I have not cared for 'rational' (one can), but this would do
(digging in old files around the Maple usenet group ...):
ExponentsToFloats:=proc(expr)
local simplePower;
simplePower:=proc(expr)
# convert x^a to x^(evalf(a))
# except division and pure sqrt
local x,a;
Digits:=Digits+3;
# validate input
if op(0,expr) = `^` then
x:= op(1,expr);
a:= op(2,expr);
else
return(expr);
end if;
# do not change for division and sqrt
if ( a = -1 ) then
return ( 1/x );
elif ( a = 1/2 ) then
return ( sqrt(x) );
elif ( a = -1/2 ) then
return ( 1/sqrt(x) );
end if;
return x^evalf(a);
end proc;
return evalindets( q, '`^`', f -> simplePower(f) );
end proc;
Examples:
q:= (x^(41/39))^(I*(3/7)), ExponentsToFloats(q) gives
(x^1.051282051282)^(.4285714286*I)
q:=hypergeom([Pi,b^(99/100)],[c],exp(1)/sqrt(x^(201/199)/2)) gives
hypergeom([Pi, b^.9900000000000],[c],exp(1)*2^(1/2)/(x^1.010050251256)^(1/2));
I have not cared for 'rational' (one can), but this would do
(digging in old files around the Maple usenet group ...):
ExponentsToFloats:=proc(expr)
local simplePower;
simplePower:=proc(expr)
# convert x^a to x^(evalf(a))
# except division and pure sqrt
local x,a;
Digits:=Digits+3;
# validate input
if op(0,expr) = `^` then
x:= op(1,expr);
a:= op(2,expr);
else
return(expr);
end if;
# do not change for division and sqrt
if ( a = -1 ) then
return ( 1/x );
elif ( a = 1/2 ) then
return ( sqrt(x) );
elif ( a = -1/2 ) then
return ( 1/sqrt(x) );
end if;
return x^evalf(a);
end proc;
return evalindets( q, '`^`', f -> simplePower(f) );
end proc;
Examples:
q:= (x^(41/39))^(I*(3/7)), ExponentsToFloats(q) gives
(x^1.051282051282)^(.4285714286*I)
q:=hypergeom([Pi,b^(99/100)],[c],exp(1)/sqrt(x^(201/199)/2)) gives
hypergeom([Pi, b^.9900000000000],[c],exp(1)*2^(1/2)/(x^1.010050251256)^(1/2));
I like Joe Riel's approach - how does one modify you solution
to get a tab spacing of 2 (which i prefer)?
no, not to the general question around the Geddes method (still
waiting that Fred finds time for some answers), but for your original task
as this board seems to allow no uploads within threads i put it here, 36Kb:
http://www.axelvogt.de/maplekram/integral_for_Bayesian_short_example.zip
PS: i can no longer find the way to get informed via mail if
someone answers to a specific thread - where is?
no, not to the general question around the Geddes method (still
waiting that Fred finds time for some answers), but for your original task
as this board seems to allow no uploads within threads i put it here, 36Kb:
http://www.axelvogt.de/maplekram/integral_for_Bayesian_short_example.zip
PS: i can no longer find the way to get informed via mail if
someone answers to a specific thread - where is?
Hi Fred,
Thx, I try to follow your sketch ... symmetric ...
Hm - may I ask some questions?
Why 'splitting'? A Taylor series already does it, no? What is the
practical advantage of the more general approach?
Then: I did not quite get the algo for splitting in Carvajal's paper,
page 11. It depends on the splitting points ... and one has to have
the supremums norm (which might difficult to get).
How would you do that for the example, ie sqrt( symmetric polynom ),
which is still symmetric (it has only degree 2)?
Hi Fred,
Thx, I try to follow your sketch ... symmetric ...
Hm - may I ask some questions?
Why 'splitting'? A Taylor series already does it, no? What is the
practical advantage of the more general approach?
Then: I did not quite get the algo for splitting in Carvajal's paper,
page 11. It depends on the splitting points ... and one has to have
the supremums norm (which might difficult to get).
How would you do that for the example, ie sqrt( symmetric polynom ),
which is still symmetric (it has only degree 2)?
grrrr ... this idiotic forum software ... I hate it! Here is the rest:
using assuming (0 LT a,r LT 1, 0 LT b,s LT 1)
So in the transformed problem one can replace infinity by maxNum,
now living in a large, but compact rectangle.
There are 1/quadratic terms in the sqrt, which still cause trouble
for the numerics. Again split the rectangle in 4 parts (but here i
got no analytical solution) according to sqrt(maxNum). Or use the
hybrid power of Maple for the 3 'ugly' parts.
Now it should be possible to evaluate the thing ... but I lost
patients to carry it out, for some bad parameters one still would
have to work a bit. Certainly not very elegant ...
And one has to care for the error estimates ...
grrrr ... this idiotic forum software ... I hate it! Here is the rest:
using assuming (0 LT a,r LT 1, 0 LT b,s LT 1)
So in the transformed problem one can replace infinity by maxNum,
now living in a large, but compact rectangle.
