N. J. Higham. Computing the polar decomposition--with applications. SIAM Journal on Scientific and Statistical Computing, 7:1160--1174, 1986. 6  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the proof, generalizations and references to related work. Lemma 2.2. Let A; E 2 M n be such that oe n (A) oe 1 (E) Then for any unitarily invariant norm k Delta k kU(A) Gamma U(A E)k 1 oe n (A) Gamma oe 1 (E) kEk: 2. 5) There are many ways to compute U(A) for a general nonsingular A [5, 6]. However, we are only interested in the case where A is almost unitary, in which case there are two simple methods. Let ffl be such that joe i (A) Gamma 1j ffl for i = 1; n. Then one can show that for B = A (A Gamma1 ) 2 or B = A[I Gamma (I Gamma A A) 2] 2.6) we have ....

N. J. Higham. Computing the polar decomposition - with applications. SIAM J. Sci. Stat. Comp., 7(4):1160--1174, 1986.

....(4) must be post multiplied by ( to obtain ) 6) This equation allows calculation of the compliance matrix from experimental data, but the result is rarely SPD and thus a SPD approximant must be found if the method is to have practical value. Higham [9, 10] used the Frobenius norm and the symmetric component of the initial non SPD matrix as a metric to locate the nearest symmetric, positive, semi definite (SPSD, # . matrix 0 21 23 . Stated formally, he solved 46587 9, BADCE0F HGI 0 21 23 FKJ (7) Higham s method effectively ....

N. J. Higham, "Computing the polar decomposition - with applications," SIAM Journal on Science and Statistical Computing, Vol. 7, no. 4, pp. 1160--1174, 1986.

....norms later in this section. In this paper we consider bounds on kU(A) Gamma U(B)k in terms of kA Gamma Bk for unitarily invariant norms k Delta k. The perturbation theory of the polar factor in the polar decomposition is of interest as it is often necessary to compute U(A) numerically. See [4] for a variety of applications of the polar decomposition. For a more recent application see [12] where the authors use the polar decomposition in a crucial way to compute block Householder transformations. The case where k Delta k is the Frobenius norm has been studied by several authors and ....

N. J. Higham. Computing the polar decomposition - with applications. SIAM J. Sci. Stat. Comp., 7(4):1160--1174, 1986. Perturbation Bounds for the Polar Decomposition 11

....Hoffman (1955) proved that minfkA Gamma Qk : Q Q = Ig = kA Gamma Uk; where k Delta k is any unitary invariant norm, a property saying that U is the best unitary (orthogonal in the real case) approximant to A in any unitary invariant norm. Optimality results for the factor H is discussed in Higham (1986) It is well known that when A is real, the matrix U is orthogonal and H is symmetric. In the remaining part of this section, we shall restrict to the case when A is real and invertible, A 2 G ae GL(R;N ) hence H is positive definite. We recall that AA T = HUU T H T = H 2 ; from which ....

Higham, N. J. (1986), `Computing the polar decomposition---with applications', SIAM J. Sci. Stat.

....no pure imaginary eigenvalues [31, 23] The iteration fails otherwise. In finite precision arithmetic, the iteration could converge slowly or not at all if A is close to having pure imaginary eigenvalues. There are many ways to improve the accuracy and convergence rate of this basic iteration [7, 22, 25]. For example, if #A 2 I# 1, we may use the Newton Schulz iteration A j 1 = 1 2 A j (3I A 2 j ) for j = 0, 1, 2, with A 0 = A (6) to avoid the use of the matrix inverse. Although it requires twice as many floating point operations, it is more e#cient whenever matrix multiply ....

N. J. HIGHAM, Computing the polar decomposition - with applications, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 1160--1174.

....the polar decomposition : i) asymptotic bounds, which bound the variations of Q and H provided that the norm of the perturbation DeltaA is small enough. ii) absolute bounds, which are true without any assumption on the size of DeltaA. iii) condition numbers. Our bibliographical review shows that [2, 7] prove asymptotic bounds, 5, 9, 12, 13] prove absolute bounds and [8, 10, 14] give condition numbers. Asymptotic bounds and absolute bounds often yield upper bounds for the condition number. We now recall existing results from the three cases above. For each existing bound we derive, when ....

