M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17:303--311, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....point arithmetic the calculation of r is traditionally done in higher precision and the process iterated. Nowadays it is more common to carry out the whole process at the working precision. This fixed precision iterative refinement was first advocated and shown to be effective in the 1970s [91], 115] It is used in LAPACK in conjunction with the linear equation solvers based on LU factorization. Current understanding can be summarized as follows [82] Consider any linear equation solver. We require only that the computed solution b x satisfies (A DeltaA)bx = b with k DeltaAk 1 f(A; ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17:303--311, 1977.

....residual norm, in which case we can use our results to bound the backward error. The idea of using iterative refinement to obtain a small backward error with a potentially unstable solution method has been investigated for linear systems by several authors, including Jankowski and Wo zniakowski [18], Skeel [23] and Higham [17] and more recently for the algebraic Riccati equation by Ghavimi and Laub [11] The idea does not previously seem to have been applied to the generalized eigenvalue problem. In Section 3 we apply our results to linear systems and to the standard and generalized ....

....satisfying ukA kOE(A; b; n; u) 1=8. Then, the norm of the residual decreases until kbr i k max(fl n ; u) kAkkbxk kbk) so that iterative refinement yields a small normwise backward error j(bx) max(fl n ; u) Corollaries 3.1 and 3. 2 are standard normwise results in the literature [17] [18], 20] 23] 27] They show that we do not lose anything by using our general analysis. 3.2. Generalized Eigenvalue Problem. Newton s method and its variants have been considered for improving the accuracy of computed eigenvalues and eigenvectors for the standard eigenvalue problem [8] 9] ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17: 303--311, 1977.

.... that, under certain conditions, just one step of fixed precision iterative refinement is sufficient to yield a small componentwise relative backward error for Gaussian elimination with partial pivoting (GEPP) the componentwise relative backward error is defined below) Jankowski and Wo zniakowski [9] had earlier shown that, again with certain provisos, an arbitrary linear equation solver is made normwise backward stable by fixed precision iterative refinement (possibly with more than one iteration) Skeel s analysis of fixed precision iterative refinement was generalized by Higham [4] to an ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17:303--311, 1977.

....refinement may be unattractive because the solution of Ad = r in the refinement step will in general be as expensive as the computation of the original solution b x. We note that iterative refinement is identified as a means of improving the normwise backward error for iterative methods in [19]. 16 To conclude, we return to our numerical examples. For the SOR example in section 1, c(A) O(10 45 ) and kHk1 = O(10 30 ) so our error bounds for this problem are all extremely large. In this problem max i j1 Gamma i j= 1 Gamma j i j) 3, where i = i (M N ) so (3.13) is very ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17:303--311, 1977.

....residual norm, in which case we can use our results to bound the backward error. The idea of using iterative refinement to obtain a small backward error with a potentially unstable solution method has been investigated for linear systems by several authors, including Jankowski and Wozniakowski [18], Skeel [22] and Higham [17] and more recently for the algebraic Riccati equation by Ghavimi and Laub [11] The idea does not seem to have been applied previously to the generalized eigenvalue problem. In section 3 we apply our results to linear systems and to the standard and generalized ....

....u#A 1 ##(A, b, n, u) # 1 8. Then the norm of the residual decreases until ##r i # # max(# n , u) #A###x# #b#) so that iterative refinement yields a small normwise backward error #(#x) # max(# n , u) Corollaries 3.1 and 3. 2 are standard normwise results in the literature [17] [18], 20] 22] 27] They show that we do not lose anything by using our general analysis. 3.2. Generalized eigenvalue problem. Newton s method and its variants have been considered for improving the accuracy of computed eigenvalues and eigenvectors for the standard eigenvalue problem [10] 7] ....

M. Jankowski and H. Wo zniakowski, Iterative refinement implies numerical stability, BIT, 17 (1977), pp. 303--311.

....and Weak Stability. In this section we give definitions of stability and weak stability of algorithms for solving linear systems. Consider algorithms for solving a nonsingular, n Theta n linear system Ax = b. There are many definitions of numerical stability in the literature, for example [3, 4, 8, 9, 14, 24, 38, 51, 59, 61, 65]. Definitions 2.1 and 2.2 below are taken from Bunch [15] Definition 2.1. An algorithm for solving linear equations is stable for a class of matrices A if for each A in A and for each b the computed solution x to Ax = b satisfies b A x = b b, where b A is close to A and b b is close to ....

