void expm(int row, double *A)
{
int i;
int m_vals[] = {3, 5, 7, 9, 13};
double theta[] = {0.01495585217958292, 0.2539398330063230, 0.9504178996162932,
2.097847961257068, 5.371920351148152};
int lentheta = 5;
double normA = onenorm(row, row, A);
if (normA <= theta[4])
{
for (i = 0; i < lentheta; i++)
{
if (normA <= theta[i])
{
padeapprox(m_vals[i], row, A);
break;
}
}
}
else
{
int s;
double t = frexp(normA / (theta[4]), &s);
s = s - (t == 0.5);
t = pow(2, -s);
int row2 = row * row;
/* int i1 = 1;*/
// dscal_(&row2, &t, A, &i1);
dscal_3l(row2, t, A);
padeapprox(m_vals[4], row, A);
double *temp = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) temp[ii] = 0.0;
// char ta = 'n'; double alpha = 1; double beta = 0;
for (i = 0; i < s; i++)
{
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, A,
// &row, &beta, temp, &row);
dgemm_nn_3l(row, row, row, A, row, A, row, temp, row);
dmcopy(row, row, temp, row, A, row);
}
free(temp);
}
}
/************************************************
Mass-spring system: nx/2 masses connected each other with springs (in a row),
and the first and the last one to walls. nu (<=nx) controls act on the first nu
masses. The system is sampled with sampling time Ts.
************************************************/
void mass_spring_system(double Ts, int nx, int nu, double *A, double *B,
double *b, double *x0) {
int nx2 = nx * nx;
int info = 0;
int pp = nx / 2; // number of masses
/************************************************
* build the continuous time system
************************************************/
double *T;
d_zeros(&T, pp, pp);
int ii;
for (ii = 0; ii < pp; ii++) T[ii * (pp + 1)] = -2;
for (ii = 0; ii < pp - 1; ii++) T[ii * (pp + 1) + 1] = 1;
for (ii = 1; ii < pp; ii++) T[ii * (pp + 1) - 1] = 1;
double *Z;
d_zeros(&Z, pp, pp);
double *I;
d_zeros(&I, pp, pp);
for (ii = 0; ii < pp; ii++) I[ii * (pp + 1)] = 1.0; // = eye(pp);
double *Ac;
d_zeros(&Ac, nx, nx);
dmcopy(pp, pp, Z, pp, Ac, nx);
dmcopy(pp, pp, T, pp, Ac + pp, nx);
dmcopy(pp, pp, I, pp, Ac + pp * nx, nx);
dmcopy(pp, pp, Z, pp, Ac + pp * (nx + 1), nx);
free(T);
free(Z);
free(I);
d_zeros(&I, nu, nu);
for (ii = 0; ii < nu; ii++) I[ii * (nu + 1)] = 1.0; // I = eye(nu);
double *Bc;
d_zeros(&Bc, nx, nu);
dmcopy(nu, nu, I, nu, Bc + pp, nx);
free(I);
/************************************************
* compute the discrete time system
************************************************/
double *bb;
d_zeros(&bb, nx, 1);
dmcopy(nx, 1, bb, nx, b, nx);
dmcopy(nx, nx, Ac, nx, A, nx);
dscal_3l(nx2, Ts, A);
expm(nx, A);
d_zeros(&T, nx, nx);
d_zeros(&I, nx, nx);
for (ii = 0; ii < nx; ii++) I[ii * (nx + 1)] = 1.0; // I = eye(nx);
dmcopy(nx, nx, A, nx, T, nx);
daxpy_3l(nx2, -1.0, I, T);
dgemm_nn_3l(nx, nu, nx, T, nx, Bc, nx, B, nx);
int *ipiv = (int *)malloc(nx * sizeof(int));
dgesv_3l(nx, nu, Ac, nx, ipiv, B, nx, &info);
free(ipiv);
free(Ac);
free(Bc);
free(bb);
/************************************************
* initial state
************************************************/
if (nx == 4) {
x0[0] = 5;
x0[1] = 10;
x0[2] = 15;
x0[3] = 20;
} else {
int jj;
for (jj = 0; jj < nx; jj++) x0[jj] = 1;
}
}
/* computes the Pade approximation of degree m of the matrix A */
void padeapprox(int m, int row, double *A)
{
int row2 = row * row;
/* int i1 = 1;*/
/* double d0 = 0;*/
/* double d1 = 1;*/
/* double dm1 = -1;*/
double *U = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) U[ii] = 0.0;
