void Poisson1D::runPoisson1D(double vTolerance, double _chargeTolerance, double magicNumber, bool Equilibrum) {
// Relatively quicker to solve potential, because potential is continuous.
// potential == vacuum level
double chargeTolerance = _chargeTolerance / ( pow(cmNL(1), 3) );
mat potential= phin + dev1D.getEAArray() + fLnArray;
bCArrayFunct();
mat oldcDA = zeros(dev1D.getSumPoint());
mat errV = ones(1)*LARGE;
mat errCDA = ones(1)*LARGE;
// superlu_opts settings; // optional
int numConvergenceStep=0;
do {
cDA = dev1D.chargeDensityArrayFunct(phin, setPhip(phin, Equilibrum), Equilibrum);
errCDA = abs(cDA - oldcDA).max();
oldcDA = cDA;
// std::cout << cDA(51) << " " << cDA(87) << std::endl;
mat error = - (dev1D.getMatrixC() * potential - ChargeQ/E0 * cDA - bCArray);
sp_mat qCMat = ChargeQ/E0 * dev1D.qCMatFunct(phin, setPhip(phin, Equilibrum), Equilibrum);
sp_mat matrixC_plusCq = dev1D.getMatrixC() - qCMat;
mat deltaPotential = spsolve(matrixC_plusCq, error, "superlu");
/** spsolve was not recognized. solve by using full path in #include,
* then follow http://stackoverflow.com/questions/30494610/how-to-link-armadillo-with-eclipse
* then rebuild (in index)
*/
potential += magicNumber * deltaPotential;
errV = abs(deltaPotential).max();
numConvergenceStep++;
phin = potential - (dev1D.getEAArray() + fLnArray );
} while ((errV(0) > vTolerance) || (errCDA(0) > chargeTolerance) || numConvergenceStep > 1E4);
if ( numConvergenceStep > 1E4)
std::cerr << "Solution not found." << std::endl;
phip = setPhip(phin, Equilibrum);
condBand = potential - dev1D.getEAArray();
valeBand = condBand - dev1D.getBGArray();
}
int sfgmr(int n,
void (*smatvec) (float, float[], float, float[]),
void (*spsolve) (int, float[], float[]),
float *rhs, float *sol, double tol, int im, int *itmax, FILE * fits)
{
/*----------------------------------------------------------------------
| *** Preconditioned FGMRES ***
+-----------------------------------------------------------------------
| This is a simple version of the ARMS preconditioned FGMRES algorithm.
+-----------------------------------------------------------------------
| Y. S. Dec. 2000. -- Apr. 2008
+-----------------------------------------------------------------------
| on entry:
|----------
|
| rhs = real vector of length n containing the right hand side.
| sol = real vector of length n containing an initial guess to the
| solution on input.
| tol = tolerance for stopping iteration
| im = Krylov subspace dimension
| (itmax) = max number of iterations allowed.
| fits = NULL: no output
| != NULL: file handle to output " resid vs time and its"
|
| on return:
|----------
| fgmr int = 0 --> successful return.
| int = 1 --> convergence not achieved in itmax iterations.
| sol = contains an approximate solution (upon successful return).
