N. HIGHAM, The Test Matrix Toolbox for MATLAB (version 3.0), Tech. Rep. 276, Manchester Centre for Computational Mathematics, Sept. 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with the p dominant eigenvalues of A and the matrix T is the corresponding partial Schur form. More details for both methods can be found in [4, 6] The matrices used for the experiments are n by n matrices of the form A=Q( R W)Q 0 R2 The matrix Q is the unitary matrix orthog.m from [5] and the block matrices are given by cN 10 and R2 n p O2 N2, where N and N2 are strictly upper triangular random matrices of unit norm, C is a random matrix of unit norm and c, c2 and ca are real numbers. Note that c and c,are parameters for the degree of the non normality of the p by ....

N.J. Higham. The test matrix toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, September 1995.

....harmonic Ritz values. 21 80 90 100 110 120 130 140 150 110 120 130 140 150 160 170 180 190 200 Fig. 4. Experiments with IRLANB on dw 2048. Left: Convergence tolerance tol=1e 8. Right: Convergence tolerance tol=1e 12. We first experiment with matrix grcar of dimension N = 1000 [12] included in MATLAB s function gallery. Our target is to compute its 10 smallest singular values. The length of LBD was l = k p = 40 while we used p = 10 implicit shifts per step. Figure 3 illustrates the norms of the residual for each iteration. The dashed lines represent the convergence ....

N.J. Higham. The Test Matrix Toolbox for MATLAB (version 3.0). Technical Report 276, Manchester Centre for Computational Mathematics, Sept. 1995.

....GEP 1 formulation only. To illustrate, we carried out some experiments in Matlab, for which the unit roundoff is u = 2 Table 3.1: Backward errors j(ex; for the QEP with data (3.9) GEP 2 GEP 3 . We used the direct search maximization routine mdsmax of the Matlab Test Matrix Toolbox [12] and we applied it to the function f(A; B; C) j(ex; where the eigenpair ( e x) is computed using the QZ algorithm. It is easy to generate matrices A; B and C where kAk 2 = kBk 2 = kCk 2 = 1 and for which the backward error associated with the GEP 2 or GEP 3 formulation is large. As an ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 22

....missing elements, e, in P we seek min e 1 ; e r kCOV(Ln(P ) COV(Ln(P (e 1 ; e r ) k F 36 where e 1 ; e r denote the missing elements of P , the r = 9 NaNs in (3.1) for example, which form a vector for our optimization. We use mdsmax, which is part the Test Matrix Toolbox [9]. This routine aims to maximise a given function, thus we supply a function of the form kCOV(Ln(P ) COV(Ln(P (e 1 ; e r ) k F ; and note that a function value of zero corresponds to a perfect extension. 5.3 Experiments From our test data we have L= Ln(P ) see gen lnp.m in Appendix ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 66

....step. To illustrate this fact and thus provide the reader with an immediate feeling of the type of information obtained using the method, we show in Figure 1 the results from the application of PsDM to matrix kahan of order 100 which has been obtained from the Test Matrix Toolbox ([7]) and is a typical example of matrices with interesting 2 2 1.5 1 0.5 0 0.5 1 1.5 1.5 1 0.5 0 0.5 1 1.5 X Figure 1: Pseudospectrum contours and trajectories of points computed by PsDM for (A) 10 1 ; 10 3 for matrix kahan of order 100. Arrows show the directions ....

N.J. Higham. The Test Matrix Toolbox for MATLAB (version 3.0). Technical Report 276, Manchester Centre for Computational Mathematics, Sept. 1995.

.... discussion of the properties of W (C n ) and the set of the eigenvalues of C n see [20] A method to determine numerically the eld of values has been described in [21] In order to compute the eld of values we use the function fv.m of The test matrix Toolbox for MATLAB from Higham (see [18]) We study the asymptotic convergence factor of the eld of values W (C n ) de ned by : W ) lim k 1 min p2Pk ;p(0) 1 max z2W jp(z)j 1=k : For GMRES residuals this yields the bound (see [9, 10] kr (k) k 2 kr (0) k 2 4 k 1 k : Note that the asymptotic ....