There are 1/quadratic terms in the sqrt, which still cause trouble
for the numerics. Again split the rectangle in 4 parts (but here i
got no analytical solution) according to sqrt(maxNum). Or use the
hybrid power of Maple for the 3 'ugly' parts.
Now it should be possible to evaluate the thing ... but I lost
patients to carry it out, for some bad parameters one still would
have to work a bit. Certainly not very elegant ...
And one has to care for the error estimates ...
Still i do not know, whether you mean dim=2 only ...
Here some more ideas for the numerics:
'Int(Int(1/eta^(s)*S(xi,eta)^(1/2)/xi^(r),xi = 0 .. a),eta = 0 .. b)';
IntegrationTools[Change](%, {xi=1/(x-a), eta=1/(y-b)},[x,y]):
sends the 'problems' to infinity and writes as
c*Int(Int(K(x,y),x = (1+a^2)/a .. infinity),y = (1+b^2)/b .. infinity),
K:= (x,y) -> S(-1/(-x+a),-1/(-y+b))^(1/2)*(y-b)^(s-2)*(x-a)^(r-2),
a = bX^100, b = bY^1411, r = 99/100, s = 1410/1411, c = 1/141100.
Now look at the expression under the sqrt: for large x or y one can
ignore the a or b (at least for a numerical integrator), say beyond
maxNum:=10^16 for usual hardware numerics, ie set them to 0.
Beyond that there are 3 cases and in any case the integrals can be
solved analytically (I have not cared for error estimates for this)
using assuming (0<a, r<1, 0<b,s<1)
So in the transformed problem one can replace infinity by maxNum,
now living in a large, but compact rectangle.
There are 1/quadratic terms in the sqrt, which still cause trouble
for the numerics. Again split the rectangle in 4 parts (but here i
got no analytical solution) according to sqrt(maxNum). Or use the
hybrid power of Maple for the 3 'ugly' parts.
Now it should be possible to evaluate the thing ... but I lost
patients to carry it out, for some bad parameters one still would
have to work a bit. Certainly not very elegant ...
And one has to care for the error estimates ...
Still i do not know, whether you mean dim=2 only ...
Here some more ideas for the numerics:
'Int(Int(1/eta^(s)*S(xi,eta)^(1/2)/xi^(r),xi = 0 .. a),eta = 0 .. b)';
IntegrationTools[Change](%, {xi=1/(x-a), eta=1/(y-b)},[x,y]):
sends the 'problems' to infinity and writes as
c*Int(Int(K(x,y),x = (1+a^2)/a .. infinity),y = (1+b^2)/b .. infinity),
K:= (x,y) -> S(-1/(-x+a),-1/(-y+b))^(1/2)*(y-b)^(s-2)*(x-a)^(r-2),
a = bX^100, b = bY^1411, r = 99/100, s = 1410/1411, c = 1/141100.
Now look at the expression under the sqrt: for large x or y one can
ignore the a or b (at least for a numerical integrator), say beyond
maxNum:=10^16 for usual hardware numerics, ie set them to 0.
Beyond that there are 3 cases and in any case the integrals can be
solved analytically (I have not cared for error estimates for this)
using assuming (0<a, r<1, 0<b,s<1)
So in the transformed problem one can replace infinity by maxNum,
now living in a large, but compact rectangle.
There are 1/quadratic terms in the sqrt, which still cause trouble
for the numerics. Again split the rectangle in 4 parts (but here i
got no analytical solution) according to sqrt(maxNum). Or use the
hybrid power of Maple for the 3 'ugly' parts.
Now it should be possible to evaluate the thing ... but I lost
patients to carry it out, for some bad parameters one still would
have to work a bit. Certainly not very elegant ...
And one has to care for the error estimates ...
for z between 0 and 1 take sqrt(z-1)*sqrt(z-2), so in both cases
the input for sqrt is negative, while ...
for z between 0 and 1 take sqrt(z-1)*sqrt(z-2), so in both cases
the input for sqrt is negative, while ...
Certainly there are several lecture notes available (for free)
as well ...
Not really introductory, but worth to look at:
Heck, Introduction to Maple
http://www.amazon.com/gp/product/0387002308/ref=sr_11_1/104-6979152-2772754?ie=UTF8
Richards, Advanced Mathematical Methods with Maple
http://www.amazon.com/gp/product/0521779812/ref=pd_bxgy_img_b/104-6979152-2772754?ie=UTF8
and other suggestions can be found through http://www.maplesoft.com/books/
Certainly there are several lecture notes available (for free)
as well ...
Not really introductory, but worth to look at:
Heck, Introduction to Maple
http://www.amazon.com/gp/product/0387002308/ref=sr_11_1/104-6979152-2772754?ie=UTF8
Richards, Advanced Mathematical Methods with Maple
http://www.amazon.com/gp/product/0521779812/ref=pd_bxgy_img_b/104-6979152-2772754?ie=UTF8
and other suggestions can be found through http://www.maplesoft.com/books/