....small we can accept x as a reasonably accurate solution. In the rare cases that the residual is not sufficiently small we can can use the slow but stable algorithm (alternatively, if the residual is not too large, one or two iterations of iterative refinement may be sufficient and faster [51, 81]) An example of this general strategy is the solution of a Toeplitz system by Ming Gu or M. Stewart s modification of the GKO algorithm. We can use the O(n 2 ) algorithm, check the residual, and resort to iterative refinement or a stable O(n 3 ) algorithm in the (rare) cases that it is ....

M. Jankowski and M. Wozniakowski, Iterative refinement implies numerical stability, BIT 17 (1977), 303--311.

....d 1 d 2 Likewise, we evaluate = 1 Gamma f i f j ) and then r ij = u i;1 u j;1 : This guarantees r i;j = r i;j (1 c 36 ffi 7 ) 9.2. Iterative Refinement. If the factorization L L T is not too inaccurate, and if R is not too ill conditioned, then it follows from the analysis in [11] that the solution x of Rx = b can be made backward stable by iterative refinement. Algorithm 9.1 (Iterative Refinement) Set x 0 = x, r = b Gamma Rx 0 repeat until krk c 37 fflkRk kxk solve L L T ffix = r set x i = x i Gamma1 ffix r = b Gamma Rx i endrepeat 10. Enhancing the ....

M. Jankowski and M. Wozniakowski, Iterative refinement implies numerical stability, BIT, 17 (1977), pp. 303--311.

....z, then z would be the exact solution to the given system. In floating point arithmetic, the computation of r can be done in higher precision. However, it is also common to carry out a single refinement step at the standard working precision. The use of iterative refinement at working precision [JW77, Ske80] has been advocated and shown to be effective. One iterative refinement step will usually be enough to improve the accuracy of the computed solution, see [Hig96] Now, we proceed to derive a working precision iterative refinement scheme that can be applied in 12 our parallel approach. ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17:303--311, 1977.

....residual norm, in which case we can use our results to bound the backward error. The idea of using iterative refinement to obtain a small backward error with a potentially unstable solution method has been investigated for linear systems by several authors, including Jankowski and Wo zniakowski [18], Skeel [23] and Higham [17] and more recently for the algebraic Riccati equation by Ghavimi and Laub [11] The idea does not previously seem to have been applied to the generalized eigenvalue problem. In Section 3 we apply our results to linear systems and to the standard and generalized ....

....kOE(A; b; n; u) 1=8. Then the norm of the residual decreases until kbr i k max(fl n ; u) kAkkbxk kbk) 12 F. TISSEUR so that iterative refinement yields a small normwise backward error j(bx) max(fl n ; u) Corollaries 3.1 and 3. 2 are standard normwise results in the literature [17] [18], 20] 23] 27] They show that we do not lose anything by using our general analysis. 3.2. Generalized eigenvalue problem. Newton s method and its variants have been considered for improving the accuracy of computed eigenvalues and eigenvectors for the standard eigenvalue problem [8] 9] ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17: 303--311, 1977.

.... error The traditional answer to a slightly different question what to do if b x does not have a small enough forward error is to use iterative refinement, and it is usually stressed that residuals must be computed in higher precision (see, e.g. 16] Work by Jankowski and Wo zniakowski [28] and Skeel [36] shows that iterative refinement using single precision residuals is usually sufficient to yield a small backward error, although it will not necessarily produce a small forward error. Jankowski and Wo zniakowski deal with normwise backward error in their analysis and cater for ....

M. Jankowski and H. Wo'zniakowski, Iterative refinement implies numerical stability, BIT, 17 (1977), pp. 303--311.

....refinement may be unattractive because the solution of Ad = r in the refinement step will in general be as expensive as the computation of the original solution b x. We note that iterative refinement is identified as a means of improving the normwise backward error for iterative methods in [19]. To conclude, we return to our numerical examples. For the SOR example in section 1, c(A) O(10 45 ) and kHk1 = O(10 30 ) so our error bounds for this problem are all extremely large. In this problem max i j1 Gamma i j= 1 Gamma j i j) 3, where i = i (M Gamma1 N ) so (3.13) is ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17:303--311, 1977.

....of a Toeplitz matrix as a sequence of Bennett s downdating steps. Finally, he estimated the forward error in the decomposition using Fletcher and Powell s methodology [12] This paper generalizes and presents new derivations of the results obtained in [22] 7. Numerical examples. We adopt from [17] the following definitions of forward and backward stability. Definition 7.1. An algorithm for solving the equation (1.1) is forward stable if the computed solution x satisfies jjx Gamma xjj c 1 (n) ffl cond(T )jj xjj ; where cond(T ) jjT jj jjT Gamma1 jj is the condition number of T ....

