# A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations

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## Abstract

The centered difference discretization of the spatial fractional coupled nonlinear Schrödinger equations obtains a discretized linear system whose coefficient matrix is the sum of a real diagonal matrix *D* and a complex symmetric Toeplitz matrix *T̃* which is just the symmetric real Toeplitz *T* plus an imaginary identity matrix *iI*. In this study, we present a medium-shifted splitting iteration method to solve the discretized linear system, in which the fast algorithm can be utilized to solve the Toeplitz linear system. Theoretical analysis shows that the new iteration method is convergent. Moreover, the new splitting iteration method naturally leads to a preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are tighter than those of the original coefficient matrix *A*. Finally, compared with the other algorithms by numerical experiments, the new method is more effective.

## Keywords

Spatial fractional Schrödinger equations Toeplitz matrix Medium-shifting iteration method Convergence Preconditioning## Abbreviations

- CNLS
coupled non-linear Schrödinger

- HSS
Hermitian and skew-Hermitian splitting

- IHSS
inexact Hermitian and skew-Hermitian splitting

- MS
medium-shifting

- PMS
preconditioned MS

- FFT
fast Fourier transform

## 1 Introduction

*x*. Moreover, it is just equivalent to the Riesz fractional derivative [2], say the following equation:

The spatial fractional Schrödinger equations have attracted wide attention in recent years. Their classical forms describe the evolution of microscopic particles, and they arise in the path integral over the Brownian motion. In [3, 4], Laskin extended firstly the path integral method to obtain the spatial fractional Schrödinger equations, which comes from the Brownian motion to the Lévy process. Guo and Xu [5] studied some physical applications of fractional Schrödinger equations. The existence of a global smooth solution of fractional nonlinear Schrödinger equations was discussed by Guo et al. [14]. Many authors have concentrated on the solution of fractional Schrödinger equations in one dimension [15, 16, 17]. Since one of the major characteristics of the fractional differential operator is being nonlocal, it is pretty hard to gain the true solution of fractional differential equations generally. Therefore, the numerical methods (e.g., some splitting iteration methods) have become usable and powerful tools in order to understand the behaviors of fractional differential equations. Recently, another approach based on the study of Lie symmetries of fractional differential equations can be found in the papers [18, 19].

From what we known, there exist certain studies on numerical methods for the fractional Schrödinger equations. The Crank–Nicolson difference scheme for the coupled nonlinear fractional Schrödinger equations was proposed by Wang et al. [20] in 2013. Subsequently, Atangana and Cloot [21] studied its stability and convergence for the space fractional variable-order Schrödinger equations. Since the scheme is nonlinearly implicit and coupled in computation, the iteration costs at each step are expensive. Moreover, the nonlocal property of the fractional operator resulted in dense or full matrices. Consequently, with the fractional centered difference formula, Wang et al. [22, 23] proposed recently an implicit conservative difference scheme, which is unconditionally stable, to discretize the spatial fractional CNLS equations. The coefficient matrix *A* of the discretized linear system about Eqs. (1.1) is \(D+\widetilde{T}\), the sum of a real diagonal matrix *D* and a complex symmetric Toeplitz matrix *T̃*, which is just the real symmetric Toeplitz matrix *T* plus an imaginary-identity matrix *Ĩ*, \(\widetilde{T}=T+iI\). In view of the coefficient matrix being full or dense, the Gaussian elimination method is unfeasible because of its heavy computational and storage load. The computational costs of the Krylov subspace iterations [24, 25, 26, 27] when solving the discretized linear system are quite expensive. Moreover, their convergence rates tend to be considerably worse.

It is appropriated that the coefficient matrix \(A=D+\widetilde{T}\) is non-Hermitian but positive definite. And then those derived iteration methods based on the famous HSS can be used to solve this complex symmetric and non-Hermitian positive definite linear system \(Au=b\), introduced by Bai et al. [28, 29] and so on. For example, the Hermitian and skew-Hermitian splitting (HSS) iteration method [30] was directly applied to solve the discretized linear system, and the following two variants were designed [31]:

*Case I*

*ω*is a given positive constant.

*Case II*

*ω*is a given positive constant.

