the error-estimates are algebraically added after each step.
Adding the magnitudes could be a less optimistic alternative:
Simply add OBJ-> EVAL ->LIST ABS OBJ-> ->ARRY
just before 1 0 PUT 'ERROR' near the end
I was worried when I saw that some coefficients
of this Runge-Kutta-Verner method were of the order of 600:
I thought it was going to produce great roundoff errors,
but fortunately, the corresponding k-value ( namely k12 )
is weighted with a small 3/7280 and indeed,
the results are usually satisfactory, even in the last decimal:
For instance: y'= -2xy , y(0) = 1
h = 0.1 gives y(1) = 0.367879441185
h = 0.05 ----- y(1) = 0.367879441171 ( exact )
h = 0.025 ---- y(1) = 0.367879441171 ( still exact! )
At least, 'RKV8E9' gives an idea of the obtained accuracy
with 16 evaluations of the function per step,
instead of 11+2*11=33 evaluations if one uses 'RK8' with h and then with h/2.
This may be an advantage if the functions are complicated.
In "Numerical Recipes" they don't seem to like high-order Runge-Kutta formulas.
On the other hand, they praise Bulirsch-Stoer methods
( "Runge-Kutta is for ploughing the fields,
Burlisch-Stoer is a high-strung racehorse" )
and it is a little contradictory:
though quite fantastic, these methods require more than 11
evaluations per step to achieve an 8th-order formula!
Moreover, roundoff errors are also greater.
I've always been fascinated by Runge-Kutta methods,
especially if they are compared to the Taylor's method
and the huge amount of calculus they require!
And the size of the non-linear systems that must be solved to find high-order formulae!
It seems like a touch of magic!
Of course, the classical 'RK4' is probably enough for an HP-41,
but we can try more accurate formulae with a 49G or 5OG.
In fact, I wrote 'RK8' and 'RKV8E9' for completeness, perhaps it will be useful.
However, I must say that, even now, I still prefer the HP-41 programming language
with its direct arithmetic in the 4-level stack,
and all the trickeries we can ( must? ) find to create neat programs.
Speed may be a problem, but thanks to Warren Furlows and his excellent V41,
HP-41 programs can run faster than HP-49 programs!
But I stop here my "philosophical' point of view...