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Computing the matrix square root with the doubling algorithm
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DOI：10.1007/s10915-025-03040-7.png)
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Abstract
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The matrix quare root is a very important matrix computation problem. The structure-preserving doubling algorithm (SDA) is quadratically convergent. We demonstrate how the matrix square root is computed by SDA, where the Cayley transform is deployed and the original equation is rewritten in terms of the invariant subspace. Further, we introduce a positive parameter $$\alpha $$ when using the Cayley transform and get SDA- $$\alpha $$ for $$A^{1/2}$$ . If the appropriate parameter $$\alpha $$ is given, SDA- $$\alpha $$ for $$A^{1/2}$$ can outperform SDA for $$A^{1/2}$$ in some examples. The convergence properties of the two algorithms are given. The algorithms in this paper are not numerically sensitive when the original matrix is ill-conditioned and the algorithms work when the original matrix is a singular matrix but has a square root.
Keywords:
Structure-preserving doubling algorithm
Cayley transform
Matrix square root
Quadratic convergence
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