+}
+
+ex clifford_star_bar(const ex & e, bool do_bar, unsigned options)
+{
+ pointer_to_map_function_2args<bool, unsigned> fcn(clifford_star_bar, do_bar, options | 1);
+
+ // is a child, no need to expand
+ ex e1= (options & 1 ? e : e.expand());
+
+ if (is_a<ncmul>(e1) ) { // reversing order of clifford units
+ exvector ev, cv;
+ ev.reserve(e1.nops());
+ cv.reserve(e1.nops());
+ // separate clifford and non-clifford entries
+ for (int i= 0; i < e1.nops(); ++i) {
+ if (is_a<clifford>(e1.op(i)) && is_a<cliffordunit>(e1.op(i).op(0)))
+ cv.push_back(e1.op(i));
+ else
+ ev.push_back(e1.op(i));
+ }
+ for (auto i=cv.rbegin(); i!=cv.rend(); ++i) { // reverse order of Clifford units
+ ev.push_back(i->conjugate());
+ }
+ // For clifford_bar an odd number of clifford units reverts the sign
+ if (do_bar && (cv.size() % 2 == 1))
+ return -dynallocate<ncmul>(std::move(ev));
+ else
+ return dynallocate<ncmul>(std::move(ev));
+ } else if (is_a<clifford>(e1) && is_a<cliffordunit>(e1.op(0))) {
+ if (do_bar)
+ return -e;
+ else
+ return e;
+ } else if (is_a<power>(e1)) {
+ // apply the procedure to the base of a power
+ return pow(clifford_star_bar(e1.op(0), do_bar, 0), e1.op(1));
+ } else if (is_a<add>(e1) || is_a<mul>(e1) || e.info(info_flags::list)) {
+ // recurse into subexpressions
+ return e1.map(fcn);
+ } else // nothing meaningful can be done
+ return e;
+}
+
+ex clifford_prime(const ex & e)
+{
+ pointer_to_map_function fcn(clifford_prime);
+ if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
+ return -e;
+ } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
+ || is_a<matrix>(e) || e.info(info_flags::list)) {
+ return e.map(fcn);
+ } else if (is_a<power>(e)) {
+ return pow(clifford_prime(e.op(0)), e.op(1));
+ } else
+ return e;
+}
+
+ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
+{
+ pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
+ bool need_reevaluation = false;
+ ex e1 = e;
+ if (! (options & 1) ) { // is not a child
+ if (options & 2)
+ e1 = expand_dummy_sum(e, true);
+ e1 = canonicalize_clifford(e1);
+ }
+
+ if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
+ if (is_a<diracone>(e1.op(0)))
+ return 1;
+ else
+ throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
+ } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
+ || is_a<matrix>(e1) || e1.info(info_flags::list)) {
+ if (options & 3) // is a child or was already expanded
+ return e1.map(fcn);
+ else
+ try {
+ return e1.map(fcn);
+ } catch (std::exception &p) {
+ need_reevaluation = true;
+ }
+ } else if (is_a<power>(e1)) {
+ if (options & 3) // is a child or was already expanded
+ return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
+ else
+ try {
+ return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
+ } catch (std::exception &p) {
+ need_reevaluation = true;
+ }
+ }
+ if (need_reevaluation)
+ return remove_dirac_ONE(e, rl, options | 2);
+ return e1;
+}
+
+int clifford_max_label(const ex & e, bool ignore_ONE)
+{
+ if (is_a<clifford>(e))
+ if (ignore_ONE && is_a<diracone>(e.op(0)))
+ return -1;
+ else
+ return ex_to<clifford>(e).get_representation_label();
+ else {
+ int rl = -1;
+ for (size_t i=0; i < e.nops(); i++)
+ rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
+ return rl;
+ }
+}
+
+ex clifford_norm(const ex & e)
+{
+ return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
+}
+
+ex clifford_inverse(const ex & e)
+{
+ ex norm = clifford_norm(e);
+ if (!norm.is_zero())
+ return clifford_bar(e) / pow(norm, 2);
+ else
+ throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
+}
+
+ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
+{
+ if (!ex_to<idx>(mu).is_dim_numeric())
+ throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
+ ex e = clifford_unit(mu, metr, rl);
+ return lst_to_clifford(v, e);
+}
+
+ex lst_to_clifford(const ex & v, const ex & e) {
+ unsigned min, max;
+
+ if (is_a<clifford>(e)) {
+ ex mu = e.op(1);
+ ex mu_toggle
+ = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
+ unsigned dim = get_dim_uint(mu);
+
+ if (is_a<matrix>(v)) {
+ if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
+ min = ex_to<matrix>(v).rows();
+ max = ex_to<matrix>(v).cols();
+ } else {
+ min = ex_to<matrix>(v).cols();
+ max = ex_to<matrix>(v).rows();
+ }
+ if (min == 1) {
+ if (dim == max)
+ return indexed(v, mu_toggle) * e;
+ else if (max - dim == 1) {
+ if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
+ return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 0, 1, 1, dim), mu_toggle) * e;
+ else
+ return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 1, dim, 0, 1), mu_toggle) * e;
+ } else
+ throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
+ } else
+ throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
+ } else if (v.info(info_flags::list)) {
+ if (dim == ex_to<lst>(v).nops())
+ return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
+ else if (ex_to<lst>(v).nops() - dim == 1)
+ return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(matrix(dim+1, 1, ex_to<lst>(v)), 1, dim, 0, 1), mu_toggle) * e;
+ else
+ throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
+ } else
+ throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
+ } else
+ throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
+}
+
+/** Auxiliary structure to define a function for striping one Clifford unit
+ * from vectors. Used in clifford_to_lst(). */
+static ex get_clifford_comp(const ex & e, const ex & c, bool root=true)
+{
+ // make expansion on the top-level call only
+ ex e1=(root? e.expand() : e);
+
+ pointer_to_map_function_2args<const ex &, bool> fcn(get_clifford_comp, c, false);
+ int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
+ int rl=ex_to<clifford>(c).get_representation_label();
+
+ if ( (is_a<add>(e1) || e1.info(info_flags::list) || is_a<matrix>(e1))) {
+ return e1.map(fcn);
+ } else if (is_a<ncmul>(e1) || is_a<mul>(e1)) {
+ // searches are done within products only
+ exvector ev, all_dummy=get_all_dummy_indices(e1);
+ bool found=false, same_value_found=false;
+ ex dummy_ind=0;
+ ev.reserve(e1.nops());
+ for (int i=0; i < e1.nops();++i) {
+ // look for a Clifford unit with the same metric and representation label,
+ // if found remember its index
+ if (is_a<clifford>(e1.op(i)) && ex_to<clifford>(e1.op(i)).get_representation_label() == rl
+ && is_a<cliffordunit>(e1.op(i).op(0)) && ex_to<clifford>(e1.op(i)).same_metric(c)) { // same Clifford unit
+ if (found)
+ throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
+ found=true;
+ if (ex_to<idx>(e1.op(i).op(1)).is_numeric() &&
+ (ival == ex_to<numeric>(ex_to<idx>(e1.op(i).op(1)).get_value()).to_int())) {
+ same_value_found = true; // desired index value is found
+ } else if ((std::find(all_dummy.begin(), all_dummy.end(), e1.op(i).op(1)) != all_dummy.end())
+ || (is_a<varidx>(e1.op(i).op(1))
+ && std::find(all_dummy.begin(), all_dummy.end(),
+ ex_to<varidx>(e1.op(i).op(1)).toggle_variance()) != all_dummy.end())) {
+ dummy_ind=(e1.op(i).op(1)); // suitable dummy index found
+ } else
+ ev.push_back(e.op(i)); // another index value
+ } else
+ ev.push_back(e1.op(i));
+ }
+
+ if (! found) // no Clifford units found at all
+ throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
+
+ ex res=dynallocate<ncmul>(std::move(ev));
+ if (same_value_found) {
+ return res;
+ } else if (! dummy_ind.is_zero()) { // a dummy index was found
+ if (is_a<varidx>(dummy_ind))
+ dummy_ind = ex_to<varidx>(dummy_ind).toggle_variance();
+ return res.subs(dummy_ind==ival, subs_options::no_pattern);
+ } else // found a Clifford unit with another index
+ return 0;
+ } else if (e1.is_zero()) {
+ return 0;
+ } else if (is_a<clifford>(e1) && is_a<cliffordunit>(e1.op(0)) && ex_to<clifford>(e1).same_metric(c)) {
+ if (ex_to<idx>(e1.op(1)).is_numeric() &&
+ (ival == ex_to<numeric>(ex_to<idx>(e1.op(1)).get_value()).to_int()) )
+ return 1;
+ else
+ return 0;
+ } else
+ throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
+}
+
+lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
+{
+ GINAC_ASSERT(is_a<clifford>(c));
+ ex mu = c.op(1);
+ if (! ex_to<idx>(mu).is_dim_numeric())
+ throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
+ unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
+
+ if (algebraic) // check if algebraic method is applicable
+ for (unsigned int i = 0; i < D; i++)
+ if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
+ || (! is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
+ algebraic = false;
+ lst V;
+ ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)))/2;
+ if (! v0.is_zero())
+ V.append(v0);
+ ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
+ if (algebraic) {
+ for (unsigned int i = 0; i < D; i++)
+ V.append(remove_dirac_ONE(
+ simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e1))
+ / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
+ } else {
+ try {
+ for (unsigned int i = 0; i < D; i++)
+ V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
+ } catch (std::exception &p) {
+ /* Try to expand dummy summations to simplify the expression*/
+ e1 = canonicalize_clifford(expand_dummy_sum(e, true));
+ V.remove_all();
+ v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)))/2;
+ if (! v0.is_zero()) {
+ V.append(v0);
+ e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
+ }
+ for (unsigned int i = 0; i < D; i++)
+ V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
+ }
+ }
+ return V;
+}
+
+
+ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl)
+{
+ ex x, D, cu;
+
+ if (! is_a<matrix>(v) && ! v.info(info_flags::list))
+ throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
+
+ if (is_a<clifford>(G)) {
+ cu = G;
+ } else {
+ if (is_a<indexed>(G)) {
+ D = ex_to<idx>(G.op(1)).get_dim();
+ varidx mu(dynallocate<symbol>(), D);
+ cu = clifford_unit(mu, G, rl);
+ } else if (is_a<matrix>(G)) {
+ D = ex_to<matrix>(G).rows();
+ idx mu(dynallocate<symbol>(), D);
+ cu = clifford_unit(mu, G, rl);
+ } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
+
+ }
+
+ x = lst_to_clifford(v, cu);
+ ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
+ return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
+}
+
+ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
+{
+ if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
+ return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
+ else
+ throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));