int
lls_complex_solve(const double lambda, gsl_vector_complex *c, lls_complex_workspace *w)
{
if (c->size != w->p)
{
fprintf(stderr, "lls_complex_solve: coefficient vector has wrong size\n");
return GSL_EBADLEN;
}
else
{
int s = 0;
/* solve (AHA + lambda^2 I) c = AHb and estimate condition number */
s = lls_lapack_zposv(lambda, c, w);
/* compute residual || AHA c - AHb || */
gsl_vector_complex_memcpy(w->work_b, w->AHb);
gsl_blas_zhemv(CblasUpper, GSL_COMPLEX_ONE, w->AHA, c, GSL_COMPLEX_NEGONE, w->work_b);
w->residual = gsl_blas_dznrm2(w->work_b);
/* compute chi^2 = b^H b - 2 c^H A^H b + c^H A^H A c */
{
gsl_complex negtwo = gsl_complex_rect(-2.0, 0.0);
gsl_complex val;
/* compute: AHA c - 2 AHb */
gsl_vector_complex_memcpy(w->work_b, w->AHb);
gsl_blas_zhemv(CblasUpper, GSL_COMPLEX_ONE, w->AHA, c, negtwo, w->work_b);
/* compute: c^H ( AHA c - 2 AHb ) */
gsl_blas_zdotc(c, w->work_b, &val);
w->chisq = w->bHb + GSL_REAL(val);
}
/* save coefficient vector for future robust iterations */
gsl_vector_complex_memcpy(w->c, c);
++(w->niter);
return s;
}
} /* lls_complex_solve() */
/**
* C++ version of gsl_blas_zhemv().
* @param Uplo Upper or lower triangular
* @param alpha A constant
* @param A A matrix
* @param X A vector
* @param beta Another constant
* @param Y A vector
* @return Error code on failure
*/
int zhemv( CBLAS_UPLO_t Uplo, complex const& alpha, matrix_complex const& A,
vector_complex const& X, complex const& beta, vector_complex& Y ){
return gsl_blas_zhemv( Uplo, alpha.get(), A.get(), X.get(), beta.get(), Y.get() ); }
int
lls_complex_lcurve(gsl_vector *reg_param, gsl_vector *rho, gsl_vector *eta,
lls_complex_workspace *w)
{
const size_t N = rho->size; /* number of points on L-curve */
if (N != reg_param->size)
{
GSL_ERROR("size of reg_param and rho do not match", GSL_EBADLEN);
}
else if (N != eta->size)
{
GSL_ERROR("size of eta and rho do not match", GSL_EBADLEN);
}
else
{
int s;
const gsl_complex negtwo = gsl_complex_rect(-2.0, 0.0);
/* smallest regularization parameter */
const double smin_ratio = 16.0 * GSL_DBL_EPSILON;
double s1, sp, ratio, tmp;
size_t i;
/* compute eigenvalues of A^H A */
gsl_matrix_complex_transpose_memcpy(w->work_A, w->AHA);
s = gsl_eigen_herm(w->work_A, w->eval, w->eigen_p);
if (s)
return s;
/* find largest and smallest eigenvalues */
gsl_vector_minmax(w->eval, &sp, &s1);
/* singular values are square roots of eigenvalues */
s1 = sqrt(s1);
if (sp > GSL_DBL_EPSILON)
sp = sqrt(fabs(sp));
tmp = GSL_MAX(sp, s1*smin_ratio);
gsl_vector_set(reg_param, N - 1, tmp);
/* ratio so that reg_param(1) = s(1) */
ratio = pow(s1 / tmp, 1.0 / (N - 1.0));
/* calculate the regularization parameters */
for (i = N - 1; i > 0 && i--; )
{
double rp1 = gsl_vector_get(reg_param, i + 1);
gsl_vector_set(reg_param, i, ratio * rp1);
}
for (i = 0; i < N; ++i)
{
double r2;
double lambda = gsl_vector_get(reg_param, i);
gsl_complex val;
lls_complex_solve(lambda, w->c, w);
/* store ||c|| */
gsl_vector_set(eta, i, gsl_blas_dznrm2(w->c));
/* compute: A^H A c - 2 A^H b */
gsl_vector_complex_memcpy(w->work_b, w->AHb);
gsl_blas_zhemv(CblasUpper, GSL_COMPLEX_ONE, w->AHA, w->c, negtwo, w->work_b);
/* compute: c^T A^T A c - 2 c^T A^T b */
gsl_blas_zdotc(w->c, w->work_b, &val);
r2 = GSL_REAL(val) + w->bHb;
gsl_vector_set(rho, i, sqrt(r2));
}
return GSL_SUCCESS;
}
} /* lls_complex_lcurve() */
int
gsl_linalg_hermtd_decomp (gsl_matrix_complex * A, gsl_vector_complex * tau)
{
if (A->size1 != A->size2)
{
GSL_ERROR ("hermitian tridiagonal decomposition requires square matrix",
GSL_ENOTSQR);
}
else if (tau->size + 1 != A->size1)
{
GSL_ERROR ("size of tau must be (matrix size - 1)", GSL_EBADLEN);
}
else
{
const size_t N = A->size1;
size_t i;
const gsl_complex zero = gsl_complex_rect (0.0, 0.0);
const gsl_complex one = gsl_complex_rect (1.0, 0.0);
const gsl_complex neg_one = gsl_complex_rect (-1.0, 0.0);
for (i = 0 ; i < N - 1; i++)
{
gsl_vector_complex_view c = gsl_matrix_complex_column (A, i);
gsl_vector_complex_view v = gsl_vector_complex_subvector (&c.vector, i + 1, N - (i + 1));
gsl_complex tau_i = gsl_linalg_complex_householder_transform (&v.vector);
/* Apply the transformation H^T A H to the remaining columns */
if ((i + 1) < (N - 1)
&& !(GSL_REAL(tau_i) == 0.0 && GSL_IMAG(tau_i) == 0.0))
{
gsl_matrix_complex_view m =
gsl_matrix_complex_submatrix (A, i + 1, i + 1,
N - (i+1), N - (i+1));
gsl_complex ei = gsl_vector_complex_get(&v.vector, 0);
gsl_vector_complex_view x = gsl_vector_complex_subvector (tau, i, N-(i+1));
gsl_vector_complex_set (&v.vector, 0, one);
/* x = tau * A * v */
gsl_blas_zhemv (CblasLower, tau_i, &m.matrix, &v.vector, zero, &x.vector);
/* w = x - (1/2) tau * (x' * v) * v */
{
gsl_complex xv, txv, alpha;
gsl_blas_zdotc(&x.vector, &v.vector, &xv);
txv = gsl_complex_mul(tau_i, xv);
alpha = gsl_complex_mul_real(txv, -0.5);
gsl_blas_zaxpy(alpha, &v.vector, &x.vector);
}
/* apply the transformation A = A - v w' - w v' */
gsl_blas_zher2(CblasLower, neg_one, &v.vector, &x.vector, &m.matrix);
gsl_vector_complex_set (&v.vector, 0, ei);
}
gsl_vector_complex_set (tau, i, tau_i);
}
return GSL_SUCCESS;
}
}
