A NUMERICALLY STABLE BLOCK MODIFIED GRAM-SCHMIDT ALGORITHM FOR SOLVING STIFF WEIGHTED LEAST SQUARES PROBLEMS

Recently, Wei in proved that perturbed stiff weighted pseudoinverses and stiff weighted least squares problems are stable, if and only if the original and perturbed coefficient matrices A and A^- satisfy several row rank preservation conditions. According to these conditions, in this paper we show that in general, ordinary modified Gram-Schmidt with column pivoting is not numerically stable for solving the stiff weighted least squares problem. We then propose a row block modified Gram-Schmidt algorithm with column pivoting, and show that with appropriately chosen tolerance, this algorithm can correctly determine the numerical ranks of these row partitioned sub-matrices, and the computed QR factor R^- contains small roundoff error which is row stable. Several numerical experiments are also provided to compare the results of the ordinary Modified Gram-Schmidt algorithm with column pivoting and the row block Modified Gram-Schmidt algorithm with column pivoting.     

近红外光谱技术快速鉴别地沟油与食用植物油的研究

地沟油检测是我国食品安全最为关注的话题之一,它给人们的生活健康带来了极大的危害。国内现有的检测手段也仅停留在定性检测水平上,只能确定地沟油的有无,还难以进行定量检测。本实验利用近红外光谱技术与光纤传感技术相结合的新方法对勾兑混合油中地沟油的含量进行了定量分析。将煎炸老油与九三大豆油按照一定的体积比进行勾兑,共计50个样本,采集其近红外透射光谱,分别采用偏最小二乘法(PLS)和BP人工神经网络建立了煎炸老油含量的定量分析模型,校正集决定系数分别为0.908和0.934,验证集决定系数分别为0.961和0.952,均方估计残差(RMSEC)为0.184和0.136,预测均方根误差(RMSEP)都为0.111 6,符合应用要求,同时还结合主成分分析法(PCA)对煎炸老油与食用植物油进行了鉴别,识别准确率为100%。实验研究证明近红外光谱技术不仅可以准确快速的定性分析地沟油, 还能定量的检测地沟油的含量,在油脂的检测方面具有很大的应用前景。     

非线性最小二乘问题的信赖域方法

本文提出一个新的非线性最小二乘的信赖域方法,在该方法中每个信赖域子问题只需要一次求解,而且每次迭代的一维搜索步长因子是给定的,避开一维搜索的环节,大大地提高了算法效率.文中证明了在一定的条件下算法的全局收敛性.     

Analysis of large experimental datasets in electrochemical impedance spectroscopy

An approach for the analysis of large experimental datasets in electrochemical impedance spectroscopy (EIS) has been developed. The approach uses the idea of successive Bayesian estimation and splits the multidimensional EIS datasets into parts with reduced dimensionality. Afterwards, estimation of the parameters of the EIS-models is performed successively, from one part to another, using complex nonlinear least squares (CNLS) method. The results obtained on the previous step are used as a priori values (in the Bayesian form) for the analysis of the next part. To provide high stability of the sequential CNLS minimisation procedure, a new hybrid algorithm has been developed. This algorithm fits the datasets of reduced dimensionality to the selected EIS models, provides high stability of the fitting and allows semi-automatic data analysis on a reasonable timescale. The hybrid algorithm consists of two stages in which different zero-order optimisation strategies are used, reducing both the computational time and the probability to overlook the global optimum. The performance of the developed approach has been evaluated using (i) simulated large EIS dataset which represents a possible output of a scanning electrochemical impedance microscopy experiments, and (ii) experimental dataset, where EIS spectra were acquired as a function of the electrode potential and time. The developed data analysis strategy showed promise and can be further extended to other electroanalytical EIS applications which require multidimensional data analysis.     

Computing the conditioning of the components of a linear least‐squares solution

In this paper, we address the accuracy of the results for the overdetermined full rank linear least‐squares problem. We recall theoretical results obtained in (SIAM J. Matrix Anal. Appl. 2007; 29 (2):413–433) on conditioning of the least‐squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities when the regression matrix and the right‐hand side are perturbed. In particular, we show that in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right‐hand side is exactly the condition number of this solution component when only perturbations on the right‐hand side are considered. We explain how to compute the variance–covariance matrix and the least‐squares conditioning using the libraries LAPACK (LAPACK Users' Guide (3rd edn). SIAM: Philadelphia, 1999) and ScaLAPACK (ScaLAPACK Users' Guide. SIAM: Philadelphia, 1997) and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace (Théorie Analytique des Probabilités. Mme Ve Courcier, 1820; 497–530) for computing the mass of Jupiter and a physical application if the area of space geodesy. Copyright © 2008 John Wiley & Sons, Ltd.     

