Matrix and permutation precomputation. More...
Functions | |
| std::vector< std::size_t > | prepare_permutation (const std::vector< std::size_t > &perm) |
| template<typename E > | |
| void | apply_permutation (const std::vector< std::size_t > &perm, const xtl::span< E > &data, std::size_t offset=0, std::size_t block_size=1) |
| template<typename E > | |
| void | apply_permutation_to_transpose (const std::vector< std::size_t > &perm, const xtl::span< E > &data, std::size_t offset=0, std::size_t block_size=1) |
| std::tuple< std::vector< std::size_t >, std::vector< double >, xt::xtensor< double, 2 > > | prepare_matrix (const xt::xtensor< double, 2 > &matrix) |
| template<typename T , typename E > | |
| void | apply_matrix (const std::tuple< std::vector< std::size_t >, std::vector< T >, xt::xtensor< T, 2 >> &matrix, const xtl::span< E > &data, std::size_t offset=0, std::size_t block_size=1) |
| template<typename T , typename E > | |
| void | apply_matrix_to_transpose (const std::tuple< std::vector< std::size_t >, std::vector< T >, xt::xtensor< T, 2 >> &matrix, const xtl::span< E > &data, std::size_t offset=0, std::size_t block_size=1) |
Matrix and permutation precomputation.
| void basix::precompute::apply_matrix | ( | const std::tuple< std::vector< std::size_t >, std::vector< T >, xt::xtensor< T, 2 >> & | matrix, |
| const xtl::span< E > & | data, | ||
| std::size_t | offset = 0, |
||
| std::size_t | block_size = 1 |
||
| ) |
Apply a (precomputed) matrix
This uses the representation returned by prepare_matrix() to apply a matrix without needing any temporary memory.
In pseudo code, this function does the following:
If block_size is set, this will apply the permutation to every block. The offset is set, this will start applying the permutation at the offsetth block.
As an example, consider the matrix \(A = \) [[-1, 0, 1], [1, 1, 0], [2, 0, 2]]. In the documentation of prepare_matrix(), we saw that the precomputed representation of this matrix is the identity permutation,
\begin{align*} D &= \begin{bmatrix}-1\\1\\4\end{bmatrix},\\ \quad M &= \begin{bmatrix} 0&0&1\\ -1&0&1\\ -2&0&0 \end{bmatrix}. \end{align*}
In this example, we look at how this representation can be used to apply this matrix to the vector \(v = \) [3, -1, 2].
No permutation is necessary, so first, we multiply \(v_0\) by \(D_0=-1\). After this, \(v\) is [-3, -1, 2].
Next, we add \(M_{0,i}v_i\) to \(v_0\) for all \(i\): in this case, we add \(0\times-3 + 0\times-1 + 1\times2 = 2\). After this, \(v\) is [-1, -1, 2].
Next, we multiply \(v_1\) by \(D_1=1\). After this, \(v\) is [-1, -1, 2].
Next, we add \(M_{1,i}v_i\) to \(v_1\) for all \(i\): in this case, we add \(-1\times-1 + 0\times-1 + 1\times2 = 3\). After this, \(v\) is [-1, 2, 2].
Next, we multiply \(v_2\) by \(D_2=4\). After this, \(v\) is [-1, 2, 8].
Next, we add \(M_{2,i}v_i\) to \(v_2\) for all \(i\): in this case, we add \(-2\times-1 + 0\times2 + 0\times8 = 2\). After this, \(v\) is [-1, 2, 10]. This final value of \(v\) is what the result of \(Av\)
| [in] | matrix | A matrix in precomputed form (as returned by prepare_matrix()) |
| [in,out] | data | The data to apply the permutation to |
| [in] | offset | The position in the data to start applying the permutation |
| [in] | block_size | The block size of the data |
| void basix::precompute::apply_matrix_to_transpose | ( | const std::tuple< std::vector< std::size_t >, std::vector< T >, xt::xtensor< T, 2 >> & | matrix, |
| const xtl::span< E > & | data, | ||
| std::size_t | offset = 0, |
||
| std::size_t | block_size = 1 |
||
| ) |
Apply a (precomputed) matrix to some transposed data.
