Newton's Method for the Matrix Square Root

Higham, Nicholas J.

doi:10.2307/2007992

Cited by 82 publications

(88 citation statements)

References 16 publications

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“…The problem of computing the principal square root of a matrix is associated with a well-known problem of estimation theory, namely the problem of solving a continuous time Riccati equation. The Riccati equation arises in linear estimation [3], namely in the implementation of the Kalman-Bucy filter and it is formulated as…”

Section: Algorithm 5(a) This Algorithm Is Proposed In [5]mentioning

confidence: 99%

“…Due to the importance of the problem, many iterative algorithms have been proposed and successfully employed for calculating the principal square root of a matrix [1][2][3][4][5][6][7] without seeking the eigenvalues and eigenvectors of the matrix; these algorithms require matrix inversion at every iteration. Blocked Schur Algorithms for Computing the Matrix Square Root are proposed in [8] where the matrix is reduced to upper triangular form and a recurrence relation enables the square root of the triangular matrix to be computed a column or superdiagonal at a time.…”

Section: Introductionmentioning

confidence: 99%

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Inversion Free Algorithms for Computing the Principal Square Root of a Matrix

Assimakis

Αδάμ

2014

International Journal of Mathematics and Mathematical Sciences

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New algorithms are presented about the principal square root of ann×nmatrixA. In particular, all the classical iterative algorithms require matrix inversion at every iteration. The proposed inversion free iterative algorithms are based on the Schulz iteration or the Bernoulli substitution as a special case of the continuous time Riccati equation. It is certified that the proposed algorithms are equivalent to the classical Newton method. An inversion free algebraic method, which is based on applying the Bernoulli substitution to a special case of the continuous time Riccati equation, is also proposed.

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Section: Algorithm 5(a) This Algorithm Is Proposed In [5]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inversion Free Algorithms for Computing the Principal Square Root of a Matrix

Assimakis

Αδάμ

2014

International Journal of Mathematics and Mathematical Sciences

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“…Next, we seek for local optimization methods that approximate (18b) using first-order Taylor series, so as to make problem (18) able to be solved by a sequence of convex problems. Applying Newton's method [19] to the quadratic matrix equation…”

Section: Design Of Time and Color Multiplexing Codesmentioning

confidence: 99%

“…where ∆F denotes a small change in F, and F k denotes the estimate of F at iteration k. The details of how we derive (19) is provided in the SM 1 . Since the newly updated F k+1 should be feasible to (18), F k + ∆F should also fulfill (18a).…”

Section: Design Of Time and Color Multiplexing Codesmentioning

confidence: 99%

Towards Optimal Design of Time and Color Multiplexing Codes

Chan

Jia

Wycoff

et al. 2012

Computer Vision – ECCV 2012

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Abstract. Multiplexed illumination has been proved to be valuable and beneficial, in terms of noise reduction, in wide applications of computer vision and graphics, provided that the limitations of photon noise and saturation are properly tackled. Existing optimal multiplexing codes, in the sense of maximum signal-to-noise ratio (SNR), are primarily designed for time multiplexing, but they only apply to a multiplexing system requiring the number of measurements (M ) equal to the number of illumination sources (N ). In this paper, we formulate a general code design problem, where M ≥ N , for time and color multiplexing, and develop a sequential semi-definite programming to deal with the formulated optimization problem. The proposed formulation and method can be readily specialized to time multiplexing, thereby making such optimized codes have a much broader application. Computer simulations will discover the main merit of the method-a significant boost of SNR as M increases. Experiments will also be presented to demonstrate the effectiveness and superiority of the method in object illumination.

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“…The orthogonal case has been studied by Schonemann [13] and the Higham [8] and the symmetric case by Higham [10]. These problems are known in the literature as the orthogonal and symmetric Procrustes problems respectively.…”

Section: Introductionmentioning

confidence: 99%

Finding the closest Toeplitz matrix

Eberle¹,

Maciel²

2003

Mat. apl. comput.

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Abstract. The constrained least-squares n × n-matrix problem where the feasibility set is the subspace of the Toeplitz matrices is analyzed. The general, the upper and lower triangular cases are solved by making use of the singular value decomposition. For the symmetric case, an algorithm based on the alternate projection method is proposed. The implementation does not require the calculation of the eigenvalue of a matrix and still guarantees convergence. Encouraging preliminary results are discussed.Mathematical subject classification: 65F20, 65F30, 65H10.

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Newton's Method for the Matrix Square Root

Cited by 82 publications

References 16 publications

Inversion Free Algorithms for Computing the Principal Square Root of a Matrix

Inversion Free Algorithms for Computing the Principal Square Root of a Matrix

Towards Optimal Design of Time and Color Multiplexing Codes

Finding the closest Toeplitz matrix

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