Errors in implementing gamma function about approxfun.jl

[MikaelSlevinsky]

Hi Jiahao,

An implementation of the gamma function in ApproxFun would look something like this:

Gamma100(x) = sum(Fun(t->t^(x-1)*exp(-t),JacobiWeightSpace(x-1-ceil(x-1),0.,UltrasphericalSpace{0}([0.,100.]))))
@show [Gamma100(5) factorial(4)]
@show [Gamma100(0.5) sqrt(π)]

At this time, functions defined on the Ray type don't have an integration method, but to take care of the algebraic singularity at the origin, JacobiWeightSpace will do the trick.

[dlfivefifty]

Forgot sum() was implemented.. that was a pull request from Mikael, right?

Just wanted to add the other implementation we talked about yesterday

x=Fun(identity,[0.,1000.])
α=0.123
f=x^(α-1)*exp(-x)
D=Derivative(space(f))
u=D\f
last(u)

When both endpoints have singularities cumsum will be more difficult. Chebfun subdivides the interval in two, which I guess is sufficient.

On 26 Nov 2014, at 5:08 am, Richard Mikael Slevinsky [email protected] wrote:

Hi Jiahao,

An implementation of the gamma function in ApproxFun would look something like this:

Gamma100(x) = sum(Fun(t->t.^(x-1).*exp(-t),JacobiWeightSpace(x-1-ceil(x-1),0.0,UltrasphericalSpace{0}([0.0,100.0]))))
@show [Gamma100(5) factorial(4)]
@show [Gamma100(0.5) sqrt(π)]
At this time, the Ray type doesn't have an integration method, but to take care of the algebraic singularity at the origin, JacobiWeightSpace will do the trick.

—
Reply to this email directly or view it on GitHub #46 (comment).

[dlfivefifty]

Here’s an alternative approach for indefinite integral for functions with singularities at both ends:

x=Fun(identity)
α=2.4;β=1.3;
f=(1+x)^(α-1)(1-x)^(β-1)
w1=exp(-40x^2)(1+x)^(α-1)(1-x)^(β-1)
w2=Fun(x->exp(-40x^2)(1+x)^(α-1)_(1-x)^(β-1))
c=sum(f)/sum(w1)
v=f-w1_c
D=Derivative(space(v))
u=D\v;
u=c*cumsum(w2)⊕u

u[.1]-0.554308241151084127 #-1.1102230246251565e-16

The result is accurate to 16 digits compared to NIntegrate with high precision. (The implementation beat NIntegrate’s default!) ⊕ is a just added syntax for adding functions from two different spaces (by interlacing coefficients). At some point this might make sense as what + should do if spaces are not compatible. But right now “not compatible” means an error is thrown…so hard to check

On 26 Nov 2014, at 10:38 am, Sheehan Olver [email protected] wrote:

Forgot sum() was implemented.. that was a pull request from Mikael, right?

Just wanted to add the other implementation we talked about yesterday

x=Fun(identity,[0.,1000.])
α=0.123
f=x^(α-1)*exp(-x)
D=Derivative(space(f))
u=D\f
last(u)

When both endpoints have singularities cumsum will be more difficult. Chebfun subdivides the interval in two, which I guess is sufficient.

On 26 Nov 2014, at 5:08 am, Richard Mikael Slevinsky <[email protected] mailto:[email protected]> wrote:

Hi Jiahao,

An implementation of the gamma function in ApproxFun would look something like this:

Gamma100(x) = sum(Fun(t->t.^(x-1).*exp(-t),JacobiWeightSpace(x-1-ceil(x-1),0.0,UltrasphericalSpace{0}([0.0,100.0]))))
@show [Gamma100(5) factorial(4)]
@show [Gamma100(0.5) sqrt(π)]
At this time, the Ray type doesn't have an integration method, but to take care of the algebraic singularity at the origin, JacobiWeightSpace will do the trick.

—
Reply to this email directly or view it on GitHub #46 (comment).

[dlfivefifty]

Just checked in updates so now the following work. Still need to do singularities + ray. @ajt60gaibb

Gaussian on the half line

f=Fun(x->exp(-x^2),[0.,Inf])
cf=cumsum(f)
sum(f)-sqrt(π)/2 # 3.0989533161829286e-10

PyPlot.plt.hist(sample(f,1000))

Singularity

x=Fun(identity,[0.,30.])
α=0.123
f=(x^(α-1)*exp(-x))
u=cumsum(f)
last(u)-gamma(α) # -3.4638958368304884e-14

Beta functions

x=Fun(identity,[0.,1.])
α=2.4;β=1.3;
f=x^(α-1)*(1-x)^(β-1)
cf=cumsum(f)
cf[.1]-0.0016226562099643412 # 3.0357660829594124e-18

x=Fun(identity,[0.,1.])
α=0.5123;β=0.423;
f=x^(α-1)*(1-x)^(β-1)
cumsum(f)[.1]-0.6123555368287131 # -1.5543122344752192e-15

Sampling Beta function

setplotter(“PyPlot”)
plot(f/sum(f))
PyPlot.plt.hist(sample(f,2000),normed=true,bins=[0.:.01:1.])

On 26 Nov 2014, at 10:39 am, Sheehan Olver [email protected] wrote:

Here’s an alternative approach for indefinite integral for functions with singularities at both ends:

x=Fun(identity)
α=2.4;β=1.3;
f=(1+x)^(α-1)(1-x)^(β-1)
w1=exp(-40x^2)(1+x)^(α-1)(1-x)^(β-1)
w2=Fun(x->exp(-40x^2)(1+x)^(α-1)_(1-x)^(β-1))
c=sum(f)/sum(w1)
v=f-w1_c
D=Derivative(space(v))
u=D\v;
u=c*cumsum(w2)⊕u

u[.1]-0.554308241151084127 #-1.1102230246251565e-16

The result is accurate to 16 digits compared to NIntegrate with high precision. (The implementation beat NIntegrate’s default!) ⊕ is a just added syntax for adding functions from two different spaces (by interlacing coefficients). At some point this might make sense as what + should do if spaces are not compatible. But right now “not compatible” means an error is thrown…so hard to check

On 26 Nov 2014, at 10:38 am, Sheehan Olver <[email protected] mailto:[email protected]> wrote:

Forgot sum() was implemented.. that was a pull request from Mikael, right?

Just wanted to add the other implementation we talked about yesterday

x=Fun(identity,[0.,1000.])
α=0.123
f=x^(α-1)*exp(-x)
D=Derivative(space(f))
u=D\f
last(u)

When both endpoints have singularities cumsum will be more difficult. Chebfun subdivides the interval in two, which I guess is sufficient.

On 26 Nov 2014, at 5:08 am, Richard Mikael Slevinsky <[email protected] mailto:[email protected]> wrote:

Hi Jiahao,

An implementation of the gamma function in ApproxFun would look something like this:

Gamma100(x) = sum(Fun(t->t.^(x-1).*exp(-t),JacobiWeightSpace(x-1-ceil(x-1),0.0,UltrasphericalSpace{0}([0.0,100.0]))))
@show [Gamma100(5) factorial(4)]
@show [Gamma100(0.5) sqrt(π)]
At this time, the Ray type doesn't have an integration method, but to take care of the algebraic singularity at the origin, JacobiWeightSpace will do the trick.

—
Reply to this email directly or view it on GitHub #46 (comment).

[dlfivefifty]

The following now works to 12 digits (not sure how to fix the extra digits of error..)

sum(Fun(x->x^0.1*exp(-x),JacobiWeightSpace(0.1,0.,[0.,Inf])))-gamma(1.1)