Sparse BLAS Functionality

Intel® oneAPI Math Kernel Library (oneMKL) - Data Parallel C++ Developer Reference

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Date 12/16/2022

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Sparse BLAS Functionality

In the following table for functionality, sm = sparse matrix, dm = dense matrix, sv = sparse vector, dv = dense vector, sc = scalar.

In the following table for operations, dense vectors = x, y, sparse vectors = w,v, dense matrices = X,Y, sparse matrices = A, B, C, and scalars = alpha, beta, d.

Level 1

Functionality	Operations	CPU	Intel GPU
Sparse Vector - Dense Vector addition (AXPY)	y <- alpha * w + y	No	No
Sparse Vector - Sparse Vector Dot product (SPDOT) (sv.sv -> sc)	d <- dot(w,v)	N/A	N/A
	dot(w,v) = sum(w_i* v_i)	No	No
	dot(w,v) = sum(conj(w_i) * v_i)	No	No
Sparse Vector - Dense Vector Dot product (SPDOT) (sv.dv -> sc)	d <- dot(w,x)	N/A	N/A
	dot(w,v) = sum(w_i* v_i)	No	No
	dot(w,v) = sum(conj(w_i) * v_i)	No	No
Dense Vector - Sparse Vector Conversion (sv <-> dv)	—	N/A	N/A
	x = scatter(w)	No	No
	w = gather(x,windx)	No	No

In the following table for functionality, sm = sparse matrix, dm = dense matrix, sv = sparse vector, dv = dense vector, sc = scalar.

In the following table for operations, dense vectors = x, y, sparse vectors = w,v, dense matrices = X,Y, sparse matrices = A, B, C, and scalars = alpha, beta, d.

Level 2

Functionality	Operations	CPU	Intel GPU
General Matrix-Vector multiplication (GEMV) (sm*dv->dv)	y <- betay + alpha op(A)*x	N/A	N/A
	op(A) = A	Yes	Yes
	op(A) = A^T	Yes	Yes
	op(A) = A^H	No	No
Symmetric Matrix-Vector multiplication (SYMV) (sm*dv->dv)	y <- betay + alpha op(A)*x	N/A	N/A
	op(A) = A	Yes	Yes
	op(A) = A^T	Yes	Yes
	op(A) = A^H	No	No
Triangular Matrix-Vector multiplication (TRMV) (sm*dv->dv)	y <- betay + alpha op(A)*x	N/A	N/A
	op(A) = A	Yes	No
	op(A) = A^T	Yes	No
	op(A) = A^H	No	No
General Matrix-Vector mult with dot product (GEMVDOT) (sm*dv -> dv, dv.dv->sc)	y <- betay + alpha op(A)*x, d = dot(x,y)	N/A	N/A
	op(A) = A	Yes	Yes
	op(A) = A^T	Yes	Yes
	op(A) = A^H	No	No
Triangular Solve (TRSV) (inv(sm)*dv -> dv)	solve for y, op(A)y = alphax	N/A	N/A
	op(A) = A	Yes	Yes
	op(A) = A^T	Yes	Yes
	op(A) = A^H	No	No

In the following table for functionality, sm = sparse matrix, dm = dense matrix, sv = sparse vector, dv = dense vector, sc = scalar.

In the following table for operations, dense vectors = x, y, sparse vectors = w,v, dense matrices = X,Y, sparse matrices = A, B, C, and scalars = alpha, beta, d.

Level 3

Functionality	Operations	CPU	Intel GPU
General Sparse Matrix - Dense Matrix Multiplication (GEMM) (sm*dm->dm)	Y <- alphaop(A)op(X) + beta*Y	N/A	N/A
	op(A) = A, op(X) = X	Yes	Yes
	op(A) = A^T, op(X) = X	Yes	Yes
	op(A) = A^H, op(X) = X	Yes	Yes
	op(A) = A, op(X) = X^T	No	No
	op(A) = A^T, op(X) = X^T	No	No
	op(A) = A, op(X) = X^H	No	No
	op(A) = A^H	No	No
	op(A) = A^T, op(X) = X^H	No	No
	op(A) = A^H, op(X) = X^H	No	No
General Dense Matrix - Sparse Matrix Multiplication (GEMM) (dm*sm->dm)	Y <- alphaop(X)op(A) + beta*Y	N/A	N/A
	op(X) = X, op(A)=A	No	No
	op(X) = X^H, op(A)=A	No	No
	op(X) = X^H, op(A)=A	No	No
	op(X) = X, op(A)=A^H	No	No
	op(X) = X^H, op(A)=A^H	No	No
	op(X) = X^H, op(A)=A^H	No	No
	op(X) = X, op(A)=A^H	No	No
	op(X) = X^H, op(A)=A^H	No	No
	op(X) = X^H, op(A)=A^H	No	No
General Sparse Matrix - Sparse Matrix Multiplication (GEMM) (sm*sm->sm)	C <- alphaop(A)op(B) + beta*C	N/A	N/A
	op(A)=A, op(B)=B	No	No
	op(A)=A^T, op(B)=B	No	No
	op(A)=A^H, op(B)=B	No	No
	op(A)=A, op(B)=B^T	No	No
	op(A)=A^T, op(B)=B^T	No	No
	op(A)=A^H, op(B)=B^T	No	No
	op(A)=A, op(B)=B^H	No	No
	op(A)=A^T, op(B)=B^H	No	No
	op(A)=A^H, op(B)=B^H	No	No
General Sparse Matrix - Sparse Matrix Multiplication (GEMM) (sm*sm->dm)	Y <- alphaop(A)op(B) + beta*Y	N/A	N/A
	op(A)=A, op(B)=B	No	No
	op(A)=A^T, op(B)=B	No	No
	op(A)=A^H, op(B)=B	No	No
	op(A)=A, op(B)=B^T	No	No
	op(A)=A^T, op(B)=B^T	No	No
	op(A)=A^H, op(B)=B^T	No	No
	op(A)=A, op(B)=B^H	No	No
	op(A)=A^T, op(B)=B^H	No	No
	op(A)=A^H, op(B)=B^H	No	No
Symmetric Rank-K update (SYRK) (sm*sm->sm)	C <- op(A)*op(A)^H	N/A	N/A
	op(A)=A	No	No
	op(A)=A^T	No	No
	op(A)=A^H	No	No
Symmetric Rank-K update (SYRK) (sm*sm->dm)	Y <- op(A)*op(A)^H	N/A	N/A
	op(A)=A	No	No
	op(A)=A^T	No	No
	op(A)=A^H	No	No
Symmetric Triple Product (SYPR) (op(sm)smsm -> sm)	C <- op(A)Bop(A)^H	N/A	N/A
	op(A)=A	No	No
	op(A)=A^T	No	No
	op(A)=A^H	No	No
Triangular Solve (TRSM) (inv(sm)*dm -> dm)	solve for Y, op(A)Y = alphaX	N/A	N/A
	op(A)=A	No	No
	op(A)=A^T	No	No
	op(A)=A^H	No	No

