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,....,xn) and V(y1,y2,....,yn)
,
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 = xn (
These 2n registers are to be initialized before executing "DOT" )
• Rbb' = y1 , • Rbb'+1 = y2
, .......... , • Ree' = yn
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,....,xn) and V(y1,y2,....,yn) are 2 vectors.
- In n-dimensional spaces, the cross product is actually an antisymmetric
Tensor
P = [Pi,j] where Pi,j = xi.yj
- xj.yi
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 = xn
ee - bb = ee' - bb' = n - 1
• Rbb' = y1 , • Rbb'+1 = y2
, .......... , • Ree' = yn
>>>> 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)