Solving a rectangular system of nonlinear equations with GSL

Question

I want to program a C++ code able to solve a set of equation systems which are generally nonlinear.

I am already able to solve a system of m equations with n variables using GSL when m = n by following its documentation.

However, often I have fewer equations than variables. Can I use the same functionality of GSL, or do I have to opt for a different approach or another library? Why is there not a section about solving rectangular systems in the documentation?

This is what I've tried:

#include <cmath>
#include <gsl/gsl_vector.h>
#include <iostream>
#include <gsl/gsl_multiroots.h>

void print_vector(gsl_vector *gsl) {
  printf("[");
  for (int r = 0; r < gsl->size; r++) {
    if (r != gsl->size - 1) {
      printf("%+.5f; ", gsl->data[r]);
    } else {
      printf("%+.5f", gsl->data[r]);
    }
  }
  printf("]\n");
}

int evaluate(const gsl_vector *x, void *params, gsl_vector *f) {
  const double x0 = gsl_vector_get(x, 0);
  const double x1 = gsl_vector_get(x, 1);
  const double x2 = gsl_vector_get(x, 2);

  const double y0 = 0.1 * pow(x0, 2) + 0.1 * pow(x1, 2) + 2 - x2;

  // alone not working
  gsl_vector_set(f, 0, y0);

  // with this not working
  // gsl_vector_set(f, 1, 0);
  // gsl_vector_set(f, 2, 0);

  // with this not working
  // gsl_vector_set(f, 1, y0);
  // gsl_vector_set(f, 2, y0);

  return GSL_SUCCESS;
}

int main() {
  const size_t n = 3;

  const gsl_multiroot_fsolver_type *T;
  T = gsl_multiroot_fsolver_hybrids; // "iteration is not making progress towards solution"
//  T = gsl_multiroot_fsolver_hybrid; // "iteration is not making progress towards solution"
//  T = gsl_multiroot_fsolver_broyden; // "gsl: lu.c:266: ERROR: matrix is singular"
//  T = gsl_multiroot_fsolver_dnewton; // "gsl: lu.c:147: ERROR: matrix is singular"
  gsl_multiroot_fsolver *s = gsl_multiroot_fsolver_alloc(T, n);

  gsl_multiroot_function f_multiroot_function = {evaluate, n, nullptr};

  gsl_vector *x_gsl = gsl_vector_alloc(n);
  gsl_vector_set(x_gsl, 0, 3.4);
  gsl_vector_set(x_gsl, 1, 2.7);
  gsl_vector_set(x_gsl, 2, 3.8);

  std::cout << "x_gsl: ";
  print_vector(x_gsl);

  gsl_multiroot_fsolver_set(s, &f_multiroot_function, x_gsl);

  int status;
  size_t iter = 0;
  do {
    iter++;
    status = gsl_multiroot_fsolver_iterate(s);

    std::cout << "s->x=" << std::endl;
    print_vector(s->x);

    if (status) {
      std::cout << "STOPPING: " << gsl_strerror(status) << "." << std::endl;
      break;
    }

    status = gsl_multiroot_test_residual(s->f, 1e-5);
  } while (status == GSL_CONTINUE && iter < 20);

  return 0;
}

And this

x_gsl: [+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
s->x=
[+3.40000; +2.70000; +3.80000]
STOPPING: iteration is not making progress towards solution.

Process finished with exit code 0

or this

x_gsl: [+3.40000; +2.70000; +3.80000]
gsl: lu.c:147: ERROR: matrix is singular
Default GSL error handler invoked.

Process finished with exit code 134 (interrupted by signal 6: SIGABRT)

is what I get.