A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM J. Numer. Anal., 16:368--375, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....singular, the minimum stretch is zero and the condition number is defined to be infinite. It is certainly not operational to compute the condition number of A by the formula kA kkAk. This number can be estimated by other procedures when the 1 norm is used. A classical reference is Cline et al. [23] and the LAPACK library. A.3 Complexity of Algorithms The analysis of the computational complexity of algorithms is a very sophisticated and difficult topic in computer science. Our goal is simply to present some terminology and distinctions that are of interest in our work. The solution of most ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An Estimate for the Condition Number of a Matrix. SIAM J. Numer. Anal., 16:368--375, 1979.

....to that of approximating fi fi fi A Gamma1 fi fi fi fi , has been largely debated. Since 1979, when LINPACK package was released, the same authors proposed an approximating algorithm involving O(n floating point operations and assuring a reliable indication for the order of magnitude of k(A) [30]. After such a proposal, several researchers have attempted to enhance the previous result [31, 32] until, in 1984, W. Hager suggested a new technique for estimating the l 1 condition number with an error under 5 = oo and a probability higher than 0.97 [33] Finally, it is worth mentioning that, ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson, "An estimate for the condition number of a matrix," SIAM Journal of Numerical Analysis, vol. 16, no. 2, pp. 368--375, 1979.

....and an estimate of is computed and placed in DPARM(3) The condition number of the matrix is the product DPARM(2) DPARM(3) The condition number is a measure of the reliability of the solution of a given linear system and can be used to compute error bounds. The algorithm described in [2] is used to estimate . If IPARM(25) 1, then 1 norms are computed, if IPARM(25) 2, then Frobenius norms are computed, and if IPARM(25) 3, then infinity norms are computed. WSSMP can be made to do this computation either with factorization, or with solution, or with iterative refinement. ....

A. K. Cline, Cleve B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM Journal on Numerical Analysis, 16:368--375, 1979. 44

....number (A) kAkkA k. Armed with this estimate, the user could estimate the accuracy of a computed solution. LINPACK s condition estimation algorithm estimates kA k 1 by constructing a vector x for which the lower bound kA xk 1 =kxk 1 is actually a good approximation to kA k 1 [28]. It makes explicit use of an LU factorization of A, but requires only O(n ) operations beyond the O(n ) operations required to factorize the n Theta n matrix A. The algorithm produces an approximate null vector (namely, A x) and it is this rather than the kA k estimate that is ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM J. Numer. Anal., 16(2):368-- 375, 1979.

....began in the 1970s with the problem of cheaply estimating the condition number (A) kAkkA and an approximate null vector of a square matrix A given some factorization of it. The earliest algorithm is one of Gragg and Stewart [10] It was improved by Cline, Moler, Stewart and Wilkinson [4], leading to the 1 norm condition estimation algorithm used in LINPACK [8] and later included in Matlab (function rcond) During the 1980s, attention was drawn to various componentwise condition numbers and it was recognized that most condition estimation problems can be reduced to the estimation ....

....power method. 1. How does the accuracy and reliability of the norm estimates vary with t 2. How good are the norm estimates in general 3. How does the number of iterations behave for t 1 Note that we are not searching for counterexamples, as was done for previous condition estimators [4], 5] 14] We know that for a fixed starting matrix and any t n there must be families of matrices whose norm is underestimated by an arbitrarily large factor, since the algorithm samples the behaviour of the n Theta n matrix A on fewer than n vectors. But since the algorithm uses a random ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM J. Numer. Anal., 16(2):368--375, 1979.

....of kA Gamma1 k is computed and placed in DPARM(3) The condition number of the matrix A is the product DPARM(2) Theta DPARM(3) The condition number is a measure of the reliability of the solution of a given linear system and can be used to compute error bounds. The algorithm described in [2] is used to estimate kA Gamma1 k. If IPARM(25) 1, then 1 norms are computed, if IPARM(25) 2, then Frobenius norms are computed, and if IPARM(25) 3, then infinity norms are computed. WSSMP can be made to do this computation either with factorization, or with solution, or with iterative ....

