Sunday, December 20, 2020

Review: Swiss Micros DM41X

 Review: Swiss Micros DM41X






General Information


Company:  SwissMicros, https://www.swissmicros.com

Type:  Programmable Scientific

Memory:  319 registers (at default) - calculator; 32 MB external flash

Battery:  CR 2032

Date:  September 2020 - current

Cost:  229 CHF, about $257.34 according to currency converter on December 12, 2020 (https://www.xe.com/currencyconverter/convert/?Amount=229&From=CHF&To=USD)

Current Software Version:  2.1 (November 13, 2020)

Case Included?  Yes


The DM41X is a long waited calculator, and joins the DM 41L and DM41 family.  The family runs HP-41CX firmware, including a clock, time, and date functions.    On February 23, 2018, I reviewed the DM 41L which you can see here:


http://edspi31415.blogspot.com/2018/02/review-swissmicros-dm41l.html


I am going to address the features that are specific to the DM41X, which is added to all the great features of the DM41 and DM41L.  


A Return to the Original Layout





The DM41X's keyboard is in a portrait design, like the original Hewlett Packard HP-41C family (HP-41C, HP-41CV, HP-41CX)


Instead of the row of ON, USER, PRGM, and ALPHA keys on the top row, the DM41X adds a column of keys to the right of the main keyboard:


ON/CLK:  ON/OFF button, the shifted CLK displays the current time


USR/SETUP:  The user keyboard.  SETUP shows the DM41X's system menu where stve states and RAW progams can be saved, USB communication can be activated, modules can be saved and loaded, system settings can be changed, and accessing the About screen.


Blue Key:  This is the ALPHA key


Backspace Key


Up Arrow: for program scrolling and debugging


Down Arrow: for program scrolling and debugging


PRG/SIZE:  The PRGM button.  The shifted function allows the user set the number of registers to be set for numerical constants.


R/S/VIEW:  Run/Stop Key, View prompts for a view of a register's value without affecting the stack.


The Big Screen


The DM41X has three screen settings that is set by the DSP key:


1.  All four stacks, T, Z, Y, and X are displayed.  This is my preferred display. 


2.  The X stack and the Alpha Register are displayed.  On the top of the screen:  alarms, the number of storage registers designated for numerical constants, free memory, the current program loaded and it's pointer location.  


3.  Two stacks, Y and X are displayed. On the top of the screen:  alarms, the number of storage registers designated for numerical constants, free memory, the current program loaded and it's pointer location.  


Connectivity


The DM41X's connectivity interface is in line with the DM42's connectivity interface.  Simply plug the DM41X with a USB-Micro B port, then on the calculator press [shift], [USR] (SETUP), 1 for File, 5 for Activate USB Disk.  The DM41X now acts as a USB storage device like a smart phone.  


You can take screenshots by pressing and holding [SHIFT] and [DSP].  Screen shots are saved as .bmp files under the disk's SCREEN folder.


Black and white images of 400 x 240 pixels, 1 bit depth, .bmp format, can be saved to the disk's OFFIMG folder.  Every picture in the folder will be shown on a rotated basis every time the calculator is turned off.  


Custom Menu


Pressing the CST key will give you access to a custom menu 16 available slots for commands and programs.  CST also gives access to Help, ROM Map, and USB Disk activation.  To learn about how to set up the custom menu, click here:  https://edspi31415.blogspot.com/2020/10/swiss-micros-dm41x-custom-menus.html.


Programming


The DM41X operates like the HP-41CX in its programming language except the DM41X has a lot more room for ROM modules.


Verdict


Like the DM41L, the DM41X is a winner.  The keys feel good, I love seeing all four stack levels on the screen, and I am fan of the custom menu built into the DM41X.  This is definitely worth a purchase.


If the price tag of DM41X is too high, consider the DM41 or DM41L.  They may not have a four line display or a custom menu, but the high-quality calculator software and firmware is still present.  Currently only the DM41L is in stock from Swiss Micros, at CHF 129.  


