Sparse BLAS Functionality

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 11/07/2023

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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	OpenMP Offload Intel GPU
Sparse Vector - Dense Vector addition (AXPY)	y <- alpha * w + y	Yes	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)	Yes	No
	dot(w,v) = sum(conj(w_i) * v_i)	Yes	No
Dense Vector - Sparse Vector Conversion (sv <-> dv)	—	N/A	N/A
	x = scatter(w)	Yes	No
	w = gather(x,windx)	Yes	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	OpenMP Offload Intel GPU
General Matrix-Vector multiplication (GEMV) (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	Yes	No
Symmetric Matrix-Vector multiplication (SYMV) (sm*dv->dv)	y <- betay + alpha op(A)*x	N/A	N/A
	op(A) = A	Yes	No
	op(A) = A^T	No	No
	op(A) = A^H	Yes	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	Yes	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	No
	op(A) = A^T	Yes	No
	op(A) = A^H	Yes	No
Triangular Solve (TRSV) (inv(sm)*dv -> dv)	solve for y, op(A)y = alphax	N/A	N/A
	op(A) = A	Yes	No
	op(A) = A^T	Yes	No
	op(A) = A^H	Yes	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	OpenMP Offload 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	No
	op(A) = A^T, op(X) = X	Yes	No
	op(A) = A^H, op(X) = X	Yes	No
	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	Yes	No
	op(A)=A^T, op(B)=B	Yes	No
	op(A)=A^H, op(B)=B	Yes	No
	op(A)=A, op(B)=B^T	Yes	No
	op(A)=A^T, op(B)=B^T	Yes	No
	op(A)=A^H, op(B)=B^T	Yes	No
	op(A)=A, op(B)=B^H	Yes	No
	op(A)=A^T, op(B)=B^H	Yes	No
	op(A)=A^H, op(B)=B^H	Yes	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	Yes	No
	op(A)=A^T, op(B)=B	Yes	No
	op(A)=A^H, op(B)=B	Yes	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	Yes	No
	op(A)=A^T	Yes	No
	op(A)=A^H	Yes	No
Symmetric Rank-K update (SYRK) (sm*sm->dm)	Y <- op(A)*op(A)^H	N/A	N/A
	op(A)=A	Yes	No
	op(A)=A^T	Yes	No
	op(A)=A^H	Yes	No
Symmetric Triple Product (SYPR) (op(sm)smsm -> sm)	C <- op(A)Bop(A)^H	N/A	N/A
	op(A)=A	Yes	No
	op(A)=A^T	Yes	No
	op(A)=A^H	Yes	No
Triangular Solve (TRSM) (inv(sm)*dm -> dm)	solve for Y, op(A)Y = alphaX	N/A	N/A
	op(A)=A	Yes	No
	op(A)=A^T	Yes	No
	op(A)=A^H	Yes	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	OpenMP Offload 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	Yes	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	Yes	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	Yes	No
Sparse Matrix Add (ADD)	C <- alpha*op(A) + B	Yes	No
	op(A) = A^T	Yes	No
	op(A) = A^H	Yes	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	OpenMP Offload Intel GPU
Sort Indices of Matrix (ORDER)	N/A	Yes	No
Transpose of Sparse Matrix (TRANSPOSE)	A <- op(A) with op=trans or conjtrans	N/A	N/A
	transpose CSR/CSC matrix	Yes	No
	transpose BSR matrix	Yes	No
Sparse Matrix Format Converter (CONVERT)	N/A	Yes	No
Dense to Sparse Matrix Format Converter (CONVERT)	N/A	Yes	No
Copy Matrix Handle (COPY)	N/A	Yes	No
Create CSR Matrix Handle	N/A	Yes	No
Create CSC Matrix Handle	N/A	Yes	No
Create COO Matrix Handle	N/A	Yes	No
Create BSR Matrix Handle	N/A	Yes	No
Export CSR Matrix	Allows access to internal data in the CSR Matrix handle	Yes	No
Export CSC Matrix	Allows access to internal data in the CSC Matrix handle	Yes	No
Export COO Matrix	Allows access to internal data in the COO Matrix handle	Yes	No
Export BSR Matrix	Allows access to internal data in the BSR Matrix handle	Yes	No
Set Value in Matrix	N/A	Yes	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.

Optimize Stages
Functionality	Operations	CPU	OpenMP Offload Intel GPU
add MEMORY hint and optimize	Chooses to allow larger memory requiring optimizations or not.	Yes	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	Yes	No
add DOTMV hint and optimize	N/A	Yes	No
add SYMGS hint and optimize	N/A	Yes	No
add SYMGS_MV hint and optimize	N/A	Yes	No
add LU_SMOOTHER hint and optimize	N/A	Yes	No

Parent topic: Appendix F: oneMKL Functionality

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

Sparse BLAS Functionality