San José State University


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Thayer Watkins
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The Discrete Approximations of the 2nd
Derivatives of a Function of Two Variables

Let B(n, p) be a function known only for integral values of n and p. An approximation of ∂B/∂n at (n, p) is [B(n+1,p)−B(n-1,p)]/2 and likewise [B(n,p+1)−B(n,p-1)]/2 is an approximation of ∂B/∂p at (n, p).

To get an estimate of ∂²B/∂n² which is centered at (n,p) first estimates of ∂B/∂n are obtained at two different points; i.e.,

[B(n+1,p)−B(n,p)] for ∂B/∂n at (n+½,p)
and
[B(n,p)−B(n-1,p)] for ∂B/∂n at (n-½,p)

The difference of these two divided by their separation distance of 1 gives

∂²B/∂n² ≅ [B(n+1,p)−B(n,p)]− [B(n,p)−B(n-1,p)]
which reduces to
∂²B/∂n² ≅ [B(n+1,p)+B(n-1,p)−2B(n,p)]

Likewise

∂²B/∂p² ≅ [B(n,p+1)+B(n,p-1)−2B(n,p)]

The computation can be depicted visually as follows:


for ∂²B/∂n²

and


for ∂²B/∂p²

In both cases the second derivative is the sum of the red values less twice the blue value.

The reason for the particular designations of the variables is that the motivation for this material came from a study concerning the binding energis of nuclides. That study found a remarkable correspondence between ∂²B/∂n² and ∂²B/∂p² along the line where n=p=2a, where a is the potential number of alpha particles in the nuclude. The purpose of this study was to make sure this was an empirical relation rather than something mathematically required from the estimating procedure.

While there is no mathematical requirement that ∂²B/∂n² be equal to ∂²B/∂p² there is such a mathematical requirement for the cross derivatives.

The Cross Derivatives

Consider first ∂²B/∂p∂n. First estimates of ∂B/∂n can be obtained at two different levels of p as

[B(n+1,p+1)−B(n-1,p+1)]/2 as ∂B/∂n at (n,p+1)
and
[B(n+1,p-1)−B(n-1,p-1)]/2 as ∂B/∂n at (n,p-1)

Then the difference of these two divided by their separation distance of 2 is the estimate of the cross derivative ∂²B/∂p∂n; i.e.,

∂²B/∂p∂n ≅ [(B(n+1,p+1)−B(n+1,p-1))−(B(n-1,p+1)−B(n-1,p-1))]/4
or, equivalently
∂²B/∂p∂n ≅ [(B(n+1,p+1)−B(n-1,p+1))−(B(n+1,p-1)−B(n-1,p-1))]/4

This latter formula is the one which arises in the process of obtaining an estimate of ∂²B/∂n∂p. Thus the estimates of the cross derivatives are exactly equal as are the true values.

The cross derivatives may also be expressed as

∂²B/∂p∂n = ∂²B/∂p∂n ≅ ≅ {[(B(n+1,p+1)+B(n-1,p-1)] − [B(n-1,p+1)+B(n-1,p+1)]}/4

The visual representation of the computation of the cross derivative is as follows.

Thus the cross derivatives are both the sum of the red values less the sum of the blue values with the result divided by 4.

(To be continued.)


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