See apply_matrix().
| void basix::precompute::apply_permutation | ( | const std::vector< std::size_t > & | perm, |
| const xtl::span< E > & | data, | ||
| std::size_t | offset = 0, |
||
| std::size_t | block_size = 1 |
||
| ) |
Apply a (precomputed) permutation
This uses the representation returned by prepare_permutation() to apply a permutation without needing any temporary memory.
In pseudo code, this function does the following:
If block_size is set, this will apply the permutation to every block. The offset is set, this will start applying the permutation at the offsetth block.
As an example, consider the permutation P = [1, 4, 0, 5, 2, 3]. In the documentation of prepare_permutation(), we saw that the precomputed representation of this permutation is P2 = [1, 4, 4, 5, 4, 5]. In this example, we look at how this representation can be used to apply this permutation to the array A = [a, b, c, d, e, f].
P2[0] is 1, so we swap A[0] and A[1]. After this, A = [b, a, c, d, e, f].
P2[1] is 4, so we swap A[1] and A[4]. After this, A = [b, e, c, d, a, f].
P2[2] is 4, so we swap A[2] and A[4]. After this, A = [b, e, a, d, c, f].
P2[3] is 5, so we swap A[3] and A[5]. After this, A = [b, e, a, f, c, d].
P2[4] is 4, so we swap A[4] and A[4]. This changes nothing.
P2[5] is 5, so we swap A[5] and A[5]. This changes nothing.
Therefore the result of applying this permutation is [b, e, a, f, c, d] (which is what we get if we apply the permutation directly).
| [in] | perm | A permutation in precomputed form (as returned by prepare_permutation()) |
| [in,out] | data | The data to apply the permutation to |
| [in] | offset | The position in the data to start applying the permutation |
| [in] | block_size | The block size of the data |
| void basix::precompute::apply_permutation_to_transpose | ( | const std::vector< std::size_t > & | perm, |
| const xtl::span< E > & | data, | ||
| std::size_t | offset = 0, |
||
| std::size_t | block_size = 1 |
||
| ) |
Apply a (precomputed) permutation to some transposed data
see apply_permutation().
| std::tuple< std::vector< std::size_t >, std::vector< double >, xt::xtensor< double, 2 > > basix::precompute::prepare_matrix | ( | const xt::xtensor< double, 2 > & | matrix | ) |
Prepare a matrix
This computes a representation of the matrix that allows the matrix to be applied without any temporary memory assignment.
This function will first permute the matrix's columns so that the top left \(n\times n\) blocks are invertible (for all \(n\)). Let \(A\) be the input matrix after the permutation is applied. The output vector \(D\) and matrix \(M\) are then given by:
\begin{align*} D_i &= \begin{cases} A_{i, i} & i = 0\\ A_{i, i} - A_{i,:i}A_{:i,:i}^{-1}A_{:i,i} & i \not= 0 \end{cases},\\ M_{i,j} &= \begin{cases} A_{i,:i}A_{:i,:i}^{-1}e_j & j < i\\ 0 & j = i\\ A_{i, i} - A_{i,:i}A_{:i,:i}^{-1}A_{:i,j} & j > i = 0 \end{cases}, \end{align*}
where \(e_j\) is the \(j\)th coordinate vector, we index all the matrices and vector starting at 0, and we use numpy-slicing-stying notation in the subscripts: for example, \(A_{:i,j}\) represents the first \(i\) entries in the \(j\)th column of \(A\)
This function returns the permutation (precomputed as in prepare_permutation()), the vector \(D\), and the matrix \(M\) as a tuple.
As an example, consider the matrix \(A = \) [[-1, 0, 1], [1, 1, 0], [2, 0, 2]]. For this matrix, no permutation is needed, so the first item in the output will represent the identity permutation. We now compute the output vector \(D\) and matrix \(M\).
