Quaternionic Spheroidal Wave Functions for the HP-41
Overview
-The following program generalizes a routine that is listed in "Spheroidal
Harmonics for the HP-41" to quaternionic variables.
-However, the parameters m , n , c2 , Lmn
remain real.
• m & n must be non-negative integers, n = m
, m+1 , m+2 , ... , c2 may be positive or negative or
equal to zero,
and the eigenvalues Lmn must
be calculated first ( see for example the 1st routine listed in "Spheroidal
Harmonics for the HP-41" )
• If c = 0 , the eigenvalues are simply n(n+1)
Program Listing
-We assume that | q | <= 1 and Smn(q) is computed by Smn(q) = ( 1 - q2 )m/2 Sumk=0,1,.... ak qk
with (k+1)(k+2) ak+2
- [ k ( k + 2m + 1 ) - Lmn + m ( m + 1 ) ] ak
- c2 ak-2 = 0
Data Registers: R00 thru R08 & R13 thru R18: temp ( Registers R09 thru R12 are to be initialized before executing "SWFQ" )
• R09 = m • R11 = c2
• R10 = n • R12 = Lmn
Flag: F00
CF 00 Flammer's Scheme Smn(0) = Pmn(0) & S'mn(0) = P' mn(0) where Pmn(x) = Associated Legendre Functions of the first kind
a0 = [ (-1)(n+m)/2
(n+m)! ] / [ 2n ((n-m)/2)! ((n+m)/2)! ]
a1 = 0
if n - m is even
a1 = [ (-1)(n+m-1)/2
(n+m+1)! ] / [ 2n ((n-m-1)/2)! ((n+m+1)/2)! ]
a0 = 0
if n - m is odd
SF 00 Non-Normalized Functions
a0 = 1
a1 = 0
if
n - m is even
a0 = 0
a1 = 1 if
n - m is odd
Subroutines: "Q*Q" "Q^R" ( cf
"Quaternions for the HP-41" )
| 01 LBL "SWFQ"
02 STO 01 03 STO 05 04 RDN 05 STO 02 06 STO 06 07 RDN 08 STO 03 09 STO 07 10 X<>Y 11 STO 04 12 STO 08 13 XEQ "Q*Q" 14 STO 01 15 RDN 16 STO 02 17 RDN 18 STO 03 19 X<>Y 20 STO 04 21 RCL 10 22 RCL 09 23 - 24 STO 13 25 2 26 MOD 27 STO 00 28 X#0? 29 GTO 00 30 STO 06 31 STO 07 32 STO 08 |
33 SIGN
34 STO 05 35 LBL 00 36 CLX 37 STO 17 38 SIGN 39 FS? 00 40 GTO 00 41 CHS 42 RCL 09 43 RCL 10 44 + 45 STO 14 46 RCL 00 47 ST- 13 48 ST+ 14 49 - 50 2 51 / 52 Y^X 53 RCL 14 54 FACT 55 * 56 2 57 ST/ 13 58 ST/ 14 59 RCL 10 60 Y^X 61 / 62 RCL 13 63 FACT 64 RCL 14 |
65 FACT
66 * 67 / 68 LBL 00 69 STO 18 70 RCL 05 71 X<>Y 72 * 73 STO 13 74 RCL 06 75 LASTX 76 * 77 STO 14 78 RCL 07 79 LASTX 80 * 81 STO 15 82 RCL 08 83 LASTX 84 * 85 STO 16 86 LBL 01 87 RCL 11 88 RCL 17 89 * 90 RCL 09 91 ST+ X 92 RCL 00 93 ST+ Y 94 ST* Y 95 + 96 RCL 09 |
97 ST+ Y
98 X^2 99 + 100 RCL 12 101 - 102 RCL 18 103 STO 17 104 * 105 + 106 RCL 00 107 1 108 + 109 ST/ Y 110 LASTX 111 + 112 STO 00 113 / 114 STO 18 115 XEQ "Q*Q" 116 STO 05 117 RDN 118 STO 06 119 RCL 18 120 * 121 RCL 14 122 + 123 STO 14 124 LASTX 125 - 126 ABS 127 X<>Y 128 STO 07 |
129 RCL 18
130 * 131 RCL 15 132 + 133 STO 15 134 LASTX 135 - 136 ABS 137 + 138 X<>Y 139 STO 08 140 RCL 18 141 * 142 RCL 16 143 + 144 STO 16 145 LASTX 146 - 147 ABS 148 + 149 RCL 05 150 RCL 18 151 * 152 RCL 13 153 + 154 STO 13 155 LASTX 156 - 157 ABS 158 + 159 X#0? 160 GTO 01 |
161 RCL 09
162 2 163 / 164 STO 00 165 1 166 RCL 01 167 - 168 RCL 04 169 CHS 170 RCL 03 171 CHS 172 RCL 02 173 CHS 174 R^ 175 XEQ "Q^R" 176 STO 01 177 RDN 178 STO 02 179 RDN 180 STO 03 181 X<>Y 182 STO 04 183 RCL 13 184 STO 05 185 RCL 14 186 STO 06 187 RCL 15 188 STO 07 189 RCL 16 190 STO 08 191 XEQ "Q*Q" 192 END |
( 244 bytes / SIZE 019 )
| STACK | INPUTS | OUTPUTS |
| T | t | t' |
| Z | z | z' |
| Y | y | y' |
| X | x | x' |
with Smn( x + y i + z j + t k ) = x' + y' i + z' j + t' k , | x + y i + z j + t k | <= 1
Example1: m = 0 , n = 1 , c2 = 2 , Lmn = 3.172127920 These 4 numbers are to be stored in R09 thru R12
CF 00
0.4 ENTER^
0.3 ENTER^
0.2 ENTER^
0.1 XEQ "SWFQ" >>>>
0.117347028
---Execution time = 34s---
RDN 0.210312365
RDN 0.315468547
RDN 0.420624729
S( 0.1 + 0.2 i + 0.3 j + 0.4 k ) = 0.117347028 + 0.210312365 i + 0.315468547 j + 0.420624729 k
Example2: m = n = 2 , c2 = 3 , Lmn = 6.412006610 These 4 numbers are to be stored in R09 thru R12
CF 00
0.4 ENTER^
0.3 ENTER^
0.2 ENTER^
0.1 XEQ "SWFQ" >>>>
4.058321831
---Execution time = 39s---
RDN -0.160164520
RDN -0.240246780
RDN -0.320329040
S( 0.1 + 0.2 i + 0.3 j + 0.4 k ) = 4.058321831 - 0.160164520 i - 0.240246780 j - 0.320329040 k
Notes:
-If you set F00 you get the same result in the 1st example, and the
same result divided by 3 in the 2nd example.
-If it is possible, use an M-Code routine that returns 0^0 = 1 to compute
"Q^R"
-If m > 0 Smn(1) = Smn(-1) = 0
so you could add
RCL 09 X=0? GTO 00 RCL 04 ABS RCL 03 ABS + RCL 02 ABS + X#0? GTO 00 RCL 01 ABS 1 X#Y? GTO 00 CLST RTN LBL 00
after line 12
-"SWFQ" may also be used to compute Legendre Functions provided | q
| does not exceed 1:
-Store 0 in R11 and n(n+1) in R12
Reference:
[1] Abramowitz & Stegun , "Handbook of Mathematical
Functions" - Dover Publications - ISBN 0-486-61272-4