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  /* Computation of Gerschgorin interval for symmetric, tridiagonal matrix */
  
  #include <cstdio>
  #include <cstdlib>
  #include <cmath>
  #include <cfloat>
  
  #include "util.h"
  #include "gerschgorin.h"
  
  ////////////////////////////////////////////////////////////////////////////////
  //! Compute Gerschgorin interval for symmetric, tridiagonal matrix
  //! @param  d  diagonal elements
  //! @param  s  superdiagonal elements
  //! @param  n  size of matrix
  //! @param  lg  lower limit of Gerschgorin interval
  //! @param  ug  upper limit of Gerschgorin interval
  ////////////////////////////////////////////////////////////////////////////////
  void computeGerschgorin(float *d, float *s, unsigned int n, float &lg,
                          float &ug) {
    lg = FLT_MAX;
    ug = -FLT_MAX;
  
    // compute bounds
    for (unsigned int i = 1; i < (n - 1); ++i) {
      // sum over the absolute values of all elements of row i
      float sum_abs_ni = fabsf(s[i - 1]) + fabsf(s[i]);
  
      lg = min(lg, d[i] - sum_abs_ni);
      ug = max(ug, d[i] + sum_abs_ni);
    }
  
    // first and last row, only one superdiagonal element
  
    // first row
    lg = min(lg, d[0] - fabsf(s[0]));
    ug = max(ug, d[0] + fabsf(s[0]));
  
    // last row
    lg = min(lg, d[n - 1] - fabsf(s[n - 2]));
    ug = max(ug, d[n - 1] + fabsf(s[n - 2]));
  
    // increase interval to avoid side effects of fp arithmetic
    float bnorm = max(fabsf(ug), fabsf(lg));
  
    // these values depend on the implementation of floating count that is
    // employed in the following
    float psi_0 = 11 * FLT_EPSILON * bnorm;
    float psi_n = 11 * FLT_EPSILON * bnorm;
  
    lg = lg - bnorm * 2 * n * FLT_EPSILON - psi_0;
    ug = ug + bnorm * 2 * n * FLT_EPSILON + psi_n;
  
    ug = max(lg, ug);
  }
  