In the following table for functionality, sm = sparse matrix, dm = dense matrix, sv = sparse vector, dv = dense vector, sc = scalar.

In the following table for operations, dense vectors = x, y, sparse vectors = w,v, dense matrices = X,Y, sparse matrices = A, B, C, and scalars = alpha, beta, d.

Other

Functionality	Operations	CPU	Intel GPU
Symmetric Gauss-Seidel Preconditioner (SYMGS) (update A*x=b, A=L+D+U)	x0 <- xalpha; (L+D)x1=b-Ux0; (U+D)x=b-L*x1	No	No
Symmetric Gauss-Seidel Preconditioner with Matrix-Vector product (SYMGS_MV) (update A*x=b, A=L+D+U)	x0 <- xalpha; (L+D)x1=b-Ux0; (U+D)x=b-Lx1; y=Ax	No	No
LU Smoother (LU_SMOOTHER) (update A*x=b, A=L+D+U, E~inv(D) )	r=b-Ax; (L+D)E(U+D)dx=r; y=x+dr	No	No
Sparse Matrix Add (ADD)	C <- alpha*op(A) + B	No	No
	op(A) = A^T	No	No
	op(A) = A^H	No	No

In the following table for operations, dense vectors = x, y, sparse vectors = w,v, dense matrices = X,Y, sparse matrices = A, B, C, and scalars = alpha, beta, d.

Helper Functions

Functionality	Operations	CPU	Intel GPU
Sort Indices of Matrix (ORDER)	N/A	No	No
Transpose of Sparse Matrix (TRANSPOSE)	A <- op(A) with op=trans or conjtrans	N/A	N/A
	transpose CSR/CSC matrix	No	No
	transpose BSR matrix	No	No
Sparse Matrix Format Converter (CONVERT)	N/A	No	No
Dense to Sparse Matrix Format Converter (CONVERT)	N/A	No	No
Copy Matrix Handle (COPY)	N/A	No	No
Create CSR Matrix Handle	N/A	Yes	Yes
Create CSC Matrix Handle	N/A	No	No
Create COO Matrix Handle	N/A	No	No
Create BSR Matrix Handle	N/A	No	No
Export CSR Matrix	Allows access to internal data in the CSR Matrix handle	No	No
Export CSC Matrix	Allows access to internal data in the CSC Matrix handle	No	No
Export COO Matrix	Allows access to internal data in the COO Matrix handle	No	No
Export BSR Matrix	Allows access to internal data in the BSR Matrix handle	No	No
Set Value in Matrix	N/A	No	N

In the following table for functionality, sm = sparse matrix, dm = dense matrix, sv = sparse vector, dv = dense vector, sc = scalar.

In the following table for operations, dense vectors = x, y, sparse vectors = w,v, dense matrices = X,Y, sparse matrices = A, B, C, and scalars = alpha, beta, d.

Optimize Stages

Functionality	Operations	CPU	Intel GPU
add MEMORY hint and optimize	Chooses to allow larger memory requiring optimizations or not.	No	No
Add GEMV hint and optimize	N/A	Yes	No
Add SYMV hint and optimize	N/A	Yes	No
Add TRMV hint and optimize	N/A	Yes	No
add TRSV hint and optimize	N/A	Yes	No
add GEMM hint and optimize	N/A	Yes	No
add TRSM hint and optimize	N/A	No	No
add DOTMV hint and optimize	N/A	Yes	No
add SYMGS hint and optimize	N/A	No	No
add SYMGS_MV hint and optimize	N/A	No	No
add LU_SMOOTHER hint and optimize	N/A	No	No

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