Nicholas J. Higham. Stable iterations for the matrix square root. Numerical Algorithms, 15:22242, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....extended the Schur method for computing a real square root of a real matrix. The Denman and Beavers iteration (1.19) can be derived via the matrix sign function [46, Leto. 2. 1] Some other methods for the matrix square root based on the sign function are the Schulz iteration and the Pad6 iteration [46]. MATLAB has also included a function sqrtra for solving matrix square root from at least version 3. Recently Higham [47] suggested a new sqrtra, which is based on the Schur method; it is more accurate than the previous version and gives useful information about the stability and conditioning. ....

Nicholas J. Higham. Stable iterations for the matrix square root. Numerical Algorithms, 15:22242, 1997.

....eigenvalues [12] sign(A) A(A 2 ) 1=2 : 3.3) Note that A 2 has no nonpositive real eigenvalues, so that (A 2 ) 1=2 is de ned. 3.1 Denman and Beavers Iteration In this section we derive the iteration (2.6) using the sign function de nition (3. 3) This section is based closely on Higham [15]. We de ne a block 2 2 matrix B = 0 0 A I 0 1 A ; 3.4) where A 2 C n n has no nonpositive real eigenvalues and I is the n n identity matrix. Lemma 3.1.1 For A 2 C n n having no nonpositive real eigenvalues, sign(B) sign 2 4 0 0 A I 0 1 A 3 5 = 0 0 A 1=2 A 1=2 0 1 A : ....

....P k = AQ k . The case k = 0 is given. Using the fact P n Q n = Q n P n , assume that P n = AQ n . Then P n 1 = 1 2 P n (3I Q n P n ) 1 2 AQ n (3I Q n P n ) 1 2 AQ n (3I P n Q n ) 1 2 A(3I Q n P n )Q n = AQ n 1 : We now consider convergence of the iteration (3.16) Theorem 3.3. 1 [15] Let kdiag(A I; A I)k 1: Then for iterations (3.16) lim k 1 P k = A 1=2 ; lim k 1 Q k = A 1=2 : CHAPTER 3. USING THE MATRIX SIGN FUNCTION 45 Proof. The convergence condition is kI X 2 0 k = kI B 2 k 1. For X 0 = B, kI B 2 k = I 0 0 A I 0 1 A 2 ....

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N. J. Higham. Stable iterations for the matrix square root. Numerical Analysis Report No. 305, Manchester Centre for Computational Mathematics, England, April 1997. 20 pp. To appear in Numerical Algorithms.

....1 Introduction Any matrix X such that X 2 = A is said to be a square root of the matrix A. For general complex matrices A 2 C n Thetan there exists a well developed although somewhat complicated theory of matrix square roots [7, 14] and a number of algorithms for their effective computation [2, 11]. Similarly for the theory and computation of real square roots for real matrices [10, 14] By contrast structured square root problems, where both the matrix A and its square root X are required to have some extra (not necessarily the same) specified structure, have been relatively little ....

....square root problems, where both the matrix A and its square root X are required to have some extra (not necessarily the same) specified structure, have been relatively little studied. Some notable exceptions include positive (semi)definite square roots of positive (semi)definite matrices [10, 11, 13], and M matrix square roots of M matrices [1, 11] In this paper we investigate another such structured Fachbereich 3 Mathematik und Informatik, Zentrum fur Technomathematik, Universitat Bremen, D28334 Bremen, FRG, e mail : heike math.uni bremen.de y Department of Mathematics, State ....

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N.J. Higham. Stable iterations for the matrix square root. Numerical Algorithms, 15:227--242, 1997.

....per iteration, is preferable to iterating with Y k and Z k from (3.1) at the cost of two inversions per iteration, since matrix multiplication is faster than matrix inversion. In practice, it is vital to scale matrix iterations to produce reasonably fast overall convergence. Higham [12] derives a scaling for the DB iteration based on = det(Y k ) det(Z k ) it requires Y k and Z k to be multiplied by j j at the start of the (k 1)st iteration, where A is of order n. For the product form of the iteration, since det(Y k ) det(Z k ) det(M k ) and we invert and hence ....

....of a solution X = f(X) if the error matrices E k = X Gamma X k satisfy E k 1 = L(E k ) O(kE k k where L is a linear operator that has bounded powers, that is, there exists a constant c such that for all p 0 and arbitrary E of unit norm, kL (E)k c. The DB iteration is stable [9] [12]; the iteration (3.2) which is a standard Newton iteration for A is unstable unless the eigenvalues i of A satisfy max i;j fi 1 Gamma ( i = j ) fi 2 [9] It is easy to show that the product form of the DB iteration is stable. Define the error terms G k = Y k Gamma A , H k = Z k ....

Nicholas J. Higham. Stable iterations for the matrix square root. Numerical Algorithms, 15 (2):227--242, 1997.

....square root method of Denman and Beavers [5] Given a matrix M with no eigenvalues on the negative real axis, the DenmanBeavers iteration for the square root is Y n 1 = # Y n Z 1 n # 2 Z n 1 = # Z n Y 1 n # 2 where Y 0 = M and Z 0 = I . This iteration has been shown by Higham [12] to be stable and converge quadratically with lim n## Y n = M 1 2 and lim n## Z n = M 1 2 . The Denman Beavers iteration has the advantage of only requiring the matrix operation of inversion there is no need to transform to upper triangular form. Cheng et al. 4] were able to exploit this ....

....for the square root of (I cos H) 2 also generates the same iterate stream. To be specific let us define these four iterations. Note that we work with Newton s method rather than the DenmanBeavers iteration as a convenience only; in the absence of rounding errors they are equivalent. See Higham [12]. The first iteration is Newton s method for the square root of cos H i sin H . Complex Newton Square Root Iteration: M 0 = cos H i sin H M k 1 = M k M 0 M 1 k 2 The second iteration is Newton s method for the square root of (I cos H) 2. Real Newton Square Root Iteration: A 0 = I ....

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Nicholas J. Higham, "Stable Iterations for the Matrix Square Root," Numerical Algorithms, 15(2), pp. 227-- 242, 1997.

....multiplication per iteration, is preferable to iterating with Y k and Z k from (3.1) at the cost of two inversions per iteration, since matrix multiplication is faster than matrix inversion. In practice, it is vital to scale matrix iterations to produce reasonably fast overall convergence. Higham [12] derives a scaling for the DB iteration based on = 6 S. H. CHENG, N. J. HIGHAM, C. S. KENNEY, AND A. J. LAUB det(Y k ) det(Z k ) it requires Y k and Z k to be multiplied by j Gamma1= 2n) j at the start of the (k 1)st iteration, where A is of order n. For the product form of the ....

....of a solution X = f(X) if the error matrices E k = X Gamma X k satisfy E k 1 = L(E k ) O(kE k k 2 ) where L is a linear operator that has bounded powers, that is, there exists a constant c such that for all p 0 and arbitrary E of unit norm, kL s (E)k c. The DB iteration is stable [9] [12]; the iteration (3.2) which is a standard Newton iteration for A 1=2 , is unstable unless the eigenvalues i of A satisfy max i;j fi fi 1 Gamma ( i = j ) 1=2 fi fi 2 [9] It is easy to show that the product form of the DB iteration is stable. Define the error terms G k = Y k Gamma A 1=2 ....

Nicholas J. Higham. Stable iterations for the matrix square root. Numerical Algorithms, 15 (2):227--242, 1997.

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