Visible to Intel only — GUID: GUID-C438A81E-F7FC-42CC-8CEF-F492B0A1F182
Visible to Intel only — GUID: GUID-C438A81E-F7FC-42CC-8CEF-F492B0A1F182
?gemm3m_batch
Computes scalar-matrix-matrix products and adds the results to scalar matrix products for groups of general matrices.
Syntax
call cgemm3m_batch(transa_array, transb_array, m_array, n_array, k_array, alpha_array, a_array, lda_array, b_array, ldb_array, beta_array, c_array, ldc_array, group_count, group_size)
call zgemm3m_batch(transa_array, transb_array, m_array, n_array, k_array, alpha_array, a_array, lda_array, b_array, ldb_array, beta_array, c_array, ldc_array, group_count, group_size)
call cgemm3m_batch(a_array, b_array, c_array, m_array, n_array, k_array, group_size [,transa_array][,transb_array] [,alpha_array][,beta_array])
call zgemm3m_batch(a_array, b_array, c_array, m_array, n_array, k_array, group_size [,transa_array][,transb_array] [,alpha_array][,beta_array])
Include Files
- mkl.fi, blas.f90
Description
The ?gemm3m_batch routines perform a series of matrix-matrix operations with general matrices. They are similar to the ?gemm3m routine counterparts, but the ?gemm3m_batch routines perform matrix-matrix operations with groups of matrices, processing a number of groups at once. The groups contain matrices with the same parameters. The ?gemm3m_batch routines use fewer matrix multiplications than the ?gemm_batch routines, as described in the Application Notes.
The operation is defined as
idx = 1 for i = 1..group_count alpha and beta in alpha_array(i) and beta_array(i) for j = 1..group_size(i) A, B, and C matrix in a_array(idx), b_array(idx), and c_array(idx) C := alpha*op(A)*op(B) + beta*C, idx = idx + 1 end for end for
where:
op(X) is one of op(X) = X, or op(X) = XT, or op(X) = XH,
alpha and beta are scalar elements of alpha_array and beta_array,
A, B and C are matrices such that for m, n, and k which are elements of m_array, n_array, and k_array:
op(A) is an m-by-k matrix,
op(B) is a k-by-n matrix,
C is an m-by-n matrix.
A, B, and C represent matrices stored at addresses pointed to by a_array, b_array, and c_array, respectively. The number of entries in a_array, b_array, and c_array is total_batch_count = the sum of all the group_size entries.
See also gemm for a detailed description of multiplication for general matrices and gemm_batch, BLAS-like extension routines for similar matrix-matrix operations.
Error checking is not performed for Intel® oneAPI Math Kernel Library (oneMKL) Windows* single dynamic libraries for the?gemm3m_batch routines.
Input Parameters
- transa_array
-
CHARACTER*1. Array of size group_count. For the group i, transai = transa_array(i) specifies the form of op(A) used in the matrix multiplication:
if transai = 'N' or 'n', then op(A) = A;
if transai = 'T' or 't', then op(A) = AT;
if transai = 'C' or 'c', then op(A) = AH.
- transb_array
-
CHARACTER*1. Array of size group_count. For the group i, transbi = transb_array(i) specifies the form of op(Bi) used in the matrix multiplication:
if transbi = 'N' or 'n', then op(B) = B;
if transbi = 'T' or 't', then op(B) = BT;
if transbi = 'C' or 'c', then op(B) = BH.
- m_array
-
INTEGER. Array of size group_count. For the group i, mi = m_array(i) specifies the number of rows of the matrix op(A) and of the matrix C.
The value of each element of m_array must be at least zero.
- n_array
-
INTEGER. Array of size group_count. For the group i, ni = n_array(i) specifies the number of columns of the matrix op(B) and the number of columns of the matrix C.
The value of each element of n_array must be at least zero.
- k_array
-
INTEGER. Array of size group_count. For the group i, ki = k_array(i) specifies the number of columns of the matrix op(A) and the number of rows of the matrix op(B).
The value of each element of k_array must be at least zero.
- alpha_array
-
COMPLEX for cgemm3m_batch
DOUBLE COMPLEX for zgemm3m_batch
Array of size group_count. For the group i, alpha_array(i) specifies the scalar alphai.
