization. Furthermore, in the conditions of 256 simulation combinations, the minimum relative distances between aircraft position and target position are distributed from 0.103m to 5.333m, and the average value is 2.208m, which means the proposed IGC scheme possesses strong robustness against different and time-varying model uncertainties.

Minimal projections onto subspaces generated by sign-matrices

The aim of this paper is to calculate relative and absolute projection constants of certain subspaces of l1(n) and l∞(n) generated by eigenvectors of sign matrices. The main tool in our considerations is so called Chalmers–Metcalf operator (see Chalmers & Metcalf (1994) and Lewicki & Prophet (2021)). Also, some results from Castejon & Lewicki (2014) and Castejon & Lewicki (2019) will be applied.

A Globally Convergent Iterative Method for Matrix Sign Function and Its Application for Determining the Eigenvalues of a Matrix Pencil

In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method.

An efficient iterative method for matrix sign function with application in stability analysis of control systems using spectrum splitting

The goal of this study is to construct a novel iterative method to compute the matrix sign function using a different approach. It is discussed that the new method is globally convergent and asymptotically stable. It achieves the sixth order of convergence and only requires five matrix–matrix multiplications. The obtained results are extended to compute the number of eigenvalues of a matrix in a specified region of the complex plane. This is done by performing appropriate sequence of matrix sign computations. An application of this technique has been discussed in stability analysis of linear time‐invariant dynamic systems in control theory. Numerical results have been given to justify the effectual performance and superiority of the proposed method. Matrices of various sizes have been considered for this purpose.

Partial Alternating Sign Matrix Bijections and Dynamics

Partial Alternating Sign Matrix Bijections and Dynamics

A finite volume scheme for solving an alternative morphodynamic model with porosity

AbstractThis work proposes an extended numerical scheme based on the finite volume method that is intended to solve a new morphodynamic model with porosity. The St-Venant equations and the Exner equation are coupled in this model, accounting for the variation of the porosity function . In the morphodynamic system, the Jacobian matrix's sign is taken into account when reconstructing the numerical fluxes using an improved Roe's approach to assure correct numerical computations.

Frobenius-König theorem for classes of (0,±1)-matrices

The Frobenius-König Theorem has a central role in combinatorial matrix theory; it characterizes when a (0,1)-matrix X contains a permutation matrix P (meaning P≤X entrywise). Our goal is to investigate similar questions for (0,±1)-matrices, and a main result is a Frobenius-König Theorem for the class of (0,±1)-matrices with all row and column sums being 1. Moreover, some related results are shown for alternating sign matrices.

A Class of Sixth-Order Iterative Methods for Solving Nonlinear Systems: The Convergence and Fractals of Attractive Basins

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω-continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems.

Weighted Projections of Alternating Sign Matrices: Latin-Like Squares and the ASM Polytope

The weighted projection of an alternating sign matrix (ASM) was introduced by Brualdi and Dahl (2018) as a step towards characterising a generalisation of Latin squares they defined using alternating sign hypermatrices. Given row-vector $z_n = (n,\dots,2,1)$, the weighted projection of an ASM $A$ is equal to $z_nA$. Brualdi and Dahl proved that the weighted projection of an $n \times n$ ASM is majorized by the vector $z_n$, and conjectured that any positive integer vector majorized by $z_n$ is the weighted projection of some ASM. The main result of this paper presents a proof of this conjecture, via monotone triangles. A relaxation of a monotone triangle, called a row-increasing triangle, is introduced. It is shown that for any row-increasing triangle $T$, there exists a monotone triangle $M$ such that each entry of $M$ occurs the same number of times as in $T$. A construction is also outlined for an ASM with given weighted projection. The relationship of the main result to existing results concerning the ASM polytope $ASM_n$ is examined, and a characterisation is given for the relationship between elements of $ASM_n$ corresponding to the same point in the regular $n$-permutohedron. Finally, the limitations of the main result for characterising alternating sign hypermatrix Latin-like squares are considered.

Recursive quantum eigenvalue and singular-value transformation: Analytic construction of matrix sign function by Newton iteration

Quantum eigenvalue transformation (QET) and its generalization, quantum singular value transformation (QSVT), are versatile quantum algorithms that allow us to apply broad matrix functions to quantum states, which cover many significant quantum algorithms such as Hamiltonian simulation. However, finding a parameter set which realizes preferable matrix functions in these techniques is difficult for large-scale quantum systems: there is no analytical result other than trivial cases as far as we know and we often suffer also from numerical instability. In this Letter, we propose recursive QET or QSVT (R-QET or R-QSVT) in which we can execute complicated matrix functions by recursively organizing block-encoding by low-degree QET or QSVT. Owing to the simplicity of recursive relations, it works only with a few parameters with exactly determining the parameters, while its iteration results in complicated matrix functions. In particular, by exploiting the recursive relation of Newton iteration, we construct the matrix sign function, which can be applied for eigenstate filtering for example, in a tractable way. We show that an analytically obtained parameter set composed of only eight different values is sufficient for executing QET of the matrix sign function with an arbitrarily small error ɛ. Our protocol will serve as an alternative protocol for constructing QET or QSVT for some useful matrix functions without numerical instability. Published by the American Physical Society 2024

