arXiv subjects

Awad H. Al-Mohy

Publications and source records attributed to Awad H. Al-Mohy.

4 recordsLinked to original sources

arXiv2025

Computing Linear Combinations of $φ$-Function Actions for Exponential Integrators

We propose a matrix-free algorithm for evaluating linear combinations of $φ$-function actions, $w_i := \sum_{j=0}^{p} α_i^{\,j}\,φ_j(t_i A)v_j$ for $i=1\colon r$, arising in exponential integrators. The method combines the scaling and recovering method with a truncated Taylor series, choosing a spectral shift and a scaling parameter by minimizing a power-based objective of the shifted operator. Accuracy is user-controlled and ultimately limited by the working precision. The algorithm decouples the stage abscissae $t_i$ from the polynomial weights $α_i^j$, and a block variant enables simultaneous evaluation of $\{w_i\}_{i=1}^r$. Across standard benchmarks, including stiff and highly nonnormal matrices, the algorithm attains near-machine accuracy (IEEE double precision in our tests) for small step sizes and maintains reliable accuracy for larger steps where several existing Krylov-based algorithms deteriorate, providing a favorable balance of reliability and computational cost.

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arXiv2025

A scaling and recovering algorithm for the matrix $φ$-functions

A new scaling and recovering algorithm is proposed for simultaneously computing the matrix $φ$-functions that arise in exponential integrator methods for the numerical solution of certain first-order systems of ordinary differential equations. The algorithm initially scales the input matrix down by a nonnegative integer power of two, and then evaluates the $[m/m]$ diagonal Padé approximant to $φ_p$, where $p$ is the largest index of interest. The remaining $[m+p{-}j/m]$ Padé approximants to $φ_j$, $0 \le j < p$, are obtained implicitly via a recurrence relation. The effect of scaling is subsequently recovered using the double-argument formula. A rigorous backward error analysis, based on the $[m+p/m]$ Padé approximant to the exponential, enables sharp bounds on the relative backward errors. These bounds are expressed in terms of the sequence $\|A^k\|^{1/k}$, which can be much smaller than $\|A\|$ for nonnormal matrices. The scaling parameter and the degrees of the Padé approximants are selected to minimize the overall computational cost, which benefits from the sharp bounds and the optimal evaluation schemes for diagonal Padé approximants. Furthermore, if the input matrix is (quasi-)triangular, the algorithm exploits its structure in the recovering phase. Numerical experiments demonstrate the superiority of the proposed algorithm over existing alternatives in both accuracy and efficiency.

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arXiv2025

A New Algorithm for Computing the Exponential of a Block Triangular Matrix

The exponential of block triangular matrices arises in a wide range of scientific computing applications, including exponential integrators for solving systems of ordinary differential equations, Hamiltonian systems in control theory, sensitivity analysis, and option pricing in finance. We propose a novel algorithm exploiting the block triangular structure for simultaneously computing the exponentials of the diagonal blocks and the off-diagonal block of the matrix exponential without direct involvement of the full block matrix in the computations. This approach generalizes the work of Al-Mohy and Higham on the Fréchet derivative of the matrix exponential. The generalization is established through a linear operator framework, facilitating efficient evaluation schemes and rigorous backward error analysis. The algorithm employs the scaling and squaring method using diagonal Padé approximants with algorithmic parameters selected based on the backward error analysis. A key feature is that the selection of the scaling parameter relies solely on the maximal norm of the diagonal blocks with no dependence on the norm of the off-diagonal block. Numerical experiments confirm that the proposed algorithm consistently outperforms existing algorithms in both accuracy and efficiency, making it a preferred choice for computing the matrix exponential of block triangular matrices.

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arXiv2017

A New Algorithm for Computing the Actions of Trigonometric and Hyperbolic Matrix Functions

A new algorithm is derived for computing the actions $f(tA)B$ and $f(tA^{1/2})B$, where $f$ is cosine, sinc, sine, hyperbolic cosine, hyperbolic sinc, or hyperbolic sine function. $A$ is an $n\times n$ matrix and $B$ is $n\times n_0$ with $n_0 \ll n$. $A^{1/2}$ denotes any matrix square root of $A$ and it is never required to be computed. The algorithm offers six independent output options given $t$, $A$, $B$, and a tolerance. For each option, actions of a pair of trigonometric or hyperbolic matrix functions are simultaneously computed. The algorithm scales the matrix $A$ down by a positive integer $s$, approximates $f(s^{-1}tA)B$ by a truncated Taylor series, and finally uses the recurrences of the Chebyshev polynomials of the first and second kind to recover $f(tA)B$. The selection of the scaling parameter and the degree of Taylor polynomial are based on a forward error analysis and a sequence of the form $\|A^k\|^{1/k}$ in such a way the overall computational cost of the algorithm is optimized. Shifting is used where applicable as a preprocessing step to reduce the scaling parameter. The algorithm works for any matrix $A$ and its computational cost is dominated by the formation of products of $A$ with $n\times n_0$ matrices that could take advantage of the implementation of level-3 BLAS. Our numerical experiments show that the new algorithm behaves in a forward stable fashion and in most problems outperforms the existing algorithms in terms of CPU time, computational cost, and accuracy.

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