....of The Chinese University of Hong Kong . The authors would like to acknowledge R. Chan, B. Fischer and G. Steidl for numerous fruitful and enlightening discussions. Especially we would like to thank B. Fischer to let us use his MATLAB code for the computation of which based on [8] and [18]. We would also like to thank the referees for their valuable suggestions. ....

N. J. Higham, The Test Matrix Toolbox for Matlab, Numerical Analysis Report No. 237, Manchester Centre for Computational Mathematics, Manchester, England, Dec. 1993. For accompanying software, see http://www.ma.man.ac.uk/~higham/testmat.html.

....positive de nite matrices inversions per iteration required by the Denman and Beavers iteration (2.6) Chapter 6 Numerical Experiments All computations have been done using MATLAB on a Sun workstation with unit roundo u = 2 53 1:1 10 16 . We use matrices from the Test Matrix Toolbox [13]. For each iterative method we report the relative residual residual(X k ) kX 2 k Ak 2 kAk 2 ; where X k denotes the iterate converging to A 1=2 and the error error(X k ) kX X k k 2 kXk 2 ; where X is the square root computed by the Schur method. We also report the point at which the ....

N. J. Higham. The Test Matrix Toolbox for MATLAB (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....which the unit roundoff is u = 2 Gamma53 10 Table 3.1: Backward errors j(ex; e ) for the QEP with data (3.9) GEP 2 GEP 3 2:8 Theta 10 Gamma16 1:6 Theta 10 Gamma8 1:1 Theta 10 Gamma16 . We used the direct search maximization routine mdsmax of the Matlab Test Matrix Toolbox [12] and we applied it to the function f(A; B; C) j(ex; e ) where the eigenpair ( e ; e x) is computed using the QZ algorithm. It is easy to generate matrices A; B and C where kAk 2 = kBk 2 = kCk 2 = 1 and for which the backward error associated with the GEP 2 or GEP 3 formulation is large. As ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 22

....in this case, bx; b ) is not a quantity we would compute routinely in the course of solving a problem. To test the strong stability of numerical algorithms for solving structured Hamiltonian eigenproblems, we applied the direct search maximisation routine mdsmax of the Matlab Test Matrix Toolbox [11] to the function f(E; F ) max 1i2n (bx i ; b i ) where ( b i ; b x i ) are the computed eigenpairs. In this way we carried out a search for problems on which the algorithms performs unstably. As expected from the theory, we could not generate examples for which the structured backward ....

Nicholas J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, Sept. 1995.

....1.1e 2 7.3e 4 4.7e 10 4.3e 17 4 2.0e7 29.2 9e 7 1.5e 1 2.6e 3 1.0e 9 2.9e 17 3 3.3e7 48.7 7e 7 4.3e 1 4.3e 3 1.6e 9 2.7e 17 5 4.3e7 62.9 2e 7 7.4e 1 5.6e 3 1.7e 9 2.2e 17 3 large kJ k, a large ratio kAk1 =kBk1 and large eigenvalues. We denote by M the Moler matrix from the Test Matrix Toolbox [15]: m ij = ae i if i = j; min(i; j) Gamma 2 otherwise. We took n = 20, A = 10 6 I and B = 10 Gamma2 M and computed the approximate eigenpairs using the Cholesky reduction. Instabilities are expected as (B) 2 Theta 10 13 . All the eigenpairs have a large backward error and a small ....

....yields a small backward error as long as the initial guess is good enough for Newton s method to converge. Example 3 We illustrate how using extended precision in computation of the residual yields a small relative error. Let A be the Prolate matrix of size n = 10 of the Test Matrix Toolbox [15] and let B be the Moler matrix. We used the Symbolic Math Toolbox of Matlab to compute the exact eigenpairs of (A; B) and the Cholesky reduction method to approximate the eigenpairs. We give the results in Table 4.4. We refined using both working precision (u = u) and double precision (u = u 2 ) ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....the p dominant eigenvalues of A and the matrix T is the corresponding partial Schur form. More details for both methods can be found in [4, 6] The matrices used for the experiments are n by n matrices of the form A = Q R 1 W H O R 2 Q H The matrix Q is the unitary matrix orthog.m from [5] and the block matrices are given by R 1 = 0 B B B B B B n 0 : 0 0 n Gamma 1 . 0 0 : 0 n Gamma p 1 1 C C C C C C A ff 1 N 1 10 R 2 = 0 B B B B B B B n Gamma p 0 : 0 0 . ....

