Sparse arrays are widely used in many scientific applications where memory and time are strictly limited. For a considerable speedup one can use CUDA. Since Mathematica 8 we can conveniently use CUDA directly from Mathematica. However we can use only dense matrices. Neither version 9 nor version 10 did not add CUDA sparse arrays. However, CUSPARSE library of GPU-accelerated sparse matrix routines was introduced in CUDA Toolkit 3.2, November 2010, i.e. 4 years ago! Is it possible to use CUSPARSE directly from Mathematica now?

It is not a simple problem. On the one hand CUDALink doesn't allow you to use routines from the external libraries. On the other hand LibraryLink doesn't allow you to use CUDAMemory and its reference to GPU memory.


1 Answer 1


Yes, it is possible! There is a WorframGPULibrary (WGL), which I discover recently. It is undocumented, however there are beautiful examples in


It is similar to LibraryLink, but allows CUDAMemory as an argument. I wrote the code below to call main CUSPARSE routines directly from Mathematica

$$ C := \alpha (AB)^T + \beta C $$


$$ c := \alpha Ab + \beta c. $$

Where $A$ is a sparse matrix, $B$ and $C$ are dense matrices, $b$ and $c$ are dense vectors. The motivation of transposition is to improve the memory access and increase CUDA performance (approximately twice). You can always quickly transpose input and output as you want.


  • Matrix-matrix and matrix-vector operations.
  • Single and double precision.
  • Real or complex values.
  • Most popular CSR and BSR formats of sparse arrays.
  • Up to 30x speedup.


There is almost nothing interesting, just calling appropriate CUSPARSE routines (sparse.cu):

#include    <wgl.h>
#include    <wgl_cuda_runtime.h>

#define wglState                            (wglData->state)
#define wglErr                              (wglData->getError(wglData))
#define WGL_SuccessQ                        (wglErr->code == WGL_Success)
#define WGL_FailQ                           (!WGL_SuccessQ)
#define WGL_Type_RealQ(mem)                 ((mem)->type == WGL_Real_t)

#define WGL_SAFE_CALL(stmt, jmp)            stmt; if (WGL_FailQ) { goto jmp; }


#define GET(type, ptr)  CUDA_Runtime_getDeviceMemoryAs##type(ptr)

#define CSRMM(c, type)  cusparse##c##csrmm2(handle, TRA, TRB, m, n, k, nnzb, \
                                            &alpha##c, descrA, GET(type, A_val), \
                                            GET(Integer, A_ptr), GET(Integer, A_ind), \
                                            GET(type, B), n, &beta##c, GET(type, C), m)

#define CSRMV(c, type)  cusparse##c##csrmv(handle, TRA, m, k, nnzb, \
                                            &alpha##c, descrA, GET(type, A_val), \
                                            GET(Integer, A_ptr), GET(Integer, A_ind), \
                                            GET(type, B), &beta##c, GET(type, C))

#if CUDA_VERSION >= 6050

#define BSRMM(c, type)  cusparse##c##bsrmm(handle, BDR, TRA, TRB, m/bs, n, k/bs, nnzb, \
                                            &alpha##c, descrA, GET(type, A_val), \
                                            GET(Integer, A_ptr), GET(Integer, A_ind), bs, \
                                            GET(type, B), n, &beta##c, GET(type, C), m)

#define BSRMM(c, type)  CUSPARSE_STATUS_SUCCESS; return LIBRARY_VERSION_ERROR;  // CUDA 6.5 or later is required for bsrmm

#define BSRMV(c, type)  cusparse##c##bsrmv(handle, BDR, TRA, m/bs, k/bs, nnzb, \
                                            &alpha##c, descrA, GET(type, A_val), \
                                            GET(Integer, A_ptr), GET(Integer, A_ind), bs, \
                                            GET(type, B), &beta##c, GET(type, C))

#define SRM(c, type)    if (r==2) { \
                            if (bs==1) ret = CSRMM(c, type); \
                            else  ret = BSRMM(c, type); \
                        } else { \
                            if (bs==1) ret = CSRMV(c, type); \
                            else  ret = BSRMV(c, type); \

WolframGPULibraryData wglData = NULL;

