Milestones in Matrix Computation最新文献

An analysis of the total least squares problem 总体最小二乘问题的分析

Milestones in Matrix Computation Pub Date : 1980-02-01 DOI: 10.1137/0717073

G. Golub, C. Loan

引用次数: 1248

Generalized cross-validation as a method for choosing a good ridge parameter 采用广义交叉验证方法选择良好的脊参数

Milestones in Matrix Computation Pub Date : 1979-05-01 DOI: 10.1080/00401706.1979.10489751

G. Golub, M. Heath, G. Wahba

引用次数: 3015

Ill-conditioned eigensystems and the computation of the Jordan canonical form 病态本征系统及Jordan标准形式的计算

Milestones in Matrix Computation Pub Date : 1975-02-01 DOI: 10.1137/1018113

G. Golub, J. H. Wilkinson

引用次数: 122

Some modified matrix eigenvalue problems 若干修正矩阵特征值问题

Milestones in Matrix Computation Pub Date : 1973-04-01 DOI: 10.1137/1015032

G. Golub

引用次数: 507

Methods for modifying matrix factorizations 修正矩阵分解的方法

Milestones in Matrix Computation Pub Date : 1972-11-01 DOI: 10.1090/S0025-5718-1974-0343558-6

G. Golub, P. Gill, W. Murray, M. Saunders

引用次数: 266

The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate 变量分离的伪逆和非线性最小二乘问题的微分

Milestones in Matrix Computation Pub Date : 1972-02-01 DOI: 10.1137/0710036

G. Golub, V. Pereyra

{"title":"The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate","authors":"G. Golub, V. Pereyra","doi":"10.1137/0710036","DOIUrl":"https://doi.org/10.1137/0710036","url":null,"abstract":"For given data ($t_i , y_i), i=1, ldots ,m$ , we consider the least squares fit of nonlinear models of the form F($underset ~to a , underset ~to alpha ; t) = sum_{j=1}^{n} g_j (underset ~to a ) varphi_j (underset ~to alpha ; t) , underset ~to a epsilon R^s , underset ~to alpha epsilon R^k $. For this purpose we study the minimization of the nonlinear functional r($underset ~to a , underset ~to alpha ) = sum_{i=1}^{m} {(y_i - F(underset ~to a , underset ~to alpha , t_i))}^2$. It is shown that by defining the matrix ${ {Phi (underset ~to alpha} }_{i,j} = varphi_j (underset ~to alpha ; t_i)$ , and the modified functional $r_2(underset ~to alpha ) = l underset ~to y - Phi (underset ~to alpha )Phi^+(underset ~to alpha ) underset ~to y l_2^2$, it is possible to optimize first with respect to the parameters $underset ~to alpha$ , and then to obtain, a posteriori, the optimal parameters $overset ^to {underset ~to a}$. The matrix $Phi^+(underset ~to alpha$) is the Moore-Penrose generalized inverse of $Phi (underset ~to alpha$), and we develop formulas for its Frechet derivative under the hypothesis that $Phi (underset ~to alpha$) is of constant (though not necessarily full) rank. From these formulas we readily obtain the derivatives of the orthogonal projectors associated with $Phi (underset ~to alpha$), and also that of the functional $r_2(underset ~to alpha$). Detailed algorithms are presented which make extensive use of well-known reliable linear least squares techniques, and numerical results and comparisons are given. These results are generalizations of those of H. D. Scolnik [1971].","PeriodicalId":250823,"journal":{"name":"Milestones in Matrix Computation","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1972-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128970421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 882

Numerical methods for computing angles between linear subspaces 线性子空间间角度计算的数值方法

Milestones in Matrix Computation Pub Date : 1971-07-01 DOI: 10.1090/S0025-5718-1973-0348991-3

A. Bjoerck, G. Golub

{"title":"Numerical methods for computing angles between linear subspaces","authors":"A. Bjoerck, G. Golub","doi":"10.1090/S0025-5718-1973-0348991-3","DOIUrl":"https://doi.org/10.1090/S0025-5718-1973-0348991-3","url":null,"abstract":"Assume that two subspaces F and G of unitary space are defined as the ranges (or nullspaces) of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles $theta_k (F,G)$ and orthogonal sets of principal vectors $u_k epsilon F$ and $v_k epsilon G$, k = 1,2,..., q = dim(G) $leq$ dim(F). An important application in statistics is computing the canonical correlations $sigma_k = cos theta_k$ between two sets of variates. A perturbation analysis shows that the condition number for $theta_k$ essentially is max($kappa (A),kappa (B)$), where $kappa$ denotes the condition number of a matrix. The algorithms are based on a preliminary QR-factorization of A and B (or $A^H$ and $B^H$), for which either the method of Householder transformations (HT) or the modified Gram-Schmidt method (MGS) is used. Then cos $theta_k$ and sin $theta_k$ are computed as the singular values of certain related matrices. Experimental results are given, which indicates that MGS gives $theta_k$ with equal precision and fewer arithmetic operations than HT. However, HT gives principal vectors, which are orthogonal to working accuracy, which is not in general true for MGS. Finally the case when A and/or B are rank deficient is discussed.","PeriodicalId":250823,"journal":{"name":"Milestones in Matrix Computation","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1971-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122192617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 664

On direct methods for solving Poisson's equation 解泊松方程的直接方法

Milestones in Matrix Computation Pub Date : 1970-12-01 DOI: 10.1137/0707049

B. L. Buzbee, G. Golub, C. Nielson

引用次数: 287

Calculation of Gauss quadrature rules 高斯正交规则的计算

Milestones in Matrix Computation Pub Date : 1967-11-01 DOI: 10.1090/S0025-5718-69-99647-1

G. Golub, John H. Welsch

引用次数: 1203

A generalized conjugate gradient method for non-symmetric systems of linear equations 非对称线性方程组的广义共轭梯度法

Milestones in Matrix Computation Pub Date : 1900-01-01 DOI: 10.1007/978-3-642-85972-4_4

P. Concus, G. Golub

引用次数: 62