M. Jankowski and H. Wozniakowski, Iterative refinement implies numerical stability, BIT, 17 (1977), pp. 303--311.

....Weak Stability In this section we give definitions of stability and weak stability of algorithms for solving linear systems. Consider algorithms for solving a nonsingular, n Theta n linear system Ax = b, so m = n. There are many definitions of numerical stability in the literature, for example [5, 6, 9, 12, 16, 22, 36, 48, 57, 61, 70]. Definitions 1 and 2 below are taken from Bunch [17] Definition 1 An algorithm for solving linear equations is stable for a class of matrices A if for each A in A and for each b the computed solution x to Ax = b satisfies b A x = b b, where b A is close to A and b b is close to b. Definition ....

....an approximate solution x to the linear system (37) or the linear least squares problem (44) In practice this gives an accurate solution in a small number of iterations so long as the residual is computed accurately and the working precision is sufficient to ensure convergence. For details see [5, 36, 37, 48, 69, 84]. A related idea is to improve the accuracy of R b and R t by using Bj orck s Corrected SemiNormal Equations [5, 63] or Foster s scheme of iterative improvement [26] However, if the aim is simply to solve a linear system, then it is more economical to apply iterative refinement directly ....

M. Jankowski and H. Wozniakowski, "Iterative refinement implies numerical stability", BIT 17 (1977) 303--311.

....partial pivoting, or with QR, is usually quite satisfactory, recent work has shown that the guaranteed a priori bounds on the norm of the residual after just one step of iterative refinement are much 9 tighter than the bounds for the residual from the original solution method. Early work [9] [19] was based on norm wise bounds, along the lines of the bounds in the previous section. But recent work [1] 7] has adopted a component wise analysis that in many cases yields much tighter bounds on the sizes of the individual components of the residual vector. In this section, we sketch one ....

M. Jankowski, H. Wozniakowski, Iterative Refinement Implies Numerical Stability, BIT 17, pp. 303-311, 1977.

....z, then z would be the exact solution to the given system. In floating point arithmetic, the computation of r can be done in higher precision. However, it is also common to carry out a single refinement step at the standard working precision. The use of iterative refinement at working precision [JW77, Ske80] has been advocated and shown to be effective. One iterative refinement step will usually be enough to improve the accuracy of the computed solution, see [Hig96] Now, we proceed to derive a working precision iterative refinement scheme that can be applied in our parallel approach. ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17:303--311, 1977.

....of a Toeplitz matrix as a sequence of Bennett s downdating steps. Finally, he estimated the forward error in the decomposition using Fletcher and Powell s methodology [12] This paper generalizes and presents new derivations of the results obtained in [22] 7 Numerical examples We adopt from [17] the following definitions of forward and backward stability. Definition 7.1: An algorithm for solving the equation (1.1) is forward stable if the computed solution x satisfies jjx Gamma xjj c 1 (n)fflcond(T )jj xjj ; where cond(T ) jjT jj jjT Gamma1 jj is the condition number of T , ....

M. Jankowski and H. Wozniakowski, "Iterative Refinement Implies Numerical Stability", BIT , vol 17, pp 303-311, 1977.

....errors and condition numbers for large size problems (I n ; A) jAj;jbj = jAj;jbj (T ; y) cond jT j;jbj = cond jT j;jbj (T ; y) in the first order perturbation of equation (2. 8) If this process can be shown to be convergent, the backward stability of the algorithm would be guaranteed, [7], 14] Since these are linear equations we expect this to converge if u Delta (y) 1 (see [16] 4. Numerical results We present the results of several experiments carried out on a SUN Sparc 10, using MATLAB and double precision (u 1:11 Theta10 Gamma16 ) The matrix pairs ( E; A) ....

M. Jankowski and M. Wo'zniakowski, "Iterative refinement implies numerical stability," BIT, vol. 17, pp. 303--311, 1977.

.... that, under certain conditions, just one step of fixed precision iterative refinement is sufficient to yield a small componentwise relative backward error for Gaussian elimination with partial pivoting (GEPP) the componentwise relative backward error is defined below) Jankowski and Wo zniakowski [9] had earlier shown that, again with certain provisos, an arbitrary linear equation solver is made normwise backward stable by fixed precision iterative refinement (possibly with more than one iteration) Skeel s analysis of fixed precision iterative refinement was generalized by Higham [4] to an ....

M. Jankowski and H. Wo'zniakowski. Iterative refinement implies numerical stability. BIT, 17:303--311, 1977.

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