N.J. Higham. The test matrix toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, September 1995.

....to enlarge the initial set R 0 when R 0 does not contain oe C (A; D;E; ae) With A Grcar Matrix (N = 6) SVD Method SH Method Num. Eval 4225 2725 Time 1280 sec 806 sec Table 1: Performances for the Grcar Matrix (n = 64) given by the pentadiagonal Toeplitz matrix A 32 =pentoep(32,0,1 2,1,1,1) [11], oe C (A; I; I; 0:001) was calculated by the standard SVD method and our algorithm. In both cases the same initial R 0 was chosen: a rectangle given by the corners (0:5; 0) 2:5; 1:5) In the lower part of Figure 4 the result of applying the SVD method is depicted. The output of our algorithm ....

N. J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 279, Manchester Centre for Computational Mathematics, 1995.

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N.J. Higham. The Matrix Computation Toolbox. Technical report, Manchester Centre for Computational Mathematics, 2002. In www.ma.man.uc.uk/higham/mctoolbox.

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N.J. Higham, The Test Matrix Toolbox for MATLAB (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

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Nicholas J. Higham. The Test Matrix Toolbox for MATLAB (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

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Nicholas J. Higham. The Test Matrix Toolbox for MATLAB. Numerical Analysis Report No. 237, Manchester Centre for Computational Mathematics, Manchester, England, December 1993. 76 pp.

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N. J. Higham, The Test Matrix Toolbox for MATLAB (Version 3.0), Numerical Analysis Report 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

....factorization with pivoting [19] 5. Numerical experiments To give further insight into the methods, we present the results of some numerical experiments. All computations were performed in MATLAB, which has unit roundoff u = 2 53 1.1 10 16 . We use matrices from the Test Matrix Toolbox [15], 17, appendix E] For each method we monitored convergence using the relative residual res(X k ) #A X where X k denotes the iterate converging (in theory) to A . Note that res can be regarded as the backward error of X k . This is not a quantity that one would use in practice, ....

N.J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September (1995) 70 pp.

....maxl j(P)l 1 IIA1AjlIp J j 0 for any p norm. Alternatively, we could bound maxj I(P)I by the largest absolute value of a point in the numerical range of P [24] but computation of this number is itself a nontrivial problem. For much more on bounding the eigenvalues of matrix polynomials see [15]. For the 2 norm, IIP(z) lll2 = O min(P(z) 1, where O min denotes the smallest singular value. If the grid is p x p and O mi n is computed using the Golub Reinsch SVD algorithm then the whole computation requires roughly (3.1) t22(Sn3 3 n2m) flops, which is prohibitively expensive for ....

N.J. HIGHAM AND F. TISSEUR, Bounds for Eigenvalues of Matrix Polynomials, Numerical Analysis Report 371, Manchester Centre for Computational Mathematics, Manchester, 2001; Linear Algebra Appl., to appear.

.... of the behavior of several (nonnormal) matrix dependent algorithms, ranging from iterative methods for large linear systems to time stepping algorithms [27,28] We note, for instance, the inclusion of specific functions to that effect (ps and pscont) in the popular Test Matrix Toolbox of MATLAB [18]; the former uses definition (1) and the latter definition (2) In general, however, computing the pseudospectrum is significantly more expensive than computing traditional characteristics such as the condition number, the norm, the eigenvalues and the singular values. This is illustrated in ....