Equations of Motion of Mechanical Systems with Switchable Constraints

The article deals with a numerical and analytical approach to solving the equations of motion, which enables to treat the considered problems with the change of system structure or number of degrees of freedom without interrupting the numerical integration process. The described methodology allows effectively incorporate switchable constraints in the systems in accordance with their flexible structures. The crucial idea is based on the formulation of the resulting differentialalgebraic equations into a saddle point system, where the switchable constraints are represented by a sign matrix with variable rank. In connection with this property, a pseudoinversion is applied to eliminate algebraic variables and transform the problem to the first order system of ordinary differential equations. Moreover, the time independent case leads to linear autonomous systems with non-diagonalizable matrices, as is proved. The relevant numerical scheme is based on Runge-Kutta methods, that correspond to the power series of the resulting matrix exponential for time independent problems. The methodology presented is illustrated on the idealized two-mass oscillator with a switchable constraint. The numerical experiments performed range from initial stages, through simple transient cases to damped intentional control. The advanced applications can be found in robotics, active and controlled systems, and in the simulations of complex systems in biology and related areas. Moreover, the methodology can also be applied in the simulation of transport systems, especially in relation to vehicle technology, a quarter car suspension system, a vibration control mechanism, a torsion system with a clutch, and machine balancing and storage should to be highlighted.

Extraction of deflating subspaces using disk function of a matrix pencil via matrix sign function with application in generalized eigenvalue problem

The objective of this study is to extract two pairs of left and right deflating subspaces of a regular matrix pencil A−λB corresponding to the eigenvalues lying inside and outside the unit disk via computation of left and right disk functions diskL(A−λB) and diskR(A−λB) which are obtained from matrix sign evaluations of two suitable matrices. We propose a novel fifth order iterative method for evaluating matrix sign function which can be implemented to compute disk functions. Convergence analysis of the proposed method has been investigated globally along with the illustration of attraction basins. Asymptotic stability of the method is also presented. Then an application of deflating subspaces in solving generalized eigenvalue problem has been explored. Numerical test problems have been performed by considering matrices of various sizes to justify the efficiency and superiority of the proposed method.

On λ-determinants and tiling problems

We review the connections between the octahedral recurrence, λ-determinants and tiling problems. This provides in particular a direct combinatorial interpretation of the λ-determinant (and generalizations thereof) of an arbitrary matrix in terms of domino tilings of Aztec diamonds. We also reinterpret the general Robbins–Rumsey formula for the rational function of consecutive minors, given by a summation over pairs of compatible alternating sign matrices, as the partition function for tilings of Aztec diamonds equipped with a general measure.

An Optimal Eighth-Order One-Parameter Single-Root Finder: Chaotic Dynamics and Stability Analysis

In this paper, we focus on a class of optimal eighth-order iterative methods, initially proposed by Sharma et al., whose second step can choose any fourth-order iterative method. By selecting the first two steps as an optimal fourth-order iterative method, we derive an optimal eighth-order one-parameter iterative method, which can solve nonlinear systems. Employing fractal theory, we investigate the dynamic behavior of rational operators associated with the iterative method through the Scaling theorem and Möbius transformation. Subsequently, we conduct a comprehensive study of the chaotic dynamics and stability of the iterative method. Our analysis involves the examination of strange fixed points and their stability, critical points, and the parameter spaces generated on the complex plane with critical points as initial points. We utilize these findings to intuitively select parameter values from the figures. Furthermore, we generate dynamical planes for the selected parameter values and ultimately determine the range of unstable parameter values, thus obtaining the range of stable parameter values. The bifurcation diagram shows the influence of parameter selection on the iteration sequence. In addition, by drawing attractive basins, it can be seen that this iterative method is superior to the same-order iterative method in terms of convergence speed and average iterations. Finally, the matrix sign function, nonlinear equation and nonlinear system are solved by this iterative method, which shows the applicability of this iterative method.

Off-Diagonally Symmetric Domino Tilings of the Aztec Diamond

We introduce a new symmetry class of domino tilings of the Aztec diamond, called the off-diagonal symmetry class, which is motivated by the off-diagonally symmetric alternating sign matrices introduced by Kuperberg in 2002. We use the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths to enumerate our new symmetry class. The number of off-diagonally symmetric domino tilings of the Aztec diamond can be expressed as a Pfaffian of a matrix whose entries satisfy a nice and simple recurrence relation.

GOE fluctuations for the maximum of the top path in alternating sign matrices

GOE fluctuations for the maximum of the top path in alternating sign matrices

Fast Differentiable Matrix Square Root and Inverse Square Root.

Computing the matrix square root and its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root and the inverse square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Padé Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equationusing the matrix sign function. A series of numerical tests show that both methods yield considerable speed-up compared with the SVD or the NS iteration. Moreover, we validate the effectiveness of our methods in several real-world applications, including de-correlated batch normalization, second-order vision transformer, global covariance pooling for large-scale and fine-grained recognition, attentive covariance pooling for video recognition, and neural style transfer. The experiments demonstrate that our methods can also achieve competitive and even slightly better performances. Code is available at https://github.com/KingJamesSong/FastDifferentiableMatSqrt.

A general approach for improving the Padé iterations for the matrix sign function

The Padé family of iterations is a well-known set of methods used to compute the matrix sign function, which includes classical methods such as Newton’s method, the Newton–Schultz iteration, and Halley’s method. In this paper, we present a general approach to enhance the Padé iterations by choosing an arbitrary three-point family of methods based on weight functions. We determine the weight functions in a way that, for a complex quadratic with distinct roots, the three-point methods are conformally conjugate to a complex polynomial with as many parameters as desired. This approach leads to a multi-parameter family of iterations for the matrix sign function, which allows us to discover many new methods, including the Padé family of iterations as a special case. We provide a convergence and stability analysis of the multi-parameter family and conduct numerical experiments to confirm the improved performance of the new methods. Although the three-point family of methods is arbitrarily chosen, our approach can be easily extended to any other multipoint methods.