- - - - - -


Announcement


This is the last post for 2020.  I want to wish everyone Merry Christmas, Happy Holidays, and Happy New Year.  2020 has been a brutal year in every respect.  May 2021 be free from disease, war, economic, social, and spiritual struggle; and be full of joy, health, and happiness.  Thank you to my blog's followers and readers; I appreciate all of you!  Take care, and the next schedule blog will be on January 2, 2021.  


Eddie


All original content copyright, © 2011-2020.  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. 


Saturday, December 19, 2020

Sum of Increasing Series

 Sum of Increasing Series


The Sum From 1 to N - Increasing by 1


It is well known that sum from 1 to N, which each term increasing by 1 is:


1 + 2 + 3 + 4 + ... + N = ∑ x from x = 1 to N = (N + 1) * N / 2


The derivation is fairly easy.  Let S be the sum:


S = 1 + 2 + 3 + ... + N-1 + N


Add S to both sides:


2 * S = 1 + 2 + 3 + ... + N-1 + N + 1 + 2 + 3 ... + N-1 + N


Thanks to the commutative property of addition, we can arrange terms, and with a clever and creative way of arranging terms:


2 * S = ( 1 + N ) + ( 2 + N - 1 ) + ( 3 + N - 2 ) + ... + ( N - 1 + 2 ) + ( N + 1 )


Note that:


2 * S = 

1   +    2   +   3   + ... +  N-1  + N +

N  +  N-1 + N-2  + ..  +   2   +  1


Written this way, there are N "pairs".  Hence:


2 * S = ( 1 + N ) + ( N + 1 ) + ( N +1 ) + ... + ( N + 1 ) + ( N + 1 )


2 * S = ( N + 1 ) * N


Solving for S:


S =  ( N + 1 ) * N / 2


The Sum From 1 to N - Increasing by 2


Now lets consider the sum:


1 + 3 + 5 + 7 + ...


Fun fact: Adding odd numbers in this fashion will always total perfect squares.


1 + 3 = 4 = 2^2

1 + 3 + 5 = 9 = 3^2

1 + 3 + 5 + 7  = 16 = 4^2

1 + 3 + 5 + 7 + 9  = 25 = 5^2

... and so on


Let's define S as the sum:


S = 1 + 3 + 5 + ... + N-4 + N-2 + N


Note that

1 = 2 * 0 + 1

3 = 2 * 1 + 1

5 = 2 * 2 + 1

and so on...


For integer q,

N = 2 * q + 1


Then:


S = 1 + 3 + 5 + ... + ( 2*(q-2) + 1 ) + ( 2*(q-1) + 1) + ( 2*q + 1 )


Using the same strategy as last time:


2 * S = 1 + 3 + 5 + ... + ( 2*(q-2) + 1 ) + ( 2*(q-1) + 1) + ( 2*q + 1 ) 

           + 1 + 3 + 5 + ... + ( 2*(q-2) + 1 ) + ( 2*(q-1) + 1) + ( 2*q + 1 )


2 * S =

     1         +          3            +         5            + ... + (2*(q-2) + 1) + (2*(q-1) + 1) +   (2*q + 1) +

(2*q + 1) + (2*(q-1) + 1)  + (2*(q-2) + 1) +  ... +             5        +            3        +     1


Combine each pair as such:


2  * S = (2*q + 2) + (2*q + 2) + (2*q + 2) + ... + (2*q + 2) + (2*q + 2) + (2*q + 2)


Since q starts at q = 0, there are q+1 "pairs".  


2 * S = (q + 1) * (2*q + 2)


S = (q + 1) * (2*q + 2) / 2


To show that the sum is a perfect square:


S = (q + 1) * (2*q + 2) / 2


= (2*q^2 + 2*q  + 2*q + 2) / 2


= (2*q^2 + 4*q + 2) / 2


= (q^2 + 2*q + 1)


= (q + 1)^2


Example:


Calculate 7^2 (albeit the "longer" way):


7 = q+1;  q = 6:


S = (6 + 1) * (2*6 + 2) / 2 = 7 * 14 / 2 = 7 * 7 = 49


The Sum From 1 to N - Increasing by 3


Now add the series:


S = 1 + 4 + 7 + 10 + ... + N - 3 + N


where each term increases by 3.  Letting q be a positive integer and noting that


1 = 3*0 + 1

4 = 3*1 + 1

7 = 3*2 + 1

10 = 3*3 + 1

and so on...


