N. J. Higham, Fortran codes for estimating the one-norm of a real or complex matrix with applications to condition estimation, ACM Trans. Math. Software, 14 (1988), pp. 381--396.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... M N) by solving triangular matrix equations: kxk 2 kck 2 kXkF kCkF kZ SYCT k 2 = min (ZSYCT ) Sep The right hand side C is chosen such that the lower bound gets as large as possible. This leads to a Frobenius norm based estimate. For computation of 1 norm based estimates see [17, 19, 31]. The Sep functions associated with the Sylvester type matrix equations are: Sep[SYCT] inf kXkF =1 kAX XBkF = min (ZSYCT ) Sep[LYCT] inf kXkF =1 kAX X( A )k F = min (ZLYCT ) Sep[GCSY] inf k(X;Y )kF =1 k(AX Y B; DX Y E)kF = min (ZGCSY ) The same techniques as presented above ....

N.J. Higham. Fortran Codes for Estimating the One-Norm of a Real or Complex Matrix with Applications to Condition Estimation, ACM Trans. Math. Software, 14:381-396, 1988.

....between the two domains. For ease of computation B 0 was chosen to be 0 , i.e. the multigrid solution on the uniform grid corresponded to an exact solution of the transferred problem. We used Jacobi as the smoother. The condition number of BA was estimated using Hager s method (see [4]) The first column of Table 1 corresponds to the case of V in (11) and the second column of Table 1 corresponds to V in (15) As can be seen from Table 1, the condition number of the matrix BA grows exponentially for the case where the grids are very poorly matched. However, for the case where ....

N. J. Higham, FORTRAN Codes for Estimating the One-Norm of a Real or Complex Matrix, with Applications to Condition Estimation (Algorithm 674), ACM Trans. Math. Software, 14 (1988), pp. 381--396.

....basis following a cold start. With partial pivoting the factors SNOPT: A LARGE SCALE SQP ALGORITHM 13 display no small diagonals in U , yet the BR factors reveal a large number of dependent columns. Thus, although condition estimators are known that could tell us this B is ill conditioned (e.g. [51]) we are using LUSOL s complete pivoting option to decide which columns are causing the poor condition. 4.5. Basis repair (rectangular case) When superbasic variables are present, the permutation P in (4.2) clearly a ects the condition of B and Z. SQOPT therefore applies an occasional ....

N. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation, ACM Trans. Math. Software, 14 (1988), pp. 381-396.

....techniques to bound k(H Gamma j I) Gamma1 k 1 , instead of just bounding the growth factor on a single vector. There are many worthwhile papers that explore the subject of condition estimation (see Bischof and Tang [5, 6] Cline, Conn and Van Loan [9] Cline and Rew [10] Hager [18] Higham [26, 27, 28], Van Loan [41] and Golub and Van Loan [16] In addition, condition estimation for specific cases of the Sylvester equation (Lyapunov equations) are exploited in Byers [7] Most condition estimation techniques are based on dynamically chosing a normalized right hand side x i so that the ....

Higham, N., J., Fortran Codes for Estimating the One-Norm of a Real or Complex Matrix, with Applications to Condition Estimation, ACM Trans. Math. Soft., Vol. 14 (1988), pp. 381-396

....= Omega Gamma1 (A T c XZXA c ) where A c = I n DX) Gamma1 A. 3 Condition estimation The quantities k Omega Gamma1 k 1 ; k Thetak 1 ; k Pik 1 arising in the sensitivity analysis of Lyapunov and Riccati equations can be efficiently estimated by using the norm estimator, proposed in [13] which estimates the norm kTk 1 of a linear operator T , given the ability to compute Tv and T T w quickly for arbitrary v and w. This estimator is implemented in the LAPACK subroutine xLACON [1] which is called via a reverse communication interface, providing the products Tv and T T w. ....

N.J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674). ACM Trans. Math. Software, 14:381--396, 1988.

