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21
New perturbation bounds for unitary polar factors,”
 SIAM Journal on Matrix Analysis and Applications,
, 2003
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A new scaling for Newton’s iteration for the polar decomposition and its backward stability
 SIAM J. Matrix Anal. Appl
"... Abstract. We propose a scaling scheme for Newton’s iteration for calculating the polar decomposition. The scaling factors are generated by a simple scalar iteration in which the initial value depends only on estimates of the extreme singular values of the original matrix, which can for example be th ..."
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Abstract. We propose a scaling scheme for Newton’s iteration for calculating the polar decomposition. The scaling factors are generated by a simple scalar iteration in which the initial value depends only on estimates of the extreme singular values of the original matrix, which can for example be the Frobenius norms of the matrix and its inverse. In exact arithmetic, for matrices with condition number no greater than 10 16, with this scaling scheme, no more than 9 iterations are needed for convergence to the unitary polar factor with a convergence tolerance roughly equal to 10 −16. It is proved that if matrix inverses computed in finite precision arithmetic satisfy a backwardforward error model then the numerical method is backward stable. It is also proved that Newton’s method with Higham’s scaling or with Frobenius norm scaling is backward stable. Key words. matrix sign function, polar decomposition, singular value decomposition (SVD), Newton’s method, numerical stability, scaling AMS subject classifications. 65F05, 65G05
The Polar Decomposition  Properties, Applications And Algorithms
, 1995
"... In the paper we review the numerical methods for computing the polar decomposition of a matrix. Numerical tests comparing these methods are included. Moreover, the applications of the polar decomposition and the most important its properties are mentioned. 1 Introduction In recent years interests i ..."
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In the paper we review the numerical methods for computing the polar decomposition of a matrix. Numerical tests comparing these methods are included. Moreover, the applications of the polar decomposition and the most important its properties are mentioned. 1 Introduction In recent years interests in the polar decomposition have increased. Many interesting papers have appeared on properties, applications and numerical methods for this decomposition. In the paper we review the most important results concerning this very useful tool. Also we present numerical experiments comparing several algorithms for computing it. The polar decomposition was introduced by Autonne [1] in 1902. A thorough discussion of the history of it is given in Horn and Johnson [29, Sect. 3.0]. Let A be an arbitrary complex matrix, A 2 C m\Thetan . A polar decomposition of A is a factorization A = UH; (1) where H 2 C n\Thetan is Hermitian positive semidefinite matrix, H = H H and x H Hx 0 for every x 2 C...
On the condition numbers associated with the polar factorization of a matrix
 Numer. Linear Algebra Appl
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An ArithmeticGeometricHarmonic Mean Inequality Involving Hadamard Products
, 1998
"... Given matrices of the same size, A = [a ij ] and B = [b ij ], we define their Hadamard Product to be A ffi B = [a ij b ij ]. We show that if x i ? 0 and q p 0 then the n \Theta n matrices p x i + x j ; are positive definite and relate these facts to some matrix valued a ..."
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Given matrices of the same size, A = [a ij ] and B = [b ij ], we define their Hadamard Product to be A ffi B = [a ij b ij ]. We show that if x i ? 0 and q p 0 then the n \Theta n matrices p x i + x j ; are positive definite and relate these facts to some matrix valued arithmeticgeometricharmonic mean inequalities  some of which involve Hadamard products and others unitarily invariant norms. It is known
Relative Perturbation Bounds for the Unitary Polar Factor
, 1996
"... Let B be an m \Theta n (m n) complex matrix. It is known that there is a unique polar decomposition B = QH, where Q Q = I, the n \Theta n identity matrix, and H is positive definite, provided B has full column rank. Existing perturbation bounds for complex matrices suggest that in the worst cas ..."
