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336 lines
4.1 KiB
Markdown
336 lines
4.1 KiB
Markdown
# 稀疏矩阵
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`Scipy` 提供了稀疏矩阵的支持（`scipy.sparse`）。
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稀疏矩阵主要使用 位置 + 值 的方法来存储矩阵的非零元素，根据存储和使用方式的不同，有如下几种类型的稀疏矩阵：
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| 类型 | 描述 |
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| --- | --- |
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| `bsr_matrix(arg1[, shape, dtype, copy, blocksize])` | Block Sparse Row matrix |
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| `coo_matrix(arg1[, shape, dtype, copy])` | A sparse matrix in COOrdinate format. |
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| `csc_matrix(arg1[, shape, dtype, copy])` | Compressed Sparse Column matrix |
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| `csr_matrix(arg1[, shape, dtype, copy])` | Compressed Sparse Row matrix |
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| `dia_matrix(arg1[, shape, dtype, copy])` | Sparse matrix with DIAgonal storage |
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| `dok_matrix(arg1[, shape, dtype, copy])` | Dictionary Of Keys based sparse matrix. |
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| `lil_matrix(arg1[, shape, dtype, copy])` | Row-based linked list sparse matrix |
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在这些存储格式中：
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* COO 格式在构建矩阵时比较高效
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* CSC 和 CSR 格式在乘法计算时比较高效
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## 构建稀疏矩阵
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In [1]:
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```py
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from scipy.sparse import *
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import numpy as np
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```
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创建一个空的稀疏矩阵：
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In [2]:
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```py
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coo_matrix((2,3))
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```
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Out[2]:
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```py
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<2x3 sparse matrix of type '<type 'numpy.float64'>'
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with 0 stored elements in COOrdinate format>
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```
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也可以使用一个已有的矩阵或数组或列表中创建新矩阵：
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In [3]:
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```py
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A = coo_matrix([[1,2,0],[0,0,3],[4,0,5]])
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print A
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```
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```py
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(0, 0) 1
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(0, 1) 2
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(1, 2) 3
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(2, 0) 4
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(2, 2) 5
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```
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不同格式的稀疏矩阵可以相互转化：
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In [4]:
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```py
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type(A)
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```
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Out[4]:
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```py
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scipy.sparse.coo.coo_matrix
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```
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In [5]:
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```py
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B = A.tocsr()
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type(B)
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```
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Out[5]:
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```py
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scipy.sparse.csr.csr_matrix
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```
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可以转化为普通矩阵：
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In [6]:
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```py
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C = A.todense()
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C
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```
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Out[6]:
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```py
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matrix([[1, 2, 0],
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[0, 0, 3],
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[4, 0, 5]])
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```
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与向量的乘法：
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In [7]:
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```py
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v = np.array([1,0,-1])
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A.dot(v)
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```
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Out[7]:
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```py
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array([ 1, -3, -1])
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```
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还可以传入一个 `(data, (row, col))` 的元组来构建稀疏矩阵：
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In [8]:
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```py
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I = np.array([0,3,1,0])
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J = np.array([0,3,1,2])
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V = np.array([4,5,7,9])
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A = coo_matrix((V,(I,J)),shape=(4,4))
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```
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In [9]:
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```py
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print A
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```
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```py
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(0, 0) 4
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(3, 3) 5
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(1, 1) 7
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(0, 2) 9
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```
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COO 格式的稀疏矩阵在构建的时候只是简单的将坐标和值加到后面，对于重复的坐标不进行处理：
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In [10]:
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```py
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I = np.array([0,0,1,3,1,0,0])
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J = np.array([0,2,1,3,1,0,0])
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V = np.array([1,1,1,1,1,1,1])
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B = coo_matrix((V,(I,J)),shape=(4,4))
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print B
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```
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```py
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(0, 0) 1
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(0, 2) 1
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(1, 1) 1
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(3, 3) 1
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(1, 1) 1
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(0, 0) 1
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(0, 0) 1
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```
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转换成 CSR 格式会自动将相同坐标的值合并：
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In [11]:
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```py
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C = B.tocsr()
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print C
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```
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```py
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(0, 0) 3
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(0, 2) 1
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(1, 1) 2
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(3, 3) 1
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```
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## 求解微分方程
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In [12]:
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```py
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from scipy.sparse import lil_matrix
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from scipy.sparse.linalg import spsolve
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from numpy.linalg import solve, norm
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from numpy.random import rand
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```
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构建 `1000 x 1000` 的稀疏矩阵：
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In [13]:
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```py
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A = lil_matrix((1000, 1000))
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A[0, :100] = rand(100)
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A[1, 100:200] = A[0, :100]
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A.setdiag(rand(1000))
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```
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转化为 CSR 之后，用 `spsolve` 求解 $Ax=b$：
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In [14]:
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```py
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A = A.tocsr()
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b = rand(1000)
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x = spsolve(A, b)
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```
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转化成正常数组之后求解：
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In [15]:
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```py
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x_ = solve(A.toarray(), b)
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```
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查看误差：
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In [16]:
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```py
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err = norm(x-x_)
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err
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```
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Out[16]:
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```py
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6.4310987107687431e-13
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```
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## sparse.find 函数
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返回一个三元组，表示稀疏矩阵中非零元素的 `(row, col, value)`：
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In [17]:
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```py
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from scipy import sparse
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row, col, val = sparse.find(C)
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print row, col, val
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```
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```py
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[0 0 1 3] [0 2 1 3] [3 1 2 1]
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```
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## sparse.issparse 函数
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查看一个对象是否为稀疏矩阵：
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In [18]:
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```py
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sparse.issparse(B)
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```
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Out[18]:
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```py
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True
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```
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或者
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In [19]:
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```py
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sparse.isspmatrix(B.todense())
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```
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Out[19]:
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```py
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False
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```
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还可以查询是否为指定格式的稀疏矩阵：
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In [20]:
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```py
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sparse.isspmatrix_coo(B)
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```
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Out[20]:
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```py
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True
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```
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In [21]:
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```py
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sparse.isspmatrix_csr(B)
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```
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Out[21]:
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```py
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False
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``` |