| itmax = has changed. It now contains the number of steps required
| to converge --
+-----------------------------------------------------------------------
| internal work arrays:
|----------
| vv = work array of length [im+1][n] (used to store the Arnoldi
| basis)
| hh = work array of length [im][im+1] (Householder matrix)
| z = work array of length [im][n] to store preconditioned vectors
+-----------------------------------------------------------------------
| subroutines called :
| matvec - matrix-vector multiplication operation
| psolve - (right) preconditionning operation
| psolve can be a NULL pointer (GMRES without preconditioner)
+---------------------------------------------------------------------*/
int maxits = *itmax;
int i, i1, ii, j, k, k1, its, retval, i_1 = 1, i_2 = 2;
float beta, eps1 = 0.0, t, t0, gam;
float **hh, *c, *s, *rs;
float **vv, **z, tt;
float zero = 0.0;
float one = 1.0;
its = 0;
vv = (float **)SUPERLU_MALLOC((im + 1) * sizeof(float *));
for (i = 0; i <= im; i++) vv[i] = floatMalloc(n);
z = (float **)SUPERLU_MALLOC(im * sizeof(float *));
hh = (float **)SUPERLU_MALLOC(im * sizeof(float *));
for (i = 0; i < im; i++)
{
hh[i] = floatMalloc(i + 2);
z[i] = floatMalloc(n);
}
c = floatMalloc(im);
s = floatMalloc(im);
rs = floatMalloc(im + 1);
/*---- outer loop starts here ----*/
do
{
/*---- compute initial residual vector ----*/
smatvec(one, sol, zero, vv[0]);
for (j = 0; j < n; j++)
vv[0][j] = rhs[j] - vv[0][j]; /* vv[0]= initial residual */
beta = snrm2_(&n, vv[0], &i_1);
/*---- print info if fits != null ----*/
if (fits != NULL && its == 0)
fprintf(fits, "%8d %10.2e\n", its, beta);
/*if ( beta <= tol * dnrm2_(&n, rhs, &i_1) )*/
if ( !(beta > tol * snrm2_(&n, rhs, &i_1)) )
break;
t = 1.0 / beta;
/*---- normalize: vv[0] = vv[0] / beta ----*/
for (j = 0; j < n; j++)
vv[0][j] = vv[0][j] * t;
if (its == 0)
eps1 = tol * beta;
/*---- initialize 1-st term of rhs of hessenberg system ----*/
rs[0] = beta;
for (i = 0; i < im; i++)
{
its++;
i1 = i + 1;
/*------------------------------------------------------------
| (Right) Preconditioning Operation z_{j} = M^{-1} v_{j}
+-----------------------------------------------------------*/
if (spsolve)
spsolve(n, z[i], vv[i]);
else
scopy_(&n, vv[i], &i_1, z[i], &i_1);
/*---- matvec operation w = A z_{j} = A M^{-1} v_{j} ----*/
smatvec(one, z[i], zero, vv[i1]);
/*------------------------------------------------------------
| modified gram - schmidt...
| h_{i,j} = (w,v_{i})
| w = w - h_{i,j} v_{i}
+------------------------------------------------------------*/
t0 = snrm2_(&n, vv[i1], &i_1);
for (j = 0; j <= i; j++)
{
float negt;
tt = sdot_(&n, vv[j], &i_1, vv[i1], &i_1);
hh[i][j] = tt;
negt = -tt;
saxpy_(&n, &negt, vv[j], &i_1, vv[i1], &i_1);
}
/*---- h_{j+1,j} = ||w||_{2} ----*/
t = snrm2_(&n, vv[i1], &i_1);
while (t < 0.5 * t0)
{
t0 = t;
for (j = 0; j <= i; j++)
{
float negt;
tt = sdot_(&n, vv[j], &i_1, vv[i1], &i_1);
hh[i][j] += tt;
negt = -tt;
saxpy_(&n, &negt, vv[j], &i_1, vv[i1], &i_1);
}
t = snrm2_(&n, vv[i1], &i_1);
}
hh[i][i1] = t;
if (t != 0.0)
{
/*---- v_{j+1} = w / h_{j+1,j} ----*/
t = 1.0 / t;
for (k = 0; k < n; k++)
vv[i1][k] = vv[i1][k] * t;
}
/*---------------------------------------------------
| done with modified gram schimdt and arnoldi step
| now update factorization of hh
+--------------------------------------------------*/
/*--------------------------------------------------------
| perform previous transformations on i-th column of h
