Advances in Discrete Differential Geometry pp 151176  Cite as
Numerical Methods for the Discrete Map \(Z^a\)
Abstract
As a basic example in nonlinear theories of discrete complex analysis, we explore various numerical methods for the accurate evaluation of the discrete map \(Z^a\) introduced by Agafonov and Bobenko. The methods are based either on a discrete Painlevé equation or on the Riemann–Hilbert method. In the latter case, the underlying structure of a triangular Riemann–Hilbert problem with a nontriangular solution requires special care in the numerical approach. Complexity and numerical stability are discussed, the results are illustrated by numerical examples.
1 Introduction
In this work we study the stable and accurate numerical calculation of \(Z^a\); to the best of our knowledge for the first time in the literature. This is an interesting mathematical problem in itself, but the underlying methods should be applicable to a large set of similar discrete integrable systems. Now, the basic difficulty is that the evolution of the discrete dynamical system (1) and (2), starting from the initial values (3), is numerically highly unstable, see Fig. 2.^{1}
As can be seen from Fig. 2, the numerical instability starts spreading from the diagonal elements \(f_{n,n}\). In fact, there is an initial exponential growth of numerical errors to be found in the diagonal entries, see Fig. 3. Such a numerical instability of an evolution is the direct consequence of the instability of the underlying dynamical system, that is, of positive Lyapunov exponents.
As a remedy we suggest two different approaches to calculating \(Z^a_{n,m}\). In Sect. 2 we stabilize the calculation of the diagonal values by solving a boundary value problem for an underlying discrete Painlevé II equation and in Sects. 3–8 we explore numerical methods based on the Riemann–Hilbert method. The latter reveals an interesting structure (Sect. 4): the Riemann–Hilbert problem has triangular data but a nontriangular solution; the operator equation can thus be written as a uniquely solvable block triangular system where the infinitedimensional diagonal operators are not invertible. We discuss two different ways to prevent this particular structure from hurting finitedimensional numerical schemes: a coefficientbased spectral method with infinitedimensional linear algebra in Sect. 6 and a modified Nyström method based on least squares in Sect. 8.
2 Discrete Painlevé II Separatrix as a Boundary Value Problem
Finally, having accurate values of \(x_n\) at hand, and therefore by (8) and (9) also those of \(f_{n,n}\), \(f_{n+1,n}\) and \(f_{n,n+1}\), one can calculate the missing values of \(f_{n,m}\) row and columnwise, starting from the second sup and superdiagonal and evolving to the boundary, either by evolving the crossratio relations (1) or by evolving the discrete differential equation (2). It turns out that the first option develops numerical instabilities spreading from the boundary, see Fig. 4, whereas the second option is, for a wide range of the parameter a, numerically observed to be perfectly stable, see Fig. 5. Note that this stable algorithm only differs from the alternative direct evolution discussed in the introduction in how the values close to the diagonal, that is \(f_{n,n}\), \(f_{n+1,n}\) and \(f_{n,n+1}\), are computed.
3 The Riemann–Hilbert Method
4 Lower Triangular Jump Matrices and Indices
Impact on Numerical Methods

\(S_N\) is nonsingular, which results in a 12component \(v_N = 0\) that does not converge;

the full system is singular and therefore numerically of not much use (illconditioning and convergence issues will abound).
Such methods compute fake lower triangular solutions, are illconditioned, or both.
The deeper structural reason for this problem can be seen in the fact that the Noether index of finitedimensional square matrices is always zero, whereas the index of the infinitedimensional subproblem (14) is strictly positive.
We suggest two approaches to deal with this problem: first, an infinitedimensional discretization using sequence spaces, that is, without truncation, and using infinitedimensional numerical linear algebra, and second, using underdetermined discretizations with rectangular linear systems that are complemented by a set of explicit compatibility conditions.
5 RHPs as Integral Equations with Singular Kernels
Theorem 1
Let \(\varGamma \) be a smooth, bounded, and nonself intersecting^{6} contour system and \(G:\varGamma \rightarrow {{\mathrm{GL}}}(2)\) a system of jump matrices which continues analytically to a vicinity of \(\varGamma \). Then, \(T_{G^{1}}\) is a Fredholm regulator of \(T_G\), that is, \(T_{G^{1}} T_G = {{\mathrm{id}}} + K\) with a compact operator \(K : L^2(\varGamma ,\mathbb {C}^{2\times 2}) \rightarrow L^2(\varGamma ,\mathbb {C}^{2\times 2})\) that can be represented as a regular integral operator.
Proof
This theorem implies that the operator \(T_{G}\) is Fredholm, that is, its nullity and deficiency are finite. In fact, since in our examples \(\det G \equiv 1\), we have that the Noether index of \(T_G\) is zero. The possibility to use the Fredholm theory is extremely important in studying RHPs: it allows one to use, when proving the solvability of RiemannHilbert problems, the “vanishing lemma” [26], see also [12, Chap. 5]. For the use of Fredholm regulators in iterative methods applied to solving singular integral equations, see [23].
6 A WellConditioned Spectral Method for Closed Contours
We follow the ideas of Olver and Townsend [20] on spectral methods for differential equations, recently extended by Olver and Slevinsky [19] to singular integral equations. First, the solution u and the data \(GI\) of the singular integral equation (15b) are expanded^{7} in the Laurent bases of the circles that built up the cycle \(\varGamma \). Next, the resulting linear system is solved using the framework of infinitedimensional linear algebra [14, 21], built out of the adaptive QR factorization introduced in [20].
Remark 1
Numerical Example 1: Model problem
The run time^{10} is 2.8 seconds, the error of Y(0) is \(4.22\cdot 10^{15}\) (spectral norm), which corresponds to a loss of one digit in absolute error.
Numerical Example 2: Riemann–Hilbert Problem for the Discrete \(\mathbf Z ^{2/3}\)
Since the amplitudes of \(u_{21}\) grow exponentially with n and m, the algorithm for computing \(Z^a_{n,m}\) based on the numerical evaluation of (15) applied to the RHP (11) is numerically unstable. Even though the initial step, the spectral method in coefficient space applied to (15b) is perfectly stable, stability is destructed by the bad conditioning of the postprocessing step, that is, the evaluation of the integral in (15a). We refer to [7] for an analysis that algorithms with a badly conditioned post processing of intermediate solutions are generally prone to numerical instability.
7 RHPs as Integral Equations with Nonsingular Kernels
Lemma 1
The nullity of the associated homogeneous RHP (20), and hence, that of the Fredholm integral equation (18) is \(n+m\) in the case of the discrete map \(Z^a\) and 2m in the case of the model problem (12).
Example 1
For the \(Z^a\)RHP, similar arguments prove that the kernel of (17) is spanned by matrices whose second row extends to polynomials of degree smaller than \((n+m)/2\) to the outside of the outer circle in Fig. 9. Thus, the same form of conditions as in (23) can be applied for picking the proper solution \(\varPhi _(\zeta )\), except that one would have to replace \(\Sigma \) by that outer circle and the upper index m by \((n+m)/2\).
8 A Modified Nyström Method
Since the Fredholm integral equation (18) has a positive nullity, applying the Nyström method to it will yield, for N large enough, a numerically singular linear system. However, the theory of the last section suggests a simple modification of the Nyström method: we use the conditions (23) (after approximating them by the same quadrature formula as for the Nyström method) as additional equations and solve the resulting overdetermined linear system by the least squares method.
Numerical Example 1: Model problem
Numerical Example 2: Discrete \(Z^{2/3}\)
9 Conclusion
To summarize, there are two fundamental options for the stable numerical evaluation of the discrete map \(Z^a_{n,m}\).

