9.6.2 The Newton-Raphson method  9.6 Solution of simultaneous  9.6 Solution of simultaneous

9.6.1 Gauss-Seidel iteration

Simultaneous equations that are linear in the coefficients can be solved by the method of matrix inversion. The alternative of Gauss-Seidel iteration has two advantages:

• Errors do not accumulate during the calculation. If the procedure converges, it approaches the correct answer without rounding errors such as can occur during inversion of large matrices. As a result, very large sets of equations can be solved; the method has been applied to sets of thousands of equations.
• The method can be used for nonlinear sets of equations.

To apply the method, the set of equations must be written in a form so that each of the variables is given by one of the equations in the set. For example, the set might be written in the form

$$= f_1(x_2,x_3,\dots)$$  x₁

$$= f_2(x_1,x_3,\dots)$$ x₂ (9.52)

$$= f_3(x_1,x_2,x_4,\dots)$$ x₃

 

etc. Then the sequence of equations is solved by using the values from all the preceding steps at each step in the solution, and the procedure is repeated iteratively.

An example where the method is effective is in the calculation of the lifted condensation level (LCL) of an unsaturated air parcel. This is needed, for example, in the calculation of equivalent potential temperature for an unsaturated air parcel. The LCL is the point at which the parcel, lifted adiabatically, reaches saturation with respect to water. During this ascent, potential temperature and mixing ratio remain constant, so finding the LCL is equivalent to finding the temperature and pressure for which a saturated air parcel would have the same potential temperature and mixing ratio as the observed parcel. If the observed temperature, pressure, and dewpoint are T, p, and T_d, and the temperature and pressure at the LCL are T_L and P_L, these conditions are equivalent to

$$p_L = p\left({{T_L}\over{T}}\right)^{c_p/R}$$ (9.53)

  

$$e_s(T_L) = e_s(T_d){{p_L}\over{p}}$$ (9.54)

 

If the last equation is solved for T_L (given p_L), for example by Lagrange interpolation, then this system of equations readily converges to the conditions at the LCL.  The sequence for T=20, T_d=10, and p=800 mb is:
 
 

	TL	pL
	293.15	800.0
	283.26	709.51
	281.37	693.09
	281.03	690.13
	280.96	689.59
	280.95	689.49
	"	689.47
	"	"

 

   
 9.6.2 The Newton-Raphson method  9.6 Solution of simultaneous  9.6 Solution of simultaneous