EXTERN_C DLLEXPORT int osrmm(WolframLibraryData libData, mint Argc, MArgument * Args, MArgument Res) {
    WGL_Memory_t A_val, A_ptr, A_ind, B, C;
    int r, m, n, k, nnzb, bs;

    cusparseHandle_t handle = 0;
    cusparseMatDescr_t descrA = 0;
    cusparseStatus_t ret;

    float alphaS, betaS;
    double alphaD, betaD;
    cuComplex alphaC, betaC;
    cuDoubleComplex alphaZ, betaZ;

    WGL_SAFE_CALL(wglData->setWolframLibraryData(wglData, libData), cleanup);

    WGL_SAFE_CALL(A_val = wglData->findMemory(wglData,MArgument_getInteger(Args[0])), cleanup);
    WGL_SAFE_CALL(A_ptr = wglData->findMemory(wglData,MArgument_getInteger(Args[1])), cleanup);
    WGL_SAFE_CALL(A_ind = wglData->findMemory(wglData,MArgument_getInteger(Args[2])), cleanup);

    m = MArgument_getInteger(Args[3]);
    k = MArgument_getInteger(Args[4]);
    nnzb = A_ind->flattenedLength;
    bs = sqrt(A_val->flattenedLength/nnzb);

    WGL_SAFE_CALL(B = wglData->findMemory(wglData, MArgument_getInteger(Args[5])), cleanup);
    WGL_SAFE_CALL(C = wglData->findMemory(wglData, MArgument_getInteger(Args[6])), cleanup);

    if ((A_val->type != B->type)||(B->type != C->type)||(A_val->type != C->type)) {
        return LIBRARY_TYPE_ERROR;

    r = B->rank;
    if ((r != C->rank)||(r > 2)) return LIBRARY_DIMENSION_ERROR;
    if (r == 2) {
        n = B->dimensions[1];
        if ((m != C->dimensions[1])||(n != C->dimensions[0])||(k != B->dimensions[0]))
            return LIBRARY_DIMENSION_ERROR;
    } else if ((m != C->flattenedLength)||(k != B->flattenedLength))

    WGL_SAFE_CALL(CUDA_Runtime_setMemoryAsInput(wglState, A_val, wglErr), cleanup);
    WGL_SAFE_CALL(CUDA_Runtime_setMemoryAsInput(wglState, A_ptr, wglErr), cleanup);
    WGL_SAFE_CALL(CUDA_Runtime_setMemoryAsInput(wglState, A_ind, wglErr), cleanup);
    WGL_SAFE_CALL(CUDA_Runtime_setMemoryAsInput(wglState, B,     wglErr), cleanup);
    WGL_SAFE_CALL(CUDA_Runtime_setMemoryAsInputOutput(wglState, C, wglErr), cleanup);


        case WGL_Type_Float:
            alphaS = MArgument_getReal(Args[7]);
            betaS = MArgument_getReal(Args[9]);
            SRM(S, Float)
        case WGL_Type_Double:
            alphaD = MArgument_getReal(Args[7]);
            betaD = MArgument_getReal(Args[9]);
            SRM(D, Double);
        case WGL_Type_ComplexFloat:
            alphaC.x = MArgument_getReal(Args[7]);
            alphaC.y = MArgument_getReal(Args[8]);
            betaC.x = MArgument_getReal(Args[9]);
            betaC.y = MArgument_getReal(Args[10]);
            SRM(C, ComplexFloat);
        case WGL_Type_ComplexDouble:
            alphaZ.x = MArgument_getReal(Args[7]);
            alphaZ.y = MArgument_getReal(Args[8]);
            betaZ.x = MArgument_getReal(Args[9]);
            betaZ.y = MArgument_getReal(Args[10]);
            SRM(Z, ComplexDouble);



    if (WGL_SuccessQ) {
        CUDA_Runtime_setMemoryAsValidInputOutput(wglState, C, wglErr);
    } else {
        CUDA_Runtime_setMemoryAsInvalidInputOutput(wglState, C, wglErr);
    CUDA_Runtime_unsetMemoryAsInput(wglState, A_val, wglErr);
    CUDA_Runtime_unsetMemoryAsInput(wglState, A_ptr, wglErr);
    CUDA_Runtime_unsetMemoryAsInput(wglState, A_ind, wglErr);
    CUDA_Runtime_unsetMemoryAsInput(wglState, B, wglErr);
    if (WGL_SuccessQ && err == LIBRARY_NO_ERROR && ret == CUSPARSE_STATUS_SUCCESS) {
        return LIBRARY_NO_ERROR;
    } else {

