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//------------------------------------------------------------------------------
// GraphBLAS/Demo/Source/wathen.c: a finite-element matrix on a regular mesh
//------------------------------------------------------------------------------
// SuiteSparse:GraphBLAS, Timothy A. Davis, (c) 2017-2020, All Rights Reserved.
// http://suitesparse.com See GraphBLAS/Doc/License.txt for license.
//------------------------------------------------------------------------------
// Create a finite-element matrix on an nx-by-ny 2D mesh, as computed by
// wathen.m in MATLAB. To view the wathen.m file, use this command in MATLAB:
//
// type private/wathen
#include "GraphBLAS.h"
#undef GB_PUBLIC
#define GB_LIBRARY
#include "graphblas_demos.h"
//------------------------------------------------------------------------------
// scale by rho
//------------------------------------------------------------------------------
double r = 0 ;
void rho_scale (double *f, const double *e)
{
(*f) = r * (*e) ;
}
//------------------------------------------------------------------------------
// Wathen function
//------------------------------------------------------------------------------
GB_PUBLIC
GrB_Info wathen // construct a random Wathen matrix
(
GrB_Matrix *A_output, // output matrix
int64_t nx, // grid dimension nx
int64_t ny, // grid dimension ny
bool scale, // if true, scale the rows
int method, // 0 to 3
double *rho_given // nx-by-ny dense matrix, if NULL use random rho
)
{
//--------------------------------------------------------------------------
// check inputs
//--------------------------------------------------------------------------
if (nx < 0 || ny < 0 || A_output == NULL || method < 0 || method > 3)
{
return (GrB_INVALID_VALUE) ;
}
// macro to free all workspace. Not every method uses every object
#define FREE_ALL \
GrB_Matrix_free (&A) ; \
GrB_Matrix_free (&F) ; \
GrB_Matrix_free (&D) ; \
GrB_Matrix_free (&E) ; \
GrB_UnaryOp_free (&rho_op) ; \
if (rho_rand != NULL) free (rho_rand) ; \
if (I != NULL) free (I) ; \
if (J != NULL) free (J) ; \
if (X != NULL) free (X) ;
GrB_Info info ;
GrB_Matrix A = NULL, F = NULL, E = NULL, D = NULL ;
GrB_UnaryOp rho_op = NULL ;
double *rho_rand = NULL, *X = NULL, *rho ;
GrB_Index *I = NULL, *J = NULL ;
//--------------------------------------------------------------------------
// construct the coefficients
//--------------------------------------------------------------------------
#define d ((double) 45),
static const double e [8][8] = {
{ 6/d -6/d 2/d -8/d 3/d -8/d 2/d -6/d },
{ -6/d 32/d -6/d 20/d -8/d 16/d -8/d 20/d },
{ 2/d -6/d 6/d -6/d 2/d -8/d 3/d -8/d },
{ -8/d 20/d -6/d 32/d -6/d 20/d -8/d 16/d },
{ 3/d -8/d 2/d -6/d 6/d -6/d 2/d -8/d },
{ -8/d 16/d -8/d 20/d -6/d 32/d -6/d 20/d },
{ 2/d -8/d 3/d -8/d 2/d -6/d 6/d -6/d },
{ -6/d 20/d -8/d 16/d -8/d 20/d -6/d 32/d } } ;
//--------------------------------------------------------------------------
// A = sparse (n,n) ;
//--------------------------------------------------------------------------
int64_t n = 3*nx*ny + 2*nx + 2*ny +1 ;
OK (GrB_Matrix_new (&A, GrB_FP64, n, n)) ;
//--------------------------------------------------------------------------
// RHO = 100 * rand (nx,ny) ;
//--------------------------------------------------------------------------
// i and j are 1-based, so the same index computations from wathen.m
// can be used
#define RHO(i,j) rho [(i-1)+((j-1)*nx)]
if (rho_given == NULL)
{
// compute a random RHO matrix
rho_rand = (double *) malloc (nx * ny * sizeof (double)) ;
if (rho_rand == NULL)
{ // out of memory
FREE_ALL ;
return (GrB_OUT_OF_MEMORY) ;
}
rho = rho_rand ;
for (int j = 1 ; j <= ny ; j++)
{
for (int i = 1 ; i <= nx ; i++)
{
RHO (i,j) = 100 * simple_rand_x ( ) ;
}
}
}
else
{
// use rho_given on input
rho = rho_given ;
}
#define em(krow,kcol) (e [krow][kcol] * RHO (i,j))
//--------------------------------------------------------------------------
// nn = zeros (8,1) ;
//--------------------------------------------------------------------------
GrB_Index nn [8] ;
//--------------------------------------------------------------------------
// construct the Wathen matrix, using one of four equivalent methods
//--------------------------------------------------------------------------
switch (method)
{
//----------------------------------------------------------------------
// create tuples and use build, just like wathen.m
//----------------------------------------------------------------------
case 0:
{
// This method is fastest, but only 20% faster than methods 2 and
// 3. It is about 15% to 20% faster than the MATLAB wathen
// function, and uses the identical algorithm. The code here is
// nearly identical to the wathen.m M-file, except that here an
// adjustment to the indices must be made since GraphBLAS matrices
// are indexed starting at row and column 0, not 1.
