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71.
72.
Nicholas J. Higham 《BIT Numerical Mathematics》1993,33(1):124-136
Because of the special structure of the equationsAX–XB=C the usual relation for linear equations backward error = relative residual does not hold, and application of the standard perturbation result forAx=b yields a perturbation bound involving sep (A, B)–1 that is not always attainable. An expression is derived for the backward error of an approximate solutionY; it shows that the backward error can exceed the relative residual by an arbitrary factor. A sharp perturbation bound is derived and it is shown that the condition number it defines can be arbitrarily smaller than the sep(A, B)–1-based quantity that is usually used to measure sensitivity. For practical error estimation using the residual of a computed solution an LAPACK-style bound is shown to be efficiently computable and potentially much smaller than a sep-based bound. A Fortran 77 code has been written that solves the Sylvester equation and computes this bound, making use of LAPACK routines.Nuffield Science Research Fellow. This work was carried out while the author was a visitor at the Institute for Mathematics and its Applications, University of Minnesota. 相似文献
73.
Nicholas J. Higham. 《Mathematics of Computation》1998,67(224):1591-1599
Complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for LU factorization. This result adds to the classes of matrix for which it is known to be safe not to pivot in LU factorization. Block factorization with the pivoting strategy of Bunch and Kaufman is also considered, and it is shown that for such matrices only pivots are used and the same growth factor bound of 2 holds, but that interchanges that destroy band structure may be made. The latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in LINPACK and LAPACK.
74.
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76.
Stable iterations for the matrix square root 总被引:3,自引:0,他引:3
Nicholas J. Higham 《Numerical Algorithms》1997,15(2):227-242
Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular M-matrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root. 相似文献
77.
[reaction: see text] Novel macrocycles possessing ether linkages and 2,6-disubstituted phenolics were produced in one step and with 100% atom economy by isoaromatization of chameleon macrocyclic precursors possessing 2,6-diarylidenecyclohexanone moieties. Intramolecular hydrogen bonding of the phenolic hydrogen atoms influenced the shape of the macrocycles and dictated host-guest behavior. 相似文献
78.
An algorithm is developed for computing the matrix cosine, building on a proposal of Serbin and Blalock. The algorithm scales the matrix by a power of 2 to make the -norm less than or equal to 1, evaluates a Padé approximant, and then uses the double angle formula cos(2A)=2cos(A)2–I to recover the cosine of the original matrix. In addition, argument reduction and balancing is used initially to decrease the norm. We give truncation and rounding error analyses to show that an [8,8] Padé approximant produces the cosine of the scaled matrix correct to machine accuracy in IEEE double precision arithmetic, and we show that this Padé approximant can be more efficiently evaluated than a corresponding Taylor series approximation. We also provide error analysis to bound the propagation of errors in the double angle recurrence. Numerical experiments show that our algorithm is competitive in accuracy with the Schur–Parlett method of Davies and Higham, which is designed for general matrix functions, and it is substantially less expensive than that method for matrices of -norm of order 1. The dominant computational kernels in the algorithm are matrix multiplication and solution of a linear system with multiple right-hand sides, so the algorithm is well suited to modern computer architectures. 相似文献
79.
Desmond J. Higham 《BIT Numerical Mathematics》1997,37(1):24-36
Certain applications produce initial value ODEs whose solutions, regarded as time-dependent matrices, preserve orthonormality.
Such systems arise in the computation of Lyapunov exponents and the construction of smooth singular value decompositions of
parametrized matrices. For some special problem classes, there exist time-stepping methods that automatically inherit the
orthonormality preservation. However, a more widely applicable approach is to apply a standard integrator and regularly replace
the approximate solution by an orthonormal matrix. Typically, the approximate solution is replaced by the factorQ from its QR decomposition (computed, for example, by the modified Gram-Schmidt method). However, the optimal replacement—the
one that is closest in the Frobenius norm—is given by the orthonormal polar factor. Quadratically convergent iteration schemes
can be used to compute this factor. In particular, there is a matrix multiplication based iteration that is ideally suited
to modern computer architectures. Hence, we argue that perturbing towards the orthonormal polar factor is an attractive choice,
and we consider performing a fixed number of iterations. Using the optimality property we show that the perturbations improve
the departure from orthonormality without significantly degrading the finite-time global error bound for the ODE solution.
Our analysis allows for adaptive time-stepping, where a local error control process is driven by a user-supplied tolerance.
Finally, using a recent result of Sun, we show how the global error bound carries through to the case where the orthonormal
QR factor is used instead of the orthonormal polar factor.
This work was supported by Engineering and Physical Sciences Research Council grants GR/H94634 and GR/K80228. 相似文献
80.
Let X be a Banach space, let K be a non–empty closed subset of X and let T : K → X be a non–self mapping. The main result of this paper is that if T satisfies the contractive–type condition (1.1) below and maps ?K (?K the boundary of K) into K then T has a unique fixed point in K. 相似文献