A. K. Cline, Cleve B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM Journal on Numerical Analysis, 16:368--375, 1979.

....neither needed nor computed. However, it might become useful to know z. In this case, we suggest multiplying Tz by T H and normalizing the result. This strategy of performing one step of power iteration with an appropriately chosen vector z was originally used by the LINPACK condition estimator (Cline, Moler, Stewart and Wilkinson 1979), but we choose z differently. 2.4 Incremental norm estimation for sparse matrices The incremental condition estimator described in Section 2.1 is intended to be used with dense matrices. In its original form, ICE cannot be applied to sparse matrices as we illustrate through Example 4. Example 4 ....

Cline, A. K., Moler, C. B., Stewart, G. W. and Wilkinson, J. H. (1979), `An estimate for the condition number of a matrix', SIAM J. Numerical Analysis 16, 368-- 375.

....problem dimension becomes large. 7.2.2 Condition Number Estimation (LINPACK) The matrix condition number (A) although useful for assessing the accuracy of computed solutions to systems of linear equations, is, in practice, too expensive to compute exactly. Cline, Moler, Stewart, and Wilkinson [3] describe O(n 2 ) algorithms that use the upper triangular factor from a matrix decomposition of A to estimate 1 (A) kAk 1 1kA 01 k 1 , A 2 R n2n . One of these algorithms has been incorporated into LINPACK. This algorithm s estimate, 5 A more sophisticated restart strategy, possibly ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM Journal of Numerical Analysis, 16(2):368--375, 1979.

....means we need to know the i th row of L 1 , i.e. W 1:i 1;i = L 1 ) i;1:i 1 . At least it would be convenient to have an estimate for k(L 1 ) i;1:i 1 k 1 which could serve as a substitute for fjW ki j : k ig. To do this we use a general condition estimator for triangular matrices from [14, 9] as a helpful estimate for k(L 1 ) i;1:i 1 k 1 . This condition estimator is based on solving a system with an upper triangular matrix U where the right hand side y only consists of 1 and the signs are chosen to successively maximize the solution x of Ux = y. Another look at this condition ....

....that the components of x = U 1 y precisely estimate k(U 1 ) i;i:i:n k 1 . To adapt this estimator to our problem we will consider Lx L = y L and U x U = y U to get estimates for k(L 1 ) i;1:i 1 k 1 and k(U 1 ) 1:i 1;i k 1 . Algorithm 5 (Condition Estimator for (L 1 ) adapted from [14, 9] ) Let L = L ij ) ij 2 R n;n be unit lower triangular. Compute Lx = y, where y 2 ( 1; 1) p = p = p = x = 0; 0) 2 R n , and let = 0 be the associated 1 norm of p 5 for i = 1 : n x = 1 p i , x = 1 p i Let s be the set of nonzero components of L i 1:n;i p ....

A. Cline, C. B. Moler, G. Stewart, and J. Wilkinson. An estimate for the condition number of a matrix. SIAM J. Numer. Anal., 16:368-375, 1979.

....For purposes of assessing the reliability of the computed solution, it is useful to have an estimate of the conditioning of the discrete system. Such an estimate can be obtained, simultaneously with the factorization and solution process, by adapting the procedure described in Cline et al. [6] to our situation. We aim to compute an estimate of the quantity = kAk1 kR Gamma1 k1 ; where A is one of the coefficient matrices from (4) 7) 5) 8) and R is the upper triangular factor produced by the procedure just described. It is easy to show that 1 p (k 1)n cond1 (A) p (k ....

....and R is the upper triangular factor produced by the procedure just described. It is easy to show that 1 p (k 1)n cond1 (A) p (k 1)n: kAk1 can of course be calculated directly, and so computation of the estimate of kR Gamma1 k1 is the major part the task of finding . Following [6], we do this by first finding vectors z and v such that R T v = z; where the components of z are all Sigma1 and are chosen by a heuristic that attempts to maximize kvk1 . This is done during the factorization of A; as reduction of each partition into a single row of blocks is performed, the ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson, An estimate for the condition number of a matrix, SIAM Journal on Numerical Analysis, 16 (1979), pp. 368-- 375.

