A Schur method for the square root of a matrix
Åke Björck,Sven Hammarling +1 more
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TLDR
The method is based on the Schur factorization A = QSQ H and uses a fast recursion to compute the upper triangular square root of S and it is shown that if α = ∥ X ∥ 2 /∥ A ∥ is not large, then the computed square root is the exact square rootAbout:
This article is published in Linear Algebra and its Applications.The article was published on 1983-07-01 and is currently open access. It has received 157 citations till now. The article focuses on the topics: Functional square root & Square root of a 2 by 2 matrix.read more
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Functions of matrices
TL;DR: The most common matrix function is the matrix inverse, which is not treated specifically in this chapter, but is covered in Section~1.5 and Section~51.3 as discussed by the authors.
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Computing the polar decomposition with applications
TL;DR: Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix.
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Computing real square roots of a real matrix
TL;DR: An extension of the Schur method is presented which enables real arithmetic to be used throughout when computing a real square root of a real matrix.
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Stable iterations for the matrix square root
TL;DR: It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation.
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A Schur-Parlett Algorithm for Computing Matrix Functions
TL;DR: An algorithm for computing matrix functions that employs a Schur decomposition with reordering and blocking followed by the block form of a recurrence of Parlett, with functions of the nontrivial diagonal blocks evaluated via a Taylor series.
References
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The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
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Solution of the matrix equation AX + XB = C [F4]
R. H. Bartels,G. W. Stewart +1 more
TL;DR: The algorithm is supplied as one file of BCD 80 character card images at 556 B.P.I., even parity, on seven ~rack tape, and the user sends a small tape (wt. less than 1 lb.) the algorithm will be copied on it and returned to him at a charge of $10.O0 (U.S.and Canada) or $18.00 (elsewhere).
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A Hessenberg-Schur method for the problem AX + XB= C
TL;DR: A new method is proposed which differs from the Bartels-Stewart algorithm in that A is only reduced to Hessenberg form, and the resulting algorithm is between 30 and 70 percent faster depending upon the dimensions of the matrices A and B.