N.J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174):537--549, April 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....boundary operator. It remains to find a suitable approximation of F Gamma . The matrix F Gamma is a solution of a quadratic matrix equation. There are some iterative methods to solve such matrix equations, including Newton s method and iterations of the algebraic Ricatti equation, see e.g. [12]. The application of these methods, however, causes additional numerical effort, therefore we try to find approximations F Sigma of F Sigma by F Sigma = Gamma C Sigma C Gamma 4D Delta 1=2 9 which is motivated by the corresponding scalar quadratic equation. Clearly, this ....

Nicholas J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174):537--549, April 1986. 15

....is computationally much more expensive than the inversion of a matrix. In this section we present a Newtontype algorithm that allows a very ecient calculation of h 0 . Newton s method is well known in the context of computing matrix sign functions and square roots of nite dimensional matrices [18, 24]. Let M be an n n matrix with positive real valued eigenvalues and polar decomposition M = UH. Set M 0 = M;X 0 = M , and de ne the iterations M k 1 = 1 2 (M k [M k ] 1 M) k = 0; 1; 119) X k 1 = 1 2 (X k [X k ] 1 ) k = 0; 1; 120) Then M k converges to M 1 2 ....

....positive real valued eigenvalues and polar decomposition M = UH. Set M 0 = M;X 0 = M , and de ne the iterations M k 1 = 1 2 (M k [M k ] 1 M) k = 0; 1; 119) X k 1 = 1 2 (X k [X k ] 1 ) k = 0; 1; 120) Then M k converges to M 1 2 and X k converges to U , see [18, 24]. In Section 2 we have seen that h 0 minimizes kg hk among all windows h for which (h; a; b) is a normalized tight frame. Clearly in the same way h 0 minimizes k 0 hk. Also recall that 0 minimizes g kgk k k (121) among all dual windows . This suggests to compute h ....

N.J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174):537-549, 1986. 33

....that L f (A; E) f A E 0 A # 12 : 5. 4) The utility of this observation is that there are special methods to compute f(X) when f is a function with special properties for example the sine or cosine [16] 10 roy mathias the exponential [14] the logarithm [8] the square root [6] and the matrix sign ( 9] and the references therein) These special methods immediately yield methods for computing the directional derivative. Furthermore, one can use error analysis and perturbation theory for the function f to obtain error analysis and perturbation theory for its derivative. ....

Nicholas J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174):537--549, April 1986.

....small and the linear transformation F 0 (X) is nonsingular. Proof. The result follows from the local convergence theorem for Newton s method [22, p. 148] 2 We now consider the relationship between the Newton iteration (2.3) and the simpli ed iterations (2.4) and (2.5) Theorem 2.1. 2 [10] Consider the iterations (2.3) 2.4) and (2.5) Suppose X 0 = Y 0 = Z 0 commutes with A and that all the Newton iterates X k are wellde ned. Then i. X k commutes with A for all k, ii. X k = Y k = Z k for all k. Proof. We prove (i) and (ii) by induction. From the remarks preceding the theorem we ....

....gives Y k 1 X = 1 2 Y 1 k (Y k X) 2 : 2 Theorem 2.1.3 shows that the iterations (2.4) and (2.5) converge quadratically when the starting value is a multiple of the identity matrix. Also, Theorem 2.1.3 can be applied to the important case where A is Hermitian positive de nite. Corollary 2.1. 4 [10] Let A 2 C n n be Hermitian positive de nite. If Y 0 = mI; m 0; then the iterates Y k in (2.4) are all Hermitian positive de nite, lim k 1 Y k = X; where X is the unique Hermitian positive de nite square root of A, and (2.7) holds. CHAPTER 2. NEWTON S METHOD FOR MATRIX SQUARE ROOT 27 ....

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N. J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174):537-549, April 1986.

....decomposition, unless the condition number is very large. Key words. Matrix square root, Pad e approximation, prime fraction form AMS subject classifications. 15A15, 41A21, 65F30 1. Introduction. The numerical computation of the square root of a matrix has been studied by a number of authors [12, 14, 4, 6, 7, 5, 11, 10, 13]. A popular approach is based on the Schur decomposition of the matrix [4, 7] Iterative methods are also possible [12, 14, 6, 13] For a symmetric positive definite matrix A, there is a unique symmetric positive definite square root (denoted by p A) For the given spectral decomposition A = V ....