....V 1 as products of reflectors. zbdsqr: Computation of the SVD B = U 2 SigmaV 2 and of the left and right singular matrices U = U 1 U 2 and V = V 1 V 2 . The modules were coordinated by the LAPACK driver zgesvd. 3. 2 Test matrices For the numerical experiments we use nonnormal matrices from [18]: i) the upper triangular matrix kahan with elements a kk = s and a kj = Gammas c when j k, where s n Gamma1 = 0:1 and s c = 1 [27] ii) the pentadiagonal Toeplitz matrix grcar A = Toeplitz( Gamma1; 1; 1; 1; 1] where the underlined element is in the diagonal [15] iii) matrix ....

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N.J. Higham. The Test Matrix Toolbox for MATLAB (version 3.0). Technical Report 276, Manchester Centre for Computational Mathematics, Sept. 1995.

....issues a warning that the matrix is singular and may not have a square root and returns a matrix of Nans and Infs together with infinite values for ff and the condition number estimate. The third matrix is A = I B) 2 2 R 4 Theta4 , where B is the matrix invol from the Test Matrix Toolbox [5]. Since B is involutary (B = I) A is idempotent (A = A) so A is its own principal square root. If it were formed exactly, A would have two zero eigenvalues and two eigenvalues 1, but because of rounding errors the computed matrix This matrix is also accessible as gallery( invol , in ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....results for four problems with m = 16, n = 10, p = 6. We denote by randn a matrix or vector from the normal(0,1) distribution and by randsvd( a random matrix with 2 norm condition number and geometrically distributed singular values, generated by the routine randsvd in the Test Matrix Toolbox [11]. Problem 1 A = randn, b = randn, B = randn, d = randn. Problem 2 A = randsvd(10) b = randn, B = randsvd(10 ) d = randn. b = randn, B = randsvd(10) d = randn. b = randn, B = randsvd(10 ) d = randn. For each problem, a parameter tol 2 (0; 1] determines a row scaling applied ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....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; n)ukAk1 , where f is a scalar function depending only on A and the dimension n. Provided that f(A; n) is not too large, A is not too ill conditioned, and the ....

Nicholas J. Higham. Iterative refinement and LAPACK. Numerical Analysis Report No. 277, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 17 pp. Submitted to IMA J. Numer. Anal.

....with respect to normwise and componentwise perturbations and that the sensitivity of an eigenvector can depend strongly on how it is normalized. The second example concerns symmetric structure in A and B. The matrix A is the ipjfact matrix (a ij = i j) from The Test Matrix Toolbox [19] but with its rows and columns in reverse order, and B is the Pascal matrix from the same source; both matrices are positive definite. Results for symmetric 8 8 A and B. Backward Condition Structured normwise (#, cond) 2.1e 17 5.2e15 4.2e15 Unstructured componentwise (#, cond) 1.5e 8 1.6e6 ....

N. J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

....predicted by the theory. Example 2. We would like to test the sharpness of the residual bound in Corollary 2.5 and the backward error bound in 3.5. We consider an example with large kJ k, a large ratio kAk1 =kBk1 and large eigenvalues. We denote by M the Moler matrix from the Test Matrix Toolbox [15]: m ij = i if i = j; min(i; j) Gamma 2 otherwise. We took n = 20, A = 10 I and B = 10 M and computed the approximate eigenpair using the Cholesky reduction. Instabilities are expected as (B) 2 Theta 10 13 . All the eigenpairs have a large backward error and a small condition ....

....yields a small backward error as long as the initial guess is good enough for Newton s method to converge. Example 3 We illustrate how using extended precision in computation of the residual yields a small relative error. Let A be the Prolate matrix of size n = 10 of the Test Matrix Toolbox [15] and let B be the Moler matrix. We used the Symbolic Math Toolbox of Matlab to compute the exact eigenpairs of (A; B) and the Cholesky reduction method to approximate the eigenpairs. We give the results in Table 4.4. We refined using both working precision (u = u) and quadruple precision (u = u ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....first example is for Gaussian elimination (GE) without pivoting, applied to the scaled 15 Theta 15 orthogonal matrix A 1 with a ij = d i (2= n 1) 1=2 sin(ij= n 1) where d(1: n) ff , with ff = 10 . This matrix is a row scaled version of orthog(15) from the Test Matrix Toolbox [6], which is the eigenvector matrix for the second difference matrix. The right hand side b is generated as b = A[1; 2; 15] The second example applies Gaussian elimination with partial pivoting (GEPP) to a random 10 Theta10 matrix A 2 with 2 (A) 10 . This matrix is generated as ....

