TI 30Xa Algorithms: Greatest Common Divisor

TI 30Xa Algorithms: Greatest Common Divisor

To find the greatest common divisor between two positive integers U and V:

Let U ≥ V. Let U = A * V + R where A is the quotient of U / V and R is the remainder. If R≠0, then V becomes the new U and R become the new V. The process repeats until R=0. At that point the value of V prior to the last calculation is the greatest common divisor (GCD) of U and V.

Example:

gcd(166, 78)

U = 166, V = 78

Algorithm Loop:

1. A = int(U / V)

2. R = U – V * int(U / V)

	A	R	U	V
Start	n/a	n/a	166	78
A = int(166 / 78) = 2 R = 166 – 2 * 78 = 10	2	10	78	10
A = int(78 / 10) = 7, R = 78 – 7 * 10 = 8	7	8	10	8
A = int(10 / 8) = 1 R = 10 – 1 * 8 = 2	1	2	8	2
A = int(8 / 2) = 4 R = 8 – 4 * 2 = 0	4	0 *STOP*

Procedure

1. Store the greater of the two numbers in memory register 1: [ STO ] [ 1 ].

2. Store the lesser of the two numbers in memory register 2: [ STO ] [ 2 ].

3. Divide memory register 1 by memory register 2. Store the integer part (no fractional part) in memory register 3: [ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ], (integer part) [ STO ] [ 3 ]

4. Figure the remainder and store the result in memory 3: [ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] [ STO ] [ 3 ]

5. If the remainder is 0, stop. The GCD is stored in memory 2.

6. If the remainder is non-zero, then store memory 2 into memory 1 then memory 3 into memory 2. You need to do it in this order. [ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]. Go back to Step 3 and repeat.

Examples

Example 1: GCD(26, 14)

M1 = 26, M2 = 14

	M1	M2	M3
26 [ STO ] [ 1 ], 14 [ STO ] [ 2 ]	26	14
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 1.857142857 1 [ STO ] [ 3 ]	26	14	1
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 12 [ STO ] [ 3 ] R is not zero, so we continue.	26	14	12
[ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]	14	12	12
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 1.166666667 1 [ STO ] [ 3 ]	14	12	1
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 2 [ STO ] [ 3 ] R is not zero, so we continue.	14	12	2
[ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]	12	2	2
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 6 6 [ STO ] [ 3 ]	12	2	6
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 0 [ STO ] [ 3 ] R is zero, so we stop. GCD: [ RCL ] [ 2 ]: GCD(26, 14) = 2	12	2	0

Example 2: GCD(27, 15)

M1 = 27, M2 = 15

	M1	M2	M3
27 [ STO ] [ 1 ], 15 [ STO ] [ 2 ]	27	15
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 1.8 1 [ STO ] [ 3 ]	27	15	1
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 12 [ STO ] [ 3 ] R is not zero, so we continue.	27	15	12
[ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]	15	12	12
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 1.25 1 [ STO ] [ 3 ]	15	12	1
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 3 [ STO ] [ 3 ] R is not zero, so we continue.	15	12	3
[ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]	12	3	3
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 4 4 [ STO ] [ 3 ]	12	3	4
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 0 [ STO ] [ 3 ] R is zero, so we stop. GCD: [ RCL ] [ 2 ]: GCD(27, 15) = 3	12	3	0

I hope you find this useful. What I hope to do with the monthly series is to demonstrate various calculations with the TI-30Xa.

Note: For June and July 2024, I will be posting on Saturdays only. I plan to resume the Saturday-Sunday schedule in August.

Eddie

All original content copyright, © 2011-2024. 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.

HP 32SII: Some Algorithms For RPN Calculators

HP 32SII:  Some Algorithms For RPN Calculators

Four programs ported to the HP 32SII calculator from algorithms designated for the 1973 HP 45 calculator.  

HP 32SII: Euclid Algorithm - Greatest Common Divisor (GCD)

The HP 45 algorithm is found on page 228 in the Algorithms For RPN Calculators book. (see source below)

This algorithm takes up three labels.

E01 LBL E

E02 INPUT M

E03 ENTER

E04 ENTER

E05 ENTER

E06 INPUT N

E07 x<>y

K01 LBL 1

K02 ÷

K03 FP

K04 ×

K05 1

K06 x>y?

