问题描述
在求解广义特征值问题A * x = lambda * M * x时，如果M不是对角线，为什么下面使用的scipy.sparse.linalg中的eigh和eigsh给出错误的结果?
Why do eigh and eigsh from scipy.sparse.linalg as used below give incorrect results when solving the generalized eigenvalue problem A * x = lambda * M * x , if M is non-diagonal?
import mkl
import numpy as np
from scipy import linalg as LA
from scipy.sparse import linalg as LAsp
from scipy.sparse import csr_matrix
A = np.diag(np.arange(1.0,7.0))
M = np.array([[ 25.1, 0. , 0. , 17.3, 0. , 0. ],
[ 0. , 33.6, 16.8, 8.4, 4.2, 2.1],
[ 0. , 16.8, 3.6, 0. , 11. , 0. ],
[ 17.3, 8.4, 0. , 4.2, 0. , 9.5],
[ 0. , 4.2, 11. , 0. , 2.7, 8.3],
[ 0. , 2.1, 0. , 9.5, 8.3, 4.4]])
Asp = csr_matrix(np.matrix(A,dtype=float))
Msp = csr_matrix(np.matrix(M,dtype=float))
D, V = LA.eig(A, b=M)
eigno = 4
Dsp0, Vsp0 = LAsp.eigs(csr_matrix(np.matrix(np.dot(np.linalg.inv(M),A))),
k=eigno,which='LM',return_eigenvectors=True)
Dsp1, Vsp1 = LAsp.eigs(Asp,k=eigno,M=Msp,which='LM',return_eigenvectors=True)
Dsp2, Vsp2 = LAsp.eigsh(Asp,k=eigno,M=Msp,which='LA',return_eigenvectors=True,
maxiter=1000)
从LA.eig并使用MatLab进行检查，该带有测试矩阵A和M的广义广义特征值问题的特征值应为:
From LA.eig and checking with MatLab the eigenvalues for this small generalized eigenvalue problem with test matrices A and M should be:
D = [ 0.7208+0.j, 0.3979+0.j, -0.3011+0.j, -0.3251+0.j, 0.0357+0.j, 0.0502+0.j]
我想使用稀疏矩阵，因为涉及的实际A和M矩阵约为30,000 x 30,000. A始终是正方形，实数和对角线，M始终是正方形，实数和对称.当M是对角线时，我得到正确的结果.但是，当求解非对角M矩阵的广义特征值问题时，eigs和eigsh都给出错误的结果.
I want to use sparse matrices because the actual A and M matrices involved are around 30,000 x 30,000. A is always square, real and diagonal, M is always square, real and symmetric. When M is diagonal I get the correct results. However, both eigs and eigsh give incorrect results when solving the generalized eigenvalue problem for a non-diagonal M matrix.
Dsp1 = [-1.6526+2.3357j, -1.6526-2.3357j, -0.6243+2.7334j, -0.6243-2.7334j]
Dsp2 = [ 2.01019097, 3.09248265, 4.06799498, 7.01216316]
当我将问题转换为标准特征值形式M ^ -1 * A * x = lambda * x时，eigs给出正确的结果(Dsp0).对于大型矩阵，这不是一种选择，因为计算M的逆值会花费太长时间.
When I convert the problem to the standard eigenvalue form M^-1 * A * x = lambda * x, eigs gives the correct result (Dsp0). For large matrices this is not an option because it takes too long to compute the inverse of M.
我注意到，使用mkl或不使用mkl也会产生不同的Dsp1和Dsp2特征值.这个特征值问题可能是由我的Python安装问题引起的吗?我在Mac OS 10.10.2上运行带有SciPy 0.15.1的Python 2.7.8 anaconda-np19py27_p0 [mkl].
I noticed that using mkl or not yields different Dsp1 and Dsp2 eigenvalues as well. Could this eigenvalue problem be caused by an issue with my Python installation? I am running Python 2.7.8 anaconda with SciPy 0.15.1 - np19py27_p0 [mkl] on Mac OS 10.10.2.
推荐答案
两者 eigs 和 eigsh 要求M为正定 (有关更多详细信息，请参见文档字符串中的M描述.)
Both eigs and eigsh require that M be positive definite (see the descriptions of M in the docstrings for more details).
您的矩阵M不是正定的.注意负特征值:
Your matrix M is not positive definite. Note the negative eigenvalues:
In [212]: M
Out[212]:
array([[ 25.1, 0. , 0. , 17.3, 0. , 0. ],
[ 0. , 33.6, 16.8, 8.4, 4.2, 2.1],
[ 0. , 16.8, 3.6, 0. , 11. , 0. ],
[ 17.3, 8.4, 0. , 4.2, 0. , 9.5],
[ 0. , 4.2, 11. , 0. , 2.7, 8.3],
[ 0. , 2.1, 0. , 9.5, 8.3, 4.4]])
In [213]: np.linalg.eigvals(M)
Out[213]:
array([ 45.92443169, 33.92113421, -13.12639751, -10.6991868 ,
5.34183619, 12.23818222])
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