Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 11/07/2023
Public

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Document Table of Contents

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(wi* vi)

No

No

dot(w,v) = sum(conj(wi) * vi)

No

No

Sparse Vector - Dense Vector Dot product (SPDOT) (sv.dv -> sc)

d <- dot(w,x)

N/A

N/A

dot(w,v) = sum(wi* vi)

Yes

No

dot(w,v) = sum(conj(wi) * vi)

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 <- beta*y + alpha * op(A)*x

N/A

N/A

op(A) = A

Yes

No

op(A) = AT

Yes

No

op(A) = AH

Yes

No

Symmetric Matrix-Vector multiplication (SYMV) (sm*dv->dv)

y <- beta*y + alpha * op(A)*x

N/A

N/A

op(A) = A

Yes

No

op(A) = AT

No

No

op(A) = AH

Yes

No

Triangular Matrix-Vector multiplication (TRMV) (sm*dv->dv)

y <- beta*y + alpha * op(A)*x

N/A

N/A

op(A) = A

Yes

No

op(A) = AT

Yes

No

op(A) = AH

Yes

No

General Matrix-Vector mult with dot product (GEMVDOT) (sm*dv -> dv, dv.dv->sc)

y <- beta*y + alpha * op(A)*x, d = dot(x,y)

N/A

N/A

op(A) = A

Yes

No

op(A) = AT

Yes

No

op(A) = AH

Yes

No

Triangular Solve (TRSV) (inv(sm)*dv -> dv)

solve for y, op(A)*y = alpha*x

N/A

N/A

op(A) = A

Yes

No

op(A) = AT

Yes

No

op(A) = AH

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 <- alpha*op(A)*op(X) + beta*Y

N/A

N/A

op(A) = A, op(X) = X

Yes

No

op(A) = AT, op(X) = X

Yes

No

op(A) = AH, op(X) = X

Yes

No

op(A) = A, op(X) = XT

No

No

op(A) = AT, op(X) = XT

No

No

op(A) = A, op(X) = XH

No

No

op(A) = AH

No

No

op(A) = AT, op(X) = XH

No

No

op(A) = AH, op(X) = XH

No

No

General Dense Matrix - Sparse Matrix Multiplication (GEMM) (dm*sm->dm)

Y <- alpha*op(X)*op(A) + beta*Y

N/A

N/A

op(X) = X, op(A)=A

No

No

op(X) = XH, op(A)=A

No

No

op(X) = XH, op(A)=A

No

No

op(X) = X, op(A)=AH

No

No

op(X) = XH, op(A)=AH

No

No

op(X) = XH, op(A)=AH

No

No

op(X) = X, op(A)=AH

No

No

op(X) = XH, op(A)=AH

No

No

op(X) = XH, op(A)=AH

No

No

General Sparse Matrix - Sparse Matrix Multiplication (GEMM) (sm*sm->sm)

C <- alpha*op(A)*op(B) + beta*C

N/A

N/A

op(A)=A, op(B)=B

Yes

No

op(A)=AT, op(B)=B

Yes

No

op(A)=AH, op(B)=B

Yes

No

op(A)=A, op(B)=BT

Yes

No

op(A)=AT, op(B)=BT

Yes

No

op(A)=AH, op(B)=BT

Yes

No

op(A)=A, op(B)=BH

Yes

No

op(A)=AT, op(B)=BH

Yes

No

op(A)=AH, op(B)=BH

Yes

No

General Sparse Matrix - Sparse Matrix Multiplication (GEMM) (sm*sm->dm)

Y <- alpha*op(A)*op(B) + beta*Y

N/A

N/A

op(A)=A, op(B)=B

Yes

No

op(A)=AT, op(B)=B

Yes

No

op(A)=AH, op(B)=B

Yes

No

op(A)=A, op(B)=BT

No

No

op(A)=AT, op(B)=BT

No

No

op(A)=AH, op(B)=BT

No

No

op(A)=A, op(B)=BH

No

No

op(A)=AT, op(B)=BH

No

No

op(A)=AH, op(B)=BH

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)=AT

Yes

No

op(A)=AH

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)=AT

Yes

No

op(A)=AH

Yes

No

Symmetric Triple Product (SYPR) (op(sm)*sm*sm -> sm)

C <- op(A)*B*op(A)H

N/A

N/A

op(A)=A

Yes

No

op(A)=AT

Yes

No

op(A)=AH

Yes

No

Triangular Solve (TRSM) (inv(sm)*dm -> dm)

solve for Y, op(A)*Y = alpha*X

N/A

N/A

op(A)=A

Yes

No

op(A)=AT

Yes

No

op(A)=AH

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 <- x*alpha; (L+D)*x1=b-U*x0; (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 <- x*alpha; (L+D)*x1=b-U*x0; (U+D)*x=b-L*x1; y=A*x

Yes

No

LU Smoother (LU_SMOOTHER) (update A*x=b, A=L+D+U, E~inv(D) )

r=b-A*x; (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) = AT

Yes

No

op(A) = AH

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