Kampé de Fériet Function for the HP-41
Overview
1°) Focal Program
2°) M-Code Routine
-Kampé de Fériet function is the generalized hypergeometric
function of 2 variables.
-Given 6 non-negative integers: A , B , C , P , Q , S ,
it is defined by the double series:
A
B C / a1 ,........, aA
; b1 ,........, bB ; c1 ,........,
cC ;
\
FP
Q S |
x , y | =
SUMm=0,1,2,... SUMn=0,1,2,..
Km,n xm yn / ( m! n! )
\ p1 ,........, pP ; q1
,........, qQ ; s1 ,........, sS
;
/
where Km,n
= [ (a1)m+n ......... (aA)m+n
(b1)m ......... (bB)m
(c1)n ......... (cC)n ]
/
[ (p1)m+n ......... (pP)m+n
(q1)m ......... (qQ)m
(s1)n ......... (sS)n ]
and (a)n
= Pochhammer's symbol: (a)0 = 1
(a)n = a(a+1)(a+2) ...... (a+n-1) if
n > 0
1°) Focal Program
Data Registers: R00 = x ( Registers R01 thru R06+A+B+C+P+Q+S are to be initialized before executing "KdF" )
• R01 = A • R07 = a1
....................................... • R06+A = aA
• R02 = B • R07+A = b1
.................................... • R06+A+B = bB
• R03 = C • R07+A+B = c1
................................ • R06+A+B+C = cC
• R04 = P • R07+A+B+C =
p1 .......................... • R06+A+B+C+P = pP
• R05 = Q • R07+A+B+C+P = q1
...................... • R06+A+B+C+P+Q = qQ
• R06 = S • R07+A+B+C+P+Q =
s1 ................. • R06+A+B+C+P+Q+S = sS
Flag: F10
Subroutines: /
-Line 169 is a three-byte GTO 04 and "" is a
TEXT0 ( NOP ) instruction
| 01 LBL "KdF"
02 CLA 03 STO O 04 X<>Y 05 STO P 06 CLX 07 SIGN 08 STO Q 09 SIGN 10 GTO 04 11 LBL 01 12 ISG 00 13 FS? 30 14 RTN 15 RCL M 16 RCL N 17 + 18 RCL IND 00 19 + 20 FC? 10 21 1/X 22 * 23 GTO 01 24 LBL 02 25 ISG 00 26 FS? 30 27 RTN 28 RCL M 29 RCL IND 00 30 + 31 FC? 10 32 1/X 33 * 34 GTO 02 35 LBL 03 |
36 ISG 00
37 FS? 30 38 RTN 39 RCL N 40 RCL IND 00 41 + 42 FC? 10 43 1/X 44 * 45 GTO 03 46 LBL 04 47 LASTX 48 PI 49 INT 50 ST+ X 51 STO 00 52 RCL 01 53 + 54 PI 55 INT 56 10^X 57 / 58 ST+ 00 59 RDN 60 SF 10 61 XEQ 01 62 RCL 02 63 PI 64 INT 65 10^X 66 / 67 DSE 00 68 "" 69 ST+ 00 70 RDN |
71 XEQ 02
72 RCL 03 73 ST+ 00 74 RCL 04 75 + 76 PI 77 INT 78 10^X 79 / 80 DSE 00 81 "" 82 ST+ 00 83 RDN 84 CF 10 85 XEQ 01 86 RCL 05 87 PI 88 INT 89 10^X 90 / 91 DSE 00 92 "" 93 ST+ 00 94 RDN 95 XEQ 02 96 RCL O 97 * 98 RCL M 99 ISG X 100 "" 101 STO M 102 / 103 + 104 X#Y? 105 GTO 04 |
106 PI
107 INT 108 ST+ X 109 STO 00 110 RCL 01 111 + 112 PI 113 INT 114 10^X 115 / 116 ST+ 00 117 CLX 118 STO M 119 X<> Q 120 SF 10 121 XEQ 03 122 RCL 02 123 ST+ 00 124 RCL 03 125 + 126 PI 127 INT 128 10^X 129 / 130 DSE 00 131 "" 132 ST+ 00 133 RDN 134 XEQ 03 135 RCL 04 136 PI 137 INT 138 10^X 139 / 140 DSE 00 |
141 ""
142 ST+ 00 143 RDN 144 CF 10 145 XEQ 03 146 RCL 05 147 ST+ 00 148 RCL 06 149 + 150 PI 151 INT 152 10^X 153 / 154 DSE 00 155 "" 156 ST+ 00 157 RDN 158 XEQ 03 159 RCL P 160 * 161 RCL N 162 ISG X 163 "" 164 STO N 165 / 166 STO Q 167 + 168 X#Y? 169 GTO 04 170 CLA 171 END |
( 255 bytes / SIZE 007+A+B+C+P+Q+S )
| STACK | INPUTS | OUTPUTS |
| Y | y | F(x,y) |
| X | x | F(x,y) |
Example:
A = 2
B = 3 C = 2
a1 = 1.2 a2 = 2.3
, b1 = 1.1 b2 = 0.7
b3 = 1.4 , c1 = 1.6
c2 = 1.5
P = 3
Q = 2 S = 4
p1 = 2 p2 = 3
p3 = 4 , q1 = 5 q2
= 6 , s1 = 7 s2 = 8
s3 = 9 s4 = 10.1
-Calculate F(5,7)
-Store these 22 numbers into
R01 R02 R03
R07 R08 , R09 R10 R11 , R12 R13
R04 R05 R06
R14 R15 R16 , R17 R18 , R19 R20 R21
R22
7 ENTER^
5 XEQ "KdF" >>>> F(5,7) = 1.022466709
---Execution time = 129s---
Note:
-If A = B = C = P = Q = S = 0 , "KdF" returns
exp(x+y)
2°) M-Code Routine
-This routine is structured like the program listed above.