....right hand side b is generated as b = A[1; 2; 15] The second example applies Gaussian elimination with partial pivoting (GEPP) to a random 10 Theta10 matrix A 2 with 2 (A) 10 . This matrix is generated as randn( seed ,1) A = randsvd(10, 1e6) using the Test Matrix Toolbox [6]. The right hand side is selected as in the first example. The results are shown in Tables 6.1 6.2. For the matrix W we take 2nP U j, where PA U is the computed LU factorization (P = I for GE) We make several observations. 12 1. In the first example, GE yields a moderately large ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....the unit roundoff u = 2 5:96 Theta 10 . We report results for LSE problems with m = 10, n = 7 and p = 3, with A and B random matrices with preassigned 2 norm condition numbers and geometrically distributed singular values, generated by the routine randsvd from the Test Matrix Toolbox [14]. However, in one of the tests B was formed in this way with 2 (B) 10 and then B(1: p; 1: p) was replaced by a matrix of random numbers from the normal(0,1) distribution all multiplied by 10 ; this choice of B is denoted small oe min (B 11 ) The vectors b and d have random elements from ....

....problem in Table 5.3 the bounds fi (bx) are all large and we suspected that some of them are pessimistic. Therefore we used direct search optimization to try to choose F and g to minimize fi (bx) with starting guesses F and g from (4. 6) An implementation from the Test Matrix Toolbox [14] of the multi directional search method of Dennis and Torczon [27] 28] was used. Note that the term ffl Gamma1 M in (3.8) of [23] should be ffl M . The improved values of fi are shown in parentheses. All are of order u except for the method of weighting with u and u . The lower ....

N. J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, Sept. 1995.

....to 24 bits , so that u = 2 5:96 Theta 10 . We generated random matrices of the form A = D 1 BD 2 , where B = U Sigma V with U and V random orthogonal matrices and Sigma a given matrix of singular values; B is constructed using the routine randsvd from the Test Matrix Toolbox [8]. In each case the singular values are from an exponential distribution, oe i = ff , and the condition number (B) 10 , i = 1: 7. The matrices D 1 and D 2 are diagonal, with positive diagonal entries chosen from one of three pairs of random distributions: uniformly distributed logarithm ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp. 24

....tolerance for the difference is exceeded. 9 We have experimented with the NAG library routine G05GBF (Mark 18) 23] which is based on the code from [20] We called the code from Matlab using a Mex interface and applied direct search optimization routines from the Test Matrix Toolbox [14]. We readily found examples where the computed diagonal is very far from 1. For example, in one run the vector of eigenvalues and the computed correlation matrix were x = 0.3844 1.8365 0.7791 A = 1.0000 0.2568 0.6458 0.2568 0.9379 0.2772 0.6458 0.2772 1.0621 The eigenvalues of the ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....10 ] 1, 1] and [ 10 1] For the first range, one eigenvalue is generated from the range [ 1, 0) to ensure that A has at least one negative eigenvalue. The matrix Q is a random orthogonal matrix from the Haar distribution, generated using the routine qmult from the Test Matrix Toolbox [14], which implements an algorithm of Stewart [22] For each eigenvalue distribution we generated 30 di#erent matrices, each corresponding to a fresh sample of # and of Q. We took n = 25, 50, 100. The ratios r F and r 2 are plotted in Figures 5.1 5.3. Figure 5.4 plots the condition numbers # 2 (A ....