Then:


S = (3*0 + 1) + (3*1 + 1) + (3*2 + 1) + .... + (3(q-1) + 1) + (3q + 1)


2* S 

= (3*0 + 1) + (3*1 + 1) + (3*2 + 1) + ... + (3(q-1) + 1) + (3q + 1) 

+ (3*0 + 1) + (3*1 + 1) + (3*2 + 1) + ... + (3(q-1) + 1) + (3q + 1)


2 * S 

= (3*0 + 1) + (3*1 + 1)     + (3*2 + 1)     + ... + (3(q-1) + 1) + (3q + 1) 

+ (3*q + 1) + (3(q-1) + 1) +(3(q-2) + 1) + ...  +  (3*1 + 1)    + (3*0 + 1)


There are q+1 pairs.


2 * S 

= (3*q + 2)  + (3*q + 2) + (3*q + 2) + ... + (3*q + 2) + (3*q + 2)


2 *S = (q + 1) * (3*q + 2)


S = (q + 1) * (3*q + 2) / 2


Example:


q = 5  (terms from 0 to 5)


S = (5 + 1) * (3*5 + 2) / 2 = 6 * 17 / 2 = 51


1 + 4 + 7 + 10 + 13 + 16 = 51 


The Sum From A for q terms, increasing by D


Let A ≥ 0, D > 0 and q be a positive integer, and A is a starting term, let the sum S be:


S = A + (A + D) + (A + 2*D) + (A + 3*D) + ... + (A + (q-2)*D) + (A + (q-1)*D) + (A + q*D)


Since we start at q = 0, there are q+1 terms.


2*S =

A + (A + D) + (A + 2*D) + (A + 3*D) + ... + (A + (q-2)*D) + (A + (q-1)*D) + (A + q*D) +

A + (A + D) + (A + 2*D) + (A + 3*D) + ... + (A + (q-2)*D) + (A + (q-1)*D) + (A + q*D)


2*S = 

A + (A + D) + (A + 2*D) + ... + (A + (q-2)*D) + (A + (q-1)*D) + (A + q*D) +

(A + q*D) + (A + (q-1)*D) + (A + (q-2)*D) + ... + (A + 2*D) + (A + D) + A


2*S = (2*A + q*D) + (2*A + q*D) + (2*A + q*D) + ... + (2*A + q*D) + (2*A + q*D) + (2*A + q*D) 


2*S = (q + 1) * (2*A + q*D)


S = (q + 1) * (2*A + q*D) / 2


Example:


A = 10, D = 7; q = 4  (5 terms, increase by 7, initial term is 10)


(4 + 1) * (2*10 + 4*7) / 2 = 5 * (20 + 28) / 2 = 120


Note:  10 + 17 + 24 + 31 + 38  = 120


Source:


Knott, Dr. Ron "Proving that 1+2+3+...+n is n(n+1)/2"  February 12, 2003.  Accessed December 6, 2020.  http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/triNbProof.html


Eddie


All original content copyright, © 2011-2020.  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. 


Sunday, December 13, 2020

Numworks and DM42: Lucas-Lehmer Primality Test

 Numworks and DM42: Lucas-Lehmer Primality Test


Introduction


The Lucas-Lehmer Primality tests whether integers of the form 2^p - 1 are prime by repeating the loop p-2 times:


s_i+1 = s_i^2 - 2


with the initial value s_0 = 4.


At the end of the loop, if s = 0 mod (2^p - 1), then integer 2^p - 1 is a prime number.


Caution:  this test fails when p = 2.  The user will be asked to input an integer greater than 2.  


Numworks Python Script:  llt.py


from math import *

# Lucas-Lehmer Test

# 2020-12-06 EWS

p=float(input("p (>2)? "))

s=4

m=2**p-1

for i in range(p-2):

  # repeat p-2 times

  s=(s*s-2)%m

print("Mersenne Number:")

print(m)

if s==0:

  print("Prime")

else:

  print("Not Prime")


HP 42S/DM42 Program LLT


00 {66-Byte Prgm}

01 LBL "LLT"

02 "P (>2)?"

