I'm starting to learn Julia, and I was wondering if there is an equivalent function in Julia that is doing the same as the epsilon function in Fortran.
In Fortran, the epsilon function of variable x gives the smallest number of the same kind of x that satisfies 1+epsilon(x)>1
I thought the the function eps() in Julia would be something similar, so I tried
eps(typeof(x))
but I got the error:
MethodError: no method matching eps(::Type{Int64})
Is there any other function that resembles the Fortran one that can be used on the different variables of the code?
If you really need eps to work for both Float and Int types, you can overload a method of eps for Int by writing Base.eps(Int) = one(Int). Then your code will work. This is okay for personal projects, but not a good idea for code you intend to share.
As the docstring for eps says:
help?> eps
eps(::Type{T}) where T<:AbstractFloat
eps()
eps is only defined for subtypes of AbstractFloat i.e. floating point numbers. It seems that your variable x is an integer variable, as the error message says no method matching eps(::Type{Int64}). It doesn't really make sense to define an eps for integers since the "the smallest number of the same kind of x that satisfies 1 + epsilon(x) > 1" is always going to be 1 for integers.
If you did want to get a 1 of the specific type of integer you have, you can use the one function instead:
julia> x = UInt8(42)
0x2a
julia> one(typeof(x))
0x01
It seems like what you actually want is
eps(x)
not
eps(typeof(x))
The former is exactly equivalent to your Fortran function.
Related
I was wondering if there is a commonly used term for a function that turns a value into a tuple-2 in ML-family languages, or functional programming languages more generally?
let toTuple2 x = (x, x)
In stack-based programming languages such as Forth, dup is a core operator that does duplicate the top stack element (not exactly a tuple though).
In Haskell, various packages provide this function under names like dup, dupe or double. Notice that two-tuples are also a core element of arrows, and dup = id &&& id.
I have not found anything specific to ML.
I don't know about the name of that specific function.
However, that function can be seen as a special case of a more general one:
let applyCtorToXX c x = c x x
Indeed, you can verify that toTuple2 is equivalent to applyCtorToXX (,).
In combinatory logic, or at least in how it is presented in To Mock a Mockingbird, such a function is named a "Warbler", and the symbol W is used for it (i.e. Wxy = xyy is the definition used in the book).
Looking at it from this perspective, your toTuple2 is W (,), which is the application of a warbler to the 2-tuple constructor.
I am puzzled by the following results of typeof in the Julia 1.0.0 REPL:
# This makes sense.
julia> typeof(10)
Int64
# This surprised me.
julia> typeof(function)
ERROR: syntax: unexpected ")"
# No answer at all for return example and no error either.
julia> typeof(return)
# In the next two examples the REPL returns the input code.
julia> typeof(in)
typeof(in)
julia> typeof(typeof)
typeof(typeof)
# The "for" word returns an error like the "function" word.
julia> typeof(for)
ERROR: syntax: unexpected ")"
The Julia 1.0.0 documentation says for typeof
"Get the concrete type of x."
The typeof(function) example is the one that really surprised me. I expected a function to be a first-class object in Julia and have a type. I guess I need to understand types in Julia.
Any suggestions?
Edit
Per some comment questions below, here is an example based on a small function:
julia> function test() return "test"; end
test (generic function with 1 method)
julia> test()
"test"
julia> typeof(test)
typeof(test)
Based on this example, I would have expected typeof(test) to return generic function, not typeof(test).
To be clear, I am not a hardcore user of the Julia internals. What follows is an answer designed to be (hopefully) an intuitive explanation of what functions are in Julia for the non-hardcore user. I do think this (very good) question could also benefit from a more technical answer provided by one of the more core developers of the language. Also, this answer is longer than I'd like, but I've used multiple examples to try and make things as intuitive as possible.
As has been pointed out in the comments, function itself is a reserved keyword, and is not an actual function istself per se, and so is orthogonal to the actual question. This answer is intended to address your edit to the question.
Since Julia v0.6+, Function is an abstract supertype, much in the same way that Number is an abstract supertype. All functions, e.g. mean, user-defined functions, and anonymous functions, are subtypes of Function, in the same way that Float64 and Int are subtypes of Number.
This structure is deliberate and has several advantages.
Firstly, for reasons I don't fully understand, structuring functions in this way was the key to allowing anonymous functions in Julia to run just as fast as in-built functions from Base. See here and here as starting points if you want to learn more about this.
