hp41programs

dot&cross Dot Product and Cross Product for the HP-41

Overview

1°) Dot Product

1-a) Focal Program
1-b) M-Code Routine

2°) Cross-Product

2-a) 3D-Vectors
2-b) 7D-Vectors
2-c) n-dimensional Vectors

>>> The latest addition is paragraph 2-b)

-Synthetic registers M N O P may be replaced by any unused data registers.

1°) Dot Product

a) Focal Program

-The following program calculates the dot-product U.V of 2 vectors U(x₁,x₂,....,x_n) and V(y₁,y₂,....,y_n) ,
the norms || U || , || V || and the angle µ defined by these vectors ( between 0 and 180° )

Formula: cos µ = U.V / ( || U || . || V || )

Data Registers: • Rbb = x₁ , • Rbb+1 = x₂ , .......... , • Ree = x_n ( These 2n registers are to be initialized before executing "DOT" )
• Rbb' = y₁ , • Rbb'+1 = y₂ , .......... , • Ree' = y_n ee - bb = ee' - bb' = n - 1
Flags: /
Subroutines: /

-If you use registers R01 and R02 instead of registers N and O, replace line 02 by 0 STO 01 STO 02 RDN

01 LBL "DOT"
02 CLA
03 STO M
04 CLX
05 LBL 01
06 RCL IND M
07 X^2

08 ST+ N
09 X<> L
10 RCL IND Z
11 X^2
12 ST+ O
13 X<> L
14 *

15 +
16 ISG Y
17 ""
18 ISG M
19 GTO 01
20 STO Y
21 RCL O

22 SQRT
23 ST/ Z
24 RCL N
25 SQRT
26 ST/ T
27 R^
28 ACOS

29 X<> T
30 CLA
31 END

( 55 bytes )

STACK	INPUTS	OUTPUTS
T	/	µ
Z	/	\|\| U \|\|
Y	bbb.eee(U)	\|\| V \|\|
X	bbb.eee(V)	U.V
L	/	cos µ

Example: U(2,3,7,1) V(3,1,4,6)

-For instance,

2 STO 01 3 STO 02 7 STO 03 1 STO 04 ( control number = 1.004 )
3 STO 06 1 STO 07 4 STO 08 6 STO 09 ( control number = 6.009 )

   1.004 ENTER^
   6.009 XEQ "DOT" >>>>     U.V = 43
                                      RDN   || V || = 7.874007874
                                      RDN   || U || = 7.937253933
                                      RDN      µ    = 46.52626239°   ( if the HP-41 is set in DEG mode )

b) M-Code Routine

-"DOTN" takes the control numbers of 2 vectors bbb.eee(ii)₁ and bbb.eee(ii)₂ in registers Y and X and returns the dot-product in X-register
-bbb.eee(ii)₁ is unchanged in Y-register, bbb.eee(ii)₂ is saved in L-register
-The "alpha" register is unused but synthetic register Q is altered.

-If ii = 01 , it may be replaced by 00

Warning:

-This routine does not check for alpha data.

