求解奇异矩阵 [英] solving a singular matrix
问题描述
我正在尝试为网格编写一些解包器.这使用有限元方法来解决平整表面和原始表面之间的最小线性应力.目前，有一些顶点被固定以得到结果.没有这个，三角形就会随机旋转和平移...
I am trying to write a little unwrapper for meshes. This uses a finite-element-method to solve for minimal linear stress between flattened and the raw surface. At the moment there are some vertices pinned to get a result. Without this the triangles are rotated and translated randomly...
但是由于这种固定对于该问题不是必需的，因此更好的解决方案是直接求解奇异矩阵. Petsc确实提供了一些方法来解决单数系统，方法是提供有关零空间的一些信息. http://www.mcs.anl. gov/petsc/petsc-current/docs/manual.pdf#section.4.6 我想知道在Eigen中是否还有其他替代方法.如果没有，那么有没有其他方法可以解决此问题而无需固定/固定顶点.
But as this pinning isn't necessary for the problem, the better solution would be to directly solve the singular matrix. Petsc does provide some methodes to solve a singular system by providing some information on the nullspace. http://www.mcs.anl.gov/petsc/petsc-current/docs/manual.pdf#section.4.6 I wonder if there is any alternative for this in Eigen. If not, are there any other possibilities to solve this problem without fixing/pinning vertices.
谢谢， 问候
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推荐答案
Eigen provides an algorithm for SVD decomposition: Jacobi SVD.
SVD分解给出了空空间.按照维基百科文章的表示法，让M = U D V是M的SVD分解，其中D是奇异值的对角矩阵.然后，从范围，空空间和等级:
The SVD decomposition gives the null-space. Following the notations of the wikipedia article, let M = U D V be the SVD decomposition of M, where D is a diagonal matrix of the singular values. Then, from the Range, null space and rank:
与消失的M的奇异值对应的右奇异矢量[V]跨越M的零空间
The right-singular vectors [V] corresponding to vanishing singular values of M span the null space of M
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