Derivation of Finite Difference Using Multiple Points
Theorem
Given a differentiable function eq01, at a point eq03 where the n-th derivative’s value is eq06, can be approximated by using the same number of points eq10 from a finite set eq09 with cardinality eq08 for sufficiently small eq07 as follows: eq01 Here, eq11 is determined as follows: eq02 The points eq10 are also referred to as Stencil Points.
- eq14 is the inverse matrix.
- eq15 is the factorial.
- eq16 is the Kronecker delta.
Explanation
eq03 When only discretized data is available and there is no actual function available, or it is too difficult to differentiate, one can only refer to the average rate of change as above. However, instead of using information from two very close points, one can use two points from the left and right like the following: eq04 Generally, using more points increases accuracy, and the worse the data is, the more one has to rely on such calculations.
Proof [^1]
ref1 Strategy: Similar to the original document, it’s just shown in a brief, example-like manner. It’s over once a system of equations is set up through the Taylor theorem.
For instance, to calculate eq17, consider using five points as follows, our goal is to find eq19 that approximates the following well: eq05 Here, each eq20 is expanded as follows according to the Taylor theorem: eq06 As usual, eq21 is small enough to be neglected, and regardless of what eq22, eq23, eq24, eq25, eq17 is, for the above approximation to hold, eq07 must be satisfied. Simplifying into the form of a determinant: eq08 In the last line, the left-hand side has the form we saw in the theorem, and on the right-hand side, everything but the eq05-th row we’re interested in turns into eq28, which can be confirmed. Moving the matrix on the left to the right gives us the approximation we wanted.
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For example, if one wishes to approximate the second derivative using four points, one can use the following matrix equation: eq09 and solve it to obtain the following. Direct calculations can be tough, so let’s rely on the power of computers. eq10
Accuracy
Though not rigorously shown, the accuracy is said to be eq30, and the more stencil points used, the more accurate it becomes, and the more differentiation done, the higher the cost becomes.
Tables [^2]
ref2
