Modelica.Media.Common.OneNonLinearEquation.f_nonlinear_Data

Information

Modelica definition

replaceable record f_nonlinear_Data 
  "Data specific for function f_nonlinear"
  extends Modelica.Icons.Record;
end f_nonlinear_Data;

Modelica.Media.Common.OneNonLinearEquation.solve

Information

Inputs

Type	Name	Default	Description
Real	y_zero		Determine x_zero, such that f_nonlinear(x_zero) = y_zero
Real	x_min		Minimum value of x
Real	x_max		Maximum value of x
Real	pressure	0.0	disregaded variables (here always used for pressure)
Real	X[:]	fill(0, 0)	disregaded variables (here always used for composition)
f_nonlinear_Data	f_nonlinear_data		Additional data for function f_nonlinear
Real	x_tol	100*Modelica.Constants.eps	Relative tolerance of the result

Outputs

Type	Name	Description
Real	x_zero	f_nonlinear(x_zero) = y_zero

Modelica definition

replaceable function solve 
  "Solve f_nonlinear(x_zero)=y_zero; f_nonlinear(x_min) - y_zero and f_nonlinear(x_max)-y_zero must have different sign"
  import Modelica.Utilities.Streams.error;
  extends Modelica.Icons.Function;
  input Real y_zero "Determine x_zero, such that f_nonlinear(x_zero) = y_zero";
  input Real x_min "Minimum value of x";
  input Real x_max "Maximum value of x";
  input Real pressure = 0.0 
    "disregaded variables (here always used for pressure)";
  input Real[:] X = fill(0,0) 
    "disregaded variables (here always used for composition)";
   input f_nonlinear_Data f_nonlinear_data 
    "Additional data for function f_nonlinear";
   input Real x_tol =  100*Modelica.Constants.eps 
    "Relative tolerance of the result";
   output Real x_zero "f_nonlinear(x_zero) = y_zero";
protected 
   constant Real eps = Modelica.Constants.eps "machine epsilon";
   constant Real x_eps = 1e-10 
    "Slight modification of x_min, x_max, since x_min, x_max are usually exactly at the borders T_min/h_min and then small numeric noise may make the interval invalid";
   Real x_min2 = x_min - x_eps;
   Real x_max2 = x_max + x_eps;
   Real a = x_min2 "Current best minimum interval value";
   Real b = x_max2 "Current best maximum interval value";
   Real c "Intermediate point a <= c <= b";
   Real d;
   Real e "b - a";
   Real m;
   Real s;
   Real p;
   Real q;
   Real r;
   Real tol;
   Real fa "= f_nonlinear(a) - y_zero";
   Real fb "= f_nonlinear(b) - y_zero";
   Real fc;
   Boolean found = false;
algorithm 
   // Check that f(x_min) and f(x_max) have different sign
   fa :=f_nonlinear(x_min2, pressure, X, f_nonlinear_data) - y_zero;
   fb :=f_nonlinear(x_max2, pressure, X, f_nonlinear_data) - y_zero;
   fc := fb;
   if fa > 0.0 and fb > 0.0 or 
      fa < 0.0 and fb < 0.0 then
      error("The arguments x_min and x_max to OneNonLinearEquation.solve(..)\n" +
            "do not bracket the root of the single non-linear equation:\n" +
            "  x_min  = " + String(x_min2) + "\n" +
            "  x_max  = " + String(x_max2) + "\n" +
            "  y_zero = " + String(y_zero) + "\n" +
            "  fa = f(x_min) - y_zero = " + String(fa) + "\n" +
            "  fb = f(x_max) - y_zero = " + String(fb) + "\n" +
            "fa and fb must have opposite sign which is not the case");
   end if;

   // Initialize variables
   c :=a;
   fc :=fa;
   e :=b - a;
   d :=e;

   // Search loop
   while not found loop
      if abs(fc) < abs(fb) then
         a :=b;
         b :=c;
         c :=a;
         fa :=fb;
         fb :=fc;
         fc :=fa;
      end if;

      tol :=2*eps*abs(b) + x_tol;
      m :=(c - b)/2;

      if abs(m) <= tol or fb == 0.0 then
         // root found (interval is small enough)
         found :=true;
         x_zero :=b;
      else
         // Determine if a bisection is needed
         if abs(e) < tol or abs(fa) <= abs(fb) then
            e :=m;
            d :=e;
         else
            s :=fb/fa;
            if a == c then
               // linear interpolation
               p :=2*m*s;
               q :=1 - s;
            else
               // inverse quadratic interpolation
               q :=fa/fc;
               r :=fb/fc;
               p :=s*(2*m*q*(q - r) - (b - a)*(r - 1));
               q :=(q - 1)*(r - 1)*(s - 1);
            end if;

            if p > 0 then
               q :=-q;
            else
               p :=-p;
            end if;

            s :=e;
            e :=d;
            if 2*p < 3*m*q-abs(tol*q) and p < abs(0.5*s*q) then
               // interpolation successful
               d :=p/q;
            else
               // use bi-section
               e :=m;
               d :=e;
            end if;
         end if;

         // Best guess value is defined as "a"
         a :=b;
         fa :=fb;
         b :=b + (if abs(d) > tol then d else if m > 0 then tol else -tol);
         fb :=f_nonlinear(b, pressure, X, f_nonlinear_data) - y_zero;

         if fb > 0 and fc > 0 or 
            fb < 0 and fc < 0 then
            // initialize variables
            c :=a;
            fc :=fa;
            e :=b - a;
            d :=e;
         end if;
      end if;
   end while;
end solve;