....particular case of an absolute bound, but no explicit reference is provided. Several authors prove that if H DeltaH is the Hermitian factor associated with A DeltaA, then k DeltaHk F p 2 k DeltaAk F : 1) This inequality has been shown by Araki and Yamagami [1] We mention also the papers [7, 2] which contain the bound (1) but under the assumption that k DeltaAk F is small enough. In [3] Bhatia gives a nice survey of these results. Kittaneh proved in [11] that p 2 is therefore the best possible constant in (1) which means that C(H) p 2. One could think, as it is wrongly stated in ....

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N. J. Higham. Computing the polar decomposition with applications. SIAM J. Sci. Stat. Comput., 7:1160--1174, 1986.

....i 1 fl i A 01 i ) where ff i ; fi i and fl i are appropriately chosen scalars. Some of the known scaling schemes are Roberts [63] ff i = kA 01 i k= kA i k kA 01 i k) fi i = kA i k= kA i k kA 01 i k) Balzer [6] ff i = jdet(A i )j 1=n 1) 01 ; fi i = 1 0 ff i : Higham [47]: fl i = h kA 01 i k 1 kA 01 i k 1 = kA i k 1 kA i k 1 ) i 1=4 : Byers [12] fl i = jdet(A i )j 01=n : A comparison of different scaling schemes for different ff i and fi i is presented by Balzer [6] and for different fl i by Kenney and Laub [54] Kenney and Laub shows that the ....

N. J. Higham. Computing the polar decomposition - with application. SIAM J. Sci. Stat. Comput., 7:1160--1174, 1986.

....convergent with lim j 1 A j = sign(A) 47, 36] The iteration could fail to converge if A has pure imaginary eigenvalues (or, in finite precision, if A is close to having pure imaginary eigenvalues. There are many ways to improve the accuracy and convergence rates of this basic iteration [16, 33, 38]. The matrix sign function may also be used in the generalized eigenproblem A0B by implicitly applying (3.8) to AB 01 [29] We do not want to invert B if it is ill conditioned, which is why we want to apply the previous algorithm implicitly. This leads to the following iteration: A j 1 = 1 2 (A ....

N. J. Higham. Computing the polar decomposition - with applications. SIAM J. Sci. Stat. Comput., 7:1160--1174, 1986.

....The computations were done by the first author in his master thesis written under direction of the second author (see Zieli nski [43] In the present paper we do not deal with the perturbations bounds for the polar decomposition. This problem is considered in the papers of Barrlund [3] Higham [17], Kenney and Laub [30] and Mathias [34] In the remaining part of the paper we assume m n, although some properties and algorithms are valid also for the case m n. 2 Properties and applications of the polar decomposition The polar decomposition is an important tool in various applications. ....

....of the polar decomposition The polar decomposition is an important tool in various applications. We will mention some of them. Let A 2 C m Thetan , m n, have the polar decomposition (1) The Hermitian factor H and the unitary factor U have the following properties (see for example Higham [17]) The matrix H is the unique positive semi definite square root of A H A (the definition and properties of the square roots of matrices are given for example in Higham [18] 19] Horn and Johnson [29, p. 419] H = A H A) 1=2 : 7) This implies that the Hermitian factor H is always ....

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N. J. Higham, Computing the polar decomposition -- with applications, SIAM J. Sci. Stat. Comput. 7 (1986), 1160-1173.

....; 3) where U = U 1 ; U 2 ) and V are unitary, U 1 is m Theta n, Sigma = Sigma 1 0 and Sigma 1 = diag (oe 1 ; oe n ) is nonnegative. There are many published bounds upon how much the two factor matrices Q and H may change if entries of B are perturbed in arbitrary manner [1, 2, 3, 4, 6, 5, 7, 8, 9]. In these papers, no assumption was made on how B was perturbed unlike what we are going to do here. In this paper, we obtain some bounds for the perturbations of Q, assuming B is complex and is perturbed to e B = D 1 BD 2 , where D 1 and D 2 are two nonsingular matrices and close to the ....