....classifications. 15A15, 41A21, 65F30 1. Introduction. The numerical computation of the square root of a matrix has been studied by a number of authors [12, 14, 4, 6, 7, 5, 11, 10, 13] A popular approach is based on the Schur decomposition of the matrix [4, 7] Iterative methods are also possible [12, 14, 6, 13]. For a symmetric positive definite matrix A, there is a unique symmetric positive definite square root (denoted by p A) For the given spectral decomposition A = V V T ; 1.1) where is the diagonal matrix of the eigenvalues, V is the orthogonal matrix of the corresponding eigenvectors, the ....

N. J. Higham, Newton's method for the matrix square root, Math. Comp., 46 (1986), pp. 537550.

.... 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 unique. On the other hand the unitary factor U is unique if and only if the matrix A has full rank, rank A = n (see Higham and Schreiber [27] Using (7) Higham [17] has proposed ....

N. J. Higham, Newton's method for the matrix square root, Math. Comput. 46 (1986), 537-549.

.... (1) by Roberts in a 1971 technical report [34] which was not published until 1980 [35] Kato [23, Page 67] reports that the resolvent integral (2) goes back to 1946 [12] and 1949 [21, 22] There is some concern about the numerical stability of numerical methods based upon the matrix sign function [2, 8, 19]. In this paper, we demonstrate that evaluating the matrix sign function is a more ill conditioned computational problem than the problem of finding bases of the invariant subspaces V and V Gamma . See Example 1 in Section 3. Nevertheless, we also give perturbation and error analyses, which ....

....the problem of finding small norm solutions E 2 R n Thetan and F 2 R n Thetan to sign(A E) S F . Of course, this does not uniquely determine E and F . Common algorithms for evaluating sign(A) like Newton s method for the square root of I guarantee that S is very nearly a square root of I [19], i.e. S is a close approximation of sign(S) In the following theorem, we have arbitrarily taken F = sign(S) Gamma S. Theorem 5.2. If k sign(S)S Gamma1 Gamma Ik 1 and k sign(S)A Gamma A sign(S)k dA , then sign(A E) S F for perturbation matrices E and F satisfying kFk kS ....

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N.J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174):537--549, April 1986.

....properties. If A has no nonpositive real eigenvalues then, provided all the iterates in (1.2) are defined, X k converges quadratically Iteration (1.2) is Newton s method for the equation X A = 0, with X 0 = A, with the equations rewritten to exploit the fact that AXk = XkA for all k. to A [13]. If A is symmetric positive definite then so is each iterate X k . Unfortunately, this Newton iteration has such poor numerical stability that it is useless for practical computation. The instability was observed by Laasonen [30] and rigorously explained by Higham [13] Analysis shows that unless ....

....= XkA for all k. to A [13] If A is symmetric positive definite then so is each iterate X k . Unfortunately, this Newton iteration has such poor numerical stability that it is useless for practical computation. The instability was observed by Laasonen [30] and rigorously explained by Higham [13]. Analysis shows that unless the eigenvalues # i of A satisfy the very strong requirement that (# j # i ) 6 1, i, j = 1, n, then small errors in the iterates can be amplified by the iteration, with the result that in floating point arithmetic the iteration fails to converge. An ....

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N.J. Higham, Newton's method for the matrix square root, Math. Comp. 46(174) (1986) 537--549.

....saw in Section 3, even the correctly rounded exact square root may have a large relative residual. If the test is failed then an orthogonal similarity transformation is applied to A, the whole computation is repeated, and the inverse transformation is performed. Then one step of Newton s method [3] is applied and the relative residual computed once more. The logic behind the similarity transformation and the Newton step is not clear, since the similarity does not change the eigenvalues and so should make little difference to the accuracy of funm, and Newton s method can increase the error ....

....computed once more. The logic behind the similarity transformation and the Newton step is not clear, since the similarity does not change the eigenvalues and so should make little difference to the accuracy of funm, and Newton s method can increase the error because of its numerical instability [3]. Newton s method here refers to a method obtained from the true Newton method by making commutativity assumptions, and these assumptions destroy the self correcting nature of Newton s method. 5 A New Our suggested replacement for sqrtm, which we refer to as sqrtm to avoid confusion, is ....

