/* PCG_F77 - Fortran interface to PCG
*/
void F77(pcg_f77)(int *n,
double *x,
double *b,
double *tol,
int *maxit,
int *clvl,
int *iter,
double *relres,
int *flag,
double *work,
void (*matvec)(double *, double *),
void (*precon)(double *, double *)) {
pcg(*n,
x,
b,
*tol,
*maxit,
*clvl,
iter,
relres,
flag,
work,
matvec,
precon);
}
void pitsol(matrix_t *mat, precon_t *prec,
options_t *opts, double *d_x, double *d_b) {
/*---------------------------------------*/
double t1,t2;
/*---------------------------------------*/
t1 = wall_timer();
switch (opts->solver) {
case GMRES:
fgmres(mat, prec, opts, d_x, d_b);
break;
case CG:
pcg(mat, prec, opts, d_x, d_b);
break;
}
t2 = wall_timer();
opts->result.tm_iter = t2-t1;
}
void main () {
double *x, *b, *work;
int i;
double relres;
int iter, flag;
read_MTX_SSS("matrices/poi2d_100.mtx", &n_s, &va_s, &da_s, &ja_s, &ia_s);
x = (double *) malloc(n_s * sizeof(double));
b = (double *) malloc(n_s * sizeof(double));
work = (double *) malloc(4*n_s * sizeof(double));
assert(x != NULL && b != NULL && work != NULL);
for (i = 0; i < n_s; i ++) {
x[i] = 0.0;
b[i] = 1.0;
}
printf("Starting PCG solver...\n");
pcg(n_s, x, b, 1e-12, 2000, 1, &iter, &relres, &flag, work, matvec, NULL);
}
scs_int solveLinSys(const AMatrix * A, const Settings * stgs, Priv * p, scs_float * b, const scs_float * s, scs_int iter) {
scs_int cgIts;
scs_float cgTol = calcNorm(b, A->n)
* (iter < 0 ? CG_BEST_TOL : CG_MIN_TOL / POWF((scs_float) iter + 1, stgs->cg_rate));
tic(&linsysTimer);
/* solves Mx = b, for x but stores result in b */
/* s contains warm-start (if available) */
accumByAtrans(A, p, &(b[A->n]), b);
/* solves (I+A'A)x = b, s warm start, solution stored in b */
cgIts = pcg(A, stgs, p, s, b, A->n, MAX(cgTol, CG_BEST_TOL));
scaleArray(&(b[A->n]), -1, A->m);
accumByA(A, p, b, &(b[A->n]));
if (iter >= 0) {
totCgIts += cgIts;
}
totalSolveTime += tocq(&linsysTimer);
#if EXTRAVERBOSE > 0
scs_printf("linsys solve time: %1.2es\n", tocq(&linsysTimer) / 1e3);
#endif
return 0;
}
int main(int argc, char* argv[])
{
// Initialization of the network, the communicator and the allocation of
// the GPU is done as in previous tutorials.
agile::NetworkEnvironment environment(argc, argv);
typedef agile::GPUCommunicator<unsigned, float, float> communicator_type;
communicator_type com;
com.allocateGPU();
agile::GPUEnvironment::printInformation(std::cout);
std::cout << std::endl;
// We are interested in solving the linear problem \f$ Ax = y \f$, with a
// given matrix \f$ A \f$ and a right-hand side vector \f$ y \f$. The unknown
// is the vector \f$ x \f$.
// Now, we can generate a matrix that shall be inverted (actually we do not
// invert the matrix but use the CG algorithm). Note that CG requires a
// symmetric positive definite (SPD) matrix and it is not too trivial to
// write down a SPD matrix. If you fail to provide a SPD matrix to the CG
// algorithm there is no guarantee that it will converge. You might be lucky,
// you might be not...
const unsigned SIZE = 20;
float A_host[SIZE][SIZE];
for (unsigned row = 0; row < SIZE; ++row)
for (unsigned column = 0; column <= row; ++column)
{
A_host[row][column] = (float(SIZE) - float(row) + float(SIZE) / 2.0)
* (float(column) + 1.0);
A_host[column][row] = A_host[row][column];
if (row == column)
A_host[row][column] = 2.0 * float(SIZE) + float(row) + float(column);
}
// The matrix is still in the host's memory and has to be transfered to the
// GPU. This is done automatically by the constructor of \p GPUMatrixPitched.
agile::GPUMatrixPitched<float> A(SIZE, SIZE, (float*)A_host);
// Next we need a reference solution. We can create any vector we like at
// this place.
std::vector<float> x_reference_host(SIZE);
for (unsigned counter = 0; counter < SIZE; ++counter)
x_reference_host[counter] = float(SIZE) - float(counter) + float(SIZE/3);
// This vector has to be transfered to the GPU memory too. For vectors, this
// can be achieved by the member function \p assignFromHost.
agile::GPUVector<float> x_reference;
x_reference.assignFromHost(x_reference_host.begin(), x_reference_host.end());
// We wrap the GPU matrix from above into a forward operator called
// \p ForwardMatrix. Forward operators are simply objects that implement
// the parenthesis-operator \p operator() which takes an
// \p accumulated vector and returns a \p distributed one. In all other
// respects the operator is a black box for us.
