Sunday, August 22, 2004
Most of Ramanujan's work is not practically useful. However, two exceptions are his formulae for the approximate circumference (perimeter) of an ellipse. Here is the simpler one:
P ~= pi [ 3(a+b) - sqrt((3a+b)(a+3b)) ]
It can be calculated on a standard basic calculator with one register of memory that doesn't obey algebraic operator precedence (times before plus, etc.)
where P= "M+", R= "MR", K="MC", Q="sqrt", S="+/-". It is a little annoying that a and b need to be typed in 3 times each.
This formula can also be expressed in terms of h=[(a-b)/(a+b)]2, although it's not so helpful for the simple calculator.
P = pi (a+b) [ 3 - sqrt( 4-h ) ]
This formula has 12th order error in eccentricity and worst case relative error of 3.8% (about two decimal digits of precision)
Ramanujan's second formula is more accurate:
P ~= pi (a+b) [ 1 + 3h / ( 10+ sqrt( 4-3h )) ] or on a simple calculator:
This formula has 20th order error in eccentricity and worst case relative error of 0.40% (about 3 decimal digits of precision). David W. Cantrell has added a correction term to Ramanujan's ellipse perimeter formula
3h 12 pi(a + b) (1 + ----------------- + (4/pi - 14/11) h ). 10 + Sqrt(4 - 3h)
which decreases the worst case error to 0.0014% (about 5 decimal digits of precision). However it no longer seems possible to calculate this formula with only one register of memory.
(What expressions can be calculated with one register of memory?)
Ramanujan came up with his formulae in 1914; Cantrell's correction term is only May 2004. It's interesting that while the circumference of a circle has been laid to rest for thousands of years, improvements in the circumference of an ellipse still continue to this day.
Some matlab code for the Gauss-Kummer series
for i=1:200,n=sym(i),coeff2(i)=(prod((n+1):2*n)/prod(1:n)/((1-2*n)*(-4)^n))^2;end xac2=vpa((1+sum(h.^(1:200).*coeff2))*pi*(a+b),1000)
cay=sym(1)+x/4*(log(16/x)-1);for i=2:100,n=sym(i),cay=cay + x^n * prod(1:2:(2*n-3))^2 * (2*n-1) / (prod(2:2:(2*n-2))^2*(2*n))/2 * (log(16/x)+sum(-4./((1:2:2*n-1) .* (2:2:2*n)))+2/(2*n-1)/(2*n));end
exact value via elliptic integral
p4=vpa(pi*(a+b)*(1+3*h/(10+sqrt(4-3*h)) + sc*h^12),120)
For Halley's comet, with e=0.9673 and a=17.94AU= 2,683,786,326 km (forget significant digits, and all rounding is by truncation), its perimeter (via 200 terms of Gauss Kummer) is 11,529,383,201.15 km. Ramanujan's first formula underapproximates by 1,175,193.9 km. Second underapproximates by 2637.7 km, and Cantell's correction formula only does slightly better, being under by 2616.4 km.
For a=4, b=3, (a 3-4-5 right triangle), which has 0.66 eccentricity, its perimeter is 22.1034921607095050452855864638724607782782891. Ramanujan's second formula (with or without correction) misses by -1.8e-12 For a=12, b=5 (another right triangle), e=0.9091, perimeter=55.6959493710847386662875219855831134524807, and Ramanujan's second formula (with our without correction) misses by -2.3e-7.
For a=1 b=1/10, e=0.9949 perimeter=4.063974180100895742557793101181576, second formula -4.6e-5, with correction -3.2e-5. For a=1 b=1/100 e=0.99995, perimeter = 4.00109832972265186074746449610774074592667795549222896254442786221, no correction -9.5e-4, with correction -0.49e-4. For a=1 b=1/1000 e=0.9999995, perimeter = 4.00001558810468824461075647365734692443, no correction -15.1e-4, with correction 0.21e-4.
Things for which exact formulae exist: circle perimeter, circle area, ellipse area, ellipsoid area for ellipsoid that have circular cross sections, ellipsoid volume
No exact formulae: ellipse perimeter, general ellipsoid surface area.
Below is a picture of the approximation error of various formulae. Blue is Ramanujan's second formula, green includes Cantrell's correction term, red is Cantrell's 2001 formula involving Hoelder means, cyan is Cantrell's 2004 formula.
>> t=exp(i);b=1./t;xc=ellip(1-b.*b)*4;h=((a-b)./(a+b)).^2;v=pi*(a+b);n=1+3*h./(10+sqrt(4-3*h)); p3=v.*n;p4=v.*(n+sc*h.^12);h2=4*(a+b)-2*(4-pi)*((a+b.^-p)/2).^(1/-p);
Saturday, August 21, 2004
1 i j k * ---------- 1 | 1 i j k i | i -1 k -j j | j k -1 -i k | k -j -i -1Generalizing complex numbers to 3-vectors of the form (a,(b,c)) runs into the problem of how to define the result of X+(Y,Z) to be a scalar.
Wednesday, August 18, 2004
Wednesday, August 11, 2004
Tuesday, August 10, 2004
Monday, August 09, 2004
Sunday, August 08, 2004
Least Primitive Root of Mersenne Primes: All the more reason the C217 composite cofactor of 2^1101+1 ought to be factored. The final terms of the sequence were calculated using the Mersenne factorizations mentioned earlier. The earlier terms were calculated using znprimroot in Pari/GP. Proving a primitive root X of a prime p requires modular exponentiation of X to all powers (p-1)/q where q is a prime factor of (p-1). At first I thought it had to be to all divisors of (p-1), which would have required something like 2^50 work for 2^2281-1; fortunately, it was not.
The least primitive root of M2203 is at least 3, and it probably is 3. 3 is probably the least primitive root for M4253, M9689, M21701, as well. This comes from the observation that 2 never seems to be a primitive root. When 3 is not a primitive root, then 3^((p-1)/3)==1 (mod p) empirically. It's interesting that all mersenne primes except the first seem to have remainder 1 when divided by 3 (true for all odd exponents).
for(i=2, length(p), if((Mod(3,2^p[i]-1)^((2^p[i]-2)/3))!=Mod(1,2^p[i]-1), print(p[i])))
where p is a list of mersenne exponents.
P-1 for some of the higher Mersenne prime exponents are surprisingly smooth: 216090 = 2*3*3*5*7*7*7*7, 20996010 = 2*3*3*3*3*5*7*7*23*23. These would be more interesting targets for the ElevenSmooth project than their number 2*2*2*2*2*2*3*3*3*5*5*7*11 = 3326400
In an unrelated topic, in June I submitted a superseeker query for [7,11,32,40,75,87,136,152,215,235,312,336,427], but I no longer know the significance of the sequence. I think it might have something to do with the Pell Equation. The generating function guesser generated the continuation [455, 560, 592, 711, 747, 880], which I recall the first few terms being correct.