9.5.1 Newton's method  9. Numerical Methods  9.4.4 The cubic spline

9.5 Roots of equations

Cases often arise where a desired solution can be specified by an equation, but the equation is difficult or impossible to solve analytically. In this section, some techniques for solution of such equations will be discussed. Three examples that have arisen in the author's recent research are:

1.

Green (1975) gives an equation that relates the semi-major axis a of a raindrop to its volume-equivalent radius a₀. The equation, after some transformation, can be written as

$$a_0^2 - {{\sigma}\over{g\rho_w}}\left[\left({{a}\over{a_0}}\......left({{a}\over{a_0}}\right)^2+{{a}\over{a_0}}\right] = 0 \ \ .$$ (9.34)

  

To estimate rainrate or radar reflectivity, one needs to be able to find the volume-equivalent radius from a measurement of the semi-major axis. However, the equation involves a high-order polynomial and so is difficult to solve.

2.

For reversible adiabatic ascent of a cloud parcel, the quantity $\Theta_q$ (the wet-equivalent potential temperature; cf. Paluch 1979) is conserved. This provides a basis for finding the temperature of a cloud parcel at some altitude above cloud base (specified by a pressure and by the requirement that the humidity correspond to water saturation in cloud) if the cloud-base pressure and temperature are known. From this one can find the liquid water content corresponding to adiabatic ascent to that level. This procedure requires solution of the equation

$$f(T)=\Theta_q(p,T)- \Theta_{q,base} = 0$$ (9.35)

  

where $\Theta_q(p,T)$ involves the pressure p and temperature T in factors that appear both multiplying and in the argument of an exponential expression:

$$XXX \ .$$ (9.36)

  

3.

To construct skew-T or other thermodynamic diagrams, it is necessary to construct lines of constant $\Theta_e$ by finding the appropriate temperature corresponding to a given $\Theta_e$ and pressure:

$$f(T) = \Theta_e(T,p)-\Theta_{e,0} = 0.$$ (9.37)

  

In all these cases, the problems reduce to finding solutions to equations of the form f(x)=0. Some methods for finding such solutions numerically are discussed in this section.

An invaluable aid in choosing an appropriate method of solution is to plot the function. This preliminary step will locate approximate roots and reveal the extent to which complicated procedures will be needed, e.g., to deal with multiple roots. It should almost always be the first step in numerical solution of such equations.
 

   
 9.5.1 Newton's method  9. Numerical Methods  9.4.4 The cubic spline