// The \p ForwardMatrix operator requires a reference to the communicator
// when constructing the object so that it has access to the network.
typedef agile::ForwardMatrix<communicator_type, agile::GPUMatrixPitched<float> >
forward_type;
forward_type forward(com, A);
// What we also want to use a preconditioner, which means that we change from
// the original problem \f$ Ax = y \f$ to the equivalent one
// \f$ PAx = Py \f$, where \f$ P \f$ is a preconditioner. The rationale is
// that most often the matrix \f$ A \f$ is ill-conditioned and the CG algorithm
// does not converge properly at all or it needs many iterations. The use of
// a preconditioner makes the whole system better conditioned. The simplest
// choice is to use the identity \f$ P = I \f$ (which means no preconditioning
// at all). The best choice would be \f$ P = A^{-1} \f$ as we would have the
// solution for \f$ x \f$ in the first step already (but then we need again
// to find the inverse of \f$ A \f$ which we wanted to avoid). An
// 'intermediate' possibility is to take \f$ P = diag(A)^{-1} \f$ which is
// easy and fast to invert and gives better results than the identity.
// A preconditioner belongs to the inverse operator. All inverse operators
// implement a parenthesis-operator which takes a \p distributed vector
// as input and returns an \p accumulated one (opposite to the forward
// operators, thus).
#if JACOBI_PRECONDITIONER
typedef agile::JacobiPreconditioner<communicator_type, float>
preconditioner_type;
std::vector<float> diagonal(SIZE);
for (unsigned row = 0; row < SIZE; ++row)
diagonal[row] = A_host[row][row];
preconditioner_type preconditioner(com, diagonal);
#else
typedef agile::InverseIdentity<communicator_type> preconditioner_type;
preconditioner_type preconditioner(com);
#endif
// The last operator needed is a measure. A measure operator has again
// a parenthesis-operator. This timeis takes an \p accumulated vector as first
// input and a \p distributed one as second input and returns a scalar
// measuring somehow the size of the vectors. An example is the scalar
// product operator.
typedef agile::ScalarProductMeasure<communicator_type> measure_type;
measure_type scalar_product(com);
// Finally, generate the PCG solver. It needs the absolute and relative
// tolerances as input so that it knows when the solution is good enough for
// our purposes. Furthermore it requires the maximum amount of iterations
// after which it simply capitulates without having found a solution.
const double REL_TOLERANCE = 1e-12;
const double ABS_TOLERANCE = 1e-6;
const unsigned MAX_ITERATIONS = 100;
agile::PreconditionedConjugateGradient<communicator_type, forward_type,
preconditioner_type, measure_type>
pcg(com, forward, preconditioner, scalar_product,
REL_TOLERANCE, ABS_TOLERANCE, MAX_ITERATIONS);
// What we have not generated, yet, is the right hand side \f$ y \f$. This is
// simply one call to our forward operator.
agile::GPUVector<float> y(SIZE);
forward(x_reference, y);
// We need one more vector to hold the result of the CG algorithm. Note that
// we also supply the initial guess for the solution via this vector.
agile::GPUVector<float> x(SIZE);
// Finally, we have constructed, initialized, wrapped... everything. The only
// thing left to do is to call the CG operator.
pcg(y, x);
// Print some statistics (and hope that the operator actually converged).
if (pcg.convergence())
std::cout << "CG converged in ";
else
std::cout << "Error: CG did not converge in ";
std::cout << pcg.getIteration() + 1 << " iterations." << std::endl;
std::cout << "Initial residual = " << pcg.getRho0() << std::endl;
std::cout << "Final residual = " << pcg.getRho() << std::endl;
std::cout << "Ratio rho_k / rho_0 = " << pcg.getRho() / pcg.getRho0()
<< std::endl;
// As the vectors in this example were quite small we can even print them to
// standard output.
std::cout << "Reference: " << std::endl << " ";
for (unsigned counter = 0; counter < x_reference_host.size(); ++counter)
std::cout << x_reference_host[counter] << " ";
std::cout << std::endl;
// The solution is still on the GPU and has to be transfered to the CPU memory.