// allocate the tuples
int64_t ntriplets = nx*ny*64 ;
I = (GrB_Index *) malloc (ntriplets * sizeof (GrB_Index)) ;
J = (GrB_Index *) malloc (ntriplets * sizeof (GrB_Index)) ;
X = (double *) malloc (ntriplets * sizeof (double )) ;
if (I == NULL || J == NULL || X == NULL)
{ // out of memory
FREE_ALL ;
return (GrB_OUT_OF_MEMORY) ;
}
ntriplets = 0 ;
for (int j = 1 ; j <= ny ; j++)
{
for (int i = 1 ; i <= nx ; i++)
{
nn [0] = 3*j*nx + 2*i + 2*j + 1 ;
nn [1] = nn [0] - 1 ;
nn [2] = nn [1] - 1 ;
nn [3] = (3*j-1)*nx + 2*j + i - 1 ;
nn [4] = 3*(j-1)*nx + 2*i + 2*j - 3 ;
nn [5] = nn [4] + 1 ;
nn [6] = nn [5] + 1 ;
nn [7] = nn [3] + 1 ;
for (int krow = 0 ; krow < 8 ; krow++) nn [krow]-- ;
for (int krow = 0 ; krow < 8 ; krow++)
{
for (int kcol = 0 ; kcol < 8 ; kcol++)
{
I [ntriplets] = nn [krow] ;
J [ntriplets] = nn [kcol] ;
X [ntriplets] = em (krow,kcol) ;
ntriplets++ ;
}
}
}
}
// A = sparse (I,J,X,n,n) ;
OK (GrB_Matrix_build_FP64 (A, I, J, X, ntriplets, GrB_PLUS_FP64)) ;
}
break ;
//----------------------------------------------------------------------
// scalar assignment
//----------------------------------------------------------------------
case 1:
{
// This method takes about 1.8x the time as other three methods,
// for both small and large problems. The difference in
// performance is likely because GrB_Matrix_assign_FP64 is
// expecting to write its double scalar to a submatrix of A, not a
// single scalar. It has some extra overhead as a result, which is
// not needed. GrB_Matrix_setElement cannot be used because that
// method does not allow for an accumulator function to be
// specified; its implicit accum operator is SECOND, not PLUS.
// Future versions of SuiteSparse:GraphBLAS may correct this
// performance discrepancy, so that this method is just as fast as
// the other three methods here.
// This method is the same as the older version of wathen.m, before
// it was updated to use the sparse function in MATLAB. That older
// wathen.m function was asymptotically slower, and 300x slower in
// practice for moderate sized problems. The performance
// difference increases greatly as the problem gets larger, as
// well. By contrast, this method is asympotically just as fast as
// the other methods here, it's just a constant times slower (by a
// uniform factor of just under 2).
for (int j = 1 ; j <= ny ; j++)
{
for (int i = 1 ; i <= nx ; i++)
{
nn [0] = 3*j*nx + 2*i + 2*j + 1 ;
nn [1] = nn [0] - 1 ;
nn [2] = nn [1] - 1 ;
nn [3] = (3*j-1)*nx + 2*j + i - 1 ;
nn [4] = 3*(j-1)*nx + 2*i + 2*j - 3 ;
nn [5] = nn [4] + 1 ;
nn [6] = nn [5] + 1 ;
nn [7] = nn [3] + 1 ;
for (int krow = 0 ; krow < 8 ; krow++) nn [krow]-- ;
for (int krow = 0 ; krow < 8 ; krow++)
{
for (int kcol = 0 ; kcol < 8 ; kcol++)
{
// A (nn[krow],nn[kcol]) += em (krow,kcol)
OK (GrB_Matrix_assign_FP64 (A, NULL,
GrB_PLUS_FP64, em (krow,kcol),
(&nn [krow]), 1, (&nn [kcol]), 1, NULL)) ;
}
}
}
}
}
break ;
//----------------------------------------------------------------------
// matrix assignment, create F one entry at a time
//----------------------------------------------------------------------
case 2:
{
// This method is about 20% slower than method 0, but it has
// the advantage of not requiring the number of tuples to be
// known in advance. Method 3 is just as fast as this method.
// This method is uniformaly about 5% to 10% slower than the
// MATLAB wathen.m regardless of the problem size.
