Approximating function derivatives

[Thomas Klemm]

Following Albert's approach we can disentangle the calculation of odd and even derivatives.

\(
\begin{align}
f_{+1} &= f(x_0 + h) &= f(x_0) + {f}'(x_0) h + \frac{1}{2}{f}''(x_0) h^2 + \frac{1}{6}{f}'''(x_0) h^3 + \mathcal{O}(h^4) \\
\\
f_{-1} &= f(x_0 - h) &= f(x_0) - {f}'(x_0) h + \frac{1}{2}{f}''(x_0) h^2 - \frac{1}{6}{f}'''(x_0) h^3 + \mathcal{O}(h^4) \\
\\
f_{+2} &= f(x_0 + 2h) &= f(x_0) + 2{f}'(x_0) h + 2{f}''(x_0) h^2 + \frac{4}{3}{f}'''(x_0) h^3 + \mathcal{O}(h^4) \\
\\
f_{-2} &= f(x_0 - 2h) &= f(x_0) - 2{f}'(x_0) h + 2{f}''(x_0) h^2 - \frac{4}{3}{f}'''(x_0) h^3 + \mathcal{O}(h^4) \\
\\
\end{align}
\)

Even derivatives

\(
\begin{align}
f_{+1} + f_{-1} &= 2f(x_0) + {f}''(x_0) h^2 + \mathcal{O}(h^4) \\
f_{+2} + f_{-2} &= 2f(x_0) + 4{f}''(x_0) h^2 + \mathcal{O}(h^4) \\
\end{align}
\)

Or:

\(
\begin{bmatrix}
f_{+1} + f_{-1} \\
f_{+2} + f_{-2} \\
\end{bmatrix}
\approx
\begin{bmatrix}
2 & 1 \\
2 & 4 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
f(x_0) \\
{f}''(x_0) h^2 \\
\end{bmatrix}
\)


This leads to:

\(
\begin{bmatrix}
f(x_0) \\
{f}''(x_0) h^2 \\
\end{bmatrix}
\approx
\begin{bmatrix}
\frac{2}{3} & - \frac{1}{6} \\
- \frac{1}{3} & \frac{1}{3} \\
\end{bmatrix}
\cdot
\begin{bmatrix}
f_{+1} + f_{-1} \\
f_{+2} + f_{-2} \\
\end{bmatrix}
\)

Odd derivatives

\(
\begin{align}
f_{+1} - f_{-1} &= 2{f}'(x_0) h + \frac{1}{3}{f}'''(x_0) h^3 + \mathcal{O}(h^4) \\
f_{+2} - f_{-2} &= 4{f}'(x_0) h + \frac{8}{3}{f}'''(x_0) h^3 + \mathcal{O}(h^4) \\
\end{align}
\)

Or:

\(
\begin{bmatrix}
f_{+1} - f_{-1} \\
f_{+2} - f_{-2} \\
\end{bmatrix}
\approx
\begin{bmatrix}
2 & \frac{1}{3} \\
4 & \frac{8}{3} \\
\end{bmatrix}
\cdot
\begin{bmatrix}
{f}'(x_0) h \\
{f}'''(x_0) h^3 \\
\end{bmatrix}
\)


This leads to:

\(
\begin{bmatrix}
{f}'(x_0) h \\
{f}'''(x_0) h^3 \\
\end{bmatrix}
\approx
\begin{bmatrix}
\frac{2}{3} & - \frac{1}{12} \\
- 1 & \frac{1}{2} \\
\end{bmatrix}
\cdot
\begin{bmatrix}
f_{+1} - f_{-1} \\
f_{+2} - f_{-2} \\
\end{bmatrix}
\)

This allows to optimise the program for the HP-42S a bit:

Code:

00 { 103-Byte Prgm }
01▸LBL "DIFF"
02 STO 01
03 R↓
04 STO 00
05 RCL 01
06 -
07 XEQ 00
08 STO 02
09 RCL 00
10 RCL 01
11 +
12 XEQ 00
13 ENTER
14 X<> 02
15 STO+ 02
16 -
17 STO 03
18 RCL 00
19 RCL 01
20 2
21 ×
22 -
23 XEQ 00
24 STO 04
25 RCL 00
26 RCL 01
27 2
28 ×
29 +
30 XEQ 00
31 ENTER
32 X<> 04
33 STO+ 04
34 -
35 STO 05
36 RCL 02
37 4
38 ×
39 RCL 04
40 -
41 6
42 ÷
43 STOP
44 RCL 03
45 8
46 ×
47 RCL 05
48 -
49 12
50 ÷
51 RCL 01
52 ÷
53 STOP
54 RCL 04
55 RCL 02
56 -
57 3
58 ÷
59 RCL 01
60 X↑2
61 ÷
62 STOP
63 RCL 05
64 2
65 ÷
66 RCL 03
67 -
68 RCL 01
69 3
70 Y↑X
71 ÷
72 RTN
73▸LBL 00
74 LN
75 END

Registers

R00: \(x_0\)
R01: \(h\)
R02: \(f_{+1} + f_{-1}\)
R03: \(f_{+1} - f_{-1}\)
R04: \(f_{+2} + f_{-2}\)
R05: \(f_{+2} - f_{-2}\)