// This is accomplished using \p copyToHost.
std::vector<float> x_host;
x.copyToHost(x_host);
// Output the solution, too.
std::cout << "CG solution: " << std::endl << " ";
for (unsigned counter = 0; counter < x_host.size(); ++counter)
std::cout << x_host[counter] << " ";
std::cout << std::endl;
// Finally, we also compute the difference between the reference solution and
// the true solution (of course, we do this on the GPU).
agile::GPUVector<float> difference(SIZE);
subVector(x_reference, x, difference);
// To measure the distance, we use the scalar product measure we have
// introduced above. Note, that this operator wants the first vector in
// accumulated format and the second one in distributed format. The solution
// we got from the CG algorithm is accumulated (because CG is an inverse
// operator). This means, we have to distribute the solution to have mixed
// formats.
agile::GPUVector<float> difference_dist(difference);
com.distribute(difference_dist);
std::cout << "L2 of difference: "
<< std::sqrt(std::abs(scalar_product(difference, difference_dist)))
<< std::endl;
// So, that's it.
return 0;
}
/** \brief Compute search direction using pcg method.
*
*/
void compute_searchdir_pcg(problem_data_t * pdat, variables_t * vars,
double t, double s, double gap, pcg_status_t * pcgstat,
adata_t * adata, mdata_t * mdata, double *precond,
double *tmp_m1, double *A2h, double *tmp_x1)
{
int i, m, n, nz;
double *p0, *p1, *p2, *p3;
double normg, pcgtol, pcgmaxi, multfact;
dmatrix *matX1, *matX2;
double lambda, tinv;
double *g, *h, *z, *expz, *expmz, *ac, *ar, *b, *d1, *d2, *Aw;
double *x, *v, *w, *u, *dx, *dv, *dw, *du, *gv, *gw, *gu, *gx;
static double pcgtol_factor = 1.0;
get_problem_data(pdat, &matX1, &matX2, &ac, &ar, &b, &lambda);
get_variables(vars, &x, &v, &w, &u, &dx, &dv, &dw, &du, &gx, &gv,
&gw, &gu, &g, &h, &z, &expz, &expmz, &d1, &d2, &Aw);
m = matX1->m;
n = matX1->n;
nz = matX1->nz;
tinv = 1.0 / t;
p0 = &precond[0];
p1 = &precond[1];
p2 = &precond[1+n];
p3 = &precond[1+n+n];
/* dmat_vset(n+n+1, 0, dx); */
dmat_yATx(matX2, h, A2h); /* A2h = A2'*h */
multfact = 0.0;
if (ac != NULL)
{
/* h.*ac */
dmat_elemprod(m, h, ac, tmp_m1);
dmat_vset(n, 0, tmp_x1);
dmat_yAmpqTx(matX1, NULL, NULL, tmp_m1, tmp_x1);
dmat_elemprod(n, ar, tmp_x1, tmp_x1);
for (i = 0; i < m; i++)
{
multfact += h[i] * ac[i] * ac[i];
}
}
p0[0] = 0;
for (i = 0; i < m; i++)
{
p0[0] += b[i] * b[i] * h[i];
}
/* complete forming gradient and d1, d2, precond */
for (i = 0; i < n; i++)
{
double q1, q2, d3, div;
q1 = 1.0 / (u[i] + w[i]);
q2 = 1.0 / (u[i] - w[i]);
gw[i] -= (q1 - q2) * tinv; /* A'*g - (q1-q2) */
gu[i] = lambda - (q1 + q2) * tinv; /* lambda - (q1+q2) */
d1[i] = (q1 * q1 + q2 * q2) * tinv;
d2[i] = (q1 * q1 - q2 * q2) * tinv;
if (ac != NULL)
{
d3 = A2h[i] + d1[i] + multfact*ar[i]*ar[i] - 2*tmp_x1[i];
}
else
{
d3 = A2h[i] + d1[i];
}
div = 1 / (d3 * d1[i] - d2[i] * d2[i]);
p1[i] = d1[i] * div;
p2[i] = d2[i] * div;
p3[i] = d3 * div;
}
normg = dmat_norm2(n+n+1, gx);
pcgtol = min(1e-1, 0.3*gap/min(1.0,normg));
/*
pcgtol = min(1e-1, 0.3*gap/min(1.0,sqrt(normg)));
*/
pcgmaxi = MAX_PCG_ITER;
if (s < 1e-5)
{
pcgtol_factor *= 0.5;
}
else
{
pcgtol_factor = 1.0;
}
pcgtol = pcgtol*pcgtol_factor;
dmat_waxpby(n+n+1, -1, gx, 0, NULL, tmp_x1);
pcg(dx, pcgstat, afun, adata, mfun, mdata, tmp_x1, pcgtol, pcgmaxi, n+n+1);
}