double *V = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) V[ii] = 0.0;
if (m == 3)
{
double c[] = {120, 60, 12, 1};
double *A0 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A0[ii] = 0.0;
for (int ii = 0; ii < row; ii++) A0[ii * (row + 1)] = 1.0;
double *A2 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A2[ii] = 0.0;
// char ta = 'n'; double alpha = 1; double beta = 0;
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, A, &row,
// &beta, A2, &row);
dgemm_nn_3l(row, row, row, A, row, A, row, A2, row);
double *temp = malloc(row * row * sizeof(double));
// dscal_(&row2, &d0, temp, &i1);
dscal_3l(row2, 0, temp);
// daxpy_(&row2, &c[3], A2, &i1, temp, &i1);
daxpy_3l(row2, c[3], A2, temp);
// daxpy_(&row2, &c[1], A0, &i1, temp, &i1);
daxpy_3l(row2, c[1], A0, temp);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, temp,
// &row, &beta, U, &row);
dgemm_nn_3l(row, row, row, A, row, temp, row, U, row);
// dscal_(&row2, &d0, V, &i1);
dscal_3l(row2, 0, V);
// daxpy_(&row2, &c[2], A2, &i1, V, &i1);
daxpy_3l(row2, c[2], A2, V);
// daxpy_(&row2, &c[0], A0, &i1, V, &i1);
daxpy_3l(row2, c[0], A0, V);
free(A0);
free(A2);
free(temp);
}
else if (m == 5)
{
double c[] = {30240, 15120, 3360, 420, 30, 1};
double *A0 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A0[ii] = 0.0;
for (int ii = 0; ii < row; ii++) A0[ii * (row + 1)] = 1.0;
double *A2 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A2[ii] = 0.0;
double *A4 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A4[ii] = 0.0;
// char ta = 'n'; double alpha = 1; double beta = 0;
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, A, &row,
// &beta, A2, &row);
dgemm_nn_3l(row, row, row, A, row, A, row, A2, row);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A2, &row, A2, &row,
// &beta, A4, &row);
dgemm_nn_3l(row, row, row, A2, row, A2, row, A4, row);
dmcopy(row, row, A4, row, V, row);
double *temp = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) temp[ii] = 0.0;
dmcopy(row, row, A4, row, temp, row);
// daxpy_(&row2, &c[3], A2, &i1, temp, &i1);
daxpy_3l(row2, c[3], A2, temp);
// daxpy_(&row2, &c[1], A0, &i1, temp, &i1);
daxpy_3l(row2, c[1], A0, temp);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, temp,
// &row, &beta, U, &row);
dgemm_nn_3l(row, row, row, A, row, temp, row, U, row);
// dscal_(&row2, &c[4], V, &i1);
dscal_3l(row2, c[4], V);
// daxpy_(&row2, &c[2], A2, &i1, V, &i1);
daxpy_3l(row2, c[2], A2, V);
// daxpy_(&row2, &c[0], A0, &i1, V, &i1);
daxpy_3l(row2, c[0], A0, V);
free(A0);
free(A2);
free(A4);
free(temp);
}
else if (m == 7)
{
double c[] = {17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1};
double *A0 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A0[ii] = 0.0;
for (int ii = 0; ii < row; ii++) A0[ii * (row + 1)] = 1.0;
double *A2 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A2[ii] = 0.0;
double *A4 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A4[ii] = 0.0;
double *A6 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A6[ii] = 0.0;
// char ta = 'n'; double alpha = 1; double beta = 1;
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, A, &row,
// &beta, A2, &row);
dgemm_nn_3l(row, row, row, A, row, A, row, A2, row);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A2, &row, A2, &row,
// &beta, A4, &row);