....matrix A. Other condition numbers of A are also of interest, but we will concentrate on this standard condition number. The best known condition estimator is the one used in LINPACK [17] it makes use of an LU factorization of A and works with the 1 norm. Its development is described in [12] 3 . Several years after [12] was published several counter examples to the LINPACK condition estimator were discovered by Cline and Rew [13] by a counter example we mean a parametrized matrix for which the quotient condition estimate divided by true condition number can be made arbitrarily ....

....numbers of A are also of interest, but we will concentrate on this standard condition number. The best known condition estimator is the one used in LINPACK [17] it makes use of an LU factorization of A and works with the 1 norm. Its development is described in [12] 3 . Several years after [12] was published several counter examples to the LINPACK condition estimator were discovered by Cline and Rew [13] by a counter example we mean a parametrized matrix for which the quotient condition estimate divided by true condition number can be made arbitrarily small (or large, depending on ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson, An estimate for the condition number of a matrix, SIAM J. Numer. Anal., 16 (1979), pp. 368--375.

.... (steps 4 6 in ADTF1) All steps in 12 ADTF2 include products of unitary matrices and Wilkinson s backward analysis for products of unitary matrices is applicable [19] In [10] 9] dif Gamma1 estimators (lower bounds on oe Gamma1 min (Z) in the style of the Linpack condition estimator [3]) that are incorporated into the generalized Schur algorithm for solving the generalized Sylvester equation (4.3) are presented. The heuristic condition estimators need O(n 2 1 n 2 n 1 n 2 2 ) flops, which is the same magnitude of work as solving the generalized Sylvester equation (4.3) see ....

A. Cline, C. Moler, G.W. Stewart, and J. Wilkinson. An estimate of the condition number of a matrix. SIAM J. Num. Anal., 16:368--375, 1979.

....the minimum distance to a singular matrix. 6.1 Estimating the distance from above. To estimate from above the minimum distance from the matrix A to a singular matrix or, equivalently, to estimate N = k U Gamma1 L Gamma1 k from below, we may apply the simple iterative algorithm of [CMSW79], which, according to [GL96] produces a good order of magnitude lower bound N Gamma on N at the cost of performing O(jn 2 ) arithmetic operations in j iterations (practically, j is much smaller than n) If (3.5) does not hold even for N replaced by N Gamma , then we may apply the ....

A. K. Cline, C. B. Moler, G. W. Stewart, J. H. Wilkinson, An Estimate for the Condition Number of a Matrix, SIAM J. Numerical Analysis, 16, 368--375, 1979.

....in the 1970s with the problem of cheaply estimating the condition number (A) kAkkA Gamma1 k and an approximate null vector of a square matrix A given some factorization of it. The earliest algorithm is one of Gragg and Stewart [10] It was improved by Cline, Moler, Stewart and Wilkinson [4], leading to the 1 norm condition estimation algorithm used in LINPACK [8] and later included in Matlab (function rcond) During the 1980s, attention was drawn to various componentwise condition numbers and it was recognized that most condition estimation problems can be reduced to the estimation ....

....1 norm power method. 1. How does the accuracy and reliability of the norm estimates vary with t 2. How good are the norm estimates in general 3. How does the number of iterations behave for t 1 Note that we are not searching for counterexamples, as was done for previous condition estimators [4], 5] 14] We know that for a fixed starting matrix and any t n there must be families of matrices whose norm is underestimated by an arbitrarily large factor, since the algorithm samples the behaviour of the n Theta n matrix A on fewer than n vectors. But since the algorithm uses a random ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM J. Numer. Anal., 16(2):368--375, 1979.