Data Registers: R00 = unused ( Registers R01 thru R06+A+B+C+P+Q+S are to be initialized before executing "KdF" )
• R01 = A • R07 = a1
....................................... • R06+A = aA
• R02 = B • R07+A = b1
.................................... • R06+A+B = bB
• R03 = C • R07+A+B = c1
................................ • R06+A+B+C = cC
• R04 = P • R07+A+B+C =
p1 .......................... • R06+A+B+C+P = pP
• R05 = Q • R07+A+B+C+P = q1
...................... • R06+A+B+C+P+Q = qQ
• R06 = S • R07+A+B+C+P+Q =
s1 ................. • R06+A+B+C+P+Q+S = sS
Flags: /
Subroutines: /
-There are 9 absolute jumps ( one ?C GO and 8
?NC XQ ) written in red & green.
-You'll have to change these instructions according to the location
of the corresponding lines ( written in the same colors ) in your own ROM.
086 "F"
at the address FF06 in my ROM
044 "d"
00B "K"
378 READ 13(c)
03C RCR 3
070 N=C ALL
106 A=C S&X
130 LDI S&X
006 006
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 R06 doesn't exist, the routine stops after displaying "NONEXISTENT"
248 SETF 9
06E A<>B ALL
130 LDI S&X
002 002
158 M=C ALL
0B0 C=N ALL
226 C=C+1 S&X
070 N=C ALL
270 RAMSLCT
038 READDATA
38D ?NCXQ
008 02E3
the different numbers in registers R01 thru R06 are converted to hexadecimal
numbers
106 A=C S&X
and the corresponding addresses are stored in CPU-register N and in synthetic
register Q
0CE C=B ALL
1BC RCR 11
0C6 C=B S&X
206 C=C+A S&X
0EE C<>B ALL
198 C=M ALL
266 C=C-1 S&X
383 JNC -16d=-10h
046 C=0 S&X
270 RAMSLCT
0CE C=B ALL
24C ?FSET 9
023 JNC +04
244 CLRF 9
268 WRIT 9(Q)
333 JNC -26d=-1Ah
228 WRIT 8(P)
106 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 any additional memory module, replace 200
by 100
381 ?NC GO
00A 02E0
if register R06+A+B+C+P+Q+S doesn't exist, the routine stops after displaying
"NONEXISTENT"
2A0 SETDEC
0F8 READ 3(X)
128 WRIT 4(L)
04E C=0 ALL
1A8 WRIT 6(N)
35C PT=12
C=
050 LD@PT- 1
1
1E8 WRIT 7(O)
0E8 WRIT 3(X)
068 WRIT 1(Z)
04E C=0 ALL
Loop
this word is at the address FF43
in my ROM
168 WRIT 5(M)
278 READ 9(Q)
If you don't have a "newest" HP-41, insert the 2 words 3CC
?KEY 360 ?C RTN in this area
17C RCR 6
You'll simply have to press a key to stop an infinite loop
104 CLRF 8
if the series is divergent or converges too slowly.
288 CETF 7
In this case however, the address of the subroutine will have to be changed...
2BD ?NXQ
?ncxq
3FC FFAF
sub
But with a "newest" HP-41, simply press ENTER^ ON
to stop the routine.