dgemm_nn_3l(row, row, row, A2, row, A2, row, A4, row);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A4, &row, A2, &row,
// &beta, A6, &row);
dgemm_nn_3l(row, row, row, A4, row, A2, row, A6, row);
double *temp = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) temp[ii] = 0.0;
// dscal_(&row2, &d0, temp, &i1);
dscal_3l(row2, 0, temp);
// daxpy_(&row2, &c[3], A2, &i1, temp, &i1);
daxpy_3l(row2, c[3], A2, temp);
// daxpy_(&row2, &c[1], A0, &i1, temp, &i1);
daxpy_3l(row2, c[1], A0, temp);
// daxpy_(&row2, &c[5], A4, &i1, temp, &i1);
daxpy_3l(row2, c[5], A4, temp);
// daxpy_(&row2, &c[7], A6, &i1, temp, &i1);
daxpy_3l(row2, c[7], A6, temp);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, temp,
// &row, &beta, U, &row);
dgemm_nn_3l(row, row, row, A, row, temp, row, U, row);
// dscal_(&row2, &d0, V, &i1);
dscal_3l(row2, 0, V);
// daxpy_(&row2, &c[2], A2, &i1, V, &i1);
daxpy_3l(row2, c[2], A2, V);
// daxpy_(&row2, &c[0], A0, &i1, V, &i1);
daxpy_3l(row2, c[0], A0, V);
// daxpy_(&row2, &c[4], A4, &i1, V, &i1);
daxpy_3l(row2, c[4], A4, V);
// daxpy_(&row2, &c[6], A6, &i1, V, &i1);
daxpy_3l(row2, c[6], A6, V);
free(A0);
free(A2);
free(A4);
free(A6);
free(temp);
}
else if (m == 9)
{
double c[] = {17643225600, 8821612800, 2075673600, 302702400, 30270240,
2162160, 110880, 3960, 90, 1};
double *A0 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A0[ii] = 0.0;
for (int ii = 0; ii < row; ii++) A0[ii * (row + 1)] = 1.0;
double *A2 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A2[ii] = 0.0;
double *A4 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A4[ii] = 0.0;
double *A6 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A6[ii] = 0.0;
double *A8 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A8[ii] = 0.0;
// char ta = 'n'; double alpha = 1; double beta = 0;
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, A, &row,
// &beta, A2, &row);
dgemm_nn_3l(row, row, row, A, row, A, row, A2, row);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A2, &row, A2, &row,
// &beta, A4, &row);
dgemm_nn_3l(row, row, row, A2, row, A2, row, A4, row);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A4, &row, A2, &row,
// &beta, A6, &row);
dgemm_nn_3l(row, row, row, A4, row, A2, row, A6, row);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A6, &row, A2, &row,
// &beta, A8, &row);
dgemm_nn_3l(row, row, row, A6, row, A2, row, A8, row);
dmcopy(row, row, A8, row, V, row);
double *temp = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) temp[ii] = 0.0;
dmcopy(row, row, A8, row, temp, row);
// daxpy_(&row2, &c[3], A2, &i1, temp, &i1);
daxpy_3l(row2, c[3], A2, temp);
// daxpy_(&row2, &c[1], A0, &i1, temp, &i1);
daxpy_3l(row2, c[1], A0, temp);
// daxpy_(&row2, &c[5], A4, &i1, temp, &i1);
daxpy_3l(row2, c[5], A4, temp);
// daxpy_(&row2, &c[7], A6, &i1, temp, &i1);
daxpy_3l(row2, c[7], A6, temp);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, temp,
// &row, &beta, U, &row);
dgemm_nn_3l(row, row, row, A, row, temp, row, U, row);
// dscal_(&row2, &c[8], V, &i1);
dscal_3l(row2, c[8], V);
// daxpy_(&row2, &c[2], A2, &i1, V, &i1);
daxpy_3l(row2, c[2], A2, V);
// daxpy_(&row2, &c[0], A0, &i1, V, &i1);
daxpy_3l(row2, c[0], A0, V);
// daxpy_(&row2, &c[4], A4, &i1, V, &i1);
daxpy_3l(row2, c[4], A4, V);
// daxpy_(&row2, &c[6], A6, &i1, V, &i1);
daxpy_3l(row2, c[6], A6, V);
free(A0);
free(A2);
free(A4);
free(A6);
free(A8);
free(temp);
}
else if (m == 13)
