托普利兹矩阵的托普利兹矩阵 [英] Toeplitz matrix of toeplitz matrix
问题描述
我想创建一个 toeplitz 矩阵的 toeplitz 矩阵.H1、H2 和 H3 已经是托普利兹矩阵.我的结果应该是这样的:<代码>H1 0 0H2 H1 0H3 H2 H10 H3 H20 0 H3
I want to create a toeplitz matrix of toeplitz matrix.
H1, H2 and H3 are toeplitz matrices already. My result should look like that:
H1 0 0
H2 H1 0
H3 H2 H1
0 H3 H2
0 0 H3
现有的 toeplitz 函数只接受向量,所以我不能将它用于矩阵.目前我使用 vstack
来创建第一列,然后是第二列等等,然后我使用 hstack
来合并所有列.这需要付出很多努力,因为我必须在某些地方专门添加 np.zeros
矩阵.我想不出更好的方法来连接 numpy 数组,因为只有几个函数可以用来解决我的问题.
The existing toeplitz-function only accepts vector, so I can't use it for matrix. Currently I'm using vstack
to create the first column, then second column etc. and then I use hstack
to merge all columns. This takes a lot of effort, since I have to specifically add np.zeros
matrices at certain places. I can't think of a better way to concatenate numpy arrays, since there are only a few functions for that and none of them really fits my problem.
推荐答案
代替对 vstack
和 hstack
的嵌套调用,预分配最终数组会更有效率,然后使用嵌套循环填充数组.最初可以使用更高维的数组来保持代码整洁.
Instead of nested calls to vstack
and hstack
, it will be more efficient to preallocate the final array, and then use a nested loop to fill in the array. You can initially use a higher dimensional array to keep the code clean.
例如这个脚本
import numpy as np
H1 = np.array([[11, 11], [11, 11]])
H2 = np.array([[22, 22], [22, 22]])
H3 = np.array([[33, 33], [33, 33]])
inputs = (H1, H2, H3)
# This assumes all the arrays in `inputs` have the same shape,
# and that the data type of all the arrays is the same as H1.dtype.
nh = len(inputs)
nrows = 2*nh - 1
m, n = H1.shape
# T is a 4D array. For a given i and j, T[i, :, j, :] is a 2D array
# with shape (m, n). T can be intepreted as a 2D array of 2D arrays.
T = np.zeros((nrows, m, nh, n), dtype=H1.dtype)
for i, H in enumerate(inputs):
for j in range(nh):
T[i + j, :, j, :] = H
# Partially flatten the 4D array to a 2D array that has the desired
# block structure.
T.shape = (nrows*m, nh*n)
print(T)
印刷品
[[11 11 0 0 0 0]
[11 11 0 0 0 0]
[22 22 11 11 0 0]
[22 22 11 11 0 0]
[33 33 22 22 11 11]
[33 33 22 22 11 11]
[ 0 0 33 33 22 22]
[ 0 0 33 33 22 22]
[ 0 0 0 0 33 33]
[ 0 0 0 0 33 33]]
(注意,结果不是托普利茨矩阵;它是一个块托普利茨矩阵.)
(Note that the result is not a Toeplitz matrix; it is a block Toeplitz matrix.)
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