static VALUE rb_gsl_blas_zdotc(int argc, VALUE *argv, VALUE obj)
{
gsl_complex *r;
int status;
gsl_vector_complex *x = NULL, *y = NULL;
get_vector_complex2(argc, argv, obj, &x, &y);
r = ALLOC(gsl_complex);
status = gsl_blas_zdotc(x, y, r);
return Data_Wrap_Struct(cgsl_complex, 0, free, r);
}
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_zdotc().
* @param X First vector
* @param Y Second vector
* @param dotc Vector product
* @return Error code on failure
*/
int zdotc( vector_complex const& X, vector_complex const& Y, complex* dotc ){
return gsl_blas_zdotc( X.get(), Y.get(), &dotc->get() ); }
void
test_eigen_herm_results (const gsl_matrix_complex * A,
const gsl_vector * eval,
const gsl_matrix_complex * evec,
size_t count,
const char * desc,
const char * desc2)
{
const size_t N = A->size1;
size_t i, j;
gsl_vector_complex * x = gsl_vector_complex_alloc(N);
gsl_vector_complex * y = gsl_vector_complex_alloc(N);
/* check eigenvalues */
for (i = 0; i < N; i++)
{
double ei = gsl_vector_get (eval, i);
gsl_vector_complex_const_view vi =
gsl_matrix_complex_const_column(evec, i);
gsl_vector_complex_memcpy(x, &vi.vector);
/* compute y = m x (should = lambda v) */
gsl_blas_zgemv (CblasNoTrans, GSL_COMPLEX_ONE, A, x,
GSL_COMPLEX_ZERO, y);
for (j = 0; j < N; j++)
{
gsl_complex xj = gsl_vector_complex_get (x, j);
gsl_complex yj = gsl_vector_complex_get (y, j);
gsl_test_rel(GSL_REAL(yj), ei * GSL_REAL(xj), 1e8*GSL_DBL_EPSILON,
"%s, eigenvalue(%d,%d), real, %s", desc, i, j, desc2);
gsl_test_rel(GSL_IMAG(yj), ei * GSL_IMAG(xj), 1e8*GSL_DBL_EPSILON,
"%s, eigenvalue(%d,%d), imag, %s", desc, i, j, desc2);
}
}
/* check eigenvectors are orthonormal */
for (i = 0; i < N; i++)
{
gsl_vector_complex_const_view vi = gsl_matrix_complex_const_column(evec, i);
double nrm_v = gsl_blas_dznrm2(&vi.vector);
gsl_test_rel (nrm_v, 1.0, N * GSL_DBL_EPSILON, "%s, normalized(%d), %s",
desc, i, desc2);
}
for (i = 0; i < N; i++)
{
gsl_vector_complex_const_view vi = gsl_matrix_complex_const_column(evec, i);
for (j = i + 1; j < N; j++)
{
gsl_vector_complex_const_view vj
= gsl_matrix_complex_const_column(evec, j);
gsl_complex vivj;
gsl_blas_zdotc (&vi.vector, &vj.vector, &vivj);
gsl_test_abs (gsl_complex_abs(vivj), 0.0, 10.0 * N * GSL_DBL_EPSILON,
"%s, orthogonal(%d,%d), %s", desc, i, j, desc2);
}
}
gsl_vector_complex_free(x);
gsl_vector_complex_free(y);
} /* test_eigen_herm_results() */
int
gsl_linalg_complex_cholesky_decomp(gsl_matrix_complex *A)
{
const size_t N = A->size1;
if (N != A->size2)
{
GSL_ERROR("cholesky decomposition requires square matrix", GSL_ENOTSQR);
}
else
{
size_t i, j;
gsl_complex z;
double ajj;
for (j = 0; j < N; ++j)
{
z = gsl_matrix_complex_get(A, j, j);
ajj = GSL_REAL(z);
if (j > 0)
{
gsl_vector_complex_const_view aj =
gsl_matrix_complex_const_subrow(A, j, 0, j);
gsl_blas_zdotc(&aj.vector, &aj.vector, &z);
ajj -= GSL_REAL(z);
}
if (ajj <= 0.0)
{
GSL_ERROR("matrix is not positive definite", GSL_EDOM);
}
ajj = sqrt(ajj);
GSL_SET_COMPLEX(&z, ajj, 0.0);