EXTERN_C DLLEXPORT int WolframGPULibrary_initialize(WolframGPULibraryData wglData0) {
    wglData = wglData0;
    return LIBRARY_NO_ERROR;

EXTERN_C DLLEXPORT int WolframLibrary_initialize(WolframLibraryData libData) {

EXTERN_C DLLEXPORT void WolframLibrary_uninitialize( ) {


  • Data storage in Mathematica and CUSPARSE are different: row-major and column-major respectively. So you have to keep in mind a virtual transposition of all dense matrices.
  • #define TRB CUSPARSE_OPERATION_TRANSPOSE means: apply additional virtual transposition to the matrix $B$ to increase memory access. It requires CUDA architecture 2.0 or later. You can switch it to #define TRB CUSPARSE_OPERATION_NON_TRANSPOSE and perform $C := \alpha (AB^T)^T + \beta C$.
  • CUDA 6.5 or later is required for bsrmm (matrix-matrix multiplication with BSR format).


Now you can try to compile this code. I say "try" because the proper compilation even more difficult then writing the code. I use Mathematica 10 under Linux with the system installation of CUDA 6.5. My magic code:

$HistoryLength = 0;
CreateLibrary[{"sparse.cu"}, "libsparse.so",
    "Compiler" -> NVCCCompiler,
    "CUDAArchitecture" -> 
     CUDALink`Private`getCUDAArchitecture /@ Range@Length@CUDAInformation[],
    "CompilerInstallation" -> "/opt/cuda/bin",
    "TargetDirectory" -> ".",
      If[#, "CONFIG_USE_DOUBLE_PRECISION", ## &[]]},
    "LibraryDirectories" -> $CUDALinkLibraryPath,
    "Libraries" -> {If[#, "CUDALink_Double", "CUDALink_Single"], cusparse"},
    "ShellCommandFunction" -> Print,
    "ShellOutputFunction" -> Print] &@
  • "CUDAArchitecture" -> ... put current CUDA architecture to nvcc. It is necessary because CreateLibrary automatically choose all possible architectures including 1.0. However CUDA 6.5 can't compile for architecture 1.0.

  • Off[OptionValue::nodef] hide meaningless warning about CUDA architecture.

  • "CompilerInstallation" -> ... location of your nvcc.

  • "TargetDirectory" -> "." create library in the same folder

  • CONFIG_USE_CUSPARSE includes CUSPARSE headers. This flag is exist in WGL but disabled by default!

  • If[#, "CONFIG_USE_DOUBLE_PRECISION", ## &[]] turns on CONFIG_USE_DOUBLE_PRECISION if your GPU supports double precision. It is necessary for proper type detection.

  • If[#, "CUDALink_Double", "CUDALink_Single"] is necessary for proper WGL initialization. Also depends on the double precision support.

  • "ShellCommandFunction" -> Print and "ShellOutputFunction" -> Print show full compilation command and output from nvcc.


Now you can load the library

srmm = CUDAFunctionLoad[{FileNameJoin@{NotebookDirectory[], "libsparse.so"}}, 
   "osrmm", {{_Integer, "Input"}, {_Integer, "Input"}, {_Integer, "Input"}, 
    _Integer, _Integer,
    {_Integer, "Input"}, {_Integer, "InputOutput"}, 
    _Real, _Real, _Real, _Real}, {16, 16}];

If you compiled the library properly then you will see no errors. There are 10 arguments of the function srmm. All {_Integer, ...} means CUDAMemory because it contains an integer ID.

  • 3 CUDAMemory arguments for the sparse matrix $A$ in CSR or BSR format.
  • 2 integer arguments are the height and the width of the matrix $A$.
  • 2 CUDAMemory arguments are matrices $B$ and $C$ or vectors $b$ and $c$.
  • 4 real arguments for the real and the imaginary part of the $\alpha$ and $\beta$.


