BLAS functions

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1. Data types
2. Matrix operations
3. Utilities

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Basic linear algebra data types and operations are described in this file. The data types are private, so that HPC implementations can use whatever data structures they prefer.

These data types are expected to be used internally in QMCkl. They are not intended to be passed to external codes.

1 Data types

1.1 Vector

Variable	Type	Description
`size`	`int64_t`	Dimension of the vector
`data`	`double*`	Elements

qmckl_vector
qmckl_vector_alloc( qmckl_context context,
                    const int64_t size);

Allocates a new vector. If the allocation failed the size is zero.

qmckl_exit_code
qmckl_vector_free( qmckl_context context,
                   qmckl_vector* vector);

1.2 Matrix

Variable	Type	Description
`size`	`int64_t[2]`	Dimension of each component
`data`	`double*`	Elements

The dimensions use Fortran ordering: two elements differing by one in the first dimension are consecutive in memory.

qmckl_matrix
qmckl_matrix_alloc( qmckl_context context,
                    const int64_t size1,
                    const int64_t size2);

Allocates a new matrix. If the allocation failed the sizes are zero.

qmckl_exit_code
qmckl_matrix_free( qmckl_context context,
                   qmckl_matrix* matrix);

1.3 Tensor

Variable	Type	Description
`order`	`int32_t`	Order of the tensor
`size`	`int64_t[QMCKL_TENSOR_ORDER_MAX]`	Dimension of each component
`data`	`double*`	Elements

The dimensions use Fortran ordering: two elements differing by one in the first dimension are consecutive in memory.

qmckl_tensor
qmckl_tensor_alloc( qmckl_context context,
                    const int32_t order,
                    const int64_t* size);

Allocates memory for a tensor. If the allocation failed, the size is zero.

qmckl_exit_code
qmckl_tensor_free (qmckl_context context,
                   qmckl_tensor* tensor);

1.4 Reshaping

Reshaping occurs in-place and the pointer to the data is copied.

1.4.1 Vector -> Matrix

qmckl_matrix
qmckl_matrix_of_vector(const qmckl_vector vector,
                       const int64_t size1,
                       const int64_t size2);

Reshapes a vector into a matrix.

1.4.2 Vector -> Tensor

qmckl_tensor
qmckl_tensor_of_vector(const qmckl_vector vector,
                       const int32_t order,
                       const int64_t* size);

Reshapes a vector into a tensor.

1.4.3 Matrix -> Vector

qmckl_vector
qmckl_vector_of_matrix(const qmckl_matrix matrix);

Reshapes a matrix into a vector.

1.4.4 Matrix -> Tensor

qmckl_tensor
qmckl_tensor_of_matrix(const qmckl_matrix matrix,
                       const int32_t order,
                       const int64_t* size);

Reshapes a matrix into a tensor.

1.4.5 Tensor -> Vector

qmckl_vector
qmckl_vector_of_tensor(const qmckl_tensor tensor);

Reshapes a tensor into a vector.

1.4.6 Tensor -> Matrix

qmckl_matrix
qmckl_matrix_of_tensor(const qmckl_tensor tensor,
                       const int64_t size1,
                       const int64_t size2);

Reshapes a tensor into a vector.

1.5 Access macros

Macros are provided to ease the access to vectors, matrices and tensors. Matrices use column-major ordering, as in Fortran.

double qmckl_vec (qmckl_vector v, int64_t i);           //  v[i]
double qmckl_mat (qmckl_matrix m, int64_t i, int64_t j) // m[j][i]

double qmckl_ten3(qmckl_tensor t, int64_t i, int64_t j, int64_t k); // t[k][j][i]
double qmckl_ten4(qmckl_tensor t, int64_t i, int64_t j, int64_t k, int64_t l);
double qmckl_ten5(qmckl_tensor t, int64_t i, int64_t j, int64_t k, int64_t l, int64_t m);
double qmckl_ten6(qmckl_tensor t, int64_t i, int64_t j, int64_t k, int64_t l, int64_t m, int64_t n);
...