N. J. Higham, The Test Matrix Toolbox for Matlab (Version 3.0), Numerical Analysis report 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

....at most n Gamma 1 iterations so the second stage costs at most 12n flops, which is negligible compared with the first stage. Finally, software for the first stage is readily available, in Matlab 5 as gallery( randsvd , or, equivalently, as routine randsvd of the Test Matrix Toolbox [9]) and in Fortran 77 in directory lapack testing matgen of the LAPACK distribution (see [4] for documentation) so the algorithm is very easy to implement. Since Algorithm 1 is based solely on orthogonal transformations, it is normwise backward stable, that is, the computed b A satisfies b A = ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....whether the cost of this part of the computation can be reduced by computing approximate square roots. The second question concerns the effect of errors in computing the square roots on the accuracy of the computed logarithm. In [15] the square roots are computed using the Schur method [4] 10] [13], which has essentially optimal accuracy and stability properties, but the effects of rounding errors are not analyzed. In partial answer to these questions we develop an extension of the inverse scaling and squaring method with two key properties. 1. It aims for a specified accuracy in the ....

....A = QTQ is same as matrix 2, but with = 0. For the tests we needed the exact logarithm, which we approximated by X computed using our own implementation of the inverse scaling and squaring method. Our code computes a Schur decomposition, computes square roots by the Schur method [4] 10] [13] and uses the [8 8] Pad e approximation once kA Gamma Ik 1 0:25 (then the Pad e approximation has error safely less than u [16, Sec. 3] For each matrix we applied Algorithm 7.1 with tolerance ffi = fflkX kF =4, with ffl ranging from 10 to 10 . The results are shown in Figure 8.1. In ....

Nicholas J. Higham. A new sqrtm for Matlab. Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999. 11 pp.

....the usual complex absolute value jx iyj = Instead they use jx iyj : jxj jyj; 4.3) which is less expensive to evaluate and less prone to overflow and underflow in floating point arithmetic. The usual absolute value is used in the routine diagpiv in the Matlab Test Matrix Toolbox [11]. We will consider both choices of absolute value in Algorithm 4.1. We assume that the 1 norm utilizes whichever absolute value has been chosen) We write abs(x iy) abs 1 (x iy) jxj jyj; and we write jzj only when z is real or when we wish to make statements holding for both ....

Nicholas J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995.

....the number of nodes or the size of A. Given that the cost of computing s(z) is at least O(n 2 ) and that a typical grid could easily contain O(10 4 ) points, the cost can be dramatic, even for matrices of small size. To get an idea of the expense, we note that the MATLAB function pscont from [9] that implements this algorithm took more than 2000 sec to compute the pseudospectrum of a matrix of size 200 on a 100 100 grid. Cost formula (2) readily indicates two major classes of methods for accelerating the computation, based on the mathematics and numerics of the problem: a) Reducing ....

....running Windows 2000, connected over fast a Ethernet network. We developed our programs using MATLAB and conducted our experiments over a novel environment that allows the concurrent operation of MATLAB; see the next subsection. We use standard test matrices, drawn from the Test Matrix Toolbox ([9]) and the Harwell Boeing collection ( 7] throughout our discussion. 1.1.1 Programming environment We used the Cornell Multitask Toolbox [19] developed by J.A. Zollweg and A. Verma at the Cornell Theory Center, for MATLAB (version 5.3) the widely used problem solving environment from ....

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N.J. Higham. The Test Matrix Toolbox for MATLAB (version 3.0). Tech. Rep. 276, Manchester Centre for Computational Mathematics, Sept. 1995.

....conditions (3.5) and (3.6) for hyperbolicity are identical in this case and are not satis ed. On applying Algorithm 2.3 to the pair (A 1 ; B 1 ) in Theorem 3.6, the pair is diagnosed de nite, and hence the QEP hyperbolic, after just one iteration. We used direct search (function mdsmax from [10]) to compute d(A; B; C) via the unconstrained minimization problem min x f(x=kxk 2 ) we took the convergence tolerance of order the unit roundo to obtain the best possible accuracy. We also evaluated (A; B; C) in a similar way. We found that d(A; B; C) 2:0, with optimal perturbations of a ....