First, we set \(D_0 = A_{0,0}=-1\), set the diagonal of \(M\) to be 0 and set \(M_{0, 1:} = A_{0, 1:}=\begin{bmatrix}0&1\end{bmatrix}\). The output so far is
\begin{align*} D &= \begin{bmatrix}-1\\?\\?\end{bmatrix},\\ \quad M &= \begin{bmatrix} 0&0&1\\ ?&0&?\\ ?&?&0 \end{bmatrix}. \end{align*}
Next, we set:
\begin{align*} D_1 &= A_{1,1} - A_{1, :1}A_{:1,:1}^{-1}A_{:1, 1}\\ &= 1 - \begin{bmatrix}-1\end{bmatrix}\cdot\begin{bmatrix}0\end{bmatrix}\\ &= 1,\\ M_{2,0} &= A_{1, :1}A_{:1,:1}^{-1}e_0\\ &= \begin{bmatrix}1\end{bmatrix}\begin{bmatrix}-1\end{bmatrix}^{-1} \begin{bmatrix}1\end{bmatrix}\\ &= \begin{bmatrix}-1\end{bmatrix} M_{2,3} &= A_{1,2}-A_{1, :1}A_{:1,:1}^{-1}A_{:1, 1}\\ &= 0-\begin{bmatrix}1\end{bmatrix}\begin{bmatrix}-1\end{bmatrix}^{-1} \begin{bmatrix}1\end{bmatrix},\\ &= 1. \end{align*}
The output so far is
\begin{align*} D &= \begin{bmatrix}-1\\1\\?\end{bmatrix},\\ \quad M &= \begin{bmatrix} 0&0&1\\ -1&0&1\\ ?&?&0 \end{bmatrix}. \end{align*}
Next, we set:
\begin{align*} D_2 &= A_{2,2} - A_{2, :2}A_{:2,:2}^{-1}A_{:2, 2}\\ &= 2 - \begin{bmatrix}2&0\end{bmatrix} \begin{bmatrix}-1&0\\1&1\end{bmatrix}^{-1} \begin{bmatrix}1\\0\end{bmatrix}\\ &= 4,\\ M_{2,0} &= A_{2, :2}A_{:2,:2}^{-1}e_0\\ &= -2.\\ M_{2,1} &= A_{2, :2}A_{:2,:2}^{-1}e_1\\ &= 0.\\ \end{align*}
The output is
\begin{align*} D &= \begin{bmatrix}-1\\1\\4\end{bmatrix},\\ \quad M &= \begin{bmatrix} 0&0&1\\ -1&0&1\\ -2&0&0 \end{bmatrix}. \end{align*}
For an example of how the permutation in this form is applied, see apply_matrix().
| [in] | matrix | A matrix |
| std::vector< std::size_t > basix::precompute::prepare_permutation | ( | const std::vector< std::size_t > & | perm | ) |
Prepare a permutation
This computes a representation of the permutation that allows the permutation to be applied without any temporary memory assignment.
In pseudo code, this function does the following:
As an example, consider the permutation P = [1, 4, 0, 5, 2, 3].
First, we look at the 0th entry. P[0] is 1. This is greater than 0, so the 0th entry of the output is 1.
Next, we look at the 1st entry. P[1] is 4. This is greater than 1, so the 1st entry of the output is 4.
Next, we look at the 2nd entry. P[2] is 0. This is less than 2, so we look at P[0].P[0]is 1. This is less than 2, so we look atP[1].P[1]` is 4. This is greater than 2, so the 2nd entry of the output is 4.
Next, we look at the 3rd entry. P[3] is 5. This is greater than 3, so the 3rd entry of the output is 5.
Next, we look at the 4th entry. P[4] is 2. This is less than 4, so we look at P[2]. P[2] is 0. This is less than 4, so we look at P[0]. P[0] is 1. This is less than 4, so we look at P[1]. P[1] is 4. This is greater than (or equal to) 4, so the 4th entry of the output is 4.
Next, we look at the 5th entry. P[5] is 3. This is less than 5, so we look at P[3]. P[3] is 5. This is greater than (or equal to) 5, so the 5th entry of the output is 5.
Hence, the output of this function in this case is [1, 4, 4, 5, 4, 5].
For an example of how the permutation in this form is applied, see apply_permutation().
| [in] | perm | A permutation |