+-------------------------------------------------------*/
for (k = 1; k <= i; k++)
{
k1 = k - 1;
tt = hh[i][k1];
hh[i][k1] = c[k1] * tt + s[k1] * hh[i][k];
hh[i][k] = -s[k1] * tt + c[k1] * hh[i][k];
}
gam = sqrt(pow(hh[i][i], 2) + pow(hh[i][i1], 2));
/*---------------------------------------------------
| if gamma is zero then any small value will do
| affect only residual estimate
+--------------------------------------------------*/
/* if (gam == 0.0) gam = epsmac; */
/*---- get next plane rotation ---*/
if (gam == 0.0)
{
c[i] = one;
s[i] = zero;
}
else
{
c[i] = hh[i][i] / gam;
s[i] = hh[i][i1] / gam;
}
rs[i1] = -s[i] * rs[i];
rs[i] = c[i] * rs[i];
/*----------------------------------------------------
| determine residual norm and test for convergence
+---------------------------------------------------*/
hh[i][i] = c[i] * hh[i][i] + s[i] * hh[i][i1];
beta = fabs(rs[i1]);
if (fits != NULL)
fprintf(fits, "%8d %10.2e\n", its, beta);
if (beta <= eps1 || its >= maxits)
break;
}
if (i == im) i--;
/*---- now compute solution. 1st, solve upper triangular system ----*/
rs[i] = rs[i] / hh[i][i];
for (ii = 1; ii <= i; ii++)
{
k = i - ii;
k1 = k + 1;
tt = rs[k];
for (j = k1; j <= i; j++)
tt = tt - hh[j][k] * rs[j];
rs[k] = tt / hh[k][k];
}
/*---- linear combination of v[i]'s to get sol. ----*/
for (j = 0; j <= i; j++)
{
tt = rs[j];
for (k = 0; k < n; k++)
sol[k] += tt * z[j][k];
}
/* calculate the residual and output */
smatvec(one, sol, zero, vv[0]);
for (j = 0; j < n; j++)
vv[0][j] = rhs[j] - vv[0][j]; /* vv[0]= initial residual */
/*---- print info if fits != null ----*/
beta = snrm2_(&n, vv[0], &i_1);
/*---- restart outer loop if needed ----*/
/*if (beta >= eps1 / tol)*/
if ( !(beta < eps1 / tol) )
{
its = maxits + 10;
break;
}
if (beta <= eps1)
break;
} while(its < maxits);
retval = (its >= maxits);
for (i = 0; i <= im; i++)
SUPERLU_FREE(vv[i]);
SUPERLU_FREE(vv);
for (i = 0; i < im; i++)
{
SUPERLU_FREE(hh[i]);
SUPERLU_FREE(z[i]);
}
SUPERLU_FREE(hh);
SUPERLU_FREE(z);
SUPERLU_FREE(c);
SUPERLU_FREE(s);
SUPERLU_FREE(rs);
*itmax = its;
return retval;
} /*----end of fgmr ----*/
SolverResult KojimaSolver::solve(const LCP & lcp,
vec & x,
vec & y) const{
superlu_opts opts;
opts.equilibrate = true;
opts.permutation = superlu_opts::COLAMD;
opts.refine = superlu_opts::REF_SINGLE;
vec q = lcp.q;
sp_mat M = lcp.M + regularizer * speye(size(lcp.M));
uint N = q.n_elem;
assert(N == x.n_elem);
assert(N == y.n_elem);
// Figure out what is free (-inf,inf)
// and what is bound to be non-negative [0,inf)
bvec free_vars = lcp.free_vars;
uvec bound_idx = find(0 == free_vars);
uvec free_idx = find(1 == free_vars);
assert(N == bound_idx.n_elem + free_idx.n_elem);
uint NB = bound_idx.n_elem; // number of bound vars
uint NF = free_idx.n_elem; // number of free vars
/*
In what follows, the primal variables x are partitioned into
free variables and bound variables x = [f;b]'
Likewise, the dual variables are partitioned into y = [0,s]'
*/
/* The Newton system:
[M_ff M_fb 0][df] [M_f x + q_f]
[M_bf M_bb -I][db] + [M_b x + q_b - s]
[0 S B][dv] [u1 + VBe]
Where "M_ff" is the free-free block
Overwrite the S,B diagonals every iteration*/
// Split M matrix into blocks based on free and bound indicies
block_sp_mat M_part = sp_partition(M,free_idx,bound_idx);
sp_mat M_recon = block_mat(M_part);
assert(PRETTY_SMALL > norm(M_recon - M));
vec qf = q(free_idx);
vec qb = q(bound_idx);
vec b = x(bound_idx);
vec f = x(free_idx);
vec s = y(bound_idx);