....If B z were explicitly available, it would be trivial to calculate kB z k est . However, in our case, B z is generally only available as the product of matrices and the calculation of kB z k est is out of the question. Wemay nonetheless estimate kB z k est using the algorithms of Hager (1984) or Higham (1988) to estimate kB z k 1 . We must be cautious here as such algorithms may underestimate the infinity norm. Unless we compute an accurate approximation of min , or a lower bound for it, we may have to perform a number of conjugate gradient cycles. Each cycle ends when a negative eigenvalue of the ....

N. J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation. ACM Transactions on Mathematical Software, 14:381--396, 1988.

....and A (B) involve B , B B, A ) etc, and computing these matrices can be relatively expensive. Fortunately, it is possible to compute inexpensive estimates of B (A) and A (B) without forming B , B B or (A ) This can be done using a method of Hager (1984) and Higham (1988) [5] that computes a lower bound for kBk1 , where B is a matrix, given a mean for evluating matrix vector products Bu and B T u. Typically, 4 or 5 products are required, and the lower bound is almost always within a factor 3 of kBk1 . The corresponding subroutine, named as S ACO , is available in ....

N. J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation, ACM TOMS, 14: 381-396(1988).

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N.J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674). ACM Trans. Math. Software, 14:381--396, 1988.

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N. J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674), ACM Trans. Math. Software, 14 (1988), pp. 381--396.

....is clearly applicable, just requiring the solution of linear systems with coefficient matrices A and A and multiplication of a vector by D. LAPACK makes extensive use of condition estimation to provide both condition estimates and forward error bounds. It implements an algorithm of Higham [80], which incorporates modifications to Hager s algorithm that make it more robust, reliable and efficient. The norm estimates are nearly always within a factor 3 of the true norm. Counterexamples are known for all existing condition estimators, that is, classes of matrix are known where the ....

Nicholas J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674). ACM Trans. Math. Software, 14(4):381--396, December 1988.

....when matrix vector products Ax and A x can be cheaply computed [2] 16, Sec. 14.1] Hager [12] derived an algorithm for the 1 norm that is a special case of the more general p norm power method proposed by Boyd [3] and later investigated by Tao [18] Hager s algorithm was modified by Higham [14] and incorporated in LAPACK (routine xLACON) 1] and Matlab (function condest) The LINPACK and LAPACK estimators both produce estimates that in practice are almost always within a factor 10 and 3, respectively, of the quantities they are estimating [13] 14] 15] This has been entirely ....

.... algorithm was modified by Higham [14] and incorporated in LAPACK (routine xLACON) 1] and Matlab (function condest) The LINPACK and LAPACK estimators both produce estimates that in practice are almost always within a factor 10 and 3, respectively, of the quantities they are estimating [13] [14], 15] This has been entirely adequate for applications where only an order of magnitude estimate is required, such as the evaluation of error bounds. However, in some applications an estimate with one or more correct digits is required (see, for example, the pseudospectra application described ....

[Article contains additional citation context not shown here]

Nicholas J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674). ACM Trans. Math. Software, 14(4):381--396, December 1988.

....(2.1) these quantities may be expressed as (see (4.2) and (4. 8) k 2 = fl fl fl fl fl ; k(AP ) k 2 ; kAB To avoid the possibly expensive formation of the matrices involving L 22 and S the norms of these matrices are estimated using the LAPACK norm estimator [1] [14], which estimates kBk 1 given only the ability to form matrix vector products Bx and B y. We can compute the required products by solving triangular systems and we accept the 1 norm estimate as an approximation to the 2 norm. A further computational saving is to compute kAkF and kBkF as kAkF ....

Nicholas J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674). ACM Trans. Math. Software, 14(4):381--396, December 1988.

....GMW , SE MC . perturbations explicitly, so it is trivial to evaluate their norms. For Algorithm MC, the perturbation to A is (see (1.2) E = P L(D F )L A, which would require O(n ) operations to form explicitly. However, we can estimate #E## using the norm estimator from [13] (which is implemented in LAPACK) The estimator requires the formation of products Ex for certain vectors x, and these can be computed in O(n operations; the estimate produced is a lower bound that is nearly always within a factor 3 of the true norm. For all three algorithms, then, we can ....