....Since kAk 1 = 13:8 we have the estimate 1 (A) 1833:6. The actual condition number is 2249:4 is within 17 of our estimate. This example should only is seen as giving the flavour of the calculation to be found in good quality software. The actual methods use more elaborate techniques (see [CMSW79]) 5.7 Error Analysis for Gaussian Elimination When using finite arithmetic, the process of Gaussian elimination will produce matrix factors L and U which are not the exact factors of A, but are the exact factors of a perturbed matrix A E. That is, if L and U are the calculated lower and upper ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. Siam Num. Anal., 16:368--375, 1979.

.... L ffi L) U ffi U) x = f : If nu is sufficiently small, the term ffi L ffi U is small with respect to the other error matrices and, in first order approximation, one has: DeltaA = ffi A ffi L U L ffi U (6) while the relative error on the solution may be bounded through [2] jj x Gamma xjj jj xjj jjA Gamma1 jj jj DeltaAjj: 7) 4 Stability of the factorization To study the stability we use an approach which exploits the sparsity structure of the error matrix ffi A. The factorization of the matrix A may be expressed through a difference equation in matrix ....

A. K. Cline, C. B. Moler, G. W. Stewart, J. H. Wilkinson, An Estimate for the Condition Number of a Matrix, SIAM J. Numer. Anal. 16 (1979), 368-375.

....[21] 30] 31] Then Theorem 1 implies that the largest M condition number of a (0,1) matrix lies in the range cn (2.274) n M(A) 2n 4 n n 2 , 22) with a similar result for the N condition number. Although ill conditioned matrices have been studied by many authors ([5], 12] 36] these results appear to be new. Weighing designs and spectroscopy If an invertible (0,1) matrix A is used as a weighing design (for weighing small objects, or in spectroscopy) then under suitable conditions the mean squared error in the measurements is reduced by a factor of n (A) ....

A. K. Cline, C. B. Moler, G. W. Stewart and J. H. Wilkinson, An estimate for the condition number of a matrix, SIAM J. Numer. Anal., 16 (1979), 368-375.

....of jjA Gamma1 jj is computed and placed in DPARM(3) The condition number of the matrix A is the product DPARM(2) Theta DPARM(3) The condition number is a measure of the reliability of the solution of a given linear system and can be used to compute error bounds. The algorithm described in [2] is used to estimate jjA Gamma1 jj. If IPARM(25) 1, then 1 norms are computed, if IPARM(25) 2, then Frobenius norms are computed, and if IPARM(25) 3, then infinity norms are computed. WSSMP can be made to do this computation either with factorization, or with solution, or with iterative ....

A. K. Cline, Cleve B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM Journal on Numerical Analysis, 16:368--375, 1979.

....b k = 1 if a T k x x 1 # 0 1 if a T k x x 1 0. 16 C.D. Meyer and D. Pierce This strategy produces a vector b and a solution x to Ax = b such that #b# # = 1 and #x# 2 = v u u t n X i=1 # 2 i # A 1 2 . Alternatively, borrowing from the work in [2] 5] [6], we will construct a vector b to have Euclidean norm 1. We do this at the k 1st step by scaling the previous k components of b by c and setting the k 1st component to s, where c 2 s 2 = 1. Notice that the e#ect of scaling the previous components of b by c corresponds to x k being ....

A.K. Cline, C.B. Moler, G.W. Stewart, and J.H. Wilkinson, An estimate for the condition number of a matrix, SIAM J. Numer. Anal., 16(1979), pp. 312--322.

....in [26] and [22, Section 10.4] Appling V costs O(n 2 p) operations. A condition estimator (possibly combined with inverse iteration) applied to the resulting triangular matrix U [A Gamma I ; B]V can approximate oe min (U [A Gamma I ; B]V ) oe min ( A Gamma I ; B] in O(n 2 ) operations [13, 12, 28, 30, 41]. After the initial reduction to Hessenberg form, each value of h( costs only O(n 2 p) operations. So, we may allow a numerical method to evaluate h( many times. Now, consider the problem of finding the minimum value of h( The WielandtHoffman [32] theorem applied to the singular value ....