Nicholas J. Higham. Newton's method for the matrix square root. Math. Comp., 46 (174):537--549, April 1986.

....b and c are equal to a non principal square root of a, and hence are not in the halfplane of a . 3. Denman Beavers Square Root Iteration. The Denman Beavers (DB) iteration [5] for the square root of a matrix A with no eigenvalues on R (3. 1) The iteration has the properties [9] lim ; lim and, for all k, Y k = AZ k ; Y k Z k = Z k Y k ; Y k 1 = Y k AY k ) 2: 3.2) The next lemma is the basis for our use of the DB iteration for computing the logarithm. Lemma 3.1. The DB iterates satisfy the splitting relations log A = log Y k Gamma log Z k = 2 log ....

....By (3.2) the individual eigenvalues of Y k follow the scalar iteration y k 1 = y k ay k ) 2; y 0 = a; where a is an eigenvalue of A. This is just the scalar Newton iteration for the square root of a and it has the property that the iterates y k remain in the halfplane of a (see, e.g. [9]) Similar arguments show that log Y k Z k = log Y k log Z k , which yields the remaining two equalities. To see how to use Lemma 3.1, note that since Y k A Y k Z k I and log Y k Z k 0. Suppose we terminate the DB iteration after k iterations; we can write log A = 2 log Y k Gamma E ....

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Nicholas J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174): 537--549, April 1986.

....Square Root Iteration. The Denman Beavers (DB) iteration [5] for the square root of a matrix A with no eigenvalues on R Gamma is Y k 1 = Gamma Y k Z Gamma1 k Delta =2; Y 0 = A; Z k 1 = Gamma Z k Y Gamma1 k Delta =2; Z 0 = I : 3. 1) The iteration has the properties [9] lim k 1 Y k = A 1=2 ; lim k 1 Z k = A Gamma1=2 and, for all k, Y k = AZ k ; Y k Z k = Z k Y k ; Y k 1 = Y k AY Gamma1 k ) 2: 3.2) The next lemma is the basis for our use of the DB iteration for computing the logarithm. Lemma 3.1. The DB iterates satisfy the splitting relations ....

....scalar iteration y k 1 = y k ay Gamma1 k ) 2; y 0 = a; APPROXIMATING THE LOGARITHM OF A MATRIX 5 where a is an eigenvalue of A. This is just the scalar Newton iteration for the square root of a and it has the property that the iterates y k remain in the halfplane of a 1=2 (see, e.g. [9]) Similar arguments show that log Y k Z k = log Y k log Z k , which yields the remaining two equalities. To see how to use Lemma 3.1, note that since Y k A 1=2 and Z k A Gamma1=2 , Y k Z k I and log Y k Z k 0. Suppose we terminate the DB iteration after k iterations; we can write ....

[Article contains additional citation context not shown here]

Nicholas J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174): 537--549, April 1986.

....saw in Section 3, even the correctly rounded exact square root may have a large relative residual. If the test is failed then an orthogonal similarity transformation is applied to A, the whole computation is repeated, and the inverse transformation is performed. Then one step of Newton s method [3] is applied and the relative residual computed once more. The logic behind the similarity transformation and the Newton step is not clear, since the similarity does not change the eigenvalues and so should make little difference to the accuracy of funm, and Newton s method can increase the error ....

....computed once more. The logic behind the similarity transformation and the Newton step is not clear, since the similarity does not change the eigenvalues and so should make little difference to the accuracy of funm, and Newton s method can increase the error because of its numerical instability [3]. Newton s method here refers to a method obtained from the true Newton method by making commutativity assumptions, and these assumptions destroy the self correcting nature of Newton s method. 5 A New sqrtm Our suggested replacement for sqrtm, which we refer to as sqrtm to avoid confusion, ....

Nicholas J. Higham. Newton's method for the matrix square root. Math. Comp., 46 (174):537--549, April 1986.

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N.J. Higham. Newton's method for the matrix square root. Math. Comp., 46(174):537--549, April 1986.

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N. J. Higham, Newton's method for the matrix square root, Math. Comp., 46 (1986), pp. 537-- 549.

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Higham, N.J., "Newton's method for the matrix square root", Math. Comp. 46, 537-550, 1986.

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