Here is how to use TaylorSeries.jl. In the next version, the syntax
taylor1_variable will change to just Taylor1:
julia> t = taylor1_variable(40)
1.0 t + 𝒪(t⁴¹)
julia> f(x) = sin(exp(x))
f (generic function with 1 method)
julia> f(t)
0.8414709848078965 + 0.5403023058681398 t - 0.15058433946987837 t² -
0.4207354924039483 t³ - 0.3229307265911699 t⁴ - 0.13861923299172246 t⁵ -
0.016963650188307994 t⁶ + 0.027151311649628966 t⁷ + 0.028082409487798127 t⁸
+ 0.016534675299332297 t⁹ + 0.006744265666861196 t¹⁰ +
0.0016199249536140723 t¹¹ - 0.00020257707395350685 t¹² -
0.0005026787959383626 t¹³ - 0.00034829060136926197 t¹⁴ -
0.00016780164241401782 t¹⁵ - 6.0803736936094484e-5 t¹⁶ -
1.4645171902851875e-5 t¹⁷ - 1.1346823483603049e-8 t¹⁸ +
2.515198118591517e-6 t¹⁹ + 1.817031683619697e-6 t²⁰ + 8.873827219817266e-7
t²¹ + 3.383697539084375e-7 t²² + 9.90873953752995e-8 t²³ +
1.7331465080481204e-8 t²⁴ - 2.8377431603022023e-9 t²⁵ -
4.478125328289587e-9 t²⁶ - 2.633876550034189e-9 t²⁷ - 1.1444925030062194e-9
t²⁸ - 4.013808037099447e-10 t²⁹ - 1.1102527569232767e-10 t³⁰ -
1.95623844080032e-11 t³¹ + 1.8017173073914597e-12 t³² +
3.715400951669623e-12 t³³ + 2.1778749843678503e-12 t³⁴ +
9.347841028230405e-13 t³⁵ + 3.279923193009391e-13 t³⁶ +
9.423306246867712e-14 t³⁷ + 1.997073344234444e-14 t³⁸ +
1.3114196619133462e-15 t³⁹ - 1.5394356646368331e-15 t⁴⁰ + 𝒪(t⁴¹)
julia> get_coeff(f(t), 40)
-1.5394356646368331e-15
Note that this coefficient is the 40th derivative divided by 40!. (The nth
Taylor coefficient is f^{(n)} / n!. )
El domingo, 17 de julio de 2016, 21:15:33 (UTC+2), David P. Sanders
escribió:
>
>
>
> El domingo, 17 de julio de 2016, 17:13:38 (UTC+2), Young Chun escribió:
>>
>> I have an optimization problem that involves series of high order
>> derivatives of f(x).
>> So, to get a decent value, I need to calculate higher order derivative at
>> least to 40~50th order.
>>
>
> I suggest you use TaylorSeries.jl.
>
> Note that ForwardDiff from version 0.2 removed ForwardDiff.derivative(f);
> this must now be written as the anonymous function x ->
> ForwardDiff.derivative(f, x).
>
>
>>
>> Initially, I tried to use ForwardDiff package with automatic
>> differentiation but kept facing the following error message
>>
>> <div id="inner-editor"><br class="Apple-interchange-newline">using
>> DualNumbers using ForwardDiff f(x::Vector) = sum(sin, x) + prod(tan, x) *
>> sum(sqrt, x); x = rand(5) g = ForwardDiff.derivative(f);</div>
>>
>> using DualNumbers
>>
>> using ForwardDiff
>>
>> f(x::Vector) = sum(sin, x) + prod(tan, x) * sum(sqrt, x);
>>
>> x = rand(5)
>>
>> g = ForwardDiff.derivative(f);
>>
>> LoadError: MethodError: `derivative` has no method matching
>> derivative(::Function)
>> Closest candidates are:
>> derivative(::Any, !Matched::Any)
>> while loading In[28], in expression starting on line 4
>>
>>
>>
>> So, I moved on to D(f) instead from Root package and it works quite well
>> for a small order (like 10th order derivative)
>>
>> <div id="inner-editor"><br class="Apple-interchange-newline">using
>> DualNumbers using ForwardDiff using Roots using ForwardDiff using JuMP #
>> Need to say it whenever we use JuMP using Gadfly function hder(f, n::Real)
>> # subroutine to calculate higher order derivative temp = f; if n==0 return
>> temp else for i=1:1:n temp = D(temp) end return temp end end</div>
>>
>> using DualNumbers
>>
>> using ForwardDiff
>>
>> using Roots
>>
>> using ForwardDiff
>>
>>
>>
>> using JuMP
>>
>> using Gadfly
>>
>>
>>
>> function hder(f, n::Real) # subroutine to calculate n-th order derivative
>>
>> temp = f;
>>
>> if n==0
>>
>> return temp
>>
>> else
>>
>> for i=1:1:n
>>
>> temp = D(temp)
>>
>> end
>>
>> return temp
>>
>> end
>>
>> end
>>
>> Out[7]:
>>
>> hder (generic function with 1 method)
>>
>> f(x) = (cos(x)) * exp(-1/5 * x)
>> g0 = hder(f, 0)
>> g1 = hder(f, 1)
>> g2 = hder(f, 2)
>> g3 = hder(f, 3)
>> g4 = hder(f, 4)
>> g5 = hder(f, 5)
>> g6 = hder(f, 6)
>> g7 = hder(f, 7)
>> g8 = hder(f, 8)
>>
>> plot([g0, g1, g2, g3, g4, g5, g6, g7, g8], 0, 5pi)
>>
>>
>> <https://lh3.googleusercontent.com/-RCrcnLVFPhE/V4tfcG9uSEI/AAAAAAAAGUU/unxjN7x60jgPaEcCmcRlykx8WpbMxJBnACLcB/s1600/Screen%2BShot%2B2016-07-17%2Bat%2B11.35.09.png>
>> However, if I push the limit and ask to calculate, like 30th order
>> derivative, the program never ends and keep calculating for hours.
>> Is there a better way to do this type of task? either by using
>> ForwardDiff package or by modifying my function?
>> I just learned Julia last week, so please understand if my question
>> sounds stupid.
>>
>>
>>
>>