03  PROMPT

04  STO 01

05  ENTER

06  2

07  X<>Y

08  Y↑X

09  1

10  -

11 STO 04

12 R↓

13 2

14 -

15 STO 02

16 4

17 STO 03

18 LBL 00

19 RCL 03

20 X↑2

21 2

22 -

23 RCL 04

24 MOD

25 STO 03

26 DSE 02

27 GTO 00

28 RCL 04

29 "M= "

30 ARCL ST X

31 AVIEW

32 STOP

33 RCL 03

34 X=0?

35 "PRIME"

36 X=0?

37 AVIEW

38 END


Registers


R01: p

R02: p - 2

R03: s

R04:  m = 2^p - 1


Examples


p = 3

m = 7

Prime


p = 4 

m = 15

Not Prime


p = 5 

m = 31

Not Prime


p = 15

m = 32767

Not Prime


p = 20

m = 1048575

Not Prime


Source:


Cook, John D.  "Searching for Mersenne Primes" John D. Cook Consulting  November 28, 2018.  Retrieved December 5, 2020.  https://www.johndcook.com/blog/2018/11/28/searching-for-mersenne-primes/


"Lucas-Lehmer primality test"  Wikipedia.   Last edited June 29, 2020.  Retrieved December 5, 2020.

https://en.wikipedia.org/wiki/Lucas–Lehmer_primality_test


Eddie


Saturday, December 12, 2020

DM41 and DM42: Self Timing Programs

 DM41 and DM42: Self Timing Programs


Timing a Program's Performance


The Swiss Micros calculators DM41 (including the DM41L, DM41X, Hewlett Packard HP 41CX, HP41C with a Time Module installed) and DM42 have an internal clock which allows us to set the date and time on the calculator.  With the use of the TIME function we can determine the how long a program takes to execute.


One structure that can be used:


[ preliminary storage]

TIME

HR (DM 41), →HR (DM 42)

STO Rxx  (register xx)

[ loop/main program starts here ]

TIME

HR (DM 41), →HR (DM 42)

RCL Rxx

-

ABS 

3600

*


The routine TIME, HR, RCL xx, -, ABS, 3600, * determines the amount of time in term of seconds.  


Let's demonstrate this technique on three examples.  The time record is the calculator ran on batteries.  


Example 1:  40th Fibonacci Numbers 


Registers:  

R01:  counter

R02:  sum

R03:  time

(counter of 39 is set because the first Fibonacci number is loaded at the beginning)


DM41 Program TFIB


01  LBL ^T TFIB

02  39

03  STO 01

04  0

05  STO 02

06  TIME

07  HR

08  STO 03

09  1

10  0

11  +

12  LBL 00

13  LASTX

14  X<>Y

15  +

16  DSE 01

17  GTO 00

18  TIME

19  HR

20  RCL 03

21  -

22  ABS

23  3600

24  *

25  STO 03

26  END


Y:  Results:  102334155.0

X: Seconds:  0.5100


DM42 Program TFIB


00  {45-Byte Prgm}

01  LBL "TFIB"

02  39

03  STO 01

04  0

05  STO 02

06  TIME

07  →HR

08  STO 03

09  1

10  0

11  +

12  LBL 00

13  LASTX

14  X<>Y

15  +

16  DSE 01

17  GTO 00

18  TIME

19  →HR

20  RCL 03

21  -

22  ABS

23  3600

24  ×

25  STO 03

26  END


Y:  Results:  102334155.0

X: Seconds:  0.0200


Example 2:  The Sum of Cubes Using a Loop from 1 to 250


Registers:  

