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What's going wrong with my implementation of a quartic equation solver?

Here is my code on GitHub. I followed this:

In fact the real implementation starts at line 271, where I create the monic poly.

Does your cubic root solver work? You should have it reduced to only returning a single real root for use within the quartic solver. Run a number of test cases on the cubic solver first, including equations with both 1 and 3 real roots. Only after this is verified should you try to debug the quartic solver.

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I have 34 variables and 34 equations. What does this mean? I played around with guess values but none seem to be making an effect. One of my variables is clearly a sum of two positive values and it's giving me a negative.

where \(\left|\psi\right>\) is the state vector and \(H\) is the matrix representation of the Hamiltonian. This matrix equation can, in principle, be solved by diagonalizing the Hamiltonian matrix \(H\). In practice, however, it is difficult to perform this diagonalization unless the size of the Hilbert space (dimension of the matrix \(H\)) is small. Analytically, it is a formidable task to calculate the dynamics for systems with more than two states. If, in addition, we consider dissipation due to the inevitable interaction with a surrounding environment, the computational complexity grows even larger, and we have to resort to numerical calculations in all realistic situations. This illustrates the importance of numerical calculations in describing the dynamics of open quantum systems, and the need for efficient and accessible tools for this task.

The Schrdinger equation, which governs the time-evolution of closed quantum systems, is defined by its Hamiltonian and state vector. In the previous section, Using Tensor Products and Partial Traces, we showed how Hamiltonians and state vectors are constructed in QuTiP. Given a Hamiltonian, we can calculate the unitary (non-dissipative) time-evolution of an arbitrary state vector \(\left|\psi_0\right>\) (psi0) using the QuTiP function qutip.sesolve. It evolves the state vector and evaluates the expectation values for a set of operators expt_ops at the points in time in the list times, using an ordinary differential equation solver.

The function qutip.sesolve returns an instance of qutip.solver.Result, as described in the previous section Dynamics Simulation Results. The attribute expect in result is a list of expectation values for the operators that are included in the list in the fourth argument. Adding operators to this list results in a larger output list returned by the function (one array of numbers, corresponding to the times in times, for each operator).

The standard approach for deriving the equations of motion for a system interacting with its environment is to expand the scope of the system to include the environment. The combined quantum system is then closed, and its evolution is governed by the von Neumann equation

includes the original system Hamiltonian \(H_{\rm sys}\), the Hamiltonian for the environment \(H_{\rm env}\), and a term representing the interaction between the system and its environment \(H_{\rm int}\). Since we are only interested in the dynamics of the system, we can at this point perform a partial trace over the environmental degrees of freedom in Eq. (2), and thereby obtain a master equation for the motion of the original system density matrix. The most general trace-preserving and completely positive form of this evolution is the Lindblad master equation for the reduced density matrix \(\rho = {\rm Tr}_{\rm env}[\rho_{\rm tot}]\)

where the \(C_n = \sqrt{\gamma_n} A_n\) are collapse operators, and \(A_n\) are the operators through which the environment couples to the system in \(H_{\rm int}\), and \(\gamma_n\) are the corresponding rates. The derivation of Eq. (3) may be found in several sources, and will not be reproduced here. Instead, we emphasize the approximations that are required to arrive at the master equation in the form of Eq. (3) from physical arguments, and hence perform a calculation in QuTiP:

Secular approximation Stipulates that elements in the master equation corresponding to transition frequencies satisfy \(|\omega_{ab}-\omega_{cd}| \ll 1/\tau_{\rm sys}\), i.e., all fast rotating terms in the interaction picture can be neglected. It also ignores terms that lead to a small renormalization of the system energy levels. This approximation is not strictly necessary for all master-equation formalisms (e.g., the Block-Redfield master equation), but it is required for arriving at the Lindblad form (3) which is used in qutip.mesolve.

For systems with environments satisfying the conditions outlined above, the Lindblad master equation (3) governs the time-evolution of the system density matrix, giving an ensemble average of the system dynamics. In order to ensure that these approximations are not violated, it is important that the decay rates \(\gamma_n\) be smaller than the minimum energy splitting in the system Hamiltonian. Situations that demand special attention therefore include, for example, systems strongly coupled to their environment, and systems with degenerate or nearly degenerate energy levels.