278 READ 9(Q)
03C RCR 3
244 CLRF 9
284 CLRF 7
2BD ?NXQ
?ncxq
3FC FFAF
sub
238 READ 8(P)
17C RCR 6
108 SETF 8
288 SETF 7
2BD ?NXQ
?ncxq
3FC FFAF
sub
238 READ 8(P)
03C RCR 3
284 CLRF 7
2BD ?NXQ
?ncxq
3FC FFAF
sub
2A0 SETDEC
078 READ 1(Z)
10E A=C ALL
138 READ 4(L)
135 ?NCXQ
C=
060 184D
A*C
070 N=C ALL
178 READ 5(M)
00E A=0 ALL
A
35C PT=12
=
162 A=A+1@PT
1
01D ?NCXQ
C=
060 1807
A+C
168 WRIT 5(M)
0B0 C=N ALL
10E A=C ALL
178 READ 5(M)
261 ?NCXQ
C=
060 1898
A/C
068 WRIT 1(Z)
10E A=C ALL
0F8 READ 3(X)
01D ?NCXQ
C=
060 1807
A+C
10E A=C ALL
0F8 READ 3(X)
0AE A<>C ALL
0E8 WRIT 3(X)
36E ?A#C ALL
267 JC-34h=-52d
1F8 READ 7(O)
068 WRIT 1(Z)
278 READ 9(Q)
17C RCR 6
104 CLRF 8
248 CLFR 9
2BD ?NXQ
?ncxq
3FC FFAF
sub
278 READ 9(Q)
2BD ?NXQ
?ncxq
3FC FFAF
sub
108 SETF 8
238 READ 8(P)
17C RCR 6
2BD ?NXQ
?ncxq
3FC FFAF
sub
238 READ 8(P)
2BD ?NXQ
?ncxq
3FC FFAF
sub
2A0 SETDEC
078 READ 1(Z)
10E A=C ALL
0B8 READ 2(Y)
135 ?NCXQ
C=
060 184D
A*C
070 N=C ALL
1B8 READ 6(N)
00E A=0 ALL
A
35C PT=12
=
162 A=A+1@PT
1
01D ?NCXQ
C=
060 1807
A+C
1A8 WRIT 6(N)
0B0 C=N ALL
10E A=C ALL
1B8 READ 6(N)
261 ?NCXQ
C=
060 1898
A/C
068 WRIT 1(Z)
1E8 WRIT 7(O)
10E A=C ALL
0F8 READ 3(X)
01D ?NCXQ
C=
060 1807
A+C
10E A=C ALL
0F8 READ 3(X)
0AE A<>C ALL
0E8 WRIT 3(X)
36E ?A#C ALL
10D ?C GO
?c go
3FF FF43
loop
345 ?NCGO
?ncgo
042 10D1
CLA
the routine stops after clearing the alpha register
260 SETHEX
sub
The first word of this subroutine is at the address FFAF in
my ROM
23A C=C+1 M
070 N=C ALL
106 A=C S&X
03C RCR 3
306 ?A<C S&X
360 ?C RTN
270 RAMSLCT
038 READDATA
10E A=C ALL
046 C=0 S&X
270 RAMSLCT
2A0 SETDEC
178 READ 5(M)
28C ?FSET 7
033 JNC+06
01D ?NCXQ
C=
060 1807
A+C
10E A=C ALL
1B8 READ 6(N)
023 JNC+04
24C ?FSET 9
013 JNC+02
1B8 READ 6(N)
01D ?NCXQ
C=
060 1807
A+C
10E A=C ALL
078 READ 1(Z)
0AE A<>C ALL
10C ?FSET 8
135 ?NCXQ
C=
060 184D
A*C
10C ?FSET 8
261 ?C XQ
C=
061 1898
A/C
068 WRIT 1(Z)
0B0 C=N ALL
2DB JNC-25h=-37d
( at the address FFD4 in my ROM )
| STACK | INPUTS | OUTPUTS |
| Y | y | y |
| X | x | F(x,y) |
| L | / | x |
Example1: With the same parameters,
7 ENTER^
5 XEQ "KdF" returns
F(5,7) = 1.022466709
---Execution time = 15s---
Example2: Appell hypergeometric function F1 is a special case of Kampé de Fériet functions with A = B = C = P = 1 and Q = S = 0
-For instance, to compute F1( 1 , 2 , 3 , 7 ; 0.5 , 0.7 )
-Store 1 into registers R01 R02 R03
R04
0 into registers R05 R06
and 1 2 3 7
into R07 R08 R09 R10
-Then,
0.7 ENTER^
0.5 XEQ "KdF" gives
F1( 1 , 2 , 3 , 7 , 0.5 , 0.7 ) = 1.856874937
---Execution time = 2mn20s
Warning: This program does not check for alpha data.
-Therefore, be sure that registers R01 thru R06+A+B+C+P+Q+S only contain
numbers!
Notes:
-If A = B = C = P = Q = S = 0 , "KdF" returns
exp(x+y)
-Register Z contains the first neglected term.
-Register T is unchanged.
-The alpha "register" is cleared.
Reference: