np.linalg.cond

np.linalg.cond是NumPy中的一个函数，用于计算矩阵的条件数。条件数是一个衡量矩阵相对于其奇异值的大小的指标，它可以用来评估矩阵的稳定性和数值精度。

具体来说，条件数是矩阵奇异值的最大值与最小值之比。当条件数很大时，矩阵的求逆和解线性方程组等操作可能会出现数值不稳定的情况。

下面是一个使用np.linalg.cond计算矩阵条件数的例子：

import numpy as np

A = np.array([[1, 2], [3, 4]])
cond_A = np.linalg.cond(A)
print(cond_A)

输出结果为：

14.933034373659268

这表示矩阵A的条件数为14.93。

相关问题

class SVDRecommender: def __init__(self, k=50, ncv=None, tol=0, which='LM', v0=None, maxiter=None, return_singular_vectors=True, solver='arpack'): self.k = k self.ncv = ncv self.tol = tol self.which = which self.v0 = v0 self.maxiter = maxiter self.return_singular_vectors = return_singular_vectors self.solver = solver def svds(self, A): if self.which == 'LM': largest = True elif self.which == 'SM': largest = False else: raise ValueError("which must be either 'LM' or 'SM'.") if not (isinstance(A, LinearOperator) or isspmatrix(A) or is_pydata_spmatrix(A)): A = np.asarray(A) n, m = A.shape if self.k <= 0 or self.k >= min(n, m): raise ValueError("k must be between 1 and min(A.shape), k=%d" % self.k) if isinstance(A, LinearOperator): if n > m: X_dot = A.matvec X_matmat = A.matmat XH_dot = A.rmatvec XH_mat = A.rmatmat else: X_dot = A.rmatvec X_matmat = A.rmatmat XH_dot = A.matvec XH_mat = A.matmat dtype = getattr(A, 'dtype', None) if dtype is None: dtype = A.dot(np.zeros([m, 1])).dtype else: if n > m: X_dot = X_matmat = A.dot XH_dot = XH_mat = _herm(A).dot else: XH_dot = XH_mat = A.dot X_dot = X_matmat = _herm(A).dot def matvec_XH_X(x): return XH_dot(X_dot(x)) def matmat_XH_X(x): return XH_mat(X_matmat(x)) XH_X = LinearOperator(matvec=matvec_XH_X, dtype=A.dtype, matmat=matmat_XH_X, shape=(min(A.shape), min(A.shape))) #获得隐式定义的格拉米矩阵的低秩近似。 eigvals, eigvec = eigsh(XH_X, k=self.k, tol=self.tol ** 2, maxiter=self.maxiter, ncv=self.ncv, which=self.which, v0=self.v0) #格拉米矩阵有实非负特征值。 eigvals = np.maximum(eigvals.real, 0) #使用来自pinvh的小特征值的复数检测。 t = eigvec.dtype.char.lower() factor = {'f': 1E3, 'd': 1E6} cond = factor[t] * np.finfo(t).eps cutoff = cond * np.max(eigvals) #获得一个指示哪些本征对不是简并微小的掩码， #并为阈值奇异值创建一个重新排序数组。 above_cutoff = (eigvals > cutoff) nlarge = above_cutoff.sum() nsmall = self.k - nlarge slarge = np.sqrt(eigvals[above_cutoff]) s = np.zeros_like(eigvals) s[:nlarge] = slarge if not self.return_singular_vectors: return np.sort(s) if n > m: vlarge = eigvec[:, above_cutoff] ularge = X_matmat(vlarge) / slarge if self.return_singular_vectors != 'vh' else None vhlarge = _herm(vlarge) else: ularge = eigvec[:, above_cutoff] vhlarge = _herm(X_matmat(ularge) / slarge) if self.return_singular_vectors != 'u' else None u = _augmented_orthonormal_cols(ularge, nsmall) if ularge is not None else None vh = _augmented_orthonormal_rows(vhlarge, nsmall) if vhlarge is not None else None indexes_sorted = np.argsort(s) s = s[indexes_sorted] if u is not None: u = u[:, indexes_sorted] if vh is not None: vh = vh[indexes_sorted] return u, s, vh def _augmented_orthonormal_cols(U, n): if U.shape[0] <= n: return U Q, R = np.linalg.qr(U) return Q[:, :n] def _augmented_orthonormal_rows(V, n): if V.shape[1] <= n: return V Q, R = np.linalg.qr(V.T) return Q[:, :n].T def _herm(x): return np.conjugate(x.T) 将上述代码修改为使用LM，迭代器使用arpack

