HP 42S/DM42: Error Function, Bernoulli Numbers, Chebyshev Polynomials
Error Function
The program ERF calculates the error function:
erf(x) = 2 / √π * ∫ e^-(t^2) dt from t = 0 to t = x
The calculation uses the series:
erf(x) = 2 * e^(-(x^2)) / √π * Σ( 2^n * x^(2n+1) / (1 * 3 * ... * (2n+1)) from n=0 to ∞
The error function can be used to calculate the cumulative normal distributive function from 0 to x by:
ncdf(x) = 1/2 + 1/2 * erf(x/√2)
HP 42S/DM42 Program: Error Function: ERF
00 { 46-Byte Prgm }
01▸LBL "ERF"
02 STO 01
03 ENTER
04 ×
05 2
06 ×
07 STO 02
08 1
09 STO 03
10 RCL 01
11▸LBL 01
12 RCL 02
13 RCL 03
14 2
15 +
16 STO 03
17 ÷
18 RCL 01
19 ×
20 STO 01
21 +
22 X≠Y?
23 GTO 01
24 2
25 ×
26 PI
27 SQRT
28 RCL 02
29 2
30 ÷
31 E↑X
32 ×
33 ÷
34 RTN
35 END
Examples
erf(1.34) ≈ 0.9419
erf(2.43) ≈ 0.9994
Source:
Hewlett Packard HP-65 Math Pac 2. 1974.
Bernoulli Numbers (Approximation)
The program BERN calculates Bernoulli numbers for positive integers n > 2. The calculator is set to Radian mode.
The formula used:
B_n = (-cos(n * π/2))^((n+2)/2) * 2 * n! * Σ(2 * j * π)^(-n) from j=1 to ∞
HP 42S/DM42 Program: Bernoulli Numbers: BERN
00 { 70-Byte Prgm }
01▸LBL "BERN"
02 RAD
03 STO 00
04 CLX
05 STO 02
06 1
07 +
08 STO 01
09▸LBL 00
10 2
11 PI
12 ×
13 RCL× 01
14 RCL 00
15 +/-
16 Y↑X
17 STO+ 02
18 1
19 STO+ 01
20 R↓
21 ABS
22 1ᴇ-10
23 X < Y ?
24 GTO 00
25 RCL 02
26 RCL 00
27 N!
28 ×
29 2
30 ×
31 RCL 00
32 PI
33 ×
34 2
35 ÷
36 COS
37 +/-
38 2
39 RCL+ 00
40 2
41 ÷
42 Y↑X
43 ×
44 RTN
45 END
Examples
B_2 ≈ 0.1667
B_3 returns 1.1184E-85 (0)
B_4 returns -0.0333
B_6 returns 0.0238
Source:
Keith Oldhma, Jan Myland, and Jerome Spanier An Atlas of Functions 2nd Ed. Springer: New York. ISBN: 978-0-387-48806-6
Chebyshev Polynomials
The program CHEBY calculates the value of a first-order Chebyshev polynomial using the recurring formula where:
T_n(x) = 2 * x * T_n(x) - T_(n-1)(x)
where T_0(x) = 1, T_1(x) = x
HP 42S/DM42 Program: Chebyshev Polynomials: CHEBY
00 { 51-Byte Prgm }
01▸LBL "CHEBY"
02 STO 02
03 2
04 ×
05 STO 03
06 R↓
07 STO 00
08 X=0?
09 GTO 00
10 1
11 STO 01
12 STO- 00
13 X=Y?
14 GTO 01
15▸LBL 02
16 RCL 03
17 RCL× 02
18 RCL- 01
19 RCL 02
20 STO 01
21 R↓
22 STO 02
23 DSE 00
24 GTO 02
25 RTN
26▸LBL 00
27 1
28 RTN
29▸LBL 01
30 RCL 02
31 RTN
32 .END.
Instructions
Input n, [ENTER], input x, XEQ CHEBY
Examples
T_3(0.4): 3, ENTER, 0.4, XEQ CHEBY
Result: -0.9440
T_4(1.5): 4, ENTER, 1.5, XEQ CHEBY
Result: 23.5000
Source
Hewlett Packard HP-65 Math Pac 2. 1974.