gsl_matrix_complex_set(A, j, j, z);
if (j < N - 1)
{
gsl_vector_complex_view av =
gsl_matrix_complex_subcolumn(A, j, j + 1, N - j - 1);
if (j > 0)
{
gsl_vector_complex_view aj =
gsl_matrix_complex_subrow(A, j, 0, j);
gsl_matrix_complex_view am =
gsl_matrix_complex_submatrix(A, j + 1, 0, N - j - 1, j);
cholesky_complex_conj_vector(&aj.vector);
gsl_blas_zgemv(CblasNoTrans,
GSL_COMPLEX_NEGONE,
&am.matrix,
&aj.vector,
GSL_COMPLEX_ONE,
&av.vector);
cholesky_complex_conj_vector(&aj.vector);
}
gsl_blas_zdscal(1.0 / ajj, &av.vector);
}
}
/* Now store L^H in upper triangle */
for (i = 1; i < N; ++i)
{
for (j = 0; j < i; ++j)
{
z = gsl_matrix_complex_get(A, i, j);
gsl_matrix_complex_set(A, j, i, gsl_complex_conjugate(z));
}
}
return GSL_SUCCESS;
}
} /* gsl_linalg_complex_cholesky_decomp() */
int
gsl_linalg_complex_cholesky_invert(gsl_matrix_complex * LLT)
{
if (LLT->size1 != LLT->size2)
{
GSL_ERROR ("cholesky matrix must be square", GSL_ENOTSQR);
}
else
{
size_t N = LLT->size1;
size_t i, j;
gsl_vector_complex_view v1;
/* invert the lower triangle of LLT */
for (i = 0; i < N; ++i)
{
double ajj;
gsl_complex z;
j = N - i - 1;
{
gsl_complex z0 = gsl_matrix_complex_get(LLT, j, j);
ajj = 1.0 / GSL_REAL(z0);
}
GSL_SET_COMPLEX(&z, ajj, 0.0);
gsl_matrix_complex_set(LLT, j, j, z);
{
gsl_complex z1 = gsl_matrix_complex_get(LLT, j, j);
ajj = -GSL_REAL(z1);
}
if (j < N - 1)
{
gsl_matrix_complex_view m;
m = gsl_matrix_complex_submatrix(LLT, j + 1, j + 1,
N - j - 1, N - j - 1);
v1 = gsl_matrix_complex_subcolumn(LLT, j, j + 1, N - j - 1);
gsl_blas_ztrmv(CblasLower, CblasNoTrans, CblasNonUnit,
&m.matrix, &v1.vector);
gsl_blas_zdscal(ajj, &v1.vector);
}
} /* for (i = 0; i < N; ++i) */
/*
* The lower triangle of LLT now contains L^{-1}. Now compute
* A^{-1} = L^{-H} L^{-1}
*
* The (ij) element of A^{-1} is column i of conj(L^{-1}) dotted into
* column j of L^{-1}
*/
for (i = 0; i < N; ++i)
{
gsl_complex sum;
for (j = i + 1; j < N; ++j)
{
gsl_vector_complex_view v2;
v1 = gsl_matrix_complex_subcolumn(LLT, i, j, N - j);
v2 = gsl_matrix_complex_subcolumn(LLT, j, j, N - j);
/* compute Ainv[i,j] = sum_k{conj(Linv[k,i]) * Linv[k,j]} */
gsl_blas_zdotc(&v1.vector, &v2.vector, &sum);
/* store in upper triangle */
gsl_matrix_complex_set(LLT, i, j, sum);
}
/* now compute the diagonal element */
v1 = gsl_matrix_complex_subcolumn(LLT, i, i, N - i);
gsl_blas_zdotc(&v1.vector, &v1.vector, &sum);
gsl_matrix_complex_set(LLT, i, i, sum);
}
/* copy the Hermitian upper triangle to the lower triangle */
for (j = 1; j < N; j++)
{
for (i = 0; i < j; i++)
{
gsl_complex z = gsl_matrix_complex_get(LLT, i, j);
gsl_matrix_complex_set(LLT, j, i, gsl_complex_conjugate(z));
}
}
return GSL_SUCCESS;
}
} /* gsl_linalg_complex_cholesky_invert() */
int main(void) {
gsl_matrix *TS; /* the training set of real waveforms */
gsl_matrix_complex *cTS; /* the training set of complex waveforms */
size_t TSsize; /* the size of the training set (number of waveforms) */
size_t wl; /* the length of each waveform */