N. J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674), ACM Trans. Math. Software, 14 (1988), pp. 381--396.

....the computed solutions. Typically, these condition numbers are as expensive to compute as the solution itself [8] The LAPACK [1] and ScaLAPACK [2] condition numbers and error bounds are based on estimated condition numbers, using the method of Hager [5] which was subsequently improved by Higham [6]. Hager s method estimates kBk 1 given only the ability to compute matrix vector products Bx and B T y. If we take B = A 1 and compute the required products by solving linear systems with A, we obtain an estimate of the 1 norm condition number 1 (A) kAk 1 kA 1 k 1 . This work was ....

....communication interface. There are two advantages to having such an interface. First it provides exibility, as the dependence on B and its associated matrix vector operations is isolated from the computational routines xLACON and PxLACON, with the matrix vector products provided by a black box [6]. By changing these black boxes, xLACON and PxLACON can be applied to di erent matrix functions for both dense and sparse matrices. Second, as the bulk of the computational e ort is in matrix vector operations, ecient implementation of these operations ensures good overall performance of xLACON ....

[Article contains additional citation context not shown here]

Nicholas J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674). ACM Trans. Math. Software, 14(4):381-396, December 1988.

....the computed solutions. Typically, these condition numbers are as expensive to compute as the solution itself [6] The LAPACK [1] and ScaLAPACK [2] condition numbers and error bounds are based on estimated condition numbers, using the method of Hager [3] which was subsequently improved by Higham [4]. Hager s method estimates kBk 1 given only the ability to compute matrix vector products Bx and B T y. If we take B = A 1 and compute the required products by solving linear systems with A, we obtain an estimate of the 1 norm condition number 1 (A) kAk 1 kA 1 k 1 . In LAPACK and ....

....Research Council grant GR L94314. The work of the second author was also supported by a Royal Society Leverhulme Trust Senior Research Fellowship. matrix vector operations is isolated from the computational routines xLACON and PxLACON, with the matrix vector products provided by a black box [4]. By changing these black boxes, xLACON and PxLACON can be applied to di erent matrix functions for both dense and sparse matrices. Second, as the bulk of the computational e ort is in matrix vector operations, ecient implementation of these operations ensures good overall performance of xLACON ....

[Article contains additional citation context not shown here]

Nicholas J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674). ACM Trans. Math. Software, 14(4):381-396, December 1988.

.... T xj and G = diag(g i ) we have k jI Gamma A Aj Delta jA T j Delta jA T xj k1 = k jI Gamma A Ajg k1 = k jI Gamma A AjGe k1 = k jI Gamma A AjG k1 = k j(I Gamma A A)Gj k1 = k (I Gamma A A)G k1 : The latter norm can be estimated by the method of [8] and [9, 11], which estimates kBk 1 given a means for forming matrix vector products Bx and B T y. Forming these products for B T = I Gamma A A)G involves multiplying by G and Q, or their transposes, and solving triangular systems with R and R T . 10 Table 4.1 A = randsvd( 10; 16] 1e2) 2 ....

N.J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381--396.

....y = Ax z = A T dual p (y) if kzk q z T x fl = kyk p quit end x = dual q (z) end Algorithm PM requires about 4rmn flops if there are r iterations for convergence. The convergence test can be written in several different ways, as we explain below; the form chosen here is the one used in [10, 13, 14]. The power method was first derived and analysed by Boyd [3] and it was later investigated by Tao [26] Tao applies the method to an arbitrary mixed subordinate norm kAk ff;fi = max x6=0 kAxk ff kxk fi ; 2.1) while Boyd takes the ff and fi norms to be p norms (possibly different) Algorithm ....

....A T A is irreducible, 1 p 1, and x 0 has positive elements, then the x k converge and fl k kAk p . In the case p = 1, Algorithm PM is a 1 norm estimation algorithm devised by Hager [10] independently of [3] and [26] and subsequently analysed and modified by the present author [13, 14]. The algorithm given in [13] is the basis of all condition number estimation in LAPACK [2] Algorithm PM has two remarkable properties when p = 1: it almost always converges within four iterations (when x 0 = 1; 1; 1) T , say) and it frequently yields kAk 1 exactly. This rapid, finite ....

