Why are the Linearised Equations not Satisifed by a Multi-step Solution?

A simple case will make this clear.

Consider the levels equation

V = P * Q      ( eq1 )

(dollar value = price times quantity). The linearised version of this is

p_V = p_P + p_Q    ( eq2)

[Percentage change in value is the sum of the percentage changes in price and quantity.]

Suppose that in a simulation P increases by 5% and Q increases by 3%. Then the dollar value increases by a little over 8 (=3+5) percent. More precisely, in the accurate solution to the levels equation (eq1) above,

new_V = old_V*1.05*1.03 = old_V*1.0815

where the 1.05 term is because P increases by 5% and the 1.03 term is because Q increases by 3%.

Thus the accurate solution of the underlying levels equation is that V increases by 8.15%.

However the linearised equation (eq2) is not satisfied if

p_P = 5, p_Q = 3 and p_V = 8.15

The whole point of doing a multi-step calculation (with sufficently many steps, followed by extrapolation) is to obtain an accurate solution of the underliyng levels equations. Thus you see that you would not want the linearised equations to be satisifed by the resulting solution.

Another way of looking at this is that the linearised equation (eq2) omits the product term. The linearised equation is only a good approximation to the levels equation if p_P and p_Q are both very small (when you can ignore the cross term). The percentage-change equation which is identical to the levels equation is

p_V = p_P + p_Q + [p_P*p_Q/100]    ( eq3 )

It is because this this product term is missing from (eq2) that accurate solutions of (eq1) are not solutions of (eq2) [but they are solutions of (eq3)].

It may be a bit of a mystery to you why, when GEMPACK solves the linearised equations at each step of a multi-step calculation, the resulting solution is indeed a solution of the underlying levels equations. Suffice to say that the theory on which this rests is well-known and applies in many areas of application (including science and engineering).

But, from the example above, it should no longer be a mystery why the linearised equations are not satisfied by the resulting solution (and why you would not want them to be satisfied).

You may be interested in a related example.

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