将代码修改为使用LM和迭代器使用arpack的方法如下：

class SVDRecommender:
    def __init__(self, k=50, ncv=None, tol=0, which='LM', v0=None, maxiter=None, return_singular_vectors=True, solver='arpack'):
        self.k = k
        self.ncv = ncv
        self.tol = tol
        self.which = which
        self.v0 = v0
        self.maxiter = maxiter
        self.return_singular_vectors = return_singular_vectors
        self.solver = solver
    
    def svds(self, A):
        if self.which == 'LM':
            largest = True
        elif self.which == 'SM':
            largest = False
        else:
            raise ValueError("which must be either 'LM' or 'SM'.")
        
        if not (isinstance(A, LinearOperator) or isspmatrix(A) or is_pydata_spmatrix(A)):
            A = np.asarray(A)
            n, m = A.shape
        
        if self.k <= 0 or self.k >= min(n, m):
            raise ValueError("k must be between 1 and min(A.shape), k=%d" % self.k)
        
        if isinstance(A, LinearOperator):
            if n > m:
                X_dot = A.matvec
                X_matmat = A.matmat
                XH_dot = A.rmatvec
                XH_mat = A.rmatmat
            else:
                X_dot = A.rmatvec
                X_matmat = A.rmatmat
                XH_dot = A.matvec
                XH_mat = A.matmat
            dtype = getattr(A, 'dtype', None)
        else:
            if n > m:
                X_dot = X_matmat = A.dot
                XH_dot = XH_mat = _herm(A).dot
            else:
                XH_dot = XH_mat = A.dot
                X_dot = X_matmat = _herm(A).dot
            dtype = A.dtype
        
        def matvec_XH_X(x):
            return XH_dot(X_dot(x))
        
        def matmat_XH_X(x):
            return XH_mat(X_matmat(x))
        
        XH_X = LinearOperator(matvec=matvec_XH_X, dtype=dtype, matmat=matmat_XH_X, shape=(min(A.shape), min(A.shape)))
        
        # 获得隐式定义的格拉米矩阵的低秩近似。
        eigvals, eigvec = eigsh(XH_X, k=self.k, tol=self.tol ** 2, maxiter=self.maxiter, ncv=self.ncv, which=self.which, v0=self.v0, mode=self.solver)
        
        # 格拉米矩阵有实非负特征值。
        eigvals = np.maximum(eigvals.real, 0)
        
        # 使用来自pinvh的小特征值的复数检测。
        t = eigvec.dtype.char.lower()
        factor = {'f': 1E3, 'd': 1E6}
        cond = factor[t] * np.finfo(t).eps
        cutoff = cond * np.max(eigvals)
        
        # 获得一个指示哪些本征对不是简并微小的掩码，
        # 并为阈值奇异值创建一个重新排序数组。
        above_cutoff = (eigvals > cutoff)
        nlarge = above_cutoff.sum()
        nsmall = self.k - nlarge
        slarge = np.sqrt(eigvals[above_cutoff])
        s = np.zeros_like(eigvals)
        s[:nlarge] = slarge
        
        if not self.return_singular_vectors:
            return np.sort(s)
        
        if n > m:
            vlarge = eigvec[:, above_cutoff]
            ularge = X_matmat(vlarge) / slarge if self.return_singular_vectors != 'vh' else None
            vhlarge = _herm(vlarge)
        else:
            ularge = eigvec[:, above_cutoff]
            vhlarge = _herm(X_matmat(ularge) / slarge)
        
        if self.return_singular_vectors != 'u':
            u = _augmented_orthonormal_cols(ularge, nsmall) if ularge is not None else None
        else:
            u = None
        
        if self.return_singular_vectors != 'vh':
            vh = _augmented_orthonormal_rows(vhlarge, nsmall) if vhlarge is not None else None
        else:
            vh = None
        
        indexes_sorted = np.argsort(s)
        s = s[indexes_sorted]
        if u is not None:
            u = u[:, indexes_sorted]
        if vh is not None:
            vh = vh[index_sorted]
        
        return u, s, vh
    
def _augmented_orthonormal_cols(U, n):
    if U.shape[0] <= n:
        return U
    Q, R = np.linalg.qr(U)
    return Q[:, :n]