Happy programming,
Eddie
All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
Error Function
The program ERF calculates the error function:
erf(x) = 2 / √π * ∫ e^-(t^2) dt from t = 0 to t = x
The calculation uses the series:
erf(x) = 2 * e^(-(x^2)) / √π * Σ( 2^n * x^(2n+1) / (1 * 3 * ... * (2n+1)) from n=0 to ∞
The error function can be used to calculate the cumulative normal distributive function from 0 to x by:
ncdf(x) = 1/2 + 1/2 * erf(x/√2)
HP 42S/DM42 Program: Error Function: ERF
00 { 46-Byte Prgm }
01▸LBL "ERF"
02 STO 01
03 ENTER
04 ×
05 2
06 ×
07 STO 02
08 1
09 STO 03
10 RCL 01
11▸LBL 01
12 RCL 02
13 RCL 03
14 2
15 +
16 STO 03
17 ÷
18 RCL 01
19 ×
20 STO 01
21 +
22 X≠Y?
23 GTO 01
24 2
25 ×
26 PI
27 SQRT
28 RCL 02
29 2
30 ÷
31 E↑X
32 ×
33 ÷
34 RTN
35 END
Examples
erf(1.34) ≈ 0.9419
erf(2.43) ≈ 0.9994
Source:
Hewlett Packard HP-65 Math Pac 2. 1974.
Bernoulli Numbers (Approximation)
The program BERN calculates Bernoulli numbers for positive integers n > 2. The calculator is set to Radian mode.
The formula used:
B_n = (-cos(n * π/2))^((n+2)/2) * 2 * n! * Σ(2 * j * π)^(-n) from j=1 to ∞
HP 42S/DM42 Program: Bernoulli Numbers: BERN
00 { 70-Byte Prgm }
01▸LBL "BERN"
02 RAD
03 STO 00
04 CLX
05 STO 02
06 1
07 +
08 STO 01
09▸LBL 00
10 2
11 PI
12 ×
13 RCL× 01
14 RCL 00
15 +/-
16 Y↑X
17 STO+ 02
18 1
19 STO+ 01
20 R↓
21 ABS
22 1ᴇ-10
23 X < Y ?
25 RCL 02
26 RCL 00
27 N!
28 ×
29 2
30 ×
31 RCL 00
32 PI
33 ×
34 2
35 ÷
36 COS
37 +/-
38 2
39 RCL+ 00
40 2
41 ÷
42 Y↑X
43 ×
44 RTN
45 END
Examples
B_2 ≈ 0.1667
B_3 returns 1.1184E-85 (0)
B_4 returns -0.0333
B_6 returns 0.0238
Source:
Keith Oldhma, Jan Myland, and Jerome Spanier An Atlas of Functions 2nd Ed. Springer: New York. ISBN: 978-0-387-48806-6
Chebyshev Polynomials
The program CHEBY calculates the value of a first-order Chebyshev polynomial using the recurring formula where:
T_n(x) = 2 * x * T_n(x) - T_(n-1)(x)
where T_0(x) = 1, T_1(x) = x
HP 42S/DM42 Program: Chebyshev Polynomials: CHEBY
00 { 51-Byte Prgm }
01▸LBL "CHEBY"
02 STO 02
03 2
04 ×
05 STO 03
06 R↓
07 STO 00
08 X=0?
09 GTO 00
10 1
11 STO 01
12 STO- 00
13 X=Y?
14 GTO 01
15▸LBL 02
16 RCL 03
17 RCL× 02
18 RCL- 01
19 RCL 02
20 STO 01
21 R↓
22 STO 02
23 DSE 00
24 GTO 02
25 RTN
26▸LBL 00
27 1
28 RTN
29▸LBL 01
30 RCL 02
31 RTN
32 .END.
Instructions
Input n, [ENTER], input x, XEQ CHEBY
Examples
T_3(0.4): 3, ENTER, 0.4, XEQ CHEBY
Result: -0.9440
T_4(1.5): 4, ENTER, 1.5, XEQ CHEBY
Result: 23.5000
Source
Hewlett Packard HP-65 Math Pac 2. 1974.
Happy programming,
Eddie
All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