size_t k = 0, j = 0, i = 0, nbases = 0, cnbases = 0;
REAL8 *RB = NULL; /* the real reduced basis set */
COMPLEX16 *cRB = NULL; /* the complex reduced basis set */
LALInferenceREALROQInterpolant *interp = NULL;
LALInferenceCOMPLEXROQInterpolant *cinterp = NULL;
gsl_vector *freqs;
double tolerance = TOLERANCE; /* tolerance for reduced basis generation loop */
TSsize = TSSIZE;
wl = WL;
/* allocate memory for training set */
TS = gsl_matrix_calloc(TSsize, wl);
cTS = gsl_matrix_complex_calloc(TSsize, wl);
/* the waveform model is just a simple chirp so set up chirp mass range for training set */
double fmin0 = 48, fmax0 = 256, f0 = 0., m0 = 0.;
double df = (fmax0-fmin0)/(wl-1.); /* model time steps */
freqs = gsl_vector_alloc(wl); /* times at which to calculate the model */
double Mcmax = 2., Mcmin = 1.5, Mc = 0.;
gsl_vector_view fweights = gsl_vector_view_array(&df, 1);
/* set up training sets (one real and one complex) */
for ( k=0; k < TSsize; k++ ){
Mc = pow(pow(Mcmin, 5./3.) + (double)k*(pow(Mcmax, 5./3.)-pow(Mcmin, 5./3.))/((double)TSsize-1), 3./5.);
for ( j=0; j < wl; j++ ){
f0 = fmin0 + (double)j*(fmax0-fmin0)/((double)wl-1.);
gsl_complex gctmp;
COMPLEX16 ctmp;
m0 = real_model(f0, Mc);
ctmp = imag_model(f0, Mc);
GSL_SET_COMPLEX(&gctmp, creal(ctmp), cimag(ctmp));
gsl_vector_set(freqs, j, f0);
gsl_matrix_set(TS, k, j, m0);
gsl_matrix_complex_set(cTS, k, j, gctmp);
}
}
/* create reduced orthonormal basis from training set */
if ( (RB = LALInferenceGenerateREAL8OrthonormalBasis(&fweights.vector, tolerance, TS, &nbases)) == NULL){
fprintf(stderr, "Error... problem producing basis\n");
return 1;
}
if ( (cRB = LALInferenceGenerateCOMPLEX16OrthonormalBasis(&fweights.vector, tolerance, cTS, &cnbases)) == NULL){
fprintf(stderr, "Error... problem producing basis\n");
return 1;
}
/* free the training set */
gsl_matrix_free(TS);
gsl_matrix_complex_free(cTS);
gsl_matrix_view RBview = gsl_matrix_view_array(RB, nbases, wl);
gsl_matrix_complex_view cRBview = gsl_matrix_complex_view_array((double*)cRB, cnbases, wl);
fprintf(stderr, "No. nodes (real) = %zu, %zu x %zu\n", nbases, RBview.matrix.size1, RBview.matrix.size2);
fprintf(stderr, "No. nodes (complex) = %zu, %zu x %zu\n", cnbases, cRBview.matrix.size1, cRBview.matrix.size2);
/* get the interpolant */
interp = LALInferenceGenerateREALROQInterpolant(&RBview.matrix);
cinterp = LALInferenceGenerateCOMPLEXROQInterpolant(&cRBview.matrix);
/* free the reduced basis */
XLALFree(RB);
XLALFree(cRB);
/* now get the terms for the likelihood with and without the reduced order quadrature
* and do some timing tests */
/* create the model dot model weights */
REAL8 varval = 1.;
gsl_vector_view vars = gsl_vector_view_array(&varval, 1);
gsl_matrix *mmw = LALInferenceGenerateREALModelModelWeights(interp->B, &vars.vector);
gsl_matrix_complex *cmmw = LALInferenceGenerateCOMPLEXModelModelWeights(cinterp->B, &vars.vector);
/* let's create some Gaussian random data */
const gsl_rng_type *T;