Cases *I* and *II* almost shared the same computational efficiency, reported in the original paper. That is to say, at each HSS iteration step of *Case I*, the solution of a subsystem with the shifted matrix \(\omega I+\widetilde{I}\), where *ω* is the shifted parameter, made easily due to its diagonal structure. However, it is very costly and impractical in solving the shifted Hermitian subsystem with the coefficient matrix holding full and Toeplitz-like rather than Toeplitz structure. In order to reduce the complexity, the IHSS iteration or preconditioned HSS iteration [32, 33] were come up with to solve the non-Hermitian positive definite linear system. However, the tolerances (or the number of inner iterations) for the inner iterates about the first linear subsystem may be different and may be changed according to the outer iteration scheme. Therefore, the IHSS iteration is actually a nonstationary iteration method for solving the linear system, and the preconditioning matrix is not easy to choose for the preconditioned HSS iteration method. In order to overcome the disadvantages of these methods, Ran and Wang [31] proposed another HSS-like iteration method, and it is precisely stated in Case *II*.

In this paper, a new splitting iteration method with medium-shifting (MS) can be presented to solve the discretized linear system of the spatial fractional coupled nonlinear Schrödinger Eqs. (1.1), in which the inconvenience of alternation can be avoided. The shifted parameter is the average value of the maximum and minimum elements of the real diagonal matrix *D*, say medium-shifting. Theoretical analysis shows that the new iteration method is convergent. Moreover, the new splitting iteration method naturally induces a matrix splitting preconditioner. To further reduce the computational costs of the MS preconditioning matrix, we replace the real Toeplitz matrix *T* involved in *T̃* by a circulant matrix, obtaining a preconditioner for the discretized linear system from the space fractional coupled nonlinear Schrödinger equations. Thus, the linear system can be solved economically by the fast Fourier transform. Numerical results show that the PMS iteration method can significantly improve the convergent property.

Here are some essential notations. As usual, we use \(\mathbb{C}^{M\times M}\) to denote the \(M\times M\) complex matrix set and \(\mathbb{C}^{M}\) the *M*-dimensional complex vector space. \(X^{*}\) represents the conjugate transpose of a matrix or a vector *X*. The spectral radius of the matrix *A* is denoted by \(\rho (A)\). \(\lambda (\cdot )\) and \(\Sigma (\cdot )\) stand for the eigenvalues set and the singular values set of a corresponding matrix, respectively. \(\Vert \cdot \Vert _{2}\) represents the Euclidean norm of a corresponding matrix. By the way, *I* represents an identity matrix in general.

The organization of this paper is as follows. In Sect. 2, we derive the discretized linear system from spatial fractional CNLS equation (Eq. (1.1)). In Sect. 3, we present the MS iteration method and preconditioned MS iteration method, and establish the convergent theory. The numerical results are reported in Sect. 4. Finally, we end this document with some conclusions in Sect. 5.

## 2 Discretization of the spatial fractional CNLS equations

In this section, the spatial fractional coupled nonlinear Schrödinger equations were discretized to a linear system. The implicit conservative difference scheme with the fractional centered difference formula, that is unconditionally stable, will be given.

*N*and

*M*are positive integers. A temporal and spatial partition can be defined as \(t_{n}=n\tau \), \(n=0,1,\ldots ,N\), and \(x_{j}=x_{\mathrm{{L}}}+jh\), \(j=0,1,\ldots ,M+1\). The corresponding numerical solutions are defined by \(u_{j}^{n}\thickapprox u(x_{j},t_{n})\) and \(v_{j}^{n}\thickapprox v(x_{j},t_{n})\). In terms of the fractional centered difference [34], the fractional Laplacian \((-\Delta )^{\frac{\alpha }{2}}\) in the truncated bounded domain can be discretized as

*M*-dimensional vectors

*T̃*is a complex Toeplitz matrix which is just the symmetric real Toeplitz matrix

*T*plus an imaginary identity matrix

*iI*as follows:

*T*is symmetric and strictly diagonally dominant, then symmetric positive definite, and \(D^{n+1}\) is a nonnegative diagonal matrix. Thus, the matrix \(D^{n+1}+T\) is symmetric positive definite. Consequently, the coefficient matrix \(A^{n+1}\triangleq D^{n+1} +\widetilde{T}\) is a diagonal-plus-Toeplitz matrix and is symmetric and non-Hermitian positive definite.