K07 GTO L

K08 R↓

K09 ENTER

K10 ENTER

K11 R↓

K12 R↓

K13 GTO K

L01 LBL L

L02 R↓

L03 R↓

L04 RTN

Sizes and Checksums:

E:  10.5 bytes, 9D4D

K:  19.5 bytes, F8AD

L:  6.0 bytes, C304

Total:  36.0 bytes

Instructions:

Press [ XEQ ] E, enter M and N.   

Examples:

Input:  M = 36, N = 28.  Result:  4

Input:  M = 48, N = 126. Result: 6

Input:  M = 115, N = 300.  Result: 5

HP 32SII:  GCD Using One Label - John Kenney 

The program was provided by Ross Barnes, and the algorithm is from the book ENTER by J. Daniel Dodlin and Keith Jarrett (ISBN 0-9615174-2-1, pg. 84).  This is smart, one label program.

Enter both numbers in the stack before running the program.

G01 LBL G

G02 ENTER

G03 ENTER

G04 -

G05 R↓

G06 x<>y

G07 LASTx

G08 /

G09 LASTx

G10 RDN

G11 IP

G12 x

G13 -

G14 x≠0?

G15 GTO G

G16 +

G17 RTN

Size and Checksum:  25.5 bytes, 4E39

Posted with permission.  

HP 32SII:  Tetens Equation

The HP 45 algorithm is found on page 290 in the Algorithms For RPN Calculators book.    The original algorithm took the temperature in Celsius. 

Find the saturation of water vapor (e_m) in mmHg (millimeters of Mercury) given the temperature in °F.

Determined Formulas:

T (in °C) = (T°F - 32) * 5/9

α = T/(236.87 + T)

e_m = 4.579 * 10^(7.49 * α)

T01 LBL T

T02 INPUT T

T03 →°C

T04 ENTER

T05 ENTER

T06 236.87

T07 +

T08 ÷

T09 7.49

T10 ×

T11 10^x

T12 4.579

T13 ×

T14 RTN

Size and Checksum:

45.0 bytes, 404A

Examples:

T = 68 °F, Result: 17.53658 mmHg

T = 99 °F, Result:  47.63501 mmHg

HP 32SII:  Dew Point Given Relative Humidity and Air Temperature

The HP 45 algorithm is found on page 290 in the Algorithms For RPN Calculators book.    The original algorithm took the temperature in Celsius. 

Relativity humidity (F) is to be entered as a decimal.  For instance, instead of 20%, enter 0.20.

Determined Formulas:

T (in °C) = (T°F - 32) * 5/9

A = T/(T + 236.87)

B = 1/(log F/7.49 + A)

TD = 236.87/(B - 1)

TD = TD * 9/5 + 32

D01 LBL D

D02 INPUT T

D03 →°C

D04 ENTER

D05 ENTER

D06 236.87

D07 STO A

D08 +

D09 ÷

D10 INPUT F

D11 LOG

D12 7.49

D13 ÷

D14 +

D15 1/x

D16 1

D17 -

D18 RCL÷ A

D19 1/x

D20 →°F

D21 RTN

Size and Checksum:

47.5 bytes, 8677

Examples:

T = 80, F = 0.64, Result:  66.725

T = 95, F = 0.32, Result:  60.50684

HP 32SII:  Effective Temperature Due to Wind Velocity

The HP 45 algorithm is found on page 291 in the Algorithms For RPN Calculators book.    The original algorithm took the temperature in Fahrenheit. 

Wind velocity is in miles per hour (mph).  

Determined Formulas:

A = 0.634*(0.634 - log V)

ΔT = A*(T - 90)

Effective T = T - ΔT

E01 LBL E

E02 0.634

E03 ENTER

E04 ENTER

E05 INPUT V

E06 LOG

E07 -

E08 ×

E09 INPUT T

E10 ENTER

E11 90

E12 -

E13 ×

E14 RCL T

E15 x<>y

E16 -

E17 RTN

Size and Checksum:

33.5 bytes, 54F7

Examples:

V = 20 mph, T = 15 °F,  Result: -16.71728

V = 15 mph, T = 86 °F,  Result: 84.62526

Source:

Ball, John A.  Algorithms For RPN Calculators  John Wiley & Sons:  New York, NY.  1978. ISBN 0-471-03070-8

For the second GCD program:

Dodin, J. Daniel and Keith Jarrett. ENTER: Reverse Polish Notation Made Easy   Synthetix:  Berkeley, CA   ISBN 0-9612174-2-1  1984.