gsl_rng *r;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc(T);
REAL8 *data = XLALCalloc(wl, sizeof(REAL8));
COMPLEX16 *cdata = XLALCalloc(wl, sizeof(COMPLEX16));
for ( i=0; i<wl; i++ ){
data[i] = gsl_ran_gaussian(r, 1.0); /* real data */
cdata[i] = gsl_ran_gaussian(r, 1.0) + I*gsl_ran_gaussian(r, 1.0); /* complex data */
}
/* create the data dot model weights */
gsl_vector_view dataview = gsl_vector_view_array(data, wl);
gsl_vector *dmw = LALInferenceGenerateREAL8DataModelWeights(interp->B, &dataview.vector, &vars.vector);
gsl_vector_complex_view cdataview = gsl_vector_complex_view_array((double*)cdata, wl);
gsl_vector_complex *cdmw = LALInferenceGenerateCOMPLEX16DataModelWeights(cinterp->B, &cdataview.vector, &vars.vector);
/* pick a chirp mass and generate a model to compare likelihoods */
double randMc = 1.873; /* a random frequency to create a model */
gsl_vector *modelfull = gsl_vector_alloc(wl);
gsl_vector *modelreduced = gsl_vector_alloc(nbases);
gsl_vector_complex *cmodelfull = gsl_vector_complex_alloc(wl);
gsl_vector_complex *cmodelreduced = gsl_vector_complex_alloc(cnbases);
/* create models */
for ( i=0; i<wl; i++ ){
/* models at all frequencies */
gsl_vector_set(modelfull, i, real_model(gsl_vector_get(freqs, i), randMc));
COMPLEX16 cval = imag_model(gsl_vector_get(freqs, i), randMc);
gsl_complex gcval;
GSL_SET_COMPLEX(&gcval, creal(cval), cimag(cval));
gsl_vector_complex_set(cmodelfull, i, gcval);
}
/* models at interpolant nodes */
for ( i=0; i<nbases; i++ ){ /* real model */
gsl_vector_set(modelreduced, i, real_model(gsl_vector_get(freqs, interp->nodes[i]), randMc));
}
for ( i=0; i<cnbases; i++ ){ /* complex model */
COMPLEX16 cval = imag_model(gsl_vector_get(freqs, cinterp->nodes[i]), randMc);
gsl_complex gcval;
GSL_SET_COMPLEX(&gcval, creal(cval), cimag(cval));
gsl_vector_complex_set(cmodelreduced, i, gcval);
}
/* timing variables */
struct timeval t1, t2, t3, t4;
double dt1, dt2;
/* start with the real model */
/* get the model model term with the full model */
REAL8 mmfull, mmred;
gettimeofday(&t1, NULL);
XLAL_CALLGSL( gsl_blas_ddot(modelfull, modelfull, &mmfull) ); /* real model */
gettimeofday(&t2, NULL);
/* now get it with the reduced order quadrature */
gettimeofday(&t3, NULL);
mmred = LALInferenceROQREAL8ModelDotModel(mmw, modelreduced);
gettimeofday(&t4, NULL);
dt1 = (double)((t2.tv_sec + t2.tv_usec*1.e-6) - (t1.tv_sec + t1.tv_usec*1.e-6));
dt2 = (double)((t4.tv_sec + t4.tv_usec*1.e-6) - (t3.tv_sec + t3.tv_usec*1.e-6));
fprintf(stderr, "Real Signal:\n - M dot M (full) = %le [%.9lf s], M dot M (reduced) = %le [%.9lf s], time ratio = %lf\n", mmfull, dt1, mmred, dt2, dt1/dt2);
/* get the data model term with the full model */
REAL8 dmfull, dmred;
gettimeofday(&t1, NULL);
XLAL_CALLGSL( gsl_blas_ddot(&dataview.vector, modelfull, &dmfull) );
gettimeofday(&t2, NULL);
/* now get it with the reduced order quadrature */
gettimeofday(&t3, NULL);
dmred = LALInferenceROQREAL8DataDotModel(dmw, modelreduced);
gettimeofday(&t4, NULL);