R01:  counter

R02:  sum

R03:  time


DM41 Program TSUM


01  LBL ^T TSUM

02  250

03  STO 01

04  0

05  STO 02

06  TIME

07  HR

08  STO 03

09  LBL 00

10  RCL 01

11  3

12  Y↑X

13  ST+ 02

14  DSE 01

15  GTO 00

16  RCL 02

17  TIME

18  HR

19  RCL 03

20  -

21  ABS

22  3600

23  *

24  STO 03

25  END


Y:  Results:  984390625.0

X: Seconds:  12.07


DM42 Program TSUM


00  {45-Byte Prgm}

01  LBL "TSUM"

02  250

03  STO 01

04  0

05  STO 02

06  TIME

07  →HR

08  STO 03

09  LBL 00

10  RCL 01

11  3

12  Y↑X

13  STO+ 02

14  DSE 01

15  GTO 00

16  RCL 02

17  TIME

18  →HR

19  RCL 03

20  -

21  ABS

22  3600

23  ×

24  STO 03

25  END


Y:  Results:  984390625.0

X: Seconds:  0.24


Example 3:  Savage Test


Let A = 1, then for 2499 loops:

A = tan(atan(exp(ln(sqrt(A^2))))) + 1   (radians mode is used)


Registers:  

R01:  counter

R02:  sum

R03:  time


DM41 Program SVGE


01  LBL ^T SVGE

02  2499

03  STO 01

04  1

05  STO 02

06  TIME

07  HR

08  STO 03

09  LBL 00

10  RCL 02

11  X↑2

12  SQRT

13  LN

14  E↑X

15  ATAN

16  TAN

17  1

18  +

19  STO 02

20  DSE 01

21  GTO 00

22  RCL 02

23  TIME

24  HR

25  RCL 03

26  -

27  ABS

28  3600

29  *

30  STO 03

31  END


Y:  Results:  2499.970322

X: Seconds:  246.4500240  (about 4 minutes, 6.45 seconds)


DM42 Program SVGE


00  {51-Byte Prgm}

01  LBL "SVGE"

02  2499

03  STO 01

04  1

05  STO 02

06  TIME

07  →HR

08  STO 03

09  LBL 00

10  RCL 02

11  X↑2

12  SQRT

13  LN

14  E↑X

15  ATAN

16  TAN

17  1

18  +

19  STO 02

20  DSE 01

21  GTO 00

22  RCL 02

23  TIME

24  →HR

25  RCL 03

26  -

27  ABS

28  3600

29  ×

30  STO 03

31  END


Y:  Results:  2500  (just over 2500)

X: Seconds:  13.24


Source:


Burkett, John.  "The Savage Benchmark"  TI-89 / TI-92 Plus Tip List 10.0   July 20, 2002.  http://www.technicalc.org/tiplist/en/files/pdf/tips/tip6_50.pdf   Retrieved November 28, 2020 


Eddie 


Sunday, December 6, 2020

Calculus of the Sinc Function

Calculus of the Sinc Function


Introduction and Setup


The unnormalized Sinc function is defined as:


unsinc(x) = sin x / x


And the normalized Sinc function is defined as:


sinc(x) = sin( π x ) / ( π x )


Two things to assume about the sinc function:


1.  The function is not defined at x = 0, and


2.  The function uses radian angle measure.  


(x ≠ 0, and assume radians measure)



Let α be a real constant and define f(x) as:


f(x) = sin( α x ) / (α x)


When α = 1, f(x) becomes unsinc(x).  Likewise, when α = π, f(x) becomes sinc(x).  I am going to analyze this function f(x).


Limit


Limit of sin( α x ) / ( α x )


lim x → 0 ( sin( α x ) / ( α x ) ) → sin 0 / 0 → 0 / 0


This form of improper limit allows us to use L'Hôspital's Rule, which allows us to take the derivative of both the numerator function and denominator function:


lim x → 0 ( α * cos ( α x ) / α ) → lim x → 0 ( cos ( α x ) ) →  cos ( 0 ) → 1


Hence  lim x → 0 ( sin( α x ) / ( α x ) ) → 1


Derivative


Taking the derivative will call for use to use the quotient rule:


d/dx [n(x) / d(x)] = ( n '(x) * d(x) - n(x) * d '(x)) / (d^2(x))


Then:


d/dx [ sin( α x ) / ( α x ) ]:


n(x) = sin ( α x )

n'(x) = α * cos( α x )

d(x) = α * x 

d^2(x) = (α * x)^2

d'(x) = α



d/dx [ sin( α x ) / ( α x ) ]

= [ α * cos( α x ) * α * x - sin( α x ) * α ] / [ α^2 * x^2 ]

= [ α^2 * cos( α x ) * x - sin( α x ) * α ] / [ α^2 * x^2 ]

= [ α * cos( α x ) * x - sin( α x ) ] / [ α * x^2 ]

= cos( α x ) / x^2 - sin( α x ) / (α * x^2)


Integral


The integral of 


∫ sin( α x ) / ( α x ) dx


does not look like it can easily integrated.