Absolute structure determination using CRYSTALS

A study of post‐refinement absolute structure determination using previously published data was carried out using the CRYSTALS software package. We show that absolute structure determination may be carried out optimally using the analyses available in CRYSTALS, and that it is not necessary to have the separate procedures absolute structure determination and no interest in absolute structure as proposed by Flack [Chimia (2014), 68 , 26–30].     

Constrained moving least‐squares immersed boundary method for fluid‐structure interaction analysis

A numerical method is presented for the analysis of interactions of inviscid and compressible flows with arbitrarily shaped stationary or moving rigid solids. The fluid equations are solved on a fixed rectangular Cartesian grid by using a higher‐order finite difference method based on the fifth‐order WENO scheme. A constrained moving least‐squares sharp interface method is proposed to enforce the Neumann‐type boundary conditions on the fluid‐solid interface by using a penalty term, while the Dirichlet boundary conditions are directly enforced. The solution of the fluid flow and the solid motion equations is advanced in time by staggerly using, respectively, the third‐order Runge‐Kutta and the implicit Newmark integration schemes. The stability and the robustness of the proposed method have been demonstrated by analyzing 5 challenging problems. For these problems, the numerical results have been found to agree well with their analytical and numerical solutions available in the literature. Effects of the support domain size and values assigned to the penalty parameter on the stability and the accuracy of the present method are also discussed.     

FEATURES OF A FLEXIBLE BACKBONE IN THE COORDINATION COMPOUNDS OF A DIOXIME LIGAND: THE CHARACTERIZATION OF SUPRAMOLECULAR AND DINUCLEAR METAL COMPLEXES

Abstract

The grinding of a 2: 1 molar ratio mixture of isonitrosoacetylacetone and 1,3-diaminopropan-2-ol led to formation of the tribasic ligand (H₃L), (1) with two oxime groups and a flexible alcoholic backbone. The 1:2 molar ratio reaction of (1) with CuX₂ produced the planar dinuclear complexes LCu₂(X) nH₂O; × = acetate (2), phenylacetate (3), formate (4), monochloroacetate (5), dichloroacetate (6), trichloroacetate (7), benzoate (8), and p-hydroxybenzoate (9); n = 1 for (2) and (8); n = 2 for (3)-(7); and n = 4 for (9). The copper(II) ions are bridged by the carbox-ylate and the alcoholic oxygen. The strong antiferromagnetic interactions in (2)-(9) are impeded in (5)-(7) by the chloroacetate bridge withdrawing electron density from the carboxylate. The latter bridge is replaced by picrate in the 1:1 molar ratio reaction of (2) with picric acid (10). The 1:1 molar ratio reaction of (1) with copper(II) acetate produced the tetranuclear [HLCu]₂[LCu₂(OAc)] 5H₂O (11), whereas the 2:1 molar ratio reaction, similar to the reaction which led to (8), produced HLCu (12). The latter complex reacted (1:1 molar ratio) with either copper(ll) acetate or nickel(II) acetate to produce complexes (2) and the heterodinuclear LNi-Cu(OAc) 2H₂O (13), respectively. Similar reactions with (11) gave the same complexes (2) and (13). The acid adducts of (9) with p-hydroxybenzoic acid (14) and LCu₂(X)-HX (15); × = p-aminobenzoic acid were isolated. The cobalt(II) analogue of the mononuclear (12), HLCo 2H₂O (16) was obtained from the 1:1 molar reaction of (1) with cobalt(II) acetate. The supramolecular structure of (11), (12) and (16) took place via intermolecular hydrogen bonding of the alcoholic proton with the oximato oxygen of the adjacent molecule which mediated electron density and allowed for a magnetic exchange interaction. The suggested structures of the ligand and metal complexes are in accordance with analytical, spectral and magnetic moment data.     

Ab initio multi-reference configuration interaction of the low-lying states of the AsP molecule

Nine low-lying electronic states of the AsP molecule, including Σ , Ⅱ, and △ symmetries with singlet, triplet, and quintet spin multiplicities, are studied using multi-reference configuration interaction method.The potential energy curves and the spectroscopic constants of these nine states are determined, and compared with the experimental observed data as well as other theoretical works available at present.Three quintet states are reported for the first time.Furthermore, the analytical potential energy functions of these states are fitted using Murrell-Sorbie function and least square fitting method.     

On Generating Discrete Orthogonal Bivariate Polynomials

In this paper we present an algorithm for recursively generating orthogonal bivariate polynomials on a discrete set S ². For this purpose we employ commuting pairs of real symmetric matrices H, K ^n×n to obtain, in a certain sense, a two dimensional Hermitian Lanczos method. The resulting algorithm relies on a recurrence having a slowly growing length. Practical implementation issues an applications are considered. The method can be generalized to compute orthogonal polynomials depending on an arbitrary number of variables.