# Smoothing property of the Jacobi method

#### Content

Complete the implementation of the (weighted) Jacobi iteration.

## Code for exercise

#include <cmath>

#include <hpc/matvec/axpy.hpp>
#include <hpc/matvec/copy.hpp>
#include <hpc/matvec/densevector.hpp>
#include <hpc/matvec/iterators.hpp>
#include <hpc/matvec/print.hpp>
#include <hpc/matvec/traits.hpp>

//------------------------------------------------------------------------------

using namespace hpc;
using namespace hpc::matvec;

template <typename T>
const T &
assertEqual(const T &a, const T &b)
{
assert(a==b);
return a;
}

//------------------------------------------------------------------------------

template <
typename F, typename G,
typename T, template<typename> class VectorF,
template<typename> class VectorU,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>> > = true
>
void
discretizeProblem(const F &f, const G &g, VectorF<T> &f_h, VectorU<T> &u_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
auto h  = T(1)/(N+1);

u_h(0)   = g(0);
u_h(N+1) = g(1);

for (std::size_t i=1; i<N; ++i) {
u_h(i) = 0;
f_h(i) = f(i*h);
}
// actually we don't have to initialize f_h(0), f_h(N+1) so we store some
// good and evil numbers there:
f_h(0)   = 42;
f_h(N+1) = 666;
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class Vector,
Require< Dense<Vector<T>> > = true
>
T
lInfNorm(const Vector<T> &x_h)
{
T res = 0;

for (auto [i, xi] : x_h) {
i=i;
if (std::abs(xi) > res) {
res = std::abs(xi);
}
}
return res;
}

//------------------------------------------------------------------------------

template <
typename U,
typename T, template<typename> class VectorU,
template<typename> class VectorE,
Require< Dense<VectorU<T>>,
Dense<VectorE<T>> > = true
>
void
evalError(const U &u, const VectorU<T> &u_h, VectorE<T> &e_h)
{
auto N  = assertEqual(u_h.length(), e_h.length()) - 2;
auto h  = T(1)/(N+1);

for (auto [i, ei] : e_h) {
ei = u(i*h) - u_h(i);
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorU,
template<typename> class VectorF,
template<typename> class VectorR,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>>,
Dense<VectorR<T>> > = true
>
void
evalResidual(const VectorU<T> &u_h, const VectorF<T> &f_h, VectorR<T> &r_h)
{
auto N  = assertEqual(u_h.length(), r_h.length()) - 2;
auto h  = T(1)/(N+1);
auto h2 = h*h;

r_h(0) = r_h(N+1) = 0;
for (std::size_t i=1; i<=N; ++i) {
r_h(i) = f_h(i) - 1./h2 * (2*u_h(i) - u_h(i-1) - u_h(i+1));
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorF,
template<typename> class VectorU,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>> > = true
>
void
direct_solver(const VectorF<T> &f_h, VectorU<T> &u_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
auto h  = T(1)/(N+1);
auto h2 = h*h;

// forward substitution
u_h(1) = h2*f_h(1) + u_h(0);
for (std::size_t k=2; k<N; ++k) {
u_h(k) = h2*f_h(k) + u_h(k-1)*(k-1)/k;
}
u_h(N) = h2*f_h(N) + u_h(N+1) + u_h(N-1)*(N-1)/N;

// backward substitution
u_h(N) = u_h(N)*N/(N+1);
for (std::size_t k=N-1; k>=1; --k) {
u_h(k) = (u_h(k)+u_h(k+1))*k/(k+1);
}
}

//==============================================================================

template <
typename T, template<typename> class Vector,
Require< Dense<Vector<T>> > = true
>
void
{
auto N  = x_h.length()-2;
auto h  = T(1)/(N+1);

for (std::size_t i=1; i<=N; ++i) {
x_h(i) += sin(M_PI*k*i*h);
}
}

//------------------------------------------------------------------------------

template <
typename T, template<typename> class VectorF,
template<typename> class VectorU,
template<typename> class VectorS,
Require< Dense<VectorF<T>>,
Dense<VectorU<T>>,
Dense<VectorS<T>> > = true
>
void
jacobi_step(double omega, const VectorF<T> &f_h, const VectorU<T> &u_h,
VectorS<T> &s_h)
{
auto N  = assertEqual(f_h.length(), u_h.length()) - 2;
/*
auto h  = T(1)/(N+1);
auto h2 = h*h;
*/

s_h(0)   = u_h(0);
s_h(N+1) = u_h(N+1);
}

//------------------------------------------------------------------------------

int
main()
{
/*
auto f = [](double x) { return -2; };
auto g = [](double x) { return 1 + 3*x; };
auto u = [](double x) { return (x+1)*(x+1); };
*/
auto f = [](double x) { return 0; };
auto g = [](double x) { return 0; };
auto u = [](double x) { return 0; };

std::size_t N = 100;

DenseVector<double>  f_h(N+2), u_h(N+2), tmp_h(N+2), r_h(N+2), e_h(N+2);

discretizeProblem(f, g, f_h, u_h);

// First initialize u_h with exact solution
direct_solver(f_h, u_h);

double omega = 0.5;     // smoothing parameter

for (std::size_t it=0; it<=20; ++it) {
jacobi_step(omega, f_h, u_h, tmp_h);
jacobi_step(omega, f_h, tmp_h, u_h);

evalResidual(u_h, f_h, r_h);
evalError(u, u_h, e_h);

fmt::printf("#lInfNorm(r_h) = %.5e\n", lInfNorm(r_h));
fmt::printf("#lInfNorm(e_h) = %.5e\n", lInfNorm(e_h));
fmt::printf("it %3d\n", it);
for (auto [i,ri] : r_h) {
fmt::printf("%d %f\n", i, ri);
}
fmt::printf("\n\n");
}
}


## Code for gnuplot

set terminal svg size 900, 500
set output "smooth.svg"
set xlabel "Grid points"
set ylabel "Residual value"
set title "Smoothing"
set key outside

theon$g++ -Wall -I /home/numerik/pub/hpc/ws19/session31 -std=c++17 -o direct_solver -o smooth smooth.cpp theon$ ./smooth > smooth.data
theon\$