R package mbend was developed for bending symmetric non-positive-definite matrices to positive-definite (PD). Bending is a procedure of transforming non-PD matrices to PD. The covariance matrices used in multi-trait best linear unbiased prediction (BLUP) should be PD. Two bending methods are implemented in mbend. The first is an unweighted bending with small positive values in a descending order replacing negative eigenvalues (LRS14), and the second method is a weighted (precision-based) bending with a custom small positive value (ϵ) replacing smaller eigenvalues (HJ03). Weighted bending is beneficial, as it relaxes low precision elements to change and it reduces or prohibits the change in high precision elements. Therefore, a weighted version of LRS14 was developed in mbend. In cases where the precision of matrix elements is unknown, the package provides an unweighted version of HJ03. Another unweighted bending method (DB88) was tested, by which all eigenvalues are changed (eigenvalues less than ϵ replaced with 100 × ϵ), and it is originally designed for correlation matrices. Different bending procedures were conducted on a 5 × 5 covariance matrix (V), V converted to a correlation matrix (C) and an ill-conditioned 1000 × 1000 genomic relationship matrix (G). Considering weighted distance statistics between matrix elements before and after bending, weighting considerably improved the bending quality. For weighted and unweighted bending of V and C, HJ03–4 (HJ03, ϵ = 10−4) performed the best. HJ03–2 (HJ03, ϵ = 10−2) ranked better than LRS14 for V, but not for C. Though the differences were marginal, LRS14 performed the best for G. DB88–4 (DB88, ϵ = 10−4) was used for unweighted bending and it ranked the last. This method could perform considerably better with a lower ϵ. R package mbend provides necessary tools for transforming symmetric non-PD matrices to PD, using different methods and parameters. There were benefits in both weighted bending and small positive values in a descending order replacing negative eigenvalues. Thus, weighted LRS14 was implemented in mbend. Different bending methods might be preferable for different matrices, depending on the matrix type (covariance vs. correlation), number and the magnitude of negative eigenvalues, and the matrix size.

中文翻译：

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mbend：R包，用于将非正定对称矩阵弯曲为正定。

开发了R包mbend，用于将对称的非正定矩阵弯曲为正定（PD）。弯曲是将非PD矩阵转换为PD的过程。多特征最佳线性无偏预测（BLUP）中使用的协方差矩阵应为PD。mbend中实现了两种弯曲方法。第一种方法是按降序以较小的正值代替负特征值（LRS14）的未加权弯曲，第二种方法是用定制的小正值（ϵ）替代较小的特征值（HJ03）的加权（基于精度）弯曲。加权弯曲是有益的，因为它可以放宽低精度元素的更改，并减少或禁止高精度元素的更改。因此，mbend开发了LRS14的加权版本。如果矩阵元素的精度未知，该软件包提供了HJ03的未加权版本。测试了另一种未加权的弯曲方法（DB88），通过该方法可以更改所有特征值（小于ϵ的特征值替换为100×ϵ），并且该方法最初是为相关矩阵设计的。在5×5协方差矩阵（V）上执行不同的弯曲过程，将V转换为相关矩阵（C）和病态的1000×1000基因组关系矩阵（G）。考虑到弯曲前后矩阵元素之间的加权距离统计信息，加权可显着改善弯曲质量。对于V和C的加权弯曲和非加权弯曲，HJ03-4（HJ03，ϵ = 10-4）表现最佳。HJ03–2（HJ03，ϵ = 10−2）在V方面的排名优于LRS14，但在C方面的排名不佳。尽管差异很小，但LRS14在G方面表现最好。DB88–4（DB88，ϵ = 10−4）用于未加权弯曲，它排名最后。ϵ较低时，此方法的性能可能会好得多。R包mbend提供了使用不同方法和参数将对称非PD矩阵转换为PD的必要工具。加权弯曲和降序替换负特征值时小的正值都有好处。因此，加权LRS14在mbend中实现。根据矩阵类型（协方差与相关性），负特征值的数量和大小以及矩阵大小，对于不同的矩阵，可能需要采用不同的折弯方法。加权弯曲和降序替换负特征值时小的正值都有好处。因此，加权LRS14在mbend中实现。根据矩阵类型（协方差与相关性），负特征值的数量和大小以及矩阵大小，对于不同的矩阵，可能需要采用不同的折弯方法。加权弯曲和降序替换负特征值时小的正值都有好处。因此，加权LRS14在mbend中实现。根据矩阵类型（协方差与相关性），负特征值的数量和大小以及矩阵大小，对于不同的矩阵，可能需要采用不同的折弯方法。