- a_array
-
INTEGER*8 for Intel® 64 architecture
INTEGER*4 for IA-32 architecture
Array, size total_batch_count, of pointers to arrays used to store A matrices.
- lda_array
-
INTEGER. Array of size group_count. For the group i, ldai = lda_array(i) specifies the leading dimension of the array storing matrix A as declared in the calling (sub)program.
When transai = 'N' or 'n', then ldai must be at least max(1, mi), otherwise ldai must be at least max(1, ki).
- b_array
-
INTEGER*8 for Intel® 64 architecture
INTEGER*4 for IA-32 architecture
Array, size total_batch_count, of pointers to arrays used to store B matrices.
- ldb_array
-
INTEGER.
Array of size group_count. For the group i, ldbi = ldb_array(i) specifies the leading dimension of the array storing matrix B as declared in the calling (sub)program.
When transbi = 'N' or 'n', then ldbi must be at least max(1, ki), otherwise ldbi must be at least max(1, ni).
- beta_array
-
COMPLEX for cgemm3m_batch
DOUBLE COMPLEX for zgemm3m_batch
For the group i, beta_array(i) specifies the scalar betai.
When betai is equal to zero, then C matrices in group i need not be set on input.
- c_array
-
INTEGER*8 for Intel® 64 architecture
INTEGER*4 for IA-32 architecture
Array, size total_batch_count, of pointers to arrays used to store C matrices.
- ldc_array
-
INTEGER.
Array of size group_count. For the group i, ldci = ldc_array(i) specifies the leading dimension of all arrays storing matrix C in group i as declared in the calling (sub)program.
ldci must be at least max(1, mi).
- group_count
-
INTEGER.
Specifies the number of groups. Must be at least 0.
- group_size
-
INTEGER.
Array of size group_count. The element group_size(i) specifies the number of matrices in group i. Each element in group_size must be at least 0.
Output Parameters
- c_array
-
Overwritten by the mi-by-ni matrix (alphai*op(A)*op(B) + betai*C) for group i.
BLAS 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see BLAS 95 Interface Conventions.
Specific details for the routine gemm3m_batch interface are the following:
- a_array
-
Holds pointers to arrays containing matrices A of size (ma,ka) where
ka = k if transa='N',
ka = m otherwise,
ma = m if transa='N',
ma = k otherwise.
- b_array
-
Holds pointers to arrays containing matrices B of size (mb,kb) where
kb = n if transb_array = 'N',
kb = k otherwise,
mb = k if transb_array = 'N',
mb = n otherwise.
- c_array
-
Holds pointers to arrays containing matrices C of size (m,n).
- m_array
-
Array indicating number of rows of matrices op(A) and C for each group.
- n_array
-
Array indicating number of columns of matrices op(B) and C for each group.
- k_array
-
Array indicating number of columns of matrices op(A) and number of rows of matrices op(B) for each group.
- group_size
-
Array indicating number of matrices for each group. Each element in group_size must be at least 0.
- transa_array
-
Array with each element set to one of 'N', 'C', or 'T'.
The default values are 'N'.
- transb_array
-
Array with each element set to one of 'N', 'C', or 'T'.
The default values are 'N'.
- alpha_array
-
Array of alpha values; the default value is 1.
- beta_array
-
Array of beta values; the default value is 0.
Application Notes
These routines perform a complex matrix multiplication by forming the real and imaginary parts of the input matrices. This uses three real matrix multiplications and five real matrix additions instead of the conventional four real matrix multiplications and two real matrix additions. The use of three real matrix multiplications reduces the time spent in matrix operations by 25%, resulting in significant savings in compute time for large matrices.
If the errors in the floating point calculations satisfy the following conditions:
fl(x op y)=(x op y)(1+δ),|δ|≤u, op=×,/, fl(x±y)=x(1+α)±y(1+β), |α|,|β|≤u
then for an n-by-n matrix Ĉ=fl(C1+iC2)= fl((A1+iA2)(B1+iB2))=Ĉ1+iĈ2, the following bounds are satisfied:
║Ĉ1-C1║≤ 2(n+1)u║A║∞║B║∞+O(u2),
║Ĉ2-C2║≤ 4(n+4)u║A║∞║B║∞+O(u2),
where ║A║∞=max(║A1║∞,║A2║∞), and ║B║∞=max(║B1║∞,║B2║∞).
Thus the corresponding matrix multiplications are stable.