N. J. Higham. Computing the polar decomposition--with applications. SIAM Journal on Scientific and Statistical Computing, 7:1160--1174, 1986.

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N. J. Higham, Computing the polar decomposition---with applications, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 1160--1174.

....the iteration (4.2) can be used to compute (I . However, unless A is extremely well conditioned (# 2 (A) 3, say) convergence will be exceedingly slow. For symmetric positive definite matrices, the best alternative iteration to the iterations of section 2 is the following method of Higham [12]. Algorithm 2. Given a symmetric positive definite matrix A this algorithm computes X = A . 1. A = R R (Cholesky factorization) 2. Compute U = X# from X k 1 = X k X T ) k = 1, 2, with X 0 = R. 3. X = U R. The iteration in step 2 is a Newton iteration for ....

..... This Newton iteration requires one nonsymmetric matrix inversion per iteration, which is equivalent to the two symmetric positive definite inversions per iteration required by the DB iteration (1. 3) The iteration is stable and is readily accelerated by the incorporation of scaling parameters [12,27]. Algorithm 2 can be extended to deal with positive semidefinite matrices by using a Cholesky factorization with pivoting [19] 5. Numerical experiments To give further insight into the methods, we present the results of some numerical experiments. All computations were performed in MATLAB, ....

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N.J. Higham, Computing the polar decomposition -- with applications, SIAM J. Sci. Statist. Comput. 7(4) (1986) 1160--1174.

....factor of A A and Q = AR . We begin with an example to illustrate the versatility of QR factorization. Any matrix A 2 R with m n has a polar decomposition A = UH , where U has orthonormal columns and H is symmetric positive semidefinite. The polar decomposition has various applications [31] and can be computed using the Newton iteration X k 1 = 2X k (I X k X k ) X 0 = A; 4.1) whose iterates converge to U quadratically. By adapting an idea from [51] For square matrices, iteration (4.1) is related to the iteration Y k 1 = Y k Y k ) 2, Y 0 = A, by Y k = X k . ....

Nicholas J. Higham. Computing the polar decomposition---with applications. SIAM J. Sci. Stat. Comput., 7(4):1160--1174, October 1986.

....at the end of a square root approximation we should try to restore the unitary property by finding the nearest unitary matrix to the approximate square root. The problem of finding the nearest unitary matrix is well studied and its solution can be effected using the polar decomposition iteration [7]. We present a result showing that the polar decomposition step is equivalent to that of three other iterations for the problem of finding the logarithm (see the Ray Lemma in Section 2) Unfortunately, the question of error propagation is somewhat difficult in the sine cosine half angle ....

.... 1 = sin(H 2) cos(H 2) 1 = tan(H 2) A second question is raised when we consider that Y approx iV approx # e iH 2 should be unitary but generally is not. Should we find the nearest unitary matrix by using, say, Newton s method for the polar decomposition of Y approx iV approx (See [7], 9] 10] and [11] for background on the polar iteration. It was this question that pointed the way to the Ray Lemma: consider the scalar polar iteration for z = #e i# z k 1 = # z k 1 z k # 2 with z 0 = #e i# . Here z k denotes the complex conjugate of z k . The iterates z k stay ....

Nicholas J. Higham, "Computing the Polar Decomposition -- with Applications," SIAM J. Sci. Statist. Comput. , 7(4), pp. 1160--1174, 1986.

....in aerospace applications, for example, in which a direction cosine matrix whose columns drift from orthonormality is periodically orthonormalized by replacing it with its unitary polar factor. An alternative approach that does take advantage of a nearly unitary A is to use the Newton iteration [11] Y k 1 = 1 2 (Y k Y Gamma k ) Y 0 = A 2 C n Thetan : 1.2) This is Newton s method applied to the equation Y T Y = I. For any nonsingular A, Y k converges to U quadratically as k 1, and if A is nearly unitary only a few iterations are needed. However, the only opportunity for ....

