One very annoying feature of the 17BII+ is that at some point in length of programs, the Solver just returns 0. A straight forward example can be seen using Viktor T. Toth's Calculators and the Gamma Function:
GAMMA:
(75122.6331530+80916.6278952×Z+36308.2951477×Z^2+8687.24529705×Z^3
+1168.92649479×Z^4+83.8676043424×Z^5+2.50662827511×Z^6)
÷Z÷(Z+1)÷(Z+2)÷(Z+3)÷(Z+4)÷(Z+5)÷(Z+6)
×(Z+5.5)^(Z+.5)×EXP(-(Z+5.5))
-GAMMA
Calculated GAMMA Values
1 1.0000000001
2 1
3 2
4 6.0000000002
5 23.9999999999
6 120
7 720
8 5,039.99999998
9 40,320
10 362,880
BETA:
0×L(R:P+Q-1)×L(A:75122.6331530)×L(B:80916.6278952)×L(C:36308.2951477)
×L(D:8687.24529705)×L(E:1168.92649479)×L(F:83.8676043424)×L(G:2.50662827511)
+(G(A)+G(B)×P+G(C)×P^2+G(D)×P^3+G(E)×P^4+G(F)×P^5+G(G)×P^6)
÷P÷(P+1)÷(P+2)÷(P+3)÷(P+4)÷(P+5)÷(P+6)
×(P+5.5)^(P+.5)×EXP(-(P+5.5))
×(G(A)+G(B)×Q+G(C)×Q^2+G(D)×Q^3+G(E)×Q^4+G(F)×Q^5+G(G)×Q^6)
÷Q÷(Q+1)÷(Q+2)÷(Q+3)÷(Q+4)÷(Q+5)÷(Q+6)
×(Q+5.5)^(Q+.5)×EXP(-(Q+5.5))
÷(G(A)+G(B)×G(R)+G(C)×G(R)^2+G(D)×G(R)^3+G(E)×G(R)^4+G(F)×G(R)^5+G(G)×G(R)^6)
×G(R)×(G(R)+1)×(G(R)+2)×(G(R)+3)×(G(R)+4)×(G(R)+5)×(G(R)+6)
÷((G(R)+5.5)^(G(R)+.5))÷EXP(-(G(R)+5.5))
-BETA
p = 4,
q = 5,
Exact beta = 1÷35 = .028571428571
Calculated beta = .0285714273948 (15 seconds on 19B)
Error = 1.177E-9
beta = 0 on the 17BII+
Bob
P.S. Of course, in defense of the 17BII+, one could ask "WHY would ANYONE need these on a financial calculator?" ;-)
Hi Bob,
You can more than halve the size of your GAMMA and BETA Solver
equations by using the *original* Lanczos formula given in
Victor's article, rather than the form given as suitable for
programmable calculators.
Use a SUM list called GAMP, comprising the P1-P6 factors:
Item Value
1 76.1800917295
2 -86.5053203294
3 24.0140982408
4 -1.23173957245
5 1.20865097387E-3
6 -5.39523938495E-6
Note the GAMP list totals 12.4583333242. Hopefully these
equations will work in the 17BII+ Solver:
GAMMA:
SQRT(2*PI)/Z*(Z+5.5)^(Z+.5)/EXP(Z+5.5)
*(1+19E-11+SIGMA(N:1:6:1:ITEM(GAMP:N)/(Z+N)))
-GAMMA
BETA:
SQRT(2*PI)*(P+Q)/P/Q*EXP(-5.5)
*(P+5.5)^(P+.5)*(Q+5.5)^(Q+.5)/(P+Q+5.5)^(P+Q+.5)
*(1+19E-11+SIGMA(N:1:6:1:ITEM(GAMP:N)/(P+N)))
*(1+19E-11+SIGMA(N:1:6:1:ITEM(GAMP:N)/(Q+N)))
/(1+19E-11+SIGMA(N:1:6:1:ITEM(GAMP:N)/(P+Q+N)))
-BETA
The 1+19E-11=1.00000000019 - the Lanczos P0.
Note, BETA(4,5)=3!*4!/8!=1/280.
The HP19BII solves BETA(4,5) in 4 seconds, giving
3.57142857143E-3 (exact), so it works a lot faster for the
shorter formula.
Cheers,
Tony
The following GAMMA and BETA functions are based on Tony Hutchins' excellent suggestions.
Smaller, faster, AND more accurate!
Who says there's no free lunch.
Use a SUM list called GAMP, comprising the P1-P6 factors:
Item Value
1 76.1800917295
2 -86.5053203294
3 24.0140982408
4 -1.23173957245
5 1.20865097387E-3
6 -5.39523938495E-6
Note the GAMP list totals 12.4583333242.
GAMMA:
SQRT(2×PI)÷Z×(Z+5.5)^(Z+.5)÷EXP(Z+5.5)
×(1+19E-11+SIGMA(N:1:6:1:ITEM(GAMP:N)÷(Z+N)))
-GAMMA
Calculated GAMMA Values (17BII+)
1 1.0000000001
2 .999999999986
3 2
4 5.99999999998
5 23.9999999999
6 120
7 720.000000002
8 5,039.99999999
9 40,319.9999999
10 362,880.000001
BETA:
SQRT(2×PI)×(P+Q)÷P÷Q×EXP(-5.5)
×(P+5.5)^(P+.5)×(Q+5.5)^(Q+.5)÷(P+Q+5.5)^(P+Q+.5)
×(1+19E-11+SIGMA(N:1:6:1:ITEM(GAMP:N)÷(P+N)))
×(1+19E-11+SIGMA(N:1:6:1:ITEM(GAMP:N)÷(Q+N)))
÷(1+19E-11+SIGMA(N:1:6:1:ITEM(GAMP:N)÷(P+Q+N)))
-BETA
p = 4
q = 5
Exact Beta = 3!×4!÷8! = 1÷280
Exact Beta = 0.00357142857142857
Calculated = 0.00357142857143 (2.8 seconds on 17BII+)
Error = 0
Bob