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Table Al: Coefficients in the monomial basis (ascending order) of the denominators of the diagonal Padé approximations for the sinc(r) function for the degree n = 2m, m= 1,...,5.

Table 3 Al: Coefficients in the monomial basis (ascending order) of the denominators of the diagonal Padé approximations for the sinc(r) function for the degree n = 2m, m= 1,...,5.

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4.1.2 Padé approximants for ;F (1; 2;-) Following the results given by Luke in [12, Section 3] and again the higher order term that can be obtained from [13, Lemma 2], we can easily obtain the [n/n]-Padé approximant to the function , F’; (1; 2; x). Thus, we write
4.1.2 Padé approximants for ;F (1; 2;-) Following the results given by Luke in [12, Section 3] and again the higher order term that can be obtained from [13, Lemma 2], we can easily obtain the [n/n]-Padé approximant to the function , F’; (1; 2; x). Thus, we write
parity of the sinc function. This does not happen when rewriting the sinc function as n (16). Indeed, a repeated application of (22)-(24) leads to
parity of the sinc function. This does not happen when rewriting the sinc function as n (16). Indeed, a repeated application of (22)-(24) leads to
Thus we can approximate sinc(A)v by approximating the integral of the inverse Fourier transform
Thus we can approximate sinc(A)v by approximating the integral of the inverse Fourier transform
Fig. 3: Basis functions for the piecewise linear continuous finite elements. Va = {v€ HQ) : VRE Th, oI, € PEx]} = Span{o;(x)} 27" C VNC((0, 7]; V).
Fig. 3: Basis functions for the piecewise linear continuous finite elements. Va = {v€ HQ) : VRE Th, oI, € PEx]} = Span{o;(x)} 27" C VNC((0, 7]; V).
Table 1: Test matrices: Finite Difference discretiza- tion of the Laplacian, centered differences for the 1D case, 5-point stencil for the 2D case, and the lin- ear Finite Element Method discussed in Section 5.1. Amin; Amax are the smallest and largest eigenvalues respectively.
Table 1: Test matrices: Finite Difference discretiza- tion of the Laplacian, centered differences for the 1D case, 5-point stencil for the 2D case, and the lin- ear Finite Element Method discussed in Section 5.1. Amin; Amax are the smallest and largest eigenvalues respectively.
Fig. 4: Relative error for the computation of sinc(A)v with A the matrices described in Table 1 and v a randomly generated vector. We test all the pole choices described in Section 4. with V; the matrix whose column spans the relevant rational Krylov subspace Q;(A, v) and the matrix-argument sinc(A) function computed in MATLAB as “sinc(A) + A\funm(A,@sin)”. From Figure 4 we observe that the best results are given by poles obtained from the Padé diagonal approximant of sinc function here computed in a symbolic way; see Appendix A. Due to the parity of the function and by the fact that the diagonals approximant of odd order coincide with the previous even ones, we report only the even cases. The other satisfactory results are obtained with the poles €,, given by the expansion discussed in Section 4.1.1, which are the complex zeros of the non orthogonal Laguerre polynomials. Also observe that the choice of poles as in Ly,
Fig. 4: Relative error for the computation of sinc(A)v with A the matrices described in Table 1 and v a randomly generated vector. We test all the pole choices described in Section 4. with V; the matrix whose column spans the relevant rational Krylov subspace Q;(A, v) and the matrix-argument sinc(A) function computed in MATLAB as “sinc(A) + A\funm(A,@sin)”. From Figure 4 we observe that the best results are given by poles obtained from the Padé diagonal approximant of sinc function here computed in a symbolic way; see Appendix A. Due to the parity of the function and by the fact that the diagonals approximant of odd order coincide with the previous even ones, we report only the even cases. The other satisfactory results are obtained with the poles €,, given by the expansion discussed in Section 4.1.1, which are the complex zeros of the non orthogonal Laguerre polynomials. Also observe that the choice of poles as in Ly,