## 3 Algorithms

*A*is a complex symmetric and non-Hermitian positive definite matrix of the form \(A=D+\widetilde{T}, D \in \mathbb{R}^{M\times M}\) is a real diagonal matrix of nonnegative diagonal elements, and \(\widetilde{T}\in \mathbb{C}^{M\times M}\) is a complex symmetric Toeplitz and non-Hermitian positive definite matrix.

*A*with respect to its diagonal and Toeplitz parts

*D*and

*T̃*, for a medium-shifting, we have

*D*. Thus, a splitting iteration method with medium-splitting (MS) is precisely stated in the following.

## Algorithm 3.1

(MS)

The following theorem demonstrates the convergence of Algorithm 3.1 under the reasonable assumptions.

## Theorem 3.1

*Let*\(A=D+\widetilde{T}=D+T+iI \in \mathbb{C}^{M\times M}\),

*where*\(\widetilde{T}\in \mathbb{C}^{M\times M}\)

*is a symmetric and non*-

*Hermitian positive definite Toeplitz matrix*,

*and*\(D=(d_{1},d_{2},\ldots ,d_{M})\in \mathbb{R}^{M\times M}\)

*is a diagonal matrix with nonnegative elements*.

*Assume that*\(d_{\min }\), \(d_{\max }\)

*and*

*ω*

*are described as*

*then the spectral radius*\(\rho (G_{\omega })\)

*is bounded by*

*where*\(\lambda _{\min }\)

*is the smallest eigenvalue of T*.

*Moreover*, *it holds that*\(\rho (G_{\omega })\leq \delta _{\omega } < 1\). *That is to say*, *the MS method converges to the unique solution*\(u_{*}\)*of* (3.1).

## Proof

In Algorithm 3.1, the matrices \(\omega I+\widetilde{T}\) and \(\omega I-D\) are Toeplitz and diagonal matrices, respectively. In order to reduce the computing time and improve the efficiency of the MS method, we can employ the fast and superfast algorithms of the inverse of the Toeplitz matrix to solve the linear system.

*T*can be approximated well by a circulant matrix. In order to further reduce the computational costs and accelerate the convergence rate of the MS iteration method, we changed equivalently the symmetric linear system (3.1) by choosing a circulant matrix, \(C\in \mathbb{C}^{M\times M}\), obtaining the preconditioned MS (PMS) iteration method. More concretely, the preconditioner is generated from Strang’s circulant preconditioner. We can take

*C*to be Strang’s circulant approximation obtained by copying the central diagonals of

*T*and bringing them around to complete the circulant requirement. When

*M*is odd,

*M*is even, it can be treated similarly as follows:

*C*is symmetric and strictly diagonally dominant, resulting in it being symmetric positive definite.

## Algorithm 3.2

(PMS)

*D*.

## Remark 3.1

*A*by rough estimate. It is verified by numerical experiments in the next section.

## Remark 3.2

Because the circulant preconditioner is the approximation of the Toeplitz matrix, the proof of the convergence property for the PMS iteration method is similar to the MS iteration method. Therefore, we will not repeat the proof process of the convergence property in Algorithm 3.2 here.

## 4 Numerical experiments

^{®}Core

^{TM}2 Quad CPU), 4.00 GB memory and Windows 7 operating system.

## Example 1

Here and in the sequel, the direct methods can be used to solve the linear systems (3.3) and (3.5), named MS_{1} and PMS_{1} methods, respectively. Correspondingly, we employ the fast Zohar algorithm and the fast Fourier transform (FFT) to solve the linear systems (3.3) and (3.5), named MS_{2} and PMS_{2} methods, respectively.

Numerical results of the Example 1 for \(\alpha=1.2\)

Method | | ||||||
---|---|---|---|---|---|---|---|

800 | 1600 | 3200 | 5000 | 10,000 | 20,000 | ||

PMS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.2330 | 0.8444 | 2.7600 | 7.2175 | 25.8146 | 121.7182 | |

Error | 1.601e − 07 | 1.544e − 07 | 1.662e − 07 | 1.187e − 07 | 3.834e − 08 | 7.336e − 08 | |

PMS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.1655 | 1.1138 | 4.8821 | 11.2724 | 66.1895 | 436.4372 | |

Error | 1.605e − 07 | 1.544e − 07 | 1.622e − 07 | 1.187e − 07 | 3.893e − 08 | 7.448e − 08 | |

