An inequality for the trace of matrix product

1992 ◽  
Vol 37 (2) ◽  
pp. 239-240 ◽  
Author(s):  
J.K. Baksalary ◽  
S. Puntanen
Keyword(s):  
Matrix Product ◽  
Trace Of Matrix

Inequalities for the trace of matrix product

1994 ◽  
Vol 39 (12) ◽  
pp. 2489-2490 ◽  
Author(s):  
Yuguang Fang ◽  
K.A. Loparo ◽  
Xiangbo Feng
Keyword(s):  
Matrix Product ◽  
Trace Of Matrix

scholarly journals Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

2021 ◽  
Vol 5 (1) ◽  
pp. 8
Author(s):  
Cundi Han ◽  
Yiming Chen ◽  
Da-Yan Liu ◽  
Driss Boutat
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Governing Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
The Matrix ◽  
The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

scholarly journals Optimal sampling of dynamical large deviations via matrix product states

2021 ◽  
Vol 103 (6) ◽  
Author(s):  
Luke Causer ◽  
Mari Carmen Bañuls ◽  
Juan P. Garrahan
Keyword(s):  
Large Deviations ◽  
Matrix Product ◽  
Optimal Sampling ◽  
Matrix Product States

scholarly journals Matrix Product State Simulations of Non-Equilibrium Steady States and Transient Heat Flows in the Two-Bath Spin-Boson Model at Finite Temperatures

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 77
Author(s):  
Angus J. Dunnett ◽  
Alex W. Chin
Keyword(s):  
Finite Temperature ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Steady States ◽  
Matrix Product ◽  
Open Quantum ◽  
Matrix Product State ◽  
Wide Range ◽  
Boson Model ◽  
Non Equilibrium

Simulating the non-perturbative and non-Markovian dynamics of open quantum systems is a very challenging many body problem, due to the need to evolve both the system and its environments on an equal footing. Tensor network and matrix product states (MPS) have emerged as powerful tools for open system models, but the numerical resources required to treat finite-temperature environments grow extremely rapidly and limit their applications. In this study we use time-dependent variational evolution of MPS to explore the striking theory of Tamascelli et al. (Phys. Rev. Lett. 2019, 123, 090402.) that shows how finite-temperature open dynamics can be obtained from zero temperature, i.e., pure wave function, simulations. Using this approach, we produce a benchmark dataset for the dynamics of the Ohmic spin-boson model across a wide range of coupling strengths and temperatures, and also present a detailed analysis of the numerical costs of simulating non-equilibrium steady states, such as those emerging from the non-perturbative coupling of a qubit to baths at different temperatures. Despite ever-growing resource requirements, we find that converged non-perturbative results can be obtained, and we discuss a number of recent ideas and numerical techniques that should allow wide application of MPS to complex open quantum systems.

scholarly journals Gaussian variant of Freivalds’ algorithm for efficient and reliable matrix product verification

2020 ◽  
Vol 26 (4) ◽  
pp. 273-284
Author(s):  
Hao Ji ◽  
Michael Mascagni ◽  
Yaohang Li
Keyword(s):  
False Positive ◽  
Measure Zero ◽  
Matrix Multiplication ◽  
Numerical Linear Algebra ◽  
Floating Point ◽  
Matrix Product ◽  
Computing Environment ◽  
Deterministic Algorithms ◽  
Product Verification ◽  
Point Arithmetic

AbstractIn this article, we consider the general problem of checking the correctness of matrix multiplication. Given three n\times n matrices 𝐴, 𝐵 and 𝐶, the goal is to verify that A\times B=C without carrying out the computationally costly operations of matrix multiplication and comparing the product A\times B with 𝐶, term by term. This is especially important when some or all of these matrices are very large, and when the computing environment is prone to soft errors. Here we extend Freivalds’ algorithm to a Gaussian Variant of Freivalds’ Algorithm (GVFA) by projecting the product A\times B as well as 𝐶 onto a Gaussian random vector and then comparing the resulting vectors. The computational complexity of GVFA is consistent with that of Freivalds’ algorithm, which is O(n^{2}). However, unlike Freivalds’ algorithm, whose probability of a false positive is 2^{-k}, where 𝑘 is the number of iterations, our theoretical analysis shows that, when A\times B\neq C, GVFA produces a false positive on set of inputs of measure zero with exact arithmetic. When we introduce round-off error and floating-point arithmetic into our analysis, we can show that the larger this error, the higher the probability that GVFA avoids false positives. Moreover, by iterating GVFA 𝑘 times, the probability of a false positive decreases as p^{k}, where 𝑝 is a very small value depending on the nature of the fault on the result matrix and the arithmetic system’s floating-point precision. Unlike deterministic algorithms, there do not exist any fault patterns that are completely undetectable with GVFA. Thus GVFA can be used to provide efficient fault tolerance in numerical linear algebra, and it can be efficiently implemented on modern computing architectures. In particular, GVFA can be very efficiently implemented on architectures with hardware support for fused multiply-add operations.

scholarly journals Matrix product wave function of the ground state and elementary excitations in the spin- 12 chain

2021 ◽  
Vol 103 (12) ◽  
Author(s):  
Jintae Kim ◽  
Minsoo Kim ◽  
Pramod Padmanabhan ◽  
Jung Hoon Han ◽  
Hyun-Yong Lee
Keyword(s):  
Wave Function ◽  
Ground State ◽  
Matrix Product ◽  
Elementary Excitations

scholarly journals Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 984
Author(s):  
Regina Finsterhölzl ◽  
Manuel Katzer ◽  
Andreas Knorr ◽  
Alexander Carmele
Keyword(s):  
Time Evolution ◽  
Nearest Neighbor ◽  
Matrix Product ◽  
Body System ◽  
Many Body ◽  
Initial State ◽  
Matrix Product States ◽  
Open Quantum ◽  
Many Body Systems ◽  
The Many

This paper presents an efficient algorithm for the time evolution of open quantum many-body systems using matrix-product states (MPS) proposing a convenient structure of the MPS-architecture, which exploits the initial state of system and reservoir. By doing so, numerically expensive re-ordering protocols are circumvented. It is applicable to systems with a Markovian type of interaction, where only the present state of the reservoir needs to be taken into account. Its adaption to a non-Markovian type of interaction between the many-body system and the reservoir is demonstrated, where the information backflow from the reservoir needs to be included in the computation. Also, the derivation of the basis in the quantum stochastic Schrödinger picture is shown. As a paradigmatic model, the Heisenberg spin chain with nearest-neighbor interaction is used. It is demonstrated that the algorithm allows for the access of large systems sizes. As an example for a non-Markovian type of interaction, the generation of highly unusual steady states in the many-body system with coherent feedback control is demonstrated for a chain length of N=30.

Erratum: Generalized Inverse of a Matrix Product

SIAM Review ◽  
1967 ◽  
Vol 9 (2) ◽  
pp. 249-249 ◽  
Author(s):  
T. N. E. Greville
Keyword(s):  
Generalized Inverse ◽  
Matrix Product
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