Basix 0.6.0

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basix::precompute Namespace Reference

Matrix and permutation precomputation. More...

Functions

void prepare_permutation (const std::span< std::size_t > &perm)
 
template<typename E >
void apply_permutation (const std::span< const std::size_t > &perm, const std::span< E > &data, std::size_t offset=0, std::size_t block_size=1)
 
template<typename E >
void apply_permutation_to_transpose (const std::span< const std::size_t > &perm, const std::span< E > &data, std::size_t offset=0, std::size_t block_size=1)
 
std::vector< std::size_t > prepare_matrix (std::pair< std::vector< double >, std::array< std::size_t, 2 >> &A)
 
template<typename T , typename E >
void apply_matrix (const std::span< const std::size_t > &v_size_t, const std::experimental::mdspan< const T, std::experimental::dextents< std::size_t, 2 >> &M, const std::span< E > &data, std::size_t offset=0, std::size_t block_size=1)
 Apply a (precomputed) matrix. More...
 
template<typename T , typename E >
void apply_matrix_to_transpose (const std::span< const std::size_t > &v_size_t, const std::experimental::mdspan< const T, std::experimental::dextents< std::size_t, 2 >> &M, const std::span< E > &data, std::size_t offset=0, std::size_t block_size=1)
 Apply a (precomputed) matrix to some transposed data. More...
 

Detailed Description

Matrix and permutation precomputation.

Function Documentation

◆ apply_matrix()

template<typename T , typename E >
void basix::precompute::apply_matrix ( const std::span< const std::size_t > &  v_size_t,
const std::experimental::mdspan< const T, std::experimental::dextents< std::size_t, 2 >> &  M,
const std::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:

INPUT perm, mat, data
apply_permutation(perm, data)
FOR i IN RANGE(dim):
FOR j IN RANGE(i+1, dim):
data[i] += mat[i, j] * data[j]
FOR i IN RANGE(dim - 1, -1, -1):
data[i] *= M[i, i]
FOR j in RANGE(i):
data[i] += mat[i, j] * data[j]

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.

Note
This function is designed to be called at runtime, so its performance is critical.
Parameters
[in]v_size_tA permutation, as computed by precompute::prepare_matrix
[in]MThe vector created by precompute::prepare_matrix
[in,out]dataThe data to apply the permutation to
[in]offsetThe position in the data to start applying the permutation
[in]block_sizeThe block size of the data

◆ apply_matrix_to_transpose()

template<typename T , typename E >
void basix::precompute::apply_matrix_to_transpose ( const std::span< const std::size_t > &  v_size_t,
const std::experimental::mdspan< const T, std::experimental::dextents< std::size_t, 2 >> &  M,
const std::span< E > &  data,
std::size_t  offset = 0,
std::size_t  block_size = 1 
)

Apply a (precomputed) matrix to some transposed data.

Note
This function is designed to be called at runtime, so its performance is critical.

See apply_matrix().

◆ apply_permutation()

template<typename E >
void basix::precompute::apply_permutation ( const std::span< const std::size_t > &  perm,
const std::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:

FOR index, entry IN perm:
SWAP(data[index], data[entry])

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.

Example

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).

Note
This function is designed to be called at runtime, so its performance is critical.
Parameters
[in]permA permutation in precomputed form (as returned by prepare_permutation())
[in,out]dataThe data to apply the permutation to
[in]offsetThe position in the data to start applying the permutation
[in]block_sizeThe block size of the data

◆ apply_permutation_to_transpose()

template<typename E >
void basix::precompute::apply_permutation_to_transpose ( const std::span< const std::size_t > &  perm,
const std::span< E > &  data,
std::size_t  offset = 0,
std::size_t  block_size = 1 
)

Apply a (precomputed) permutation to some transposed data

Note
This function is designed to be called at runtime, so its performance is critical.

see apply_permutation().

◆ prepare_matrix()

std::vector< std::size_t > basix::precompute::prepare_matrix ( std::pair< std::vector< double >, std::array< std::size_t, 2 >> &  A)

Prepare a square matrix

This computes the LU decomposition of the transpose of the matrix

This function returns the permutation \(P\)P in the representation \(PA^t=LU\) (precomputed as in prepare_permutation()). The LU decomposition of \(A^t\) is computed in-place

For an example of how the permutation in this form is applied, see apply_matrix().

Parameters
[in,out]AThe matrix's data
Returns
The three parts of a precomputed representation of the matrix. These are (as described above):

◆ prepare_permutation()

void basix::precompute::prepare_permutation ( const std::span< 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:

FOR index, entry IN perm:
new_index = entry
WHILE new_index < index:
new_index = perm[new_index]
OUT[index] = new_index

Example

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().

Parameters
[in,out]permA permutation