For non-unitary evolution of a quantum systems, i.e., evolution that includesincoherent processes such as relaxation and dephasing, it is common to usemaster equations. In QuTiP, the function qutip.mesolve is used for both:the evolution according to the Schrdinger equation and to the master equation,even though these two equations of motion are very different. The qutip.mesolvefunction automatically determines if it is sufficient to use the Schrdingerequation (if no collapse operators were given) or if it has to use themaster equation (if collapse operators were given). Note that to calculatethe time evolution according to the Schrdinger equation is easier and muchfaster (for large systems) than using the master equation, so if possible thesolver will fall back on using the Schrdinger equation.

What is new in the master equation compared to the Schrdinger equation areprocesses that describe dissipation in the quantum system due to its interactionwith an environment. These environmental interactions are defined by theoperators through which the system couples to the environment, and rates thatdescribe the strength of the processes.

Okay, time for DifferentialEquations.jl. I left it for last because it is by far the most complex of the solver suites, and pulls ideas from many of them. While most of the other suite offer no more than about 15 methods on the high end (with most offering about 8 or less), DifferentialEquations.jl offers 200+ methods and is continually growing. Like the standard Python and R suites, it offers wrappers to Sundials, ODEPACK, and Hairer methods. However, since Julia code is always JIT compiled, its wrappers are more akin to PyDSTool or JiTCODE in terms of efficiency. Thus all of the standard methods mentioned before are available in this suite.

More edits today. This time for Maple. I missed some things that Maple has and updated the discussion about Maple to include its BVP/DDE solvers and its ability to handle arbitrary precision and complex numbers.

DifferentialEquations.jl hit an issue in one of the stiff DDE benchmarks. This makes the current ranking of stiff DDE solvers go RADAR5 (Hairer) > Maple > DifferenitalEquations.jl. There is some active development to fix this.

Which framework would you recommend? Does DifferentialEquations.jl / Julia include MINLP solvers (like BONMIN, NOMAD or SCIP)?

Searching the web, I came up with DAE Tools, which seems to include almost everything I want, though I have yet to test it. Maybe you could give it a look and include it in your article.

Hey,

ode15s uses finite differences unless you supply a Jacobian. In fact, the fact that the MATLAB ODE Suite relies on finite differences is the reason that ode23s is only recommended if you supply a Jacobian function because even Rosenbrock-W methods lose accuracy with more inaccurate Jacobian information (while implicit equations just use the Jacobian as a line search).

For Complex Numbers I am just looking to see that the integrators support solving equations with dependent variables that are complex numbers. Can it do it, and is there a good spread of methods that can do it (if it only has one it gets dinged).

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An overview of the discretization and solution algorithm of the DWE is described in section. The derivation begins with the discretization of continuity equation and momentum equations. The discrete form of the SWE is then obtained from the continuity and momentum equations. Finally, the solution algorithm is described.

where \Delta t// is the time step, and the velocities have been interpolated in time using the generalized Crank-Nicolson method (which is used to weight the contribution of velocities at time steps and ). Since the momentum equation is rotation invariant, it will be assumed that s_{i,k}// is the sign in the outward direction at face k. Note that the face areas are treated explicitly. This increases the stability of the solver but reduces the accuracy for large time steps and limits the wetting and drying to one cell at a time for each time step.

Once the DSW equation has been solved, the velocities can be recovered by substituting the water surface elevation back into the Diffusion Wave equation. The above equation may be written in compact form as

Where \boldsymbol{\Omega}// is the vector of all cell volumes, \boldsymbol{Z}// is the vector of all cell water surface elevations at time n+1, \boldsymbol{\Psi}// is the coefficient matrix of the system and \textbf{b}// is the right-hand-side vector.

The system of equations is mildly non-linear due to the bathymetric relationship for \Omega// as a function of \boldsymbol{Z}// . The Jacobian (derivative) of with respect to \boldsymbol{Z}// is given by another bathymetric relationship \boldsymbol { A(z_s)}// : the diagonal matrix of cell wet surface areas. If this information is known, a Newton-like technique can be applied to solve the system of equations, producing the iterative formula, 2351a5e196