....If jjAjj 2 jjBjj 2 = 1=2 and ffl = 001, then it uses 10 6 values of h( Although the BFI is too expensive to be practical, it has some attributes worth noting. The rounding error analysis of the unitary transformations is ideal [27, 46] Heuristic condition number estimation procedures [13, 12, 30, 41] have been extensively tested and are known to be reliable. Nonheuristic estimation procedures like [28] or inverse iteration are provably reliable. So, the BFI method is provably reliable. The number of mesh points does not depend on n and p, so the BFI method is 5 3 n 3 O(pn 2 ) algorithm ....

A. K. Cline, C. M. Moler, S. G. W., and J. H. Wilkinson, An estimate for the condition number of a matrix, SIAM Journal of Numerical Analysis, 16 (1979), pp. 368--375.

....its condition number is approximately 2 n . The condition number is defined as A A 1 , where A = x = 1 max Ax . Our condition estimator does not compute A 1 explicitly, but estimates the size of its largest elements. It uses the algorithm defined in [2] which solves A T x = b (2.4) and Ay = x 4 with the vector b suitably chosen so that y looks like the column of A 1 with the largest elements. Since this algorithm requires a decomposition of A, each CE subroutine returns that decomposition along with the condition estimate. Several ....

A. K. Cline, C. B. Moler, G. W. Stewart and J. H. Wilkinson, An Estimate for the Condition Number of a Matrix, SIAM J. Numer. Anal.16 April 1979, 368-375.

.... Rx = d = 1 oe min (R) kR Gamma1 k 2 kR Gamma1 dk 2 kdk 2 = kxk 2 kdk 2 ; which suggests generating a large norm solution x to a moderately sized right hand side d and then using : kdk 2 kxk 2 as an estimate for oe min (R) This idea underlies many condition estimators [12, 13, 19]. The incremental characteristic of ICE was achieved by choosing the right hand side d in a special way. As it turns out, the same estimator can also be derived by considering the following well known projection property of singular values. Let A be an n Theta n complex matrix and Y be an n ....

.... delivers reliable estimates (even though one can construct matrices where it performs arbitrarily badly [5] Second, we wish to show that ICE performs qualitatively in the same way as some of the well known condition estimators currently in use, in particular the Linpack condition estimator [13] and Higham s condition estimator [21, 22] which is being used in the LAPACK package [1, 2] Third, we wish to demonstrate that ICE is more reliable in correctly identifying the rank of triangular matrices produced by the QR factorization with column pivoting [17] than is the heuristic that is ....

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A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM Journal on Numerical Analysis, 16:368--375, 1979.

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A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. \An Estimate for the Condition Number of a Matrix." SIAM Journal on Numerical Analysis, 16(2):368-375, 1979.

....the norm of an inverse or pseudo inverse of a matrix. The matrix is typically a triangular matrix R often the result of computing a QR or LR factorization. The first condition estimator was devised by Gragg and Stewart [8] and was later improved by Cline, Moler, Stewart, and Wilkinson [4] for incorporation into linpack [5] Other condition estimators have since been proposed, many of which are treated in a survey by Higham [9] Condition estimators typically trade precision for speed. They produce an approximation to the norm of the inverse, usually a lower bound, in O(n 2 ) ....

A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson (1979). "An Estimate for the Condition Number of a Matrix." SIAM Journal on Numerical Analysis, 16, 368--375.

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A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson. An estimate for the condition number of a matrix. SIAM J. Numer. Anal., 16:368--375, 1979.

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A. Cline, C. Moler, G. Stewart and J. Wilkinson, "An Estimate for the Condition Number of a Matrix," SIAM J. Num. Anal., 16 (1979), pp. 368--375. 27

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A. K. Cline, C. B. Moler, G. W. Stewart and J. H. Wilkinson, 1979. An estimate for the condition number of a matrix. SIAM J. Numer. Anal., 16(2): 368-375.

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A. K. Cline, C. B. Moler, G. W. Stewart, and J. H. Wilkinson, An estimate for the condition number of a matrix, SIAM J. Numer. Anal., 16 (1979), pp. 368--375.

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