Numerical stability of the symplectic LLT factorization

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Authors
Bujok, Maksymilian
Rozložník, Miroslav
Smoktunowicz, Agata
Smoktunowicz, Alicja
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Date
2025-04-22
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Numerical Algorithms
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1017-1398
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https://link.springer.com/article/10.1007/s11075-025-02075-z
Access date
2025-09-18
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Abstract EN
In this paper we give the detailed error analysis of two algorithms W1 and W2 for computing the symplectic factorization of a symmetric positive definite and symplectic matrix AeR2n*2n in the form A=LLT , where LeR2n*2n is a symplectic block lower triangular matrix. We prove that Algorithm w2 is numerically stable for a broader class of symmetric positive definite matrices AeR2n*2n . It means that Algorithm W2 is producing the computed factors L in floating-point arithmetic with machine precision u such that A - LLT//2=0(n2u//A//2) . On the other hand, Algorithm W1 is unstable, in general, for symmetric positive definite and symplectic matrix A . In this paper we also give corresponding bounds for Algorithm W1 that are weaker. We show that the factorization error depends on the conditioning of the principal submatrix A11eRnxn, located in the upper-left block of A . Bounds for the loss of symplecticity of the lower block triangular matrices L for both Algorithms W1 and W2 that hold in exact arithmetic for a broader class of symmetric positive definite matrices A (but not necessarily symplectic) are also given. The tests performed in MATLAB illustrate that our error bounds for considered algorithms are reasonably sharp.
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Keywords EN
L L T factorization
Backward error analysis
Condition number
Symplectic matrix
Cholesky decomposition
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https://share.swps.edu.pl/handle/swps/1798
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