## Wednesday, July 4, 2018

### Fun with the HP 15C

Fun with the HP 15C

Happy Independence Day, United States!

Fractional Derivative:  kth Derivative of f(x) = x^n

This program calculates the kth derivative of x^n. The program allows for partial derivatives of x^n (where k is not an integer, such as half-integers).

Formula:

d^k/dx^k x^n = n!/(n-k)! * x^(n-k)

Input:
R0: point (x’)
R1:  n
R2: k

Program:

 Step Key Code 001 LBL A 42, 21, 11 002 RCL 0 45, 0 003 RCL 1 45, 1 004 RCL - 2 45, 30, 2 005 y^x 14 006 LAST x 43, 36 007 x! 42, 0 008 1/x 15 009 * 20 010 RCL 1 45, 1 011 x! 42, 0 012 * 20 013 RTN 43, 32

Example:

d/dx^4 x^8  at x = 6

R0 = x’ = 6
R1 = n = 8
R2 = k = 4

Result:  2177280

d/dx^1.5 x^3.4 at x = 1

R0 = x’ = 1
R1 = n = 3.4
R2 = k = 1.5

Result:  5.5469

Sight Reduction Table

The program calculates altitude and azimuth of a given celestial body.

Inputs:

R1: Local Hour Angle (LHA), west is positive, east is negative
R2: The observer’s latitude on Earth, north is positive, south is negative (L)
R3: Declination of the celestial’s body, north is positive, south is negative (δ)

Each entry will need to be in decimal format.  If your angels are in HMS (hours-minutes-seconds) format, convert the angels to decimals by H before storing the values.

Formulas:

Altitude:
H = asin (sin δ sin L + cos δ cos L cos LHA)

Azimuth:
Z = acos ((sin δ – sin L sin H) ÷ (cos H cos L))
If sin LHA < 0 then Z = 360° - Z

Outputs:

Altitude (in decimal), R/S, Azimuth (in decimal)

Program:

 Step Key Code 001 LBL C 42, 21, 13 002 DEG 43, 7 003 RCL 3 45, 3 004 SIN 23 005 RCL 2 45, 2 006 SIN 23 007 * 20 008 RCL 3 45, 3 009 COS 24 010 RCL 2 45, 2 011 COS 24 012 * 20 013 RCL 1 45, 1 014 COS 24 015 * 20 016 + 40 017 ASIN 43, 23 018 STO 4 44, 4 019 R/S 31 020 SIN 23 021 RCL 2 45, 2 022 SIN 23 023 * 20 024 CHS 16 025 RCL 3 45, 3 026 SIN 23 027 + 40 028 RCL 4 45, 4 029 COS 24 030 RCL 2 45, 2 031 COS 24 032 * 20 033 ÷ 10 034 ACOS 43, 24 035 STO 5 44, 5 036 RCL 1 45, 1 037 SIN 23 038 TEST 2 (x<0) 43, 30, 2 039 GTO 1 22, 1 040 3 3 041 6 6 042 0 0 043 RCL - 5 45, 30, 5 044 STO 5 44, 5 045 LBL 1 42, 21, 1 046 RCL 5 45, 5 047 RTN 43, 32

Source:  “NAV 1-19A Sight Reduction Table”    HP 65 Navigation Pac.  Hewlett Packard, 1974.

Three Point Lagrangian Interpolation

This program calculates a point (x0, y0) given three known points (x1, y1), (x2, y2), and (x3, y3) where x0 is in between min(x1, x2, x3) and max(x1, x2, x3).  Ideally, x1 < x2 < x3.

Input:

R0 = x0 (on the x stack)

Store before running the program:

R1 = x1, R4 = y1
R2 = x2, R5 = y2
R3 = x3, R6 = y3

Other registers used: R7, R8, R9

 Step Key Code 001 LBL E 42, 21, 15 002 STO 0 44, 0 003 RCL 4 45, 4 004 STO 7 44, 7 005 RCL 5 45, 5 006 STO 8 44, 8 007 RCL 6 45, 6 008 STO 9 44, 9 009 RCL 0 45, 0 010 RCL - 1 45, 30, 1 011 STO * 8 44, 20, 8 012 STO * 9 44, 20, 9 013 RCL 0 45, 0 014 RCL – 2 45, 30, 2 015 STO * 7 44, 20, 7 016 STO * 9 44, 20, 9 017 RCL 0 45, 0 018 RCL – 3 45, 30, 3 019 STO * 7 44, 20, 7 020 STO * 8 44, 20, 8 021 RCL 1 45, 1 022 RCL – 2 45, 30, 2 023 STO ÷ 7 44, 10, 7 024 CHS 16 025 STO ÷ 8 44, 10, 8 026 RCL 1 45, 1 027 RCL – 3 45, 30, 3 028 STO ÷ 7 44, 10, 7 029 STO ÷ 9 44, 10, 9 030 RCL 2 45, 2 031 RCL – 3 45, 30, 3 032 STO ÷ 8 44, 10, 8 033 STO ÷ 9 44, 10, 9 034 RCL 7 45, 7 035 RCL + 8 45, 40, 8 036 RCL + 9 45, 40, 9 037 RTN 43, 32

Example:

R1 = 5, R4 = 0.4
R2 = 10, R5 = 0.9
R3 = 15, R6 = 1.3

R0 = 12, result: 1.0720

R0 = 8, result: 0.7120

Source:  R. Woodhouse.  “Lagrangian Interpolation Routines” Datafile Summer 1984 Vol. 3 No. 3, Page 14

This program solves the equation where B and C are complex numbers:

x^2 + B*x + C = 0

where the roots are:

x = -B/2 ± (B^2/4 – C)

Store the following values before running:

R1 = real(B),  R2 = imag(B)
R3 = real(C), R4 = imag(C)

Output:

Root 1 (press [ f ], hold (i) for the complex part), R/S, Root 2

Program:

 Step Key Code 001 LBL C 42, 21, 13 002 GSB 1 32, 1 003 GSB 2 32, 2 004 + 40 005 R/S 31 006 GSB 1 32, 1 007 GSB 2 32, 2 008 - 30 009 RTN 43, 32 010 LBL 1 42, 21, 1 011 RCL 1 45, 1 012 RCL 2 45, 2 013 I 42, 25 014 2 2 015 CHS 16 016 ÷ 10 017 RTN 43, 32 018 LBL 2 42, 21, 2 019 RCL 1 45, 1 020 RCL 2 45, 2 021 I 42, 25 022 x^2 43, 11 023 4 4 024 ÷ 10 025 RCL 3 45, 3 026 RCL 4 45, 4 027 I 42, 25 028 - 30 029 √ 11 030 RTN 43, 32

Example:

x^2 + (-1 + i)*x + (3i) = 0

R1 = -1, R2 = 1
R3 = 0, R4 = 3

Results:
1.8229 – 1.8229i
-0.8229 + 0.8229i

x^2 + 3*x + (-5 + 6i) = 0

R1 = 3, R2 = 0
R3 = -5, R4 = 6

Results:
1.3862 – 1.0394i
-4.3862 + 1.0394i

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

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