Abstract
We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on Chebyshev polynomials of degrees 2, 4, 8, 12 and 18 which are computed with only 1, 2, 3, 4 and 5 matrix–matrix products, respectively. For problems of the form exp(−iA), with A a real and symmetric matrix, an improved version is presented that computes the sine and cosine of A with a reduced computational cost. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on rational Padé approximants and Taylor polynomials for all tolerances and time interval lengths. The new procedure is particularly recommended to be used in conjunction with exponential integrators for the numerical time integration of the Schrödinger equation.
Highlights
Given a skew-Hermitian matrix, X ∈ CN×N, XH = −X, we propose in this paper an algorithm to evaluate eX up to round off accuracy that is more efficient than standard procedures implemented in computing packages for dimensions N up to few hundreds or thousands.Computing exponentials of skew-Hermitian matrices is very often an intermediate step in the formulation of numerical schemes used for simulating the evolution of different problems in Quantum Mechanics
We have presented an algorithm to approximate the exponential of skew-Hermitian matrices based on an improved computation of Chebyshev polynomials of matrices and the corresponding error analysis
In [26] a polynomial of degree 16 is presented in terms of only 4 products that coincides with the Taylor expansion up to order 15
Summary
The technique we propose can be considered as a direct descent of the procedure presented in [9] for reducing the number of commutators appearing in different exponential integrators It was later generalized in [5] to reduce the number of products necessary to compute the Taylor polynomials for approximating the exponential of a generic matrix (see [6, 26] for a more detailed treatment). In many cases, when solving different quantum mechanical or quantum control problems [4] one ends up with a real and symmetric matrix, AT = A ∈ RN×N , so that e−iA = cos(A) − i sin(A), and we provide an algorithm for computing cos(A) and sin(A) simultaneously only involving products of real symmetric matrices This new algorithm is more efficient than the approach (3) since that scheme usually requires products of complex matrices, and other existing algorithms for the simultaneous computation of the matrix sine and cosine [2, 27].
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