08E "N"
014 "T"
00F "O"
004 "D"
248   SETF 9
0B8 READ 2(Y)
128   WRIT 4(L)
2A0 SETDEC
084   CLRF 5                   C
0ED ?NCXQ                   =
064   193B                      frc(C)
226   C=C+1 S&X           C
226   C=C+1 S&X           =
226   C=C+1 S&X       1000 C
070   N=C ALL
013   JNC +02
3B3 JNC -10d=-0Ah
088   SETF 5                    C
0ED ?NCXQ                   =
064   193B                     int(C)
260   SETHEX
38D ?NCXQ
008   02E3
106   A=C S&X
378   READ 13(c)
03C RCR 3
146   A=A+C S&X
130   LDI S&X
200   200h                          this value is for an HP-41CV , CX or C with a Quad memory module or 4 memory modules
306   ?A<C S&X               if you have an HP-41C without memory modules, replace 200 by 100
381   ?NC GO
00A   02E0                        if register Ree1 or Ree2 doesn't exist, the routine stops after displaying "NONEXISTENT"
0A6   A<>C S&X
268   WRIT 9(Q)
0B0   C=N ALL
2A0   SETDEC
084   CLRF 5                   C
0ED ?NCXQ                   =
064   193B                      frc(C)
226   C=C+1 S&X          C=
226   C=C+1 S&X        100 C
260   SETHEX
38D ?NCXQ
008   02E3
2E6   ?C#0 S&X
017   JC+02
226   C=C+1 S&X
106   A=C S&X
278   READ 9(Q)
1BC RCR 11
0A6 A<>C S&X
268   WRIT 9(Q)
138   READ 4(L)
2A0 SETDEC
088   SETF 5                    C
0ED ?NCXQ                   =
064   193B                     int(C)
260   SETHEX
38D ?NCXQ
008   02E3
106   A=C S&X
378   READ 13(c)
03C RCR 3
146   A=A+C S&X
278   READ 9(Q)
1BC RCR 11
0A6 A<>C S&X
24C ?FSET 9
043   JNC+ 08
244   CLRF 9
10E   A=C ALL
0F8   READ 3(X)
0AE A<>C ALL
0E8   WRIT 3(X)
0AE A<>C ALL
22B JNC -59d=-3Bh
268   WRIT 9(Q)
0F8   READ 3(X)
070   N=C ALL
04E   C=0 ALL
0E8   WRIT 3(X)
278   READ 9(Q)
106   A=C S&X
17C RCR 6
0A6 A<>C S&X
306   ?A<C S&X
360   ?C RTN                               the routine stops when the last register of one vector has been used.
270   RAMSLCT
038   READDATA
0EE C<>B ALL
0B0 C=N ALL
106   A=C S&X
17C RCR 6
0A6 A<>C S&X
306   ?A<C S&X
360   ?C RTN                               the routine stops when the last register of one vector has been used.
270   RAMSLCT
038   READDATA
10E   A=C ALL
046   C=0 S&X
270   RAMSLCT
2A0 SETDEC
0EE C<>B ALL
135   ?NCXQ                                C=
060   184D                                  A*C
10E   A=C ALL
0F8   READ 3(X)
01D ?NCXQ                                C=
060   1807                                   A+C
0E8   WRIT 3(X)
260   SETHEX
0B0   C=N ALL
106   A=C S&X
03C RCR 3
146   A=A+C S&X
1BC RCR 11
0A6 A<>C S&X
070   N=C ALL
278   READ 9(Q)
106   A=C S&X
03C RCR 3
146   A=A+C S&X
1BC RCR 11
0A6 A<>C S&X
268   WRIT 9(Q)
2AB JNC -43d=-2Bh

-The dot product of two vectors of 100 components is returned in 6.5 seconds.

2°) Cross Product

a) 3-Dimensional Vectors

-The cross product of 2 vectors U(x,y,z) and V(x',y',z') is a vector defined by UxV ( yz'-y'z , x'z-xz' , xy'-x'y )
-Since the stack has less than 6 registers, we have to store x , y , z into 3 data registers.

Data Registers: • Rbb = x • Rbb+1 = y • Rbb+2 = z ( These 3 registers are to be initialized before executing "CROSS" )
Flags: /
Subroutines: /

01 LBL "CROSS"
02 STO M
03 RDN
04 STO N
05 X<>Y
06 STO O
07 RCL IND Z

08 ST* Z
09 *
10 ISG Z
11 ""
12 RCL IND Z
13 ST* O
14 X<> M

15 ST* M
16 X<> M
17 ST- Z
18 ISG T
19 CLX
20 CLX
21 RCL IND T

22 ST* N
23 RCL M
24 *
25 X<>Y
26 -
27 RCL O
28 RCL N

29 -
30 CLA
31 END

( 59 bytes )

STACK	INPUTS	OUTPUTS
T	bbb.eee(U)	/
Z	z'	z"
Y	y'	y"
X	x'	x"

Example: U(2,3,7) V(3,1,4) Compute the cross product UxV

-For instance, 2 STO 01 3 STO 02 7 STO 03 ( control number = 1.003 )