MS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.4323 | 1.4621 | 9.6115 | 36.8597 | 187.4272 | 3436.70 | |

Error | 1.605e − 07 | 1.544e − 07 | 1.622e − 07 | 1.187e − 07 | 3.893e − 08 | 7.336e − 08 | |

MS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.1554 | 1.1751 | 4.8452 | 12.3658 | 64.8330 | 458.5189 | |

Error | 1.605e − 07 | 1.788e − 06 | 1.622e − 07 | 1.187e − 07 | 3.817e − 08 | 7.336e − 08 | |

SOR | IT | 10 | 21 | 62 | 153 | 695 | 2662 |

CPU (s) | 0.1833 | 1.6661 | 17.7699 | 94.5633 | 1033.4 | 28963.00 | |

Error | 8.598e − 07 | 5.310e − 07 | 8.710e − 07 | 9.868e − 07 | 9.984e − 07 | 9.983e − 07 |

Numerical results of the Example 1 for \(\alpha=1.4\)

Method | | ||||||
---|---|---|---|---|---|---|---|

800 | 1600 | 3200 | 5000 | 10,000 | 20,000 | ||

PMS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.2344 | 0.9332 | 3.0236 | 7.3649 | 25.5258 | 111.9227 | |

Error | 5.916e − 08 | 5.860e − 08 | 6.807e − 08 | 8.218e − 08 | 7.691e − 08 | 3.511e − 08 | |

PMS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.1598 | 1.0988 | 5.1738 | 11.5349 | 67.9240 | 347.5191 | |

Error | 5.916e − 08 | 5.860e − 08 | 6.807e − 08 | 8.218e − 08 | 7.691e − 08 | 3.563e − 08 | |

MS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.4789 | 1.7869 | 8.3820 | 33.1318 | 232.2944 | 1454.50 | |

Error | 5.916e − 08 | 5.860e − 08 | 6.807e − 08 | 8.218e − 08 | 7.691e − 08 | 3.846e − 08 | |

MS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.1652 | 1.0106 | 5.2072 | 11.7405 | 64.7608 | 360.4382 | |

Error | 5.916e − 08 | 5.860e − 08 | 6.807e − 08 | 8.218e − 08 | 7.691e − 08 | 3.846e − 08 | |

SOR | IT | 10 | 21 | 62 | 153 | 695 | − |

CPU (s) | 0.3568 | 5.2840 | 96.6897 | 701.8188 | 1476.700 | − | |

Error | 5.533e − 07 | 8.717e − 07 | 9.815e − 07 | 9.900e − 07 | 9.978e − 07 | − |

_{2}, PMS

_{1}, MS

_{2}, and MS

_{1}methods are pretty stable and always kept in 3 with the growth of the number of spatial grid points, but those of the SOR iteration method are increasing quickly. As for the computing time, that of the PMS

_{2}method is the least of all methods in the experiments, while the time of the SOR method has already reached over 1000 seconds when \(M=10\text{,}000\). Figure 1, which depicts the comparison of resulting CPU of PMS

_{2}, MS

_{2}, and SOR iteration methods for Example 1, intuitively shows that the PMS

_{2}method performs pretty well.

## Example 2

Numerical results of the Example 2 for \(\alpha=1.2\)

Method | | ||||||
---|---|---|---|---|---|---|---|

800 | 1600 | 3200 | 5000 | 10,000 | 20,000 | ||

PMS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.2348 | 0.9579 | 3.2193 | 7.1782 | 29.0157 | 129.1786 | |

Error | 2.770e − 07 | 2.686e − 07 | 2.961e − 07 | 2.079e − 07 | 7.641e − 08 | 1.286e − 07 | |

PMS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.2409 | 1.0279 | 4.9241 | 12.0993 | 67.6799 | 371.9630 | |

Error | 2.770e − 07 | 2.686e − 07 | 2.961e − 07 | 2.079e − 07 | 7.641e − 08 | 1.286e − 07 | |

MS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.4332 | 1.4621 | 9.2120 | 41.7877 | 321.7863 | 1142.70 | |

Error | 2.769e − 07 | 2.686e − 07 | 2.961e − 07 | 2.077e − 07 | 7.641e − 08 | 1.244e − 07 | |