Special thanks and gratitude to Ross Barnes.  

All original content copyright, © 2011-2022.  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. 

Swiss Micros DM41X: Applications

Swiss Micros DM41X: Applications

Swiss Micros DM41X Program:  Euclid Algorithm

Registers:

R01:  A

R02: B

R03:  C  (used)

Finds the GCD of A and B, where A and B are positive integers and A > B.

01 LBL^T EUCLID

02 ^T GCD A>B

03 AVIEW

04 PSE

05 ^T A?

06 PROMPT

07 STO 01

08 ^T B?

09 PROMPT

10 STO 02

11 LBL 00

12 RCL 01

13 ENTER

14 ENTER

15 RCL 02

16 /

17 LASTX

18 X<>Y

19 INT

20 *

21 - 

22 STO 03

23 X=0?

24 GTO 01

25 RCL 02

26 STO 01

27 X<>Y

28 STO 02

29 GTO 00

30 LBL 01

31 ^T GCD=

32 ARCL 02

33 AVIEW

34 END

Examples:

A = 100, B = 20, GCD = 20

A = 78, B =24, GCD = 6

Swiss Micros DM41X Program: Fan Laws 

The program will calculate RPM_new (Revolutions per Minute), SP_new (Static Pressure), and BHP_new (Brake Horsepower).  

Inputs:

CFM_old:  Cubic Feet of Minute - old

CFM_new:  Cubic Feet of Minute - new

RPM_old:  Revolutions per Minute - old

SP_old:  Static Pressure - old

BHP_old:  Brake Horsepower - old

Outputs:

RPM_new

SP_new

BHP_new

01 LBL^T FANLAWS

02 CLA

03 ^T CFM.OLD?

04 PROMPT

05 STO 01

06 ^T CFM.NEW?

07 PROMPT

08 STO 02

09 X<>Y

10 /

11 STO 04

12 STO 06

13 ST* 06

14 STO 08

15 ST* 08

16 ST* 08

17 ^T RPM.OLD?

18 PROMPT

19 STO 03

20 ST* 04

21 ^T SP.OLD

22 PROMPT

23 STO 05

24 ST* 06

25 ^T BHP.OLD?

26 PROMPT

27 STO 07

28 ST* 08

29 ^T RPM.NEW=

30 ARCL 04

31 AVIEW

32 STOP

33 ^T SP.NEW=

34 ARCL 06

35 AVIEW

36 STOP

37 ^T BHP.NEW=

38 ARCL 08

39 AVIEW

40 END

Variables:

R01 = CFM.OLD

R02 = CFM.NEW

R03 = RPM.OLD

R04 = RPM.NEW

R05 = SP.OLD

R06 = SP.NEW

R07 = BHP.OLD

R08 = BHP.NEW

The program uses a lot of storage arithmetic.  

Example

Inputs:

CFM.OLD:  1250 CFM

CFM.NEW: 1600 CFM

RPM.OLD:  840 RPM

SP.OLD: 4 in

BHP.OLD: 7 BHP

Results:

RPM.NEW: 1075.2 RPM

SP.NEW: 6.5536 in

BHP.OLD:  14.680064 BHP

Source:

Calculated Industries "Sheet Metal/HVAC Pro Calc User's Guide" 2021

Swiss Micros DM41X Program: Johnson-Nyquist Noise Analysis

Equations Used:

Power (in Watts):

P = kb * T * Δf

RMS Voltage (in Volts):

v_n = √(4 * R * kb * T * Δf) = √(4 * R * P)

Current (in Amps):

i_n = √((4 * T * kb * Δf / R) = v_n / R

Inputs:

T = temperature in Kelvin  (°C + 273.15)

Δf = bandwidth, difference of frequencies in Hz

R = resistance in ohms (Ω)

Constants:  Boltzmann's Constant

kb ≈ 1.380649 * 10^-23 J/K

Program:

01 LBL^T NOISE

02 ^T TEMP? <K>

03 PROMPT

04 ^T BANDWIDTH?

05 PROMPT

06 *

07 1.308649E-23

08 *

09 ^T POW=

10 ARCL X

11 AVIEW

12 STOP

13 ^T R?