    1.003 ENTER^
       4      ENTER^
       1      ENTER^
       3      XEQ "CROSS"   >>>>    5
                                           RDN   13
                                           RDN   -7      whence    UxV(5,13,-7)

Notes:

-In fact, this program will work as well if you place bbb in T register ( instead of bbb.eee )
-You may also store x , y , z in non-contiguous registers like R01 R03 R05 but in this case, place 1.00502 ( or 1.00002 ) in T

-The cross product is anticommutative: VxU = -UxV

-Alternatively, you can also use the following program that is essentially similar to the M-Code routine CROSS
-It requires that you store x' y' z' into synthetic registers M N O
( assign STO M STO N STO O to 3 keys or use it in a program )

Data Registers: /
Flags: /
Subroutines: /

01 LBL "CROSS"
02 STO P
03 RCL N
04 *
05 X<>Y
06 STO Q
07 RCL M
08 *
09 -
10 X<>Y
11 RCL M
12 *
13 RCL O
14 ST* P
15 ST* Q
16 X<> P
17 -
18 RCL N
19 R^
20 *
21 ST- Q
22 X<> Q
23 END

( 45 bytes / SIZE 000 )

STACK	INPUTS	OUTPUTS
T	bbb.eee(U)	/
Z	z'	z"
Y	y'	y"
X	x'	x"

Example: U(2,3,7) V(3,1,4) Compute the cross product UxV

-For instance, 2 STO M 3 STO N 7 STO O

       4      ENTER^
       1      ENTER^
       3      XEQ "CROSS"   >>>>    5
                                           RDN   13
                                           RDN   -7      whence    UxV(5,13,-7)

Notes:

-The 3 components x' y' z' are still in M N O at the end.
-If you don't need to save these values, add CLA before the end.

b) 7-Dimensional Vectors

- U(x₁,x₂,....,x₇) and V(y₁,y₂,....,y₇) are two 7D-vectors.

-"U*V7" calculates the cross product W(z₁,z₂,....,z₇) by the formula given here

-You store the components of U in R01 thru R07 and the components of V in R08 thru R14.
-"U*V7" stores the result in R15 thru R21

Data Registers: R00 = temp ( Registers R01 thru R14 are to be initialized before executing "U*V7" )

                                      • R01 = x₁           • R08 = y₁                                                    R15 = z₁
                                      • R02 = x₂           • R09 = y₂                                                    R16 = z₂
                                         .............               ............          When the program stops:        ...........

• R07 = x₇ • R14 = y₇ R21 = z₇
Flags: /
Subroutines: /

-Synthetic register M may be replaced by R22

01 LBL "U*V7"
02 15.021
03 STO 00
04 7
05 STO M
06 LBL 01
07 RCL 02
08 RCL 11
09 *
10 RCL 04
11 RCL 09

12 *
13 -
14 RCL 03
15 RCL 14
16 *
17 +
18 RCL 07
19 RCL 10
20 *
21 -
22 RCL 05

23 RCL 13
24 *
25 +
26 RCL 06
27 RCL 12
28 *
29 -
30 STO IND 00
31 RCL 07
32 X<> 06
33 X<> 05

34 X<> 04
35 X<> 03
36 X<> 02
37 X<> 01
38 STO 07
39 RCL 14
40 X<> 13
41 X<> 12
42 X<> 11
43 X<> 10
44 X<> 09

45 X<> 08
46 STO 14
47 ISG 00
48 CLX
49 DSE M
50 GTO 01
51 RCL 00
52 7
53 -
54 END

( 85 bytes / SIZE 022 )

STACK	INPUT	OUTPUT
X	/	15.021

Example: U( 6 2 4 7 10 13 12 ) V( 4 6 3 2 5 7 8 )

-Store these 14 numbers in R01 thru R14

XEQ "U*V7" >>>> 15.021 ---Execution time = 10s---

-And you get in R15 thru R21

W( -37 26 73 59 -40 33 -47 )

Notes:

-Registers R01 to R14 are modified during execution, but their contents is restored at the end.
-If n > 3 & n # 7 , the cross-product cannot be defined as a vector:

c) n-Dimensional Vectors

- U(x₁,x₂,....,x_n) and V(y₁,y₂,....,y_n) are 2 vectors.