dt1 = (double)((t2.tv_sec + t2.tv_usec*1.e-6) - (t1.tv_sec + t1.tv_usec*1.e-6));
dt2 = (double)((t4.tv_sec + t4.tv_usec*1.e-6) - (t3.tv_sec + t3.tv_usec*1.e-6));
fprintf(stderr, " - D dot M (full) = %le [%.9lf s], D dot M (reduced) = %le [%.9lf s], time ratio = %lf\n", dmfull, dt1, dmred, dt2, dt1/dt2);
/* check difference in log likelihoods */
double Lfull, Lred, Lfrac;
Lfull = mmfull - 2.*dmfull;
Lred = mmred - 2.*dmred;
Lfrac = 100.*fabs(Lfull-Lred)/fabs(Lfull); /* fractional log likelihood difference (in %) */
fprintf(stderr, " - Fractional difference in log likelihoods = %lf%%\n", Lfrac);
XLALFree(data);
gsl_vector_free(modelfull);
gsl_vector_free(modelreduced);
gsl_matrix_free(mmw);
gsl_vector_free(dmw);
/* check log likelihood difference is within tolerance */
if ( Lfrac > LTOL ) { return 1; }
/* now do the same with the complex model */
/* get the model model term with the full model */
COMPLEX16 cmmred, cmmfull;
gsl_complex cmmfulltmp;
gettimeofday(&t1, NULL);
XLAL_CALLGSL( gsl_blas_zdotc(cmodelfull, cmodelfull, &cmmfulltmp) ); /* complex model */
cmmfull = GSL_REAL(cmmfulltmp) + I*GSL_IMAG(cmmfulltmp);
gettimeofday(&t2, NULL);
gettimeofday(&t3, NULL);
cmmred = LALInferenceROQCOMPLEX16ModelDotModel(cmmw, cmodelreduced);
gettimeofday(&t4, NULL);
dt1 = (double)((t2.tv_sec + t2.tv_usec*1.e-6) - (t1.tv_sec + t1.tv_usec*1.e-6));
dt2 = (double)((t4.tv_sec + t4.tv_usec*1.e-6) - (t3.tv_sec + t3.tv_usec*1.e-6));
fprintf(stderr, "Complex Signal:\n - M dot M (full) = %le [%.9lf s], M dot M (reduced) = %le [%.9lf s], time ratio = %lf\n", creal(cmmfull), dt1, creal(cmmred), dt2, dt1/dt2);
COMPLEX16 cdmfull, cdmred;
gsl_complex cdmfulltmp;
gettimeofday(&t1, NULL);
XLAL_CALLGSL( gsl_blas_zdotc(&cdataview.vector, cmodelfull, &cdmfulltmp) );
cdmfull = GSL_REAL(cdmfulltmp) + I*GSL_IMAG(cdmfulltmp);
gettimeofday(&t2, NULL);
gettimeofday(&t3, NULL);
cdmred = LALInferenceROQCOMPLEX16DataDotModel(cdmw, cmodelreduced);
gettimeofday(&t4, NULL);
dt1 = (double)((t2.tv_sec + t2.tv_usec*1.e-6) - (t1.tv_sec + t1.tv_usec*1.e-6));
dt2 = (double)((t4.tv_sec + t4.tv_usec*1.e-6) - (t3.tv_sec + t3.tv_usec*1.e-6));
fprintf(stderr, " - D dot M (full) = %le [%.9lf s], D dot M (reduced) = %le [%.9lf s], time ratio = %lf\n", creal(cdmfull), dt1, creal(cdmred), dt2, dt1/dt2);
/* check difference in log likelihoods */
Lfull = creal(cmmfull) - 2.*creal(cdmfull);
Lred = creal(cmmred) - 2.*creal(cdmred);
Lfrac = 100.*fabs(Lfull-Lred)/fabs(Lfull); /* fractional log likelihood difference (in %) */
fprintf(stderr, " - Fractional difference in log likelihoods = %lf%%\n", Lfrac);
XLALFree(cdata);
gsl_vector_complex_free(cmodelfull);
gsl_vector_complex_free(cmodelreduced);
gsl_matrix_complex_free(cmmw);
gsl_vector_complex_free(cdmw);
LALInferenceRemoveREALROQInterpolant( interp );
LALInferenceRemoveCOMPLEXROQInterpolant( cinterp );
/* check log likelihood difference is within tolerance */
if ( Lfrac > LTOL ) { return 1; }
return 0;
}
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;
}
}