Numerical Derivatives
In Part 9, we will calculate numerical derivatives of f(x). Computing accurate numerical derivatives can present a challenge. Often, calculation involves a small increment, usually named h. Generally, the smaller h is, the better the calculation. However with certain methods, if h is too small, the final result may be unexpected.
This program uses a five-point formula:
f'(x) ≈ (f(x - 2h) - 8 * f(x - h) + 8 * f(x + h) - f(x + 2h)) / (12h)
The error is of the order of h^4.
Source: Burden, Richard L. and J. Douglas Faires. "Numerical Analysis 8th Edition" Thomson Brooks/Cole Belton, CA 2005
Labels Used:
Label B: Main routine
Label 0: The function f(R1). The function starts with R1 loaded on the X register.
Memory Registers Used:
R0 = the numerical derivative
R1 = X
R2 = h
Program ListingKey Codes Key
001 42 21 12 LBL B
002 44 2 STO 2 * stores h
003 33 R ↓
004 44 1 STO 1 * stores X
005 2 2 * first term calculation
006 45 20 2 RCL × 2
007 30 -
008 32 0 GSB 0 * f(x - 2h)
009 44 0 STO 0
010 45 1 RCL 1
011 45 30 2 RCL- 2
012 32 0 GSB 0
013 8 8
014 16 CHS
015 20 × * -8 * f(x - h)
016 44 40 0 STO+ 0
017 45 1 RCL 1
018 45 40 2 RCL+ 2
019 32 0 GSB 0
020 8 8
021 20 × * 8 * (f + h)
022 44 40 0 STO+ 0
023 45 1 RCL 1
024 2 2
025 45 20 2 RCL × 2
026 40 +
027 32 0 GSB 0
028 16 CHS * -f(x + h)
029 44 40 0 STO+ 0
030 45 0 RCL 0
031 45 10 2 RCL÷ 2
032 1 1
033 2 2
034 10 ÷
035 44 0 STO 0
036 43 32 RTN
Instructions:
1. Enter the main program (Label B).
2. Enter the function f(R1) (Label 0).
3. Enter X, press [ENTER].
4. Enter h, press [ f ] [e^x] (B).
5. The approximate numerical derivative is displayed.
Caution: Remember to end the function with the RTN command ([ g ] [GSB] (RTN)).
Example 1
Let f(x) = x*e^x
Estimate f'(2) with h = 0.0001Key Code Key
001 42 21 0 LBL 0
002 36 ENTER
003 12 e^x
004 20 ×
005 43 32 RTN
To find the derivative:
Press 2 [ENTER] .0001 [ f ] [e^x] (B)
Result: f(2) ≈ 22.1672
Example 2
Let f(x) = -x^2 + 2x + 3
Estimate f'(1.5) with h = 0.0001
We can rewrite f(x) as:
f(x) = -x^2 + 2x + 3
-1 * (x^2 - 2x - 3)
-1 * (x * (x - 2) - 3)
Use the last form as the function.Key Code Key
001 42 21 0 LBL 0
002 36 ENTER
003 36 ENTER
004 2 2
005 30 -
006 20 ×
007 3 3
008 30 -
009 16 CHS
010 43 32 RTN
To find the derivative:
1.5 [ENTER] .0001 [ f ] [e^x] (B)
Result: f'(1.5) ≈ -1
Next time, we will look at how we can use statistics in programming. Until next time, happy programming,
Eddie
This tutorial is property of Edward Shore. © 2011