Definition of the custom cuSparseArray

HoldPattern@uncover@SparseArray@s__ := {s};

Options[cuSparseArray] = 
  Append[Options[CUDAMemoryLoad], blockSize -> 1];
cuSparseArray[s_SparseArray, opts : OptionsPattern[]] := 
  Module[{bs = OptionValue[blockSize], list},
      CUDAMemoryCopyToDevice /@ 
       CUDAMemoryLoad @@@ {{list = Identity; Flatten@#4[[3]], 
          FilterRules[{opts}, Options@CUDAMemoryLoad]}, {#4[[2, 1]], 
          "Integer32"}, {Flatten@#4[[2, 2]] - 1, "Integer32"}}, 
      bs #2] & @@ 
    uncover@If[bs > 1, 
      Developer`PartitionMap[list@Normal@# &, s, {bs, bs}], s]];

cuDot[cuSparseArray[{val_, ptr_, ind_}, {h_, w_}], b_CUDAMemory, 
   c_CUDAMemory, α_: 1, β_: 0] := 
  srmm[val, ptr, ind, h, w, b, c, Re@α, Im@α, 
   Re@β, Im@β];

CUDAMemoryUnload@cuSparseArray[a_, _] ^:= CUDAMemoryUnload @@ a;
  • SparseArray is already in CSR format. So we have to uncover it and load the data.

  • If the option blockSize is greater then 1 then SparseArray will be splitted by blockSize*blockSize blocks and loaded in BSR format.

  • "Integer32" is necessary for indexes because CUSPARSE use 32-bit integers, not 64-bit integers (at least for my system)


a = N@Flatten[LeviCivitaTensor[4], {{1, 2}, {3, 4}}];
b = RandomReal[1, 16];
c = ConstantArray[0., 16];
ga = cuSparseArray[a];
{gb, gc} = CUDAMemoryLoad /@ {b, c};
cuDot[ga, gb, gc];
CUDAMemoryGet[gc] - a.b

{0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}


Now we can perform a benchmarking for various data types and structures. I use two types of the distribution of nonzero elements in the sparse matrix $A$: random distribution (bs = 1) and clustered distribution (bs = 16)

enter image description here

prec = "Single"; (* "Single" or "Double" *)
type = "Real"; (* "Real" or "Complex" *)
bs = 1; (* size of blocks in the sparse array *)
format = "CSR"; (* "CSR" or "BSR", GPU storage format *)
struct = "Vector"; (* "Matrix" or "Vector", multiply by what *)
scale = 2^8; (* change depending on your GPU memory, check ByteCount below *)

rnd = If[type == "Real", RandomReal[1, #] &, 
  RandomComplex[1 + I, #] &];
{m, k, n} = scale 2^{7, 6, 0};
nnz = scale 2^14/bs^2;
a = KroneckerProduct[#, rnd[bs {1, 1}]] &@
    Sort@Transpose[RandomInteger[{1, #}, nnz] & /@ ({m, k}/bs)] -> 
     rnd[nnz], {m, k}/bs];
b = rnd@If[struct == "Matrix", {k, n}, k];
c = rnd@If[struct == "Matrix", {n, m}, m];
{α, β} = rnd[2];
repeats = If[struct == "Matrix", 20, 300];
Plus @@ ByteCount /@ {a, b, c}

66 982 088 (Memory used in bytes)

ga = cuSparseArray[a, TargetPrecision -> prec, 
   blockSize -> If[format == "CSR", 1, bs]];
{gb, gc} = 
     CUDAMemoryLoad[#, TargetPrecision -> prec] & /@ {b, c};
   Do[c = α If[struct == "Matrix", (a.b)\[Transpose], 
        a.b] + β c, {repeats}]/
   Do[cuDot[ga, gb, gc, α, β], {repeats}]; 
   c2 = CUDAMemoryGet@gc]
Norm[c2 - c, 1]/Norm[c, 1]
CUDAMemoryUnload /@ {ga, gb, gc};

2.3152 (Speedup)

4.31758*10^-6 (Accuracy)

The comprehensive table of benchmarking on my system (GTX 560/Core i7 3820)

enter image description here

Up to 30x speedup! It's not bad for €125 GPU.

I will appreciate for any tests of WGL and CUSPARSE. Thank you for reading such a long post!

  • $\begingroup$ sorry to send your information again. I have tried your example code link in my windows mathematica 10.4 and cuda paclet, but it showed link errors. The details list in my question here. $\endgroup$
    – sejabs
    Commented Apr 7, 2016 at 3:39

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