14 PROMPT

15 *

16 LASTX

17 X<>Y

18 4

19 *

20 SQRT

21 ^T V=

22 ARCL X

23 AVIEW

24 STOP

25 X<>Y

26 /

27 ^T I=

28 ARCL X

29 AVIEW

30 END

Example:

Temperature:  299.68 K

Bandwidth:  10,500 Hz

Resistance:  1375 Ω

Results:

Power:  4.3444E-17 W

Volts:  4.8882E-7 V

Current:  3.5550E-10 A

"Johnson-Nyquist Noise" Wikipedia.  Retrieved February 15, 2015 https://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise

Eddie

All original content copyright, © 2011-2021.  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. 

HP 12C Programming Part I: Modulus, GCD, PITI

HP 12C Programming Part I:  Modulus, GCD, PITI 

The last stop (for now) on my 1980s tour is the Hewlett Packard HP 12C calculator, one of the most popular financial calculators of all time.  The HP 12C was first manufactured in 1982 and has been seen ever since.

Today, there are two versions of the HP 12C:  the original and the Platinum.  The original is RPN only and has 99 programming steps of memory.  The Platinum edition, first released in 2003, has room for 400 steps and includes Algebraic mode.

In this series, I'll concentrate on the 1982 classic.  Albeit, using one manufactured this decade, the processing speed in today's HP 12C's compared to those produced in the 1980s is tremendous.

HP 12C Modulus

This program takes the modulus of two numbers:

Y MOD X = X * FRAC(Y/X)

In this program, X > 0 and Y > 0.

STEP	CODE	KEY
01	10	÷
02	43, 36	LST x
03	34	X<>Y
04	43, 24	FRAC
05	20	*
06	43, 33, 00	GTO 00

Input:  Y [ENTER] X [R/S],  

Result:  Y MOD X

Test 1:  124 MOD 77  = 47

Test 2: 3862 MOD 108 = 82

HP 12C Greatest Common Divisor (GCD)

This program calculates the greatest common divisor of two integers.  You can enter the two integers in either order.

Program:

STEP	CODE	KEY	COMMENT
01	43, 34	X≤Y	Determine which integer is greater
02	34	X<>Y
03	44, 1	STO 1	R1 = max(X,Y)
04	34	X<>Y
05	44, 0	STO 0	R0 = min(X,Y)
06	45, 1	RCL 1	Begin Euclidian division routine
07	45, 1	RCL 1	Recall R1 twice
08	45, 0	RCL 0
09	10	÷
10	43, 25	INTG
11	45, 0	RCL 0
12	20	*
13	30	-
14	43, 35	X=0
15	43, 33, 21	GTO 21
16	34	X<>Y
17	44, 1	STO 1
18	34	X<>Y
19	44, 0	STO 0
20	43, 33, 06	GTO 06
21	45, 0	RCL 0
22	43, 33, 00	GTO 00

Input:  integer [ENTER] integer [R/S]

Result: GCD

Test 1:   GCD(142,25)

Input:  142 [ENTER] 25 [R/S]

Result: 1

Test 2:  GCD(2555, 1365).   Result: 35

HP 12 PITI (Principal, Interest, Taxes, Insurance)

Here is a simple program that calculates the payment (principal and interest) while taking property taxes and insurance in consideration.  Property taxes and insurance are assumed to be combined as an annual amount.  Payments are assumed to be on a monthly basis.

Program:

STEP	CODE	KEY	COMMENT
01	1	1
02	2	2
03	10	÷	(tax + ins)/12
04	44, 0	STO 0
05	14	PMT
06	14	PMT	Need to press PMT twice.  Second press calculates payment.
07	45, 0	RCL 0
08	16	CHS
09	40	+	PITI is assumed to be an outflow
10	43, 33, 00	GTO 00	Ends the program

Input:

Number of Years [ g ] [ n ] (12*)

Annual Interest Rate [ g ] [ i ] (12÷)

Loan Amount [PV]

Annual Property Insurance and Taxes [R/S]

Result:  PITI

Test:  Calculate PITI for a 30-year loan of $205,000.  The rate on the loan is 4.1%.  Annual insurance and property taxes are estimated to be $1,102.50.

Input:  30 [ g ] [ n ] (12*), 4.1 [ g ] [ i ] (12÷), 205000 [PV], 1102.50 [R/S]

Result:  -1,082.43  (PITI = $1,082.43)

This is blog is proerpty of Edward Shore, 2016.