Let's use the Taylor Series approach:


sin x = x - x^3 / 3! + x^5 / 5! - x^7 / 7! + x^9 / 9! + . . . 


sin( α x ) =  ( α x ) - ( α x )^3 / 3! + ( α x )^5 / 5! - ( α x )^7 / 7! + ( α x )^9 / 9! + ...


With x≠0


sin( α x ) / ( α x ) 

=  1 - ( α x )^2 / 3! + ( α x )^4 / 5! - ( α x )^6 / 7! + ( α x )^8 / 9! + ...

=  1 -  α^2 * x^2 / 3! + α^4 * x^4 / 5! - α^6 * x^6 / 7! + α^8 * x^8 / 9! + ...



Now integrate the series:


∫ sin( α x ) / ( α x ) dx

=  x - ( α^2 * x^3 ) / (3 * 3!) + ( α^4 * x^5 ) / (5 * 5!) - ( α^6 * x^7 ) / (7 * 7!) +  ( α^8 * x^9 ) / (9 * 9!) + ... + C

 =  x - ( α^2 * x^3 ) / 18 + ( α^4 * x^5 ) / 600 - ( α^6 * x^7 ) / 35280 +  ( α^8 * x^9 ) / 3265920 + ... + C


Eddie


All original content copyright, © 2011-2020.  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. 


Saturday, December 5, 2020

HP 71B: Sign "Graph" of Trig Functions

 HP 71B: Sign "Graph" of Trig Functions


Introduction


The program SGNTRIG builds a 22 character binary string that depends on the function:


g(x) = 

1  if  sgn(t(x/c)) > 1

0  if  sgn(t(x/c)) ≤ 0


where:


c is a scaling factor, 


sgn(x) is the sign function,  and


t(x) represents one of three trigonometric functions, sin(x), cos(x), or tan(x).   Angles are assumed to be radians.


If we set to scale to c = 1, then g(x) takes the integer values from 1 to 22.  


The HP 71B builds the result to the string S$.   This is an aid to visualize at least some of the graph.   When the bit is 1, the graph is above the x-axis, otherwise, the graph is either on or below the x-axis.  The program presented can be expanded or modified to explore other functions.


The resulting string is a "psuedo-graph".  


Below is a graphical representation of g(x) (an HP Prime is used to graph g(x)).


HP 71B Program SGNTRIG

Size:  270 bytes


100 DESTROY S,I,N,E,C

105 S$=""

110 RADIANS

115 INPUT 'SCALE? ';C

120 DISP "S:SIN C:COS T:TAN"

125 E$=KEY$

130 IF E$="S" OR E$="C" OR E$="T" THEN 200 ELSE 120


200 FOR I=1 TO 22

205 IF E$="S" THEN N=SGN(SIN(I/C))

210 IF E$="C" THEN N=SGN(COS(I/C))

215 IF E$="T" THEN N=SGN(TAN(I/C))

220 IF N=1 THEN N=1 ELSE N=0

225 S$=S$&STR$(N)

230 DISP S$ @ BEEP 589,0.1 @ WAIT 0.2

235 NEXT I

240 DISP S$ @ BEEP 661,2


Notes:


Line 105:  creates a blank string, which is allowed on the HP 71B


Line 125:  The KEY$ function calls for a key input. S for sine, C for cosine, and T for tangent.


Lines 205 to 215: determine which trigonometric function to use


Line 220:  Sets the character to 0 if the sign function returns 0 or -1


Line 225:  &, the ampersand symbol concatenates two strings


Lines 230 and 240, respectively:  589 Hz is middle D, 661 Hz is middle E


Example


Below are results for each of the trigonometric functions when scale is set to 1.  




YouTube video: https://www.youtube.com/watch?v=GL6urOGvRGY&feature=youtu.be

Eddie


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