For example:

1.6 Set all elements

1.6.1 Vector

qmckl_vector
qmckl_vector_set(qmckl_vector vector, double value);

1.6.2 Matrix

qmckl_matrix
qmckl_matrix_set(qmckl_matrix matrix, double value);

1.6.3 Tensor

qmckl_tensor
qmckl_tensor_set(qmckl_tensor tensor, double value);

1.7 Copy to/from to `double*`

qmckl_exit_code
qmckl_double_of_vector(const qmckl_context context,
                       const qmckl_vector vector,
                       double* const target,
                       const int64_t size_max);

Converts a vector to a double*.

qmckl_exit_code
qmckl_double_of_matrix(const qmckl_context context,
                       const qmckl_matrix matrix,
                       double* const target,
                       const int64_t size_max);

Converts a matrix to a double*.

qmckl_exit_code
qmckl_double_of_tensor(const qmckl_context context,
                       const qmckl_tensor tensor,
                       double* const target,
                       const int64_t size_max);

Converts a tensor to a double*.

qmckl_exit_code
qmckl_vector_of_double(const qmckl_context context,
                       const double* target,
                       const int64_t size_max,
                       qmckl_vector* vector);

Converts a double* to a vector.

qmckl_exit_code
qmckl_matrix_of_double(const qmckl_context context,
                       const double* target,
                       const int64_t size_max,
                       qmckl_matrix* matrix);

Converts a double* to a matrix.

qmckl_exit_code
qmckl_tensor_of_double(const qmckl_context context,
                       const double* target,
                       const int64_t size_max,
                       qmckl_tensor* tensor);

Converts a double* to a tensor.

1.8 Allocate and copy to `double*`

double* qmckl_alloc_double_of_vector(const qmckl_context context,
                                     const qmckl_vector vector);

double* qmckl_alloc_double_of_matrix(const qmckl_context context,
                                     const qmckl_matrix matrix);

double* qmckl_alloc_double_of_tensor(const qmckl_context context,
                                     const qmckl_tensor tensor);

2 Matrix operations

2.1 `qmckl_dgemm`

Matrix multiplication with a BLAS interface:

\[ C_{ij} = \beta C_{ij} + \alpha \sum_{k} A_{ik} \cdot B_{kj} \]

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`TransA`	`char`	in	'T' is transposed
`TransB`	`char`	in	'T' is transposed
`m`	`int64_t`	in	Number of rows of the input matrix
`n`	`int64_t`	in	Number of columns of the input matrix
`k`	`int64_t`	in	Number of columns of the input matrix
`alpha`	`double`	in	α
`A`	`double[][lda]`	in	Array containing the matrix \(A\)
`lda`	`int64_t`	in	Leading dimension of array `A`
`B`	`double[][ldb]`	in	Array containing the matrix \(B\)
`ldb`	`int64_t`	in	Leading dimension of array `B`
`beta`	`double`	in	β
`C`	`double[][ldc]`	out	Array containing the matrix \(C\)
`ldc`	`int64_t`	in	Leading dimension of array `C`

Requirements:

context is not QMCKL_NULL_CONTEXT
m > 0
n > 0
k > 0
lda >= m
ldb >= n
ldc >= n
A is allocated with at least \(m \times k \times 8\) bytes
B is allocated with at least \(k \times n \times 8\) bytes
C is allocated with at least \(m \times n \times 8\) bytes

qmckl_exit_code qmckl_dgemm (
      const qmckl_context context,
      const char TransA,
      const char TransB,
      const int64_t m,
      const int64_t n,
      const int64_t k,
      const double alpha,
      const double* A,
      const int64_t lda,
      const double* B,
      const int64_t ldb,
      const double beta,
      double* const C,
      const int64_t ldc );

2.2 `qmckl_dgemm_safe`

"Size-safe" proxy function with the same functionality as qmckl_dgemm but with 3 additional arguments. These arguments size_max_M (where M is a matix) are required primarily for the Python API, where compatibility with NumPy arrays implies that sizes of the input and output arrays are provided.