- In n-dimensional spaces, the cross product is actually an antisymmetric Tensor P = [P_i,j] where P_i,j = x_i.y_j - x_j.y_i i , j = 1 , 2 , ...... , n
-"U*V" computes and stores the n² components of this tensor ( in column order )

Data Registers: R00 = n ( Rbb thru Ree & Rbb' thru Ree' are to be initialized before executing "U*V" )

• Rbb = x₁ , • Rbb+1 = x₂ , .......... , • Ree = x_n ee - bb = ee' - bb' = n - 1
• Rbb' = y₁ , • Rbb'+1 = y₂ , .......... , • Ree' = y_n

>>>> When the program stops, Rbb" to Ree" contain the n² coefficients of the product ( in column order ) ee" - bb" = n² - 1
Flags: /
Subroutines: /

01 LBL "U*V"
02 STO M
03 RDN
04 STO O
05 X<>Y
06 STO N
07 INT
08 CHS
09 LASTX
10 FRC
11 E3
12 *
13 +
14 1
15 +
16 STO 00

17 X^2
18 1
19 -
20 .1
21 %
22 STO P
23 LBL 01
24 RCL P
25 INT
26 STO Y
27 RCL 00
28 ST/ Z
29 MOD
30 X<>Y
31 INT
32 RCL X

33 RCL O
34 +
35 RDN
36 RCL IND T
37 RCL N
38 R^
39 ST+ Y
40 R^
41 R^
42 RCL IND T
43 *
44 RCL O
45 ST+ T
46 CLX
47 RCL IND T
48 RCL N

49 ST+ T
50 CLX
51 RCL IND T
52 *
53 -
54 STO IND M
55 ISG M
56 CLX
57 ISG P
58 GTO 01
59 RCL N
60 RCL M
61 ENTER^
62 DSE X
63 E3
64 /

65 +
66 RCL 00
67 X^2
68 -
69 RCL 00
70 E5
71 /
72 +
73 RCL O
74 X<>Y
75 CLA
76 END

( 115 bytes )

STACK	INPUTS	OUTPUTS
Z	bbb.eee(U)	bbb.eee(U)
Y	bbb.eee(V)	bbb.eee(V)
X	bbb(P)	bbb.eee,nn(P)

Example1: U(2,3,7,1) V(3,1,4,6) Calculate the 16 coefficients of P = UxV

-For instance,

2 STO 01 3 STO 02 7 STO 03 1 STO 04 ( control number = 1.004 )
3 STO 06 1 STO 07 4 STO 08 6 STO 09 ( control number = 6.009 )

-If we choose register R12 as the first register for the product:

   1.004 ENTER^
   6.009 ENTER^
     12     XEQ "U*V"   >>>>   12.02704   ( in 17 seconds )   and we get:

              0   -7   -13   9                    R12   R16   R20   R24
              7    0     5    17        =        R13   R17   R21   R25      respectively.
             13 -5    0    38                   R14   R18   R22   R26
             -9 -17 -38   0                    R15   R19   R23   R27

-There are only n(n-1)/2 independent coefficients ( 6 in this example ) and the diagonal elements are always zeros.

Example2: U(2,3,7) V(3,1,4) Compute UxV

2 STO 01 3 STO 02 7 STO 03 ( control number = 1.003 )
3 STO 06 1 STO 07 4 STO 08 ( control number = 6.008 )

-If we choose again register R12 as the first register for the product:

   1.003 ENTER^
   6.008 ENTER^
     12        R/S     >>>>   12.02003   ( in 10s ) , we get:

              0   -7 -13                  R12   R15   R18
              7    0     5         =       R13   R16   R19     respectively.
             13 -5    0                   R14   R17   R20

Note:

-If n = 3 , there are only 3 independent coefficients and p₂₃ , p₃₁ , p₁₂ ( in this order )
form the components of the usual cross product ( which is actually a pseudo-vector )
-In this example, we recognize UxV = (5,13,-7)