{ // tested
double c[] = {64764752532480000,
32382376266240000,
7771770303897600,
1187353796428800,
129060195264000,
10559470521600,
670442572800,
33522128640,
1323241920,
40840800,
960960,
16380,
182,
1};
double *A0 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A0[ii] = 0.0;
for (int ii = 0; ii < row; ii++) A0[ii * (row + 1)] = 1.0;
double *A2 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A2[ii] = 0.0;
double *A4 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A4[ii] = 0.0;
double *A6 = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) A6[ii] = 0.0;
// char ta = 'n'; double alpha = 1; double beta = 0;
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, A, &row,
// &beta, A2, &row);
dgemm_nn_3l(row, row, row, A, row, A, row, A2, row);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A2, &row, A2, &row,
// &beta, A4, &row);
dgemm_nn_3l(row, row, row, A2, row, A2, row, A4, row);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A4, &row, A2, &row,
// &beta, A6, &row);
dgemm_nn_3l(row, row, row, A4, row, A2, row, A6, row);
dmcopy(row, row, A2, row, U, row);
double *temp = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) temp[ii] = 0.0;
// dscal_(&row2, &c[9], U, &i1);
dscal_3l(row2, c[9], U);
// daxpy_(&row2, &c[11], A4, &i1, U, &i1);
daxpy_3l(row2, c[11], A4, U);
// daxpy_(&row2, &c[13], A6, &i1, U, &i1);
daxpy_3l(row2, c[13], A6, U);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A6, &row, U, &row,
// &beta, temp, &row);
dgemm_nn_3l(row, row, row, A6, row, U, row, temp, row);
// daxpy_(&row2, &c[7], A6, &i1, temp, &i1);
daxpy_3l(row2, c[7], A6, temp);
// daxpy_(&row2, &c[5], A4, &i1, temp, &i1);
daxpy_3l(row2, c[5], A4, temp);
// daxpy_(&row2, &c[3], A2, &i1, temp, &i1);
daxpy_3l(row2, c[3], A2, temp);
// daxpy_(&row2, &c[1], A0, &i1, temp, &i1);
daxpy_3l(row2, c[1], A0, temp);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A, &row, temp,
// &row, &beta, U, &row);
dgemm_nn_3l(row, row, row, A, row, temp, row, U, row);
dmcopy(row, row, A2, row, temp, row);
// dscal_(&row2, &c[8], V, &i1);
dscal_3l(row2, c[8], V);
// daxpy_(&row2, &c[12], A6, &i1, temp, &i1);
daxpy_3l(row2, c[12], A6, temp);
// daxpy_(&row2, &c[10], A4, &i1, temp, &i1);
daxpy_3l(row2, c[10], A4, temp);
// dgemm_(&ta, &ta, &row, &row, &row, &alpha, A6, &row, temp,
// &row, &beta, V, &row);
dgemm_nn_3l(row, row, row, A6, row, temp, row, V, row);
// daxpy_(&row2, &c[6], A6, &i1, V, &i1);
daxpy_3l(row2, c[6], A6, V);
// daxpy_(&row2, &c[4], A4, &i1, V, &i1);
daxpy_3l(row2, c[4], A4, V);
// daxpy_(&row2, &c[2], A2, &i1, V, &i1);
daxpy_3l(row2, c[2], A2, V);
// daxpy_(&row2, &c[0], A0, &i1, V, &i1);
daxpy_3l(row2, c[0], A0, V);
free(A0);
free(A2);
free(A4);
free(A6);
free(temp);
}
else
{
printf("%s\n", "Wrong Pade approximatin degree");
exit(1);
}
double *D = malloc(row * row * sizeof(double));
for (int ii = 0; ii < row * row; ii++) D[ii] = 0.0;
// dcopy_(&row2, V, &i1, A, &i1);
dmcopy(row, row, V, row, A, row);
// daxpy_(&row2, &d1, U, &i1, A, &i1);
daxpy_3l(row2, 1.0, U, A);
// dcopy_(&row2, V, &i1, D, &i1);
dmcopy(row, row, V, row, D, row);
// daxpy_(&row2, &dm1, U, &i1, D, &i1);
daxpy_3l(row2, -1.0, U, D);
int *ipiv = (int *) calloc(row, sizeof(int));
int info = 0;
// dgesv_(&row, &row, D, &row, ipiv, A, &row, &info);
dgesv_3l(row, row, D, row, ipiv, A, row, &info);
free(ipiv);
free(D);
free(U);
free(V);
}