// Build the Newton matrix
vector<vector<sp_mat>> block_G;
block_G.push_back(block_sp_vec{sp_mat(),sp_mat(),sp_mat(NB,NB)});
block_G.push_back(block_sp_vec{-M_part[0][0],-M_part[0][1],sp_mat()});
block_G.push_back(block_sp_vec{-M_part[1][0],-M_part[1][1],speye(NB,NB)});
// Start iteration
double mean_comp, steplen;
double sigma = initial_sigma;
uint iter;
for(iter = 0; iter < max_iter; iter++){
if(verbose or iter_verbose)
cout << "---Iteration " << iter << "---" << endl;
assert(all(0 == y(free_idx)));
// Mean complementarity
mean_comp = dot(b,s) / (double) NB;
if(mean_comp < comp_thresh)
break;
block_G[0][1] = spdiag(s);
block_G[0][2] = spdiag(b);
sp_mat G = block_mat(block_G);
assert(size(N + NB,N + NB) == size(G));
// Form RHS from residual and complementarity
vec h = vec(N + NB);
vec res_f = M_part[0][0]*f + M_part[0][1]*b + qf;
vec res_b = M_part[1][0]*f + M_part[1][1]*b + qb - s;
h.head(NB) = sigma * mean_comp - b % s;
h.subvec(NB,size(res_f)) = res_f;
h.tail(NB) = res_b;
//Archiver arch;
//arch.add_sp_mat("G",G);
//arch.add_vec("h",h);
//arch.write("test.sys");
// Solve and extract directions
vec dir = spsolve(G,h,"superlu",opts);
assert((N+NB) == dir.n_elem);
vec df = dir.head(NF);
vec db = dir.subvec(NF,N-1);
assert(NB == db.n_elem);
vec ds = dir.tail(NB);
vec dir_recon = join_vert(df,join_vert(db,ds));
assert(PRETTY_SMALL > norm(dir_recon-dir));
steplen = steplen_heuristic(b,s,db,ds,0.9);
sigma = sigma_heuristic(sigma,steplen);
f += steplen * df;
b += steplen * db;
s += steplen * ds;
if(verbose){
double res = norm(join_vert(res_f,res_b));
cout <<"\t Mean complementarity: " << mean_comp
<<"\n\t Residual norm: " << res
<<"\n\t |df|: " << norm(df)
<<"\n\t |db|: " << norm(db)
<<"\n\t |ds|: " << norm(ds)
<<"\n\t Step length: " << steplen
<<"\n\t Centering sigma: " << sigma << endl;
}
}
if(verbose){
cout << "Finished"
<<"\n\t Final mean complementarity: " << mean_comp << endl;
}
x(free_idx) = f;
x(bound_idx) = b;
y(free_idx).fill(0);
y(bound_idx) = s;
return SolverResult(x,y,iter);
}
PLCP augment_plcp(const PLCP & original,
vec & x,
vec & y,
vec & w,
double scale){
uint N = original.P.n_rows;
uint K = original.P.n_cols;
assert(size(K,N) == size(original.U));
sp_mat P = sp_mat(original.P);
double I_norm = norm(speye(K,K) - P.t() * P);
if(norm(I_norm) >= PRETTY_SMALL){
cerr << "Error: P does not look orthogonal ("
<< I_norm << ")..." << endl;
}
assert(I_norm < PRETTY_SMALL);
sp_mat U = sp_mat(original.U);
vec q = vec(original.q);
assert(all(q(find(1 == original.free_vars)) <= 0));
vec q_neg = min(zeros<vec>(N),q);
vec q_pos = max(zeros<vec>(N),q);
x = ones<vec>(N) - q_neg;
y = ones<vec>(N) + q_pos;
y(find(1 == original.free_vars)).fill(0);
assert(norm(q_pos(find(1 == original.free_vars))) < ALMOST_ZERO);
assert(N == x.n_elem);
assert(N == y.n_elem);
assert(all(x >= 0));
assert(all(y >= 0));
vec res = x - y + q;
w = spsolve(P.t()*P + 1e-15*speye(K,K),P.t()*(x - y + q));
vec w_res = P * w - res;
if(norm(w_res) >= PRETTY_SMALL){
cerr << "Error: Reduced vector w residual large ("
<< w_res << ")..." << endl;
}
assert(norm(w_res) < PRETTY_SMALL);
vec b = P.t()*x - U*x - w;
assert(K == b.n_elem);
P.resize(N+1,K+1);
U.resize(K+1,N+1);
q.resize(N+1);
P(N,K) = 1.0;
U(span(0,K-1),N) = b;
U(K,N) = scale;
q(N) = 0;
x.resize(N+1);
y.resize(N+1);
w.resize(K+1);
x(N) = 1;
y(N) = scale;
w(K) = 1.0 - scale;
bvec free_vars = bvec(N+1);
free_vars.head(N) = original.free_vars;
free_vars(N) = 0;
return PLCP(P,U,q,free_vars);
}