Computing all the values of the array \(1\leqslant n,m\leqslant N\) at once by, first, computing the diagonal using a boundary value solve for the discrete Painlevé II equation (5) and, then, by recursing from the diagonal to the boundary using the discrete differential equation (2). This approach has optimal complexity \(O(N^2)\).

Computing just a single value for a given index pair (n, m) by using the RHP (11) and one of the methods discussed in Sect. 6 or 8. Since both methods suffer from an instability caused by a postprocessing quadrature for larger values of n and m, one would rather mix this approach with the asymptotics (4). For instance, using the numerical schemes for \(n,m \leqslant 10\), and the asymptotics otherwise, gives a uniform precision of about 5 digits for \(a=2/3\). Higher accuracy would require the calculation of the next order terms of the asymptotics as in Sect. 2. This mixed numericalasymptotic method has optimal complexity O(1).
Footnotes
 1.
All numerical calculations are done in hardware arithmetic using double precision.
 2.To make \(G_1\) holomorphic in the vicinity of \(\varGamma _1\) we place the branchcut of \(\zeta ^{a/2}\) at the negative imaginary axis, that is, we take, using the principal branch \({{\mathrm{Log}}}\) of the logarithm,.$$ e^{ia \pi /2} \zeta ^{a/2} = e^{ia \pi /4} e^{\frac{a}{2} {{\mathrm{Log}}}(\zeta /i)}. $$
 3.
Though the parametrix leads to a nearidentity RHP, the actually computation of the parametrix would require solving a problem that is, numerically, of similar difficulty as the SRHP itself.
 4.The standard form, see [2, p. 1124], of that orthogonal polynomial RHP would beThe model problem (12) is obtained by putting the diagonal scaling at \(z\rightarrow \infty \) into the jump matrix.$$ X_+(\zeta ) = \begin{pmatrix} 1 &{} 0\\ e^\zeta \zeta ^{m} &{} 1 \end{pmatrix} X_(\zeta )\quad (\zeta =1),\qquad X(z) = \begin{pmatrix} z^m &{} 0 \\ 0 &{} z^{m} \end{pmatrix}(I+O(z^{1}))\qquad (z\rightarrow \infty ). $$
 5.
Here, we identify a RHP with an equivalent linear operator equation \(Tu = \cdots \), see, e.g., (15) in the next section. We recall that \(\lambda = \dim \ker T\) is called the nullity, \(\mu = \dim {{\mathrm{coker}}}T\) the deficiency and \(\kappa = \lambda  \mu \) the Noether index of a linear operator T with closed range.
 6.
Points of self intersection are allowed if certain cyclic conditions are satisfied [13]: at such a point the product of the corresponding parts of the jump matrix should be the identity matrix. These conditions guarantee smoothness in the sense of [26], where the analog of Theorem 1 is proved for the general smooth Riemann–Hilbert data.
 7.
It is actually implemented this way in SingularIntegralEquations.jl, a Julia software package described in [19].
 8.
We use the Iverson bracket of a condition: \([{\mathscr {P}}] = 1\) if the predicate \({\mathscr {P}}\) is true, \([{\mathscr {P}}] = 0\) otherwise.
 9.
https://github.com/ApproxFun/SingularIntegralEquations.jl, cf. [19].
 10.
Using a MacBook Pro with a 3.0 GHz Intel Core i74578U processor and 16 GB of RAM.
Notes
Acknowledgments
The research of F.B., A.I., and G.W. was supported by the DFGCollaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics.”
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