Fig. 5: Computation of the exponential sums (31) obtained using different approxima- ions for the exponential, relative error versus time (s) graphs. To have a comparison vith the results in Figure 4 we report in both pictures the results obtained with the n/n|-Padé approximation, and the rational Krylov methods with poles €,, and Ly. Che number of poles for the [k/k]-Padé approximation of the exponential is k = 15 or the 1D FD case and & = 20 for the 2D FD and Linear FEM cases. The number of sauss-Legendre quadrature nodes goes from v = 1,..., 15 in all the cases. In the next set of experiments we compare the attained accuracy in terms of the relative error with the time needed to achieve it. In addition to rational Krylov-type approaches we consider also the exponential sums algorithm discussed in Section 4.2. To ensure that the strategy of approximating the matrix exponential within exponential sums does not reduce the overall accuracy we employ the cubic cost dense-matrix computation of the different exponential; this serves just as a sanity check of the overall procedure since it is indeed an expensive procedure. From the results in Figure 5 we
Fig. 5: Computation of the exponential sums (31) obtained using different approxima- ions for the exponential, relative error versus time (s) graphs. To have a comparison vith the results in Figure 4 we report in both pictures the results obtained with the n/n|-Padé approximation, and the rational Krylov methods with poles €,, and Ly. Che number of poles for the [k/k]-Padé approximation of the exponential is k = 15 or the 1D FD case and & = 20 for the 2D FD and Linear FEM cases. The number of sauss-Legendre quadrature nodes goes from v = 1,..., 15 in all the cases. In the next set of experiments we compare the attained accuracy in terms of the relative error with the time needed to achieve it. In addition to rational Krylov-type approaches we consider also the exponential sums algorithm discussed in Section 4.2. To ensure that the strategy of approximating the matrix exponential within exponential sums does not reduce the overall accuracy we employ the cubic cost dense-matrix computation of the different exponential; this serves just as a sanity check of the overall procedure since it is indeed an expensive procedure. From the results in Figure 5 we
Table 2: Comparison in terms of elapsed time and two-norm relative error of the Gautschi integrator and the Adams-Bashforth-Norsett scheme of stiff order 2 from [20].
Table 2: Comparison in terms of elapsed time and two-norm relative error of the Gautschi integrator and the Adams-Bashforth-Norsett scheme of stiff order 2 from [20].
Fig. 7: From left to right, we have the domain, the triangular mesh for the solution of the wave equation (32), the mass and stiffness matrix for the P!-element on the associated triangulation.
Fig. 7: From left to right, we have the domain, the triangular mesh for the solution of the wave equation (32), the mass and stiffness matrix for the P!-element on the associated triangulation.
Fig. 8: Pointwise absolute error (log; -scale) computed against the solution generated with MATLAB’s solvepde command at T = 1 with h = 10~? with respect to time, and linear FEM with largest element size of hmax = 0.03 in the generateMesh program. We used four poles both for the €,, and £,,. The exponential sum case has been computed with 6 poles for the rational approximation of the exponential and 12 terms in the sum. and a zero initial velocity, i.e., vo(x, y) = 0. Since we simply want to test the robustness of the routines for computing the different matrix functions, we also set the forcing term equal to zero. From the errors reported in Figure 8 it can be observed that also in
Fig. 8: Pointwise absolute error (log; -scale) computed against the solution generated with MATLAB’s solvepde command at T = 1 with h = 10~? with respect to time, and linear FEM with largest element size of hmax = 0.03 in the generateMesh program. We used four poles both for the €,, and £,,. The exponential sum case has been computed with 6 poles for the rational approximation of the exponential and 12 terms in the sum. and a zero initial velocity, i.e., vo(x, y) = 0. Since we simply want to test the robustness of the routines for computing the different matrix functions, we also set the forcing term equal to zero. From the errors reported in Figure 8 it can be observed that also in
We report here a depiction of the scalar bounds for the approximations (20) and (27) on the real line. As shown in the Figure B1 there is an excellent correspondence
We report here a depiction of the scalar bounds for the approximations (20) and (27) on the real line. As shown in the Figure B1 there is an excellent correspondence
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