// create a single 8-by-8 finite-element matrix F
OK (GrB_Matrix_new (&F, GrB_FP64, 8, 8)) ;
for (int j = 1 ; j <= ny ; j++)
{
for (int i = 1 ; i <= nx ; i++)
{
nn [0] = 3*j*nx + 2*i + 2*j + 1 ;
nn [1] = nn [0] - 1 ;
nn [2] = nn [1] - 1 ;
nn [3] = (3*j-1)*nx + 2*j + i - 1 ;
nn [4] = 3*(j-1)*nx + 2*i + 2*j - 3 ;
nn [5] = nn [4] + 1 ;
nn [6] = nn [5] + 1 ;
nn [7] = nn [3] + 1 ;
for (int krow = 0 ; krow < 8 ; krow++) nn [krow]-- ;
for (int krow = 0 ; krow < 8 ; krow++)
{
for (int kcol = 0 ; kcol < 8 ; kcol++)
{
// F (krow,kcol) = em (krow, kcol)
OK (GrB_Matrix_setElement_FP64 (F,
em (krow,kcol), krow, kcol)) ;
}
}
// A (nn,nn) += F
OK (GrB_Matrix_assign (A, NULL, GrB_PLUS_FP64,
F, nn, 8, nn, 8, NULL)) ;
}
}
}
break ;
//----------------------------------------------------------------------
// matrix assignment, create F all at once
//----------------------------------------------------------------------
case 3:
{
// This method is as fast as method 2. It is very flexible
// since any method can be used to construct the finite-element
// matrix. Then A(nn,nn)+=F is very efficient when F is a matrix.
// This method is uniformaly about 5% to 10% slower than the
// MATLAB wathen.m regardless of the problem size.
// create a single 8-by-8 finite-element matrix F
OK (GrB_Matrix_new (&F, GrB_FP64, 8, 8)) ;
// create a single 8-by-8 coefficient matrix E
OK (GrB_Matrix_new (&E, GrB_FP64, 8, 8)) ;
for (int krow = 0 ; krow < 8 ; krow++)
{
for (int kcol = 0 ; kcol < 8 ; kcol++)
{
double ex = e [krow][kcol] ;
OK (GrB_Matrix_setElement_FP64 (E, ex, krow, kcol)) ;
}
}
// create a unary operator to scale by RHO(i,j)
OK (GrB_UnaryOp_new (&rho_op,
(GxB_unary_function) rho_scale, GrB_FP64, GrB_FP64)) ;
for (int j = 1 ; j <= ny ; j++)
{
for (int i = 1 ; i <= nx ; i++)
{
nn [0] = 3*j*nx + 2*i + 2*j + 1 ;
nn [1] = nn [0] - 1 ;
nn [2] = nn [1] - 1 ;
nn [3] = (3*j-1)*nx + 2*j + i - 1 ;
nn [4] = 3*(j-1)*nx + 2*i + 2*j - 3 ;
nn [5] = nn [4] + 1 ;
nn [6] = nn [5] + 1 ;
nn [7] = nn [3] + 1 ;
for (int krow = 0 ; krow < 8 ; krow++) nn [krow]-- ;
// F = E * RHO(i,j)
// note that this computation on F does not force
// A to be assembled.
r = RHO (i,j) ;
OK (GrB_Matrix_apply (F, NULL, NULL, rho_op, E, NULL)) ;
// A (nn,nn) += F
OK (GrB_Matrix_assign (A, NULL, GrB_PLUS_FP64,
F, nn, 8, nn, 8, NULL)) ;
}
}
}
break ;
default:
CHECK (false, GrB_INVALID_VALUE) ;
break ;
}
//--------------------------------------------------------------------------
// scale the matrix, if requested
//--------------------------------------------------------------------------
// An alternative to multiplying by the inverse of the diagonal would be to
// compute A=A/D using the PLUS_DIV_FP64 semiring, which scales the columns
// instead of the rows, and then transposing the result, since A is
// symmetric but D\A and A/D are not. Alternatively, a user-defined
// operator z=f(x,y) that computes z=y/x could be used, along with a
// user-defined semiring.
if (scale)
{
// D = sparse (n,n)
OK (GrB_Matrix_new (&D, GrB_FP64, n, n)) ;
for (int64_t i = 0 ; i < n ; i++)
{
// D (i,i) = 1 / A (i,i) ;
double di ;
OK (GrB_Matrix_extractElement_FP64 (&di, A, i, i)) ;
OK (GrB_Matrix_setElement_FP64 (D, 1/di, i, i)) ;
}
// A = D*A
OK (GrB_mxm (A, NULL, NULL, GxB_PLUS_TIMES_FP64, D, A, NULL)) ;
}
// force completion
GrB_Index nvals ;
OK (GrB_Matrix_nvals (&nvals, A)) ;
//--------------------------------------------------------------------------
// free workspace and return the result
//--------------------------------------------------------------------------
*A_output = A ;
A = NULL ;
FREE_ALL ;
return (GrB_SUCCESS) ;
}
|