def _augmented_orthonormal_rows(V, n):
    if V.shape[1] <= n:
        return V
    Q, R = np.linalg.qr(V.T)
    return Q[:, :n].T

def _herm(x):
    return np.conjugate(x.T)

from scipy.sparse.linalg import eigsh, LinearOperator from scipy.sparse import isspmatrix, is_pydata_spmatrix class SVDRecommender: def init(self, k=50, ncv=None, tol=0, which='LM', v0=None, maxiter=None, return_singular_vectors=True, solver='arpack'): self.k = k self.ncv = ncv self.tol = tol self.which = which self.v0 = v0 self.maxiter = maxiter self.return_singular_vectors = return_singular_vectors self.solver = solver def svds(self, A): largest = self.which == 'LM' if not largest and self.which != 'SM': raise ValueError("which must be either 'LM' or 'SM'.") if not (isinstance(A, LinearOperator) or isspmatrix(A) or is_pydata_spmatrix(A)): A = np.asarray(A) n, m = A.shape if self.k <= 0 or self.k >= min(n, m): raise ValueError("k must be between 1 and min(A.shape), k=%d" % self.k) if isinstance(A, LinearOperator): if n > m: X_dot = A.matvec X_matmat = A.matmat XH_dot = A.rmatvec XH_mat = A.rmatmat else: X_dot = A.rmatvec X_matmat = A.rmatmat XH_dot = A.matvec XH_mat = A.matmat dtype = getattr(A, 'dtype', None) if dtype is None: dtype = A.dot(np.zeros([m, 1])).dtype else: if n > m: X_dot = X_matmat = A.dot XH_dot = XH_mat = _herm(A).dot else: XH_dot = XH_mat = A.dot X_dot = X_matmat = _herm(A).dot def matvec_XH_X(x): return XH_dot(X_dot(x)) def matmat_XH_X(x): return XH_mat(X_matmat(x)) XH_X = LinearOperator(matvec=matvec_XH_X, dtype=A.dtype, matmat=matmat_XH_X, shape=(min(A.shape), min(A.shape))) eigvals, eigvec = eigsh(XH_X, k=self.k, tol=self.tol ** 2, maxiter=self.maxiter, ncv=self.ncv, which=self.which, v0=self.v0) eigvals = np.maximum(eigvals.real, 0) t = eigvec.dtype.char.lower() factor = {'f': 1E3, 'd': 1E6} cond = factor[t] * np.finfo(t).eps cutoff = cond * np.max(eigvals) above_cutoff = (eigvals > cutoff) nlarge = above_cutoff.sum() nsmall = self.k - nlarge slarge = np.sqrt(eigvals[above_cutoff]) s = np.zeros_like(eigvals) s[:nlarge] = slarge if not self.return_singular_vectors: return np.sort(s) if n > m: vlarge = eigvec[:, above_cutoff] ularge = X_matmat(vlarge) / slarge if self.return_singular_vectors != 'vh' else None vhlarge = _herm(vlarge) else: ularge = eigvec[:, above_cutoff] vhlarge = _herm(X_matmat(ularge) / slarge) if self.return_singular_vectors != 'u' else None u = _augmented_orthonormal_cols(ularge, nsmall) if ularge is not None else None vh = _augmented_orthonormal_rows(vhlarge, nsmall) if vhlarge is not None else None indexes_sorted = np.argsort(s) s = s[indexes_sorted] if u is not None: u = u[:, indexes_sorted] if vh is not None: vh = vh[indexes_sorted] return u, s, vh def _augmented_orthonormal_cols(U, n): if U.shape[0] <= n: return U Q, R = np.linalg.qr(U) return Q[:, :n] def _augmented_orthonormal_rows(V, n): if V.shape[1] <= n: return V Q, R = np.linalg.qr(V.T) return Q[:, :n].T def _herm(x): return np.conjugate(x.T)这段代码中使用的scipy包太旧了，导致会出现报错信息为：cannot import name 'is_pydata_spmatrix' from 'scipy.sparse' (D:\Anaconda\lib\site-packages\scipy\sparse_init.py)，将这段代码修改为使用最新版的scipy包

好的，将使用最新版的scipy包进行修改。首先需要将代码中的`is_pydata_spmatrix`替换为`isspmatrix_csr`。同时需要将代码开头的导入语句修改为`from scipy.sparse.linalg import eigsh, LinearOperator from scipy.sparse import isspmatrix_csr`。