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Let B be an m \Theta n (m n) complex matrix. It is known that there is a unique polar decomposition B = QH, where Q Q = I, the n \Theta n identity matrix, and H is positive definite, provided B has full column rank. Existing perturbation bounds for complex matrices suggest that in the worst case, the change in Q be proportional to the reciprocal of the smallest singular value of B. However, there are situations where this unitary polar factor is much more accurately determined by the data than the existing perturbation bounds would indicate. In this paper the following question is addressed: how much may Q change if B is perturbed to e B = D 1 BD 2 ? Here D 1 and D 2 are nonsingular and close to the identity matrices of suitable dimensions. It will be proved that for such kinds of perturbations, the change in Q is bounded only by the distances from D 1 and D 2 to identity matrices, and thus independent of the singular values of B.
The Singular Values Of The Hadamard Product Of A Positive Semidefinite And A SkewSymmetric Matrix
, 1997
"... . Let oe 1 (X) : : : oe n (X) 0 denote the ordered singular values of an n \Theta n matrix X and let ff 1 (X) ff 2 (X) \Delta \Delta \Delta ff n (X) denote its ordered main diagonal entries (assuming that they are real). Let B be any complex n \Theta n skewsymmetric matrix and k \Delta k any ..."
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. Let oe 1 (X) : : : oe n (X) 0 denote the ordered singular values of an n \Theta n matrix X and let ff 1 (X) ff 2 (X) \Delta \Delta \Delta ff n (X) denote its ordered main diagonal entries (assuming that they are real). Let B be any complex n \Theta n skewsymmetric matrix and k \Delta k any unitarily invariant norm. It is shown that for any real positive semidefinite n \Theta n matrix A k X i=1 oe i (A ffi B) k X i=1 ~ ff i (A)oe i (B); k = 1; \Delta \Delta \Delta ; n; (1) where ~ ff 2i = ~ ff 2i\Gamma1 = p ff 2i\Gamma1 (A)ff 2i (A), i = 1; 2; : : : ; [n=2] (take ~ ff n (A) = 0 if n is odd), and that kA ffi Bk p ff 1 (A)ff 2 (A) kBk: It is also shown that if A is the Cauchy matrix given by a ij = (s i + s j ) \Gamma1 where s 1 s 2 \Delta \Delta \Delta s n ? 0, then for any real skewsymmetric matrix B k X i=1 oe i (A ffi B) k X i=1 s i oe i (B) ; k = 1; \Delta \Delta \Delta ; n (2) where s 2i = s 2i\Gamma1 = (s n\Gamma2i+2 + s n\Gamma2i+1 ) \Gam...
Relative perturbation bounds for positive polar factors of graded matrices
 SIAM J. Matrix Anal. Appl
"... Abstract. Let B be an m × n (m ≥ n) complex (or real) matrix. It is known that there is a unique polar decomposition B = QH, where Q∗Q = I, the n × n identity matrix, and H is positive definite, provided that B has full column rank. If B is perturbed to B̃, how do the polar factors Q and H change? T ..."
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Abstract. Let B be an m × n (m ≥ n) complex (or real) matrix. It is known that there is a unique polar decomposition B = QH, where Q∗Q = I, the n × n identity matrix, and H is positive definite, provided that B has full column rank. If B is perturbed to B̃, how do the polar factors Q and H change? This question has been investigated quite extensively, but most work so far has been on how the perturbation changed the unitary polar factor Q, with very little on the positive polar factor H, except ‖H − H̃‖F ≤ 2‖B − B̃‖F in the Frobenius norm, due to [F. Kittaneh, Comm. Math. Phys., 104 (1986), pp. 307–310], where Q ̃ and H ̃ are the corresponding polar factors of B̃. While this inequality of Kittaneh shows that H is always well behaved under perturbations, it does not tell much about smaller entries of H in the case when H’s entries vary a great deal in magnitude. This paper is intended to fill the gap by addressing the variations of H for a graded matrix B = GS, where S is a scaling matrix and usually diagonal (but may not be). The elements of S can vary wildly, while G is well conditioned. In such cases, the magnitudes of H’s entries indeed often vary a lot, and thus any bound on ‖H − H̃‖F means little, if anything, to the accuracy of H̃’s smaller entries. This paper proposes a new way of measuring the errors in the H factor via bounding the scaled difference (H ̃ − H)S−1, as well as accurately computing the factor when S is diagonal. Numerical examples are presented. The results are also extended to the matrix square root of a graded positive definite matrix.