Secondly, because each function is its own subtype, you can now dispatch on specific functions. For example:
f1(f::T, x) where {T<:typeof(mean)} = f(x)
and:
f1(f::T, x) where {T<:typeof(sum)} = f(x) + 1
are different dispatch methods for the function f1
So, given all this, why does, e.g. typeof(sum) return typeof(sum), especially given that typeof(Float64) returns DataType? The issue here is that, roughly speaking, from a syntactical perspective, sum needs to serves two purposes simultaneously. It needs to be both a value, like e.g. 1.0, albeit one that is used to call the sum function on some input. But, it is also needs to be a type name, like Float64.
Obviously, it can't do both at the same time. So sum on its own behaves like a value. You can write f = sum ; f(randn(5)) to see how it behaves like a value. But we also need some way of representing the type of sum that will work not just for sum, but for any user-defined function, and any anonymous function. The developers decided to go with the (arguably) simplest option and have the type of sum print literally as typeof(sum), hence the behaviour you observe. Similarly if I write f1(x) = x ; typeof(f1), that will also return typeof(f1).
Anonymous functions are a bit more tricky, since they are not named as such. What should we do for typeof(x -> x^2)? What actually happens is that when you build an anonymous function, it is stored as a temporary global variable in the module Main, and given a number that serves as its type for lookup purposes. So if you write f = (x -> x^2), you'll get something back like #3 (generic function with 1 method), and typeof(f) will return something like getfield(Main, Symbol("##3#4")), where you can see that Symbol("##3#4") is the temporary type of this anonymous function stored in Main. (a side effect of this is that if you write code that keeps arbitrarily generating the same anonymous function over and over you will eventually overflow memory, since they are all actually being stored as separate global variables of their own type - however, this does not prevent you from doing something like this for n = 1:largenumber ; findall(y -> y > 1.0, x) ; end inside a function, since in this case the anonymous function is only compiled once at compile-time).
Relating all of this back to the Function supertype, you'll note that typeof(sum) <: Function returns true, showing that the type of sum, aka typeof(sum) is indeed a subtype of Function. And note also that typeof(typeof(sum)) returns DataType, in much the same way that typeof(typeof(1.0)) returns DataType, which shows how sum actually behaves like a value.
Now, given everything I've said, all the examples in your question now make sense. typeof(function) and typeof(for) return errors as they should, since function and for are reserved syntax. typeof(typeof) and typeof(in) correctly return (respectively) typeof(typeof), and typeof(in), since typeof and in are both functions. Note of course that typeof(typeof(typeof)) returns DataType.
I am writing some functions in julia and want the results to be represented as rational numbers. That is, if a function returns 1/2, 1/3, 13/2571 etc I want them to be returned as written and not converted to floats. Say the functions compute some coefficients by some iterative process and I want the coefficient values to be shown as rationals. How can I do that in julia?
Rationals in Julia can be written as
1//2
These will work with functions, including user-defined ones, as you would expect:
5//7*3//5 # results in 3//7
f(x) = x^2 - 1
f(3//4) # results in -7//16
There's really not much else to it, but see also the manual section. If there's something in particular that's not working for you, post some example code and I'll take a look.
After spending some days already searching for something like this on the internet, I still couldn't manage to find anything describing this problem. Reading through the (otherwise quite recommendable) 'Writing R Extensions' dind't offer a solution as well. Thus, here's my most urgent question:
Is it possible to pass functions (for simplicity's sake, assume a simple R function - in reality, the problem is even uglier) as function/subroutine parameters to Fortran via .Fortran(...) call - and if so, how?
I wrote two simple functions in order to test this, first a Fortran subroutine (tailored to use the function I originally intended to pass, thus the kinda weird dimensions in the interface):
subroutine foo(o, x)
implicit none
interface
subroutine o(t, y, param, f)
double precision, intent(in) :: t
double precision, dimension(58), intent(in) :: y, param
double precision, dimension(22), intent(out) :: f
end subroutine
end interface
double precision, dimension(22), intent(out) :: x
double precision, dimension(58) :: yt, paramt
integer :: i
do i = 1, 58
yt(i) = rand(0)
paramt(i) = rand(1)
end do
call o(dble(4.2), yt, paramt, x)
end subroutine
and a simple R function to pass to the above function:
asdf <- function(a, s, d, f){x <- c(a, s, d, f)}
Calling .Fortran("foo", asdf, vector(mode="numeric", length=22)) yields
Error: invalid mode (closure) to pass to Fortran (arg 1) and passing "asdf" (as a string) results in a segfault, as the argument obviously doesn't fit the expected type (namely, a function).
