cblas_?gemm3m

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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cblas_?gemm3m

Computes a scalar-matrix-matrix product using matrix multiplications and adds the result to a scalar-matrix product.

Syntax

void cblas_cgemm3m (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);

void cblas_zgemm3m (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);

Include Files

mkl.h

Description

The ?gemm3m routines perform a matrix-matrix operation with general complex matrices. These routines are similar to the ?gemm routines, but they use fewer matrix multiplication operations (see Application Notes below).

The operation is defined as

C := alpha*op(A)*op(B) + beta*C,

where:

op(x) is one of op(x) = x, or op(x) = x', or op(x) = conjg(x'),

alpha and beta are scalars,

A, B and C are matrices:

op(A) is an m-by-k matrix,

op(B) is a k-by-n matrix,

C is an m-by-n matrix.

Input Parameters

Layout

Specifies whether two-dimensional array storage is row-major (CblasRowMajor) or column-major (CblasColMajor).

transa

Specifies the form of op(A) used in the matrix multiplication:

if transa=CblasNoTrans, then op(A) = A;

if transa=CblasTrans, then op(A) = A';

if transa=CblasConjTrans, then op(A) = conjg(A').

transb

Specifies the form of op(B) used in the matrix multiplication:

if transb=CblasNoTrans, then op(B) = B;

if transb=CblasTrans, then op(B) = B';

if transb=CblasConjTrans, then op(B) = conjg(B').

m

Specifies the number of rows of the matrix op(A) and of the matrix C. The value of m must be at least zero.

n

Specifies the number of columns of the matrix op(B) and the number of columns of the matrix C.

The value of n must be at least zero.

k

Specifies the number of columns of the matrix op(A) and the number of rows of the matrix op(B).

The value of k must be at least zero.

alpha

Specifies the scalar alpha.

a

transa=CblasNoTrans

transa=CblasTrans or transa=CblasConjTrans

Layout = CblasColMajor

Array, size lda*k.

Before entry, the leading m-by-k part of the array a must contain the matrix A.

Array, size lda*m.

Before entry, the leading k-by-m part of the array a must contain the matrix A.

Layout = CblasRowMajor

Array, size lda* m.

Before entry, the leading k-by-m part of the array a must contain the matrix A.

Array, size lda*k.

Before entry, the leading m-by-k part of the array a must contain the matrix A.

lda

Specifies the leading dimension of a as declared in the calling (sub)program.

	transa=CblasNoTrans	transa=CblasTrans or transa=CblasConjTrans
Layout = CblasColMajor	lda must be at least `max(1, m)`.	lda must be at least `max(1, k)`
Layout = CblasRowMajor	lda must be at least `max(1, k)`	lda must be at least `max(1, m)`.

b

	transb=CblasNoTrans	transb=CblasTrans or transb=CblasConjTrans
Layout = CblasColMajor	Array, size ldb by n. Before entry, the leading k-by-n part of the array b must contain the matrix `B`.	Array, size ldb by k. Before entry the leading n-by-k part of the array b must contain the matrix `B`.
Layout = CblasRowMajor	Array, size ldb by k. Before entry the leading n-by-k part of the array b must contain the matrix `B`.	Array, size ldb by n. Before entry, the leading k-by-n part of the array b must contain the matrix `B`.

ldb

Specifies the leading dimension of b as declared in the calling (sub)program.

	transb=CblasNoTrans	transb=CblasTrans or transb=CblasConjTrans
Layout = CblasColMajor	ldb must be at least `max(1, k)`.	ldb must be at least `max(1, n)`.
Layout = CblasRowMajor	ldb must be at least `max(1, n)`.	ldb must be at least `max(1, k)`.

beta

Specifies the scalar beta.

When beta is equal to zero, then c need not be set on input.

c

Layout = CblasColMajor	Array, size ldc by n. Before entry, the leading m-by-n part of the array c must contain the matrix `C`, except when beta is equal to zero, in which case c need not be set on entry.
Layout = CblasRowMajor	Array, size ldc by m. Before entry, the leading n-by-m part of the array c must contain the matrix `C`, except when beta is equal to zero, in which case c need not be set on entry.

ldc

Specifies the leading dimension of c as declared in the calling (sub)program.

Layout = CblasColMajor	ldc must be at least `max(1, m)`.
Layout = CblasRowMajor	ldc must be at least `max(1, n)`.

Output Parameters

c: Overwritten by the m-by-n matrix (alpha*op(A)*op(B) + beta*C).

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.

Parent topic: BLAS-like Extensions

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Developer Reference for Intel® oneAPI Math Kernel Library for C

cblas_?gemm3m