MS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.1284 | 1.2033 | 5.2434 | 11.9182 | 64.7608 | 385.2560 | |

Error | 2.769e − 07 | 2.686e − 07 | 2.961e − 07 | 2.079e − 07 | 7.641e − 08 | 1.293e − 07 | |

SOR | IT | 10 | 21 | 62 | 153 | 695 | 2643 |

CPU (s) | 0.4765 | 1.2821 | 17.4924 | 97.5482 | 2052.3 | 29904.00 | |

Error | 8.763e − 07 | 5.357e − 07 | 8.707e − 07 | 9.933e − 07 | 9.976e − 07 | 9.999e − 07 |

Numerical results of the Example 2 for \(\alpha=1.4\)

Method | | ||||||
---|---|---|---|---|---|---|---|

800 | 1600 | 3200 | 5000 | 10,000 | 20,000 | ||

PMS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.2488 | 0.9239 | 3.2826 | 7.3958 | 24.7668 | 126.0364 | |

Error | 1.353e − 07 | 1.328e − 07 | 1.586e − 07 | 2.159e − 07 | 3.254e − 07 | 5.434e − 07 | |

PMS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.1666 | 1.1425 | 5.1075 | 12.1415 | 68.6869 | 351.8482 | |

Error | 1.353e − 07 | 1.328e − 07 | 1.586e − 07 | 2.160e − 07 | 3.255e − 07 | 5.444e − 07 | |

MS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.2034 | 1.5269 | 8.6382 | 63.1743 | 221.7870 | 7645.00 | |

Error | 1.354e − 07 | 1.327e − 07 | 1.586e − 07 | 2.159e − 07 | 3.255e − 07 | 5.615e − 07 | |

MS | IT | 3 | 3 | 3 | 3 | 3 | 3 |

CPU (s) | 0.1969 | 1.0012 | 4.9433 | 12.3412 | 67.0036 | 385.5151 | |

Error | 1.354e − 07 | 1.327e − 07 | 1.586e − 07 | 2.159e − 07 | 3.255e − 07 | 5.533e − 07 | |

SOR | IT | 10 | 21 | 62 | 153 | 695 | − |

CPU (s) | 0.3228 | 4.8402 | 98.6408 | 723.7896 | 1854.10 | − | |

Error | 5.590e − 07 | 8.608e − 07 | 9.952e − 07 | 9.960e − 07 | 9.900e − 07 | − |

_{2}and MS

_{2}methods are constant no matter what the number of the spatial grid points is, but those of the SOR iteration method are increasing quickly. The PMS

_{2}and MS

_{2}methods require less computing time than the SOR iteration method. The advantage of the computing time of the PMS

_{2}method over the other methods for different values of

*α*and

*M*can be visually shown in Fig. 2.

*A*by sketching the circles in Figs. 3–6, whose centers are all the middle points of the smallest and largest eigenvalues of the preconditioner \(\widetilde{M}_{\omega }^{-1}\), and their radii are half of the distance between the two points.

## 5 Conclusion

According to the discretization of the spatial fractional coupled nonlinear Schrödinger equations, we have obtained that the coefficient matrix is the sum of a real diagonal matrix *D* and a complex symmetric Toeplitz matrix *T̃*, which is just the symmetric real Toeplitz *T* plus an imaginary identity matrix *iI*, and then established a new splitting iteration method with medium-shifting (MS). Since the structure of the coefficient matrix is Toeplitz matrix, the iteration method with medium-shifting sufficiently employs the fast and superfast direct methods of complexity \(\mathcal{O}(M^{2})\) and \(\mathcal{O}(M\log^{2}M)\), respectively. Furthermore, we replace the Toeplitz matrix *T* by an appropriate circulant matrix *C* and obtain the preconditioned MS (PMS) iteration method. In terms of the special structure of coefficient matrix in Algorithms 3.1–3.2, we need to solve the linear system associated with the shifted Toeplitz matrix and the shifted circulant matrix, respectively, by employing fast and superfast algorithms. Theoretical analysis shows that the approach is convergent, and numerical experiments have already demonstrated its efficiency.

## Notes

## Acknowledgements

The authors are very much indebted to the anonymous referees for their helpful comments and suggestions.

## Availability of data and materials

Not applicable.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

## Competing interests

The authors declare that they have no competing interests.

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