Nicholas J. Higham. The Test Matrix Toolbox for MATLAB (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....theory. Example 2. We would like to test the sharpness of the residual bound in Corollary 2.5 and the backward error bound in Corollary 3.5. We consider an example with large #J # #, a large ratio #A## #B## , and large eigenvalues. We denote by M the Moler matrix from the Test Matrix Toolbox [15]: m ij = # i if i = j, min(i, j) 2 otherwise. NEWTON S METHOD IN FLOATING POINT ARITHMETIC 1055 Table 4.3 Estimated and computed residuals and backward errors for Example 2. Before refinement From theory After refinement # i cond(# i ) #(#x i , # # i ) #r est # # est #r# #(#x ....

....yields a small backward error as long as the initial guess is good enough for Newton s method to converge. Example 3. We illustrate how using extended precision in computation of the residual yields a small relative error. Let A be the Prolate matrix of size n = 10 of the Test Matrix Toolbox [15], and let B be the Moler matrix. We used the Symbolic Math Toolbox of Matlab to compute the exact eigenpairs of (A, B) and the Cholesky reduction method to approximate the eigenpairs. We give the results in Table 4.4. We refined using both working precision (u = u) and double precision (u = u 2 ....

N. J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, UK, 1995.

....in terms of real x when Q is real (see [7] 8] 14] for example) The exclusion of n = 2 relates to a subtlety in the de nition (2.1) of de nite pair; see [17] 19] 20, p. 290] for a discussion of this issue. In computing d(A; B; C) and (A; B; C) we used direct search (function mdsmax from [12]) taking the convergence tolerance of order the unit roundo to obtain the best possible accuracy and trying di erent starting values in order to be con dent that the global minima were obtained. 3.3.1 Damped mass spring system Our rst example is from a damped mass spring system; see [21] for ....

N. J. Higham. The Test Matrix Toolbox for MATLAB (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

....results for four problems with m = 16, n = 10, p = 6. We denote by randn a matrix or vector from the normal(0,1) distribution and by randsvd(#) a random matrix with 2 norm condition number # and geometrically distributed singular values, generated by the routine randsvd in the Test Matrix Toolbox [11]. Problem 1 A = randn, b = randn, B = randn, d = randn. Problem 2 A = randsvd(10) b = randn, B = randsvd(10 4 ) d = randn. Problem 3 A = randsvd(10 6 ) b = randn, B = randsvd(10) d = randn. Problem 4 A = randsvd(10 4 ) b = randn, B = randsvd(10 4 ) d = randn. For each problem, a ....

N. J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

....user specified tolerance for the difference is exceeded. We have experimented with the NAG library routine G05GBF (Mark 18) 24] which is based on the code from [21] We called the code from Matlab using a Mex interface and applied direct search optimization routines from the Test Matrix Toolbox [14]. We readily found examples where the computed diagonal is very far from 1. For example, in one run the vector of eigenvalues and the computed correlation matrix were x = NUMERICALLY STABLE GENERATION OF CORRELATION MATRICES 9 0.3844 1.8365 0.7791 A = 1.0000 0.2568 0.6458 0.2568 0.9379 ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

....for which the unit roundoff u 1:1 Theta 10 Gamma16 . We took k j 1, p = 16 and tol = mu in Algorithm Polar, and used the QR algorithm to compute the eigensystem of H (that is, we used Matlab s eig function) Results are shown in Table 2. 1 for three types of matrix from the test collection in [8]. Vand(25) is the Vandermonde matrix of order 25 with (i; j) element ( j Gamma 1) 24) i Gamma1 ; it has rank 21 to working precision. Cycol(16) is a matrix of the form [B B B B] where B is a random 16 Theta 4 matrix, and so it has rank 4 to working precision. Randsvd( m,n] kappa) is a random ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab. Numerical Analysis Report No. 237, Manchester Centre for Computational Mathematics, Manchester, England, December 1993. 76 pp.

