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gmres.cpp
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gmres.cpp
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/*
* GMRES subroutine.
*
* copyright@ Yimin Zhong. yzhong@math.utexas.edu. All Rights Reserved.
*
*/
#include "gmres.h"
using namespace bbfmm;
inline void
Update(Vector &x, int k, Matrix &h, Vector &s, Vector v[]) {
Vector y(s);
for (int i = k; i >= 0; i--) {
y(i) /= h(i, i);
for (int j = i - 1; j >= 0; j--)
y(j) -= h(j, i) * y(i);
}
for (int j = 0; j <= k; j++)
daxpy(y(j), v[j], x);
}
void GeneratePlaneRotation(scalar_t &dx, scalar_t &dy, scalar_t &cs, scalar_t &sn) {
if (dy == 0.0) {
cs = 1.0;
sn = 0.0;
} else if (abs(dy) > abs(dx)) {
scalar_t temp = dx / dy;
sn = 1.0 / sqrt(1.0 + temp * temp);
cs = temp * sn;
} else {
scalar_t temp = dy / dx;
cs = 1.0 / sqrt(1.0 + temp * temp);
sn = temp * cs;
}
}
void ApplyPlaneRotation(scalar_t &dx, scalar_t &dy, scalar_t &cs, scalar_t &sn) {
scalar_t temp = cs * dx + sn * dy;
dy = -sn * dx + cs * dy;
dx = temp;
}
inline scalar_t abs(scalar_t x) {
return (x > 0 ? x : -x);
}
int GMRES(const std::function<Vector(Vector &)> A, Vector &x, Vector &b, int m, int max_iter,
scalar_t tol) {
/*
* for performance use
*/
std::chrono::steady_clock::time_point begin;
std::chrono::steady_clock::time_point end;
scalar_t resid;
scalar_t _tol = tol;
int _max_iter = max_iter;
int i, j = 1, k;
Vector s(m + 1), cs(m + 1), sn(m + 1), w;
scalar_t normb = nrm2(b);
Vector r = b;
Vector p = A(x);
daxpy(-1.0, p, r);
scalar_t beta = nrm2(r);
/*
* error is |Ax - b|/|b|
*/
std::cout << "=============== GMRES =================" << std::endl;
std::cout << " iter | rel error | time " << std::endl;
begin = std::chrono::steady_clock::now();
if (normb == 0.0)
normb = 1;
if ((resid = nrm2(r) / normb) <= _tol) {
_tol = resid;
_max_iter = 0;
return 0;
}
Vector *v = new Vector[m + 1];
Matrix H(x.row(), m + 1);
while (j <= _max_iter) {
v[0] = r; // ??? r / beta
dscal((1.0 / beta), v[0]);
setValue(s, 0.);
s(0) = beta;
for (i = 0; i < m && j <= _max_iter; i++, j++) {
end = std::chrono::steady_clock::now();
std::cout << std::setw(6) << j << std::setw(20) << std::scientific << resid
<< std::setw(12) << std::fixed
<< std::chrono::duration_cast<std::chrono::microseconds>(end - begin).count() / 1000000.0
<< std::fixed
<< std::endl;
begin = std::chrono::steady_clock::now();
w = A(v[i]);
for (k = 0; k <= i; k++) {
H(k, i) = ddot(w, v[k]);
daxpy(-H(k, i), v[k], w);
}
H(i + 1, i) = nrm2(w);
v[i + 1] = w;
dscal((1.0 / H(i + 1, i)), v[i + 1]);
// ??? w / H(i+1, i)
for (k = 0; k < i; k++) {
ApplyPlaneRotation((scalar_t &) H(k, i), (scalar_t &) H(k + 1, i), (scalar_t &) cs(k),
(scalar_t &) sn(k));
}
GeneratePlaneRotation((scalar_t &) H(i, i), (scalar_t &) H(i + 1, i), (scalar_t &) cs(i),
(scalar_t &) sn(i));
ApplyPlaneRotation((scalar_t &) H(i, i), (scalar_t &) H(i + 1, i), (scalar_t &) cs(i), (scalar_t &) sn(i));
ApplyPlaneRotation((scalar_t &) s(i), (scalar_t &) s(i + 1), (scalar_t &) cs(i), (scalar_t &) sn(i));
if ((resid = abs(s(i + 1)) / normb) < _tol) {
Update(x, i, H, s, v);
_tol = resid;
_max_iter = j;
std::cout << "=============== CONVERGED =============" << std::endl;
delete[] v;
return 0;
}
}
Update(x, i - 1, H, s, v);
r = b;
p = A(x);
daxpy(-1.0, p, r);
beta = nrm2(r);
if ((resid = beta / normb) < _tol) {
_tol = resid;
_max_iter = j;
std::cout << "=============== CONVERGED =============" << std::endl;
delete[] v;
return 0;
}
}
_tol = resid;
delete[] v;
std::cout << "============= NOT CONVERGED ===========" << std::endl;
return 1;
}