Solving Systems Pdf

[Justina Ky]

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Superconducting quantum circuits are a promising candidate for building scalable quantum computers. Here, we use a four-qubit superconducting quantum processor to solve a two-dimensional system of linear equations based on a quantum algorithm proposed by Harrow, Hassidim, and Lloyd [Phys. Rev. Lett. 103, 150502 (2009)], which promises an exponential speedup over classical algorithms under certain circumstances. We benchmark the solver with quantum inputs and outputs, and characterize it by nontrace-preserving quantum process tomography, which yields a process fidelity of 0.8370.006. Our results highlight the potential of superconducting quantum circuits for applications in solving large-scale linear systems, a ubiquitous task in science and engineering.

(a) False color photomicrograph and (b) simplified circuit schematic of the superconducting quantum circuit for solving 22 linear equations. Shown are the four Xmon qubits, marked from Q1 to Q4, and their corresponding readout resonators, marked from R1 to R4.

I am currently a Ph.D. student in economics, and I am looking to find a way to model and solve dynamic systems of nonlinear equations. The goal is to create a dynamic Computable General Equilibrium (CGE) model in Julia. I am currently a beginner in dynamic systems and have never programmed such a model for conducting simulations. Thank you in advance for your responses.

A more specific description of your model would help. Eg parameters, endogenous objects, relevant functional equations (equilibrium conditions, optimality), state space, whether it is continuous or discrete time, etc. And whether you just want to solve a few times, or estimate (ie solve a gazillion of times, fast).

A CGE (Computable General Equilibrium) is a mathematical model that comprises a set of equations (often nonlinear) describing the behaviors of various economic agents within an economy. These models enable the modeling of an economy and conducting simulations. For instance, they help understand how an increase in public spending affects the initial state of an economy and towards which new equilibrium state different economic aggregates will converge. As far as I understand, dynamic models like these allow predictions for the future. I hope this clarifies it.

Are these equations differential equations? or are they stochastic processes? The packages that can be helpful for you depend on this context. But in any case, to the best of my understanding, Julia seems to have the best-of-class packages for various flavours of dynamical modelling. Perhaps have a look in the functionality offered by:

While some econ models (especially continuous time) have a PDE representation, classic PDE solvers are usually not very useful outside special cases since they expect traditional boundary conditions, while in most simple macro models agents live forever. My preference is spectral methods for mid-scale models, but larger models are still tricky (some are using deep learning).

So I have found that NonlinearSolve.jl is generally pretty good for numerically solving these problems. Now keep in mind, in many cases econ papers will try to find some analytic solution to these simple model, which are easier to interpret. An analytic solution will let you see the functions for the optimal values of the quantities or variables of interest.

In the case of DSGE or such models, I think a lot of that can be done with the ControlSystems.jl package. Many DSGE models are solved using linear quadratic regulators. So the lqr() function in ControlSystems will do what you want.

A classic problem in mathematics is solving systems of polynomial equations in several unknowns. Today, polynomial models are ubiquitous and widely used across the sciences. They arise in robotics, coding theory, optimization, mathematical biology, computer vision, game theory, statistics, and numerous other areas.

This book furnishes a bridge across mathematical disciplines and exposes many facets of systems of polynomial equations. It covers a wide spectrum of mathematical techniques and algorithms, both symbolic and numerical.

The first half of the book gives a snapshot of the state of the art of the topic. Familiar themes are covered in the first five chapters, including polynomials in one variable, Grbner bases of zero-dimensional ideals, Newton polytopes and Bernstein's Theorem, multidimensional resultants, and primary decomposition.

The second half of the book explores polynomial equations from a variety of novel and unexpected angles. It introduces interdisciplinary connections, discusses highlights of current research, and outlines possible future algorithms. Topics include computation of Nash equilibria in game theory, semidefinite programming and the real Nullstellensatz, the algebraic geometry of statistical models, the piecewise-linear geometry of valuations and amoebas, and the Ehrenpreis-Palamodov theorem on linear partial differential equations with constant coefficients.

Throughout the text, there are many hands-on examples and exercises, including short but complete sessions in Maple, MATLAB, Macaulay 2, Singular, PHCpack, CoCoA, and SOSTools software. These examples will be particularly useful for readers with no background in algebraic geometry or commutative algebra. Within minutes, readers can learn how to type in polynomial equations and actually see some meaningful results on their computer screens.

Methods for solving systems of polynomial equations in the tropical semiring promise to have wide-ranging applications and have not been treated in monograph before. The book is written in an lively style with many examples, comments, computer algebra sessions and a generous dose of tempting exercises. It can be read with a reasonably good knowledge of basic algebra and is well suited for a lecture course on the graduate level. For the researcher who wants to get an accessible introduction to current techniques in polynomial system solving it can be highly recommended.

So I have come across an issue where it would be very nice to solve systems of linear equations over semirings but I have no clue how to do that. Over a field I would use Gaussian elimination but I'm at a loss of how to find solution spaces to even rings much less semirings. In fact I'm not 100% sure this is decidable. The problem demands that I be able to do this for a wide class of semirings (preferably all) and not just 1 specific kind.

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The first principle is: you can't design around a self you don't understand. This was a quote that came up from one of our clients. She's a fabulous entrepreneur, and she was talking about everything she was learning in her work with Talentism and her clarity coach. She was starting to understand: design is possible, I can change the system around me, but I can only change that to the extent I understand myself. Because the system affects me, and the system is a part of me, and I'm creating that system. So I have to start with me, and I have to understand myself, because mostly I'm creating this system around me through unconscious action and bias. It's not part of a plan. It's not part of a conscious, logical, thoughtful, slow-thinking process. It's happening many times every day in little movements that are compounding into a big system effect.

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