....4 ] 1, 1] and [ 10 4 , 1] For the first range, one eigenvalue is generated from the range [ 1, 0) to ensure that A has at least one negative eigenvalue. The matrix Q is a random orthogonal matrix from the Haar distribution, generated using the routine qmult from the Test Matrix Toolbox [14], which implements an algorithm of Stewart [22] For each eigenvalue distribution we generated 30 di#erent matrices, each corresponding to a fresh sample of # and of Q. We took n = 25, 50, 100. The ratios r F and r 2 are plotted in Figures 5.1 5.3. Figure 5.4 plots the condition numbers # 2 (A ....

N. J. Higham, The Test Matrix Toolbox for Matlab (Version 3.0), Numerical Analysis report 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

....sensitivity with respect to normwise and componentwise perturbations and that the sensitivity of an eigenvector can depend strongly on how it is normalized. The second example concerns symmetric structure in A and B. The matrix A is the ipjfact matrix (a ij = i j) from The Test Matrix Toolbox [19] but with its rows and columns in reverse order, and B is the Pascal matrix from the same source; both matrices are positive definite. 508 DESMOND J. HIGHAM AND NICHOLAS J. HIGHAM Table 5.1 Results for symmetric 8 8 A and B. Backward Condition error # x Unstructured normwise (#, #) 2.1e 17 ....

N. J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

....whether the cost of this part of the computation can be reduced by computing approximate square roots. The second question concerns the effect of errors in computing the square roots on the accuracy of the computed logarithm. In [15] the square roots are computed using the Schur method [4] 10] [13], which has essentially optimal accuracy and stability properties, but the effects of rounding errors are not analyzed. In partial answer to these questions we develop an extension of the inverse scaling and squaring method with two key properties. 1. It aims for a specified accuracy in the ....

....A = QTQ T is same as matrix 2, but with = 0. For the tests we needed the exact logarithm, which we approximated by X computed using our own implementation of the inverse scaling and squaring method. Our code computes a Schur decomposition, computes square roots by the Schur method [4] 10] [13] and uses the [8 8] Pad e approximation once kA 1=2 k Gamma Ik 1 0:25 (then the Pad e approximation has error safely less than u [16, Sec. 3] For each matrix we applied Algorithm 7.1 with tolerance ffi = fflkX kF =4, with ffl ranging from 10 Gamma16 to 10 Gamma1 . The results are ....

Nicholas J. Higham. A new sqrtm for Matlab. Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999. 11 pp.

....tolerance for the difference is exceeded. 9 We have experimented with the NAG library routine G05GBF (Mark 18) 23] which is based on the code from [20] We called the code from Matlab using a Mex interface and applied direct search optimization routines from the Test Matrix Toolbox [14]. We readily found examples where the computed diagonal is very far from 1. For example, in one run the vector of eigenvalues and the computed correlation matrix were x = 0.3844 1.8365 0.7791 A = 1.0000 0.2568 0.6458 0.2568 0.9379 0.2772 0.6458 0.2772 1.0621 The eigenvalues of the ....

Nicholas J. Higham. The Test Matrix Toolbox for Matlab (version 3.0). Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. 70 pp.

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N. HIGHAM, The Test Matrix Toolbox for MATLAB (version 3.0), Tech. Rep. 276, Manchester Centre for Computational Mathematics, Sept. 1995.

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Nicholas J. Higham. J-orthogonal matrices: Properties and generation. Numerical Analysis Report No. 408, Manchester Centre for Computational Mathematics, Manchester, England, September 2002.

No context found.

N. J. Higham, The Test Matrix Toolbox for Matlab (Version 3.0). Tech. Rep. 276, Manchester Centre for Computational Mathematics, University of Manchester, UK, 1995.

No context found.

Nicholas J. Higham, The Test Matrix Toolbox for Matlab (version 3.0), Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, Sept. 1995.

No context found.

N. J. Higham. Iterative refinement and LAPACK. Numerical Analysis Report No. 277, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

No context found.

N. J. Higham. Iterative refinement and LAPACK. Numerical Analysis Report No. 277, Manchester Centre for Computational Mathematics, Manchester, England, 1995.

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