FYI, I don't expect the code to do anything meaningful (that would be the task of another function), I mainly would like to know, whether passing functions (or function pointers) from R is possible at all or wether I better give up on this approach instantly and look for something that might work.
Thanks in advance,
Dean
You can't pass R objects via .Fortran. You would need to use the .Call or .External interface to pass the R objects to C/C++ code.
You could write a C/C++ wrapper for your R function, which you could then call from your Fortran code (see Calling-C-from-FORTRAN-and-vice-versa in Writing R Extensions).
I'm interested in building a derivative calculator. I've racked my brains over solving the problem, but I haven't found a right solution at all. May you have a hint how to start? Thanks
I'm sorry! I clearly want to make symbolic differentiation.
Let's say you have the function f(x) = x^3 + 2x^2 + x
I want to display the derivative, in this case f'(x) = 3x^2 + 4x + 1
I'd like to implement it in objective-c for the iPhone.
I assume that you're trying to find the exact derivative of a function. (Symbolic differentiation)
You need to parse the mathematical expression and store the individual operations in the function in a tree structure.
For example, x + sin²(x) would be stored as a + operation, applied to the expression x and a ^ (exponentiation) operation of sin(x) and 2.
You can then recursively differentiate the tree by applying the rules of differentiation to each node. For example, a + node would become the u' + v', and a * node would become uv' + vu'.
you need to remember your calculus. basically you need two things: table of derivatives of basic functions and rules of how to derivate compound expressions (like d(f + g)/dx = df/dx + dg/dx). Then take expressions parser and recursively go other the tree. (http://www.sosmath.com/tables/derivative/derivative.html)
Parse your string into an S-expression (even though this is usually taken in Lisp context, you can do an equivalent thing in pretty much any language), easiest with lex/yacc or equivalent, then write a recursive "derive" function. In OCaml-ish dialect, something like this:
let rec derive var = function
| Const(_) -> Const(0)
| Var(x) -> if x = var then Const(1) else Deriv(Var(x), Var(var))
| Add(x, y) -> Add(derive var x, derive var y)
| Mul(a, b) -> Add(Mul(a, derive var b), Mul(derive var a, b))
...
(If you don't know OCaml syntax - derive is two-parameter recursive function, with first parameter the variable name, and the second being mathched in successive lines; for example, if this parameter is a structure of form Add(x, y), return the structure Add built from two fields, with values of derived x and derived y; and similarly for other cases of what derive might receive as a parameter; _ in the first pattern means "match anything")
After this you might have some clean-up function to tidy up the resultant expression (reducing fractions etc.) but this gets complicated, and is not necessary for derivation itself (i.e. what you get without it is still a correct answer).
When your transformation of the s-exp is done, reconvert the resultant s-exp into string form, again with a recursive function
SLaks already described the procedure for symbolic differentiation. I'd just like to add a few things:
Symbolic math is mostly parsing and tree transformations. ANTLR is a great tool for both. I'd suggest starting with this great book Language implementation patterns
There are open-source programs that do what you want (e.g. Maxima). Dissecting such a program might be interesting, too (but it's probably easier to understand what's going on if you tried to write it yourself, first)
Probably, you also want some kind of simplification for the output. For example, just applying the basic derivative rules to the expression 2 * x would yield 2 + 0*x. This can also be done by tree processing (e.g. by transforming 0 * [...] to 0 and [...] + 0 to [...] and so on)
For what kinds of operations are you wanting to compute a derivative? If you allow trigonometric functions like sine, cosine and tangent, these are probably best stored in a table while others like polynomials may be much easier to do. Are you allowing for functions to have multiple inputs,e.g. f(x,y) rather than just f(x)?
Polynomials in a single variable would be my suggestion and then consider adding in trigonometric, logarithmic, exponential and other advanced functions to compute derivatives which may be harder to do.
Symbolic differentiation over common functions (+, -, *, /, ^, sin, cos, etc.) ignoring regions where the function or its derivative is undefined is easy. What's difficult, perhaps counterintuitively, is simplifying the result afterward.
To do the differentiation, store the operations in a tree (or even just in Polish notation) and make a table of the derivative of each of the elementary operations. Then repeatedly apply the chain rule and the elementary derivatives, together with setting the derivative of a constant to 0. This is fast and easy to implement.