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`TransA`	`char`	in	'T' is transposed
`TransB`	`char`	in	'T' is transposed
`m`	`int64_t`	in	Number of rows of the input matrix
`n`	`int64_t`	in	Number of columns of the input matrix
`k`	`int64_t`	in	Number of columns of the input matrix
`alpha`	`double`	in	α
`A`	`double[][lda]`	in	Array containing the matrix \(A\)
`size_max_A`	`int64_t`	in	Size of the matrix \(A\)
`lda`	`int64_t`	in	Leading dimension of array `A`
`B`	`double[][ldb]`	in	Array containing the matrix \(B\)
`size_max_B`	`int64_t`	in	Size of the matrix \(B\)
`ldb`	`int64_t`	in	Leading dimension of array `B`
`beta`	`double`	in	β
`C`	`double[][ldc]`	out	Array containing the matrix \(C\)
`size_max_C`	`int64_t`	in	Size of the matrix \(C\)
`ldc`	`int64_t`	in	Leading dimension of array `C`

Requirements:

context is not QMCKL_NULL_CONTEXT
m > 0
n > 0
k > 0
lda >= m
ldb >= n
ldc >= n
A is allocated with at least \(m \times k \times 8\) bytes
B is allocated with at least \(k \times n \times 8\) bytes
C is allocated with at least \(m \times n \times 8\) bytes
size_max_A >= m * k
size_max_B >= k * n
size_max_C >= m * n

qmckl_exit_code qmckl_dgemm_safe (
      const qmckl_context context,
      const char TransA,
      const char TransB,
      const int64_t m,
      const int64_t n,
      const int64_t k,
      const double alpha,
      const double* A,
      const int64_t size_max_A,
      const int64_t lda,
      const double* B,
      const int64_t size_max_B,
      const int64_t ldb,
      const double beta,
      double* const C,
      const int64_t size_max_C,
      const int64_t ldc );

2.3 `qmckl_matmul`

Matrix multiplication using the qmckl_matrix data type:

\[ C_{ij} = \beta C_{ij} + \alpha \sum_{k} A_{ik} \cdot B_{kj} \]

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`TransA`	`char`	in	'T' is transposed
`TransB`	`char`	in	'T' is transposed
`alpha`	`double`	in	α
`A`	`qmckl_matrix`	in	Matrix \(A\)
`B`	`qmckl_matrix`	in	Matrix \(B\)
`beta`	`double`	in	β
`C`	`qmckl_matrix`	out	Matrix \(C\)

qmckl_exit_code qmckl_matmul (
      const qmckl_context context,
      const char TransA,
      const char TransB,
      const double alpha,
      const qmckl_matrix A,
      const qmckl_matrix B,
      const double beta,
      qmckl_matrix* const C );

2.4 `qmckl_adjugate`

Given a matrix \(\mathbf{A}\), the adjugate matrix \(\text{adj}(\mathbf{A})\) is the transpose of the cofactors matrix of \(\mathbf{A}\).

\[ \mathbf{B} = \text{adj}(\mathbf{A}) = \text{det}(\mathbf{A}) \, \mathbf{A}^{-1} \]

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`n`	`int64_t`	in	Number of rows and columns of the input matrix
`A`	`double[][lda]`	in	Array containing the \(n \times n\) matrix \(A\)
`lda`	`int64_t`	in	Leading dimension of array `A`
`B`	`double[][ldb]`	out	Adjugate of \(A\)
`ldb`	`int64_t`	in	Leading dimension of array `B`
`det_l`	`double`	inout	determinant of \(A\)

Requirements:

context is not QMCKL_NULL_CONTEXT
n > 0
lda >= m
A is allocated with at least \(m \times m \times 8\) bytes
ldb >= m
B is allocated with at least \(m \times m \times 8\) bytes

qmckl_exit_code qmckl_adjugate (
      const qmckl_context context,
      const int64_t n,
      const double* A,
      const int64_t lda,
      double* const B,
      const int64_t ldb,
      double* det_l );

For small matrices (≤ 5× 5), we use specialized routines for performance motivations. For larger sizes, we rely on the LAPACK library.

subroutine adjugate_general(context, na, A, LDA, B, LDB, det_l)
  use qmckl
  implicit none
  integer(qmckl_context), intent(in) :: context
  integer*8, intent(in)              :: LDA
  integer*8, intent(in)              :: LDB
  integer*8, intent(in)              :: na
  double precision, intent(in)       :: A (LDA,na)
  double precision, intent(out)      :: B (LDB,na)
  double precision, intent(inout)    :: det_l

  double precision :: work(LDA*max(na,64))
  integer          :: inf
  integer          :: ipiv(LDA)
  integer          :: lwork
  integer(8)       :: i, j

We first copy the array A into array B.