[Article contains additional citation context not shown here]

N.J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381--396. 21

....I and we recover the expression for cond 1 (A; x) in (1.2) Finally, we note that when p = 1 or 1, the bounds in (3. 12) can be estimated in O(n 2 r) operations without forming A Gamma1 if a QR or LU factorization of A is available; this can be done using the method of Hager [15] and Higham [17, 18] which estimates kBk 1 or kBk1 by evaluating several matrix vector products involving B and B T . The use of this method to estimate a componentwise condition number was first suggested in [2] in connection with the condition number (1.2) and the latter condition number is estimated this way ....

N.J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381--396.

....[2] In particular, they show how to estimate the condition number (2.10) cheaply, given a factorization of A. Their approach avoids explicit computation of A Gamma1 by manipulating the condition number into a form that can be estimated using a matrix norm estimator of Hager [36] and Higham [40]. This approach can be adapted to estimate virtually any form of componentwise condition number for a linear system. Finally, we mention how the analysis of this section is reflected in LAPACK. The LAPACK expert driver routine xGESVX 1 solves a linear system Ax = b by 1 The leading x in ....

Nicholas J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674), ACM Trans. Math. Software 14 (1988), no. 4, 381--396.

....[21] for a survey. The most well known estimator is the one used in LINPACK, which provides a lower bound for 1 (A) The method underlying this estimator does not generalize to the estimation of E;f (A; b) A more versatile estimator with this capability is one developed by Hager [20] and Higham [23]. This estimator treats the general problem of estimating kBk 1 , where B is not known explicitly. The estimator assumes that B is described by a black box that can evaluate Bx or B T x given x. Typically, 4 or 5 such matrix vector products are required to produce a lower bound for kBk 1 , and ....

N.J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation, ACM Trans. Math. Soft., 14 (1988), pp. 381--396.

....SE , MC . perturbations explicitly, so it is trivial to evaluate their norms. For Algorithm MC, the perturbation to A is (see (1.2) E = P T L(D F )L T P A, which would require O(n 3 ) operations to form explicitly. However, we can estimate #E## using the norm estimator from [13] (which is implemented in LAPACK) The estimator requires the formation of products Ex for certain vectors x, and these can be computed in O(n 2 ) operations; the estimate produced is a lower bound that is nearly always within a factor 3 of the true norm. For all three algorithms, then, we can ....

N. J. Higham, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674), ACM Trans. Math. Software, 14 (1988), pp. 381--396.

....when matrix vector products Ax and A T x can be cheaply computed [2] 16, Sec. 14.1] Hager [12] derived an algorithm for the 1 norm that is a special case of the more general p norm power method proposed by Boyd [3] and later investigated by Tao [18] Hager s algorithm was modified by Higham [14] and incorporated in LAPACK (routine xLACON) 1] and Matlab (function condest) The LINPACK and LAPACK estimators both produce estimates that in practice are almost always within a factor 10 and 3, respectively, of the quantities they are estimating [13] 14] 15] This has been entirely ....

.... algorithm was modified by Higham [14] and incorporated in LAPACK (routine xLACON) 1] and Matlab (function condest) The LINPACK and LAPACK estimators both produce estimates that in practice are almost always within a factor 10 and 3, respectively, of the quantities they are estimating [13] [14], 15] This has been entirely adequate for applications where only an order of magnitude estimate is required, such as the evaluation of error bounds. However, in some applications an estimate with one or more correct digits is required (see, for example, the pseudospectra application described ....

[Article contains additional citation context not shown here]

Nicholas J. Higham. FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (Algorithm 674). ACM Trans. Math. Software, 14(4):381--396, December 1988.

No context found.

N. J. Higham, Fortran codes for estimating the one-norm of a real or complex matrix with applications to condition estimation, ACM Trans. Math. Software, 14 (1988), pp. 381--396.

No context found.

N.J. Higham. Fortran Codes for Estimating the One-Norm of a Real or Complex Matrix with Applications to Condition Estimation, ACM Trans. Math. Software, 14:381-396, 1988.

No context found.

N. J. Higham. Fortran codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation (algorithm 674). ACM Transactions on Mathematical Software, 14:381--396, 1988.

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