The Convergence of a Class of Double-rank Minimization Algorithms 2. The New Algorithm
The Convergence of a Class of Double-rank Minimization Algorithms 2. The New Algorithm
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DOI：
10.1093/imamat/6.3.222
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发表时间：
1970-09
影响因子：
1.2
通讯作者：
C. G. Broyden
中科院分区：
文献类型：
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作者：
C. G. Broyden
一类双秩极小化算法的收敛性2.其中q和ql是唯一确定的正交向量。参数1/为。在第1部分中提出，Ij的合适的冰将为零，因为如果它是负的，或大而正的，则矩阵KI因此HI可能变得不必要地条件恶劣。此外，注意到以这种方式丢失I]产生新的算法。这类算法中的两个算法已经发表，由于Davidon（1959）的修改，弗莱彻和鲍威尔（1963）是通过使P等于零和s得到的，在第一部分中表明，这通常导致负值1]。因此，我们期望通过该算法得到的矩阵序列{HI}呈现出一种arity的趋势，这种趋势已经被Broyden（1967）和on（1969）等注意到。在最近的算法中，由于Greenstadt（1967），如果H是正数，则1]的值甚至比DFP ithm中出现的值更负。这样做的一个结果是，该算法的矩阵H不能，不像DFP算法，被证明是正定的，这有严重的问题时，考虑数值稳定性。本文从理论上证明了新算法是稳定的，并证明了在极小化二次函数时，新算法是唯一能严格单调减少矩阵误差的算法。我们四舍五入的效果和不良条件的H上可达到的精度解决方案，并得出结论，提出了一个数值调查的结果，在他的性能的新算法的各种测试问题进行了比较的DFP算法。C. G.埃塞克斯大学布洛伊登计算中心，威文霍公园，科尔切斯特，埃塞克斯
The Convergence of a Class of Double-rank Minimization Algorithms 2. The New Algorithm d where q and ql are uniquely determined orthonormal vectors. The parameter 1/ is . ntially arbitrary in that it depends upon p. It was suggested in Part 1 that a suitable ice for I] would be zero since if it were negative, or large and positive, the matrix KI hence HI might become needlessly badly conditioned. It was noted moreover that osing I] in this way gives rise to a new algorithm. the two algorithms in this class already published, that due to Davidon (1959) modified by Fletcher & Powell (1963) is obtained by putting P equal to zero and s shown in Part I that this led, in general, to negative values of 1]. We thus expect quence of matrices {HI} obtained by that algorithm to exhibit a tendency to arity and this tendency has been noted by, among others, Broyden (1967) and on (1969). In a more recent algorithm, due to Greenstadt (1967), if H is positive ite the values of 1] are even more negative than those occurring in the DFP ithm. One result of this is that for this algorithm the matrices H cannot, unlike for the DFP algorithm, be proved to be positive definite and this has serious tions when considering numerical stability. this paper we show theoretically that the new algorithm is stable and we prove is the only member of the class considered for which a certain matrix error is reduced strictly monotonically when minimizing quadratic functions. We the effect of rounding and of poor conditioning of H on the attainable accuracy solution and conclude by presenting the results of a numerical survey in he performance of the new algorithm for a variety of test problem is compared t of the DFP algorithm. C. G. BROYDEN Computing Centre, University of Essex, Wivenhoe Park, Colchester, Essex