B(1:na,1:na) = A(1:na,1:na)

Then, we compute the LU factorization \(LU=A\) using the dgetrf routine.

call dgetrf(na, na, B, LDB, ipiv, inf )

By convention, the determinant of \(L\) is equal to one, so the determinant of \(A\) is equal to the determinant of \(U\), which is simply computed as the product of its diagonal elements.

det_l = 1.d0
j=0_8
do i=1,na
 j = j+min(abs(ipiv(i)-i),1)
 det_l = det_l*B(i,i)
enddo

As dgetrf returns \(PLU=A\) where \(P\) is a permutation matrix, the sign of the determinant is computed as \(-1^m\) where \(m\) is the number of permutations.

if (iand(j,1_8) /= 0_8)  then
  det_l = -det_l
endif

Then, the inverse of \(A\) is computed using dgetri:

lwork = SIZE(work)
call dgetri(na, B, LDB, ipiv, work, lwork, inf )

and the adjugate matrix is computed as the product of the determinant with the inverse:

  B(:,:) = B(:,:)*det_l

end subroutine adjugate_general

2.5 `qmckl_adjugate_safe`

"Size-safe" proxy function with the same functionality as qmckl_adjugate but with 2 additional arguments. These arguments size_max_M (where M is a matix) are required primarily for the Python API, where compatibility with NumPy arrays implies that sizes of the input and output arrays are provided.

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`n`	`int64_t`	in	Number of rows and columns of the input matrix
`A`	`double[][lda]`	in	Array containing the \(n \times n\) matrix \(A\)
`size_max_A`	`int64_t`	in	Size of the matrix \(A\)
`lda`	`int64_t`	in	Leading dimension of array `A`
`B`	`double[][ldb]`	out	Adjugate of \(A\)
`size_max_B`	`int64_t`	in	Size of the matrix \(B\)
`ldb`	`int64_t`	in	Leading dimension of array `B`
`det_l`	`double`	inout	determinant of \(A\)

Requirements:

context is not QMCKL_NULL_CONTEXT
n > 0
lda >= m
A is allocated with at least \(m \times m \times 8\) bytes
ldb >= m
B is allocated with at least \(m \times m \times 8\) bytes
size_max_A >= m * m
size_max_B >= m * m

qmckl_exit_code qmckl_adjugate_safe (
      const qmckl_context context,
      const int64_t n,
      const double* A,
      const int64_t size_max_A,
      const int64_t lda,
      double* const B,
      const int64_t size_max_B,
      const int64_t ldb,
      double* det_l );

For small matrices (≤ 5× 5), we use specialized routines for performance motivations. For larger sizes, we rely on the LAPACK library.

2.5.1 C interface

2.6 `qmckl_transpose`

Transposes a matrix: \(A^\dagger_{ji} = A_{ij}\).

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`A`	`qmckl_matrix`	in	Input matrix
`At`	`qmckl_matrix`	out	Transposed matrix

qmckl_exit_code
qmckl_transpose (qmckl_context context,
                 const qmckl_matrix A,
                 qmckl_matrix At );

3 Utilities

Trick to make MKL efficient on AMD

int mkl_serv_intel_cpu_true() {
  return 1;
}

BLAS functions

Table of Contents

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1 Data types

1.1 Vector

1.2 Matrix

1.3 Tensor

1.4 Reshaping

1.4.1 Vector -> Matrix

1.4.2 Vector -> Tensor

1.4.3 Matrix -> Vector

1.4.4 Matrix -> Tensor

1.4.5 Tensor -> Vector

1.4.6 Tensor -> Matrix

1.5 Access macros

1.6 Set all elements

1.6.1 Vector

1.6.2 Matrix

1.6.3 Tensor

1.7 Copy to/from to double*

1.8 Allocate and copy to double*

2 Matrix operations

2.1 qmckl_dgemm

2.2 qmckl_dgemm_safe

2.3 qmckl_matmul

2.4 qmckl_adjugate

2.5 qmckl_adjugate_safe

2.5.1 C interface

2.6 qmckl_transpose

3 Utilities

1.7 Copy to/from to `double*`

1.8 Allocate and copy to `double*`

2.1 `qmckl_dgemm`

2.2 `qmckl_dgemm_safe`

2.3 `qmckl_matmul`

2.4 `qmckl_adjugate`

2.5 `qmckl_adjugate_safe`

2.6 `qmckl_transpose`