....error y k Gamma 1 is only approximately halved on each step. The speed of convergence can be improved by scaling the iterates Y k fl k Y k , so that the iteration becomes Y k 1 = 1 2 (fl k Y k fl Gamma1 k Y Gamma k ) Y 0 = A 2 C n Thetan : 2. 1) This idea has been investigated in [11], where it is shown that the scaling factor fl k = Gamma oe max (Y k )oe min (Y k ) Delta Gamma1=2 (2.2) minimizes a bound on kY k 1 Gamma Uk 2 , where oe max (Y k ) and oe min (Y k ) are the largest and smallest singular values of Y k , respectively. In practice, fl k is too expensive to ....

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Nicholas J. Higham. Computing the polar decomposition---with applications. SIAM J. Sci. Stat. Comput., 7(4):1160--1174, October 1986.

....we need either a proof that the algorithm is stable or an a posteriori stability test that can be used as a fourth stage of the algorithm. 3 An operation count for the case m = n gives insight into the efficiency. If we compute the polar decomposition using the Newton iteration of Higham [7], Y k 1 = 1 2 (fl k Y k fl Gamma1 k Y Gamma k ) Y 0 = A 2 C n Thetan (2.1) for which Y k U as k 1 with a quadratic rate of convergence, and where fl k is an acceleration parameter) the cost is in practice at most 20n 3 flops, where a flop is an addition, multiplication or ....

Nicholas J. Higham. Computing the polar decomposition---with applications. SIAM J. Sci. Stat. Comput., 7(4):1160--1174, October 1986.

....Road, Oxford, OX2 8DR. nagjdc vax.oxford.ac.uk) z Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK. mbbgsnh cms.mcc.ac.uk) 1 Nevertheless, there are some applications that genuinely require computation of a matrix inverse see [1, sec. 7.5] 14, p. 342ff] and [4, 10] for example. LAPACK [3] like LINPACK before it, will include routines for matrix inversion. LAPACK will support inversion of triangular matrices and of general, symmetric indefinite, and symmetric positive definite matrices via an LU (or related) factorization. Each of these matrix inversions ....

N.J. Higham, Computing the polar decomposition---with applications, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 1160--1174.

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Nicholas J. Higham. Computing the polar decomposition---with applications. SIAM J. Sci. Stat. Comput., 7(4):1160--1174, October 1986.

....of the polar representation z = re i for complex numbers. The factor U has the property that when m n it is the nearest matrix with orthonormal columns to A for both the 2 norm and the Frobenius norm: kA Gamma Uk = minf kA Gamma Qk : Q Q = I ; Q 2 C m Thetan g: For more details see [18] or [28] ffl signm computes the matrix sign decomposition A = SN 2 C n Thetan , where S = sign(A) is the matrix sign function [25] If A has the Jordan canonical form A = XJX Gamma1 = X J 1 0 0 J 2 X Gamma1 ; where the eigenvalues of J 1 lie in the open left half plane and those of ....

Nicholas J. Higham. Computing the polar decomposition---with applications. SIAM J. Sci. Stat. Comput., 7(4):1160--1174, October 1986.

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N. J. Higham. Computing the polar decomposition--with applications. SIAM Journal on Scientific and Statistical Computing, 7:1160--1174, 1986. 6

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N. J. Higham, Computing the polar decomposition---with applications, SIAM J. Sci. Statist. Comput., 7(1986), 1160--1174.

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N. J. Higham. Computing the polar decomposition - with applications. SIAM J. Sci. Stat. Comput., 7:1160--1174, 1986.

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N. J. Higham, "Computing the polar decomposition---with applications, " SIAM J. Sci. Stat. Comput., no. 7, pp. 1160--1174, 1986.

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N.J. Higham. Computing the polar decomposition - with application. SIAM J. Sci. Stat. Comput., 7:1160--1174, 1986.

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N. Higham, Computing the polar decomposition -- with applications, SIAM J. Sci. Stat. Computing 7 (1986), 1160--1174.

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