I want to be able to allow users of my package to define functions in a more mathematical manner, and I think a macro is the right direction. The problem is as follows. The code allows the users to define functions which are then used in specialized solvers to solve PDEs. However, to make things easier for the solver, some of the inputs are "matrices" in ways that you wouldn't normally wouldn't think they would be. For example, the solvers can take in functions f(x,t), but x[:,1] is what you'd think of as x and x[:,2] is what you'd think of as y (and sometimes it is 3D).
The bigger issues is that when the PDE is nonlinear, I place everything in a u vector, when in many cases (like Reaction-Diffusion equations) these things are named. So in this general case, I'd like to be able to write
#mathdefine f(RA,RABP,RAR,x,y,t) = RA*RABP + RA*x + RAR*t
and have it translate to
f(u,x,t) = u[:,1].*u[:,2] + u[:,1].*x[:,1] + u[:,3]*t
I am not up to snuff on my macro-foo, so I was hoping someone could get me started (or if macros are not the right way to approach this, explain why).
It's not too hard if the user has to give what is being translated to what, but I'd like to have it be as clean to use as possible, so somehow know that it's to the spatial variables and so everything before is part of a u, but after is part of x.
The trick to macro "find/replace" is just to pass off processing to a recursive function that updates the expression args. Your signature will come in as a bunch of symbols, so you can loop through the call signature and add to two dicts, mapping variable name to column index. Then recursively replace the arg tree when you see any of the variables. This is untested:
function replace_vars!(expr::Expr, xd::Dict{Symbol,Int}, ud::Dict{Symbol,Int})
for (i,arg) in enumerate(expr.args)
if haskey(xd, arg)
expr.arg[i] = :(x[:,$(xd[arg])])
elseif haskey(ud, arg)
expr.arg[i] = :(u[:,$(ud[arg])])
elseif isa(arg,Expr)
replace_vars!(arg, xd, ud)
end
end
end
macro mathdefine(expr)
# todo: loop through function signature (expr.args[1]?) to build xd/ud
replace_vars!(expr)
expr
end
I left a little homework for you, but this should get you started.
Related
Background -- I was reading up on accessing shadowed functions, and started playing with builtin . I wrote a little function:
function klear(x)
% go to parent environment...
evalin('base', builtin('clear','x')) ;
end
This throws the error:
Error using clear
Too many output arguments.
I think this happens because evalin demands an output from whatever it's being fed, but clear is one of the functions which has no return value.
So two questions: am I interpreting this correctly, and if so, is there an alternative function that allows me to execute a function in the parent environment (that doesn't require an output)?
Note: I'm fully aware of the arguments against trying to access shadowed funcs (or rather, to avoid naming functions in a way that overload base funcs, etc). This is primarily a question to help me learn what can and can't be done in MATLAB.
Note 2
My original goal was to write an overload function that would require an input argument, to avoid the malware-ish behavior of clear, which defaults to deleting everything. In Q&D pseudocode,
function clear(x)
if ~exist('x','var') return
execute_in_base_env(builtin(clear(x)))
end
There's a couple issues with your clear override:
It will always clear in the base workspace regardless of where it's called from.
It doesn't support multiple inputs, which is a common use case for clear.
Instead I'd have it check for whether it was called from the base workspace, and special-case that for your check for whether it's clearing everything. If some function is calling plain clear to clear all its variables, that's bad practice, but it's still how that function's logic works, and you don't want to break that. Otherwise it could error, or worse, return incorrect results.
So, something like this:
function clear(varargin)
stk = dbstack;
if numel(stk) == 1 && (nargin == 0 || ismember('all', varargin))
fprintf('clear: balking at clearing all vars in base workspace. Nothing cleared.\n');
return;
end
% Check for quoting problems
for i = 1:numel(varargin)
if any(varargin{i} == '''')
error('You have a quote in one of your args. That''s not valid.');
end
end
% Construct a clear() call that works with evalin()
arg_strs = strcat('''', varargin, '''');
arg_strs = [{'''clear'''} arg_strs];
expr = ['builtin(' strjoin(arg_strs, ', '), ')'];
% Do it
evalin('caller', expr);
end
I hope it goes without saying that this is an atrocious hack that I wouldn't recommend in practice. :)
What happens in your code:
evalin('base', builtin('clear','x'));
is that builtin is evaluated in the current context, and because it is used as an argument to evalin, it is expected to produce an output. It is exactly the same as:
ans = builtin('clear','x');
evalin('base',ans);
The error message you see occurs in the first of those two lines of code, not in the second. It is not because of evalin, which does support calling statements that don't produce an output argument.
evalin requires a string to evaluate. You need to build this string:
str = 'builtin(''clear'',''x'')';
evalin('base',ans);
(In MATLAB, the quote character is escaped by doubling it.)
You function thus would look like this:
function clear(var)
try
evalin('base',['builtin(''clear'',''',var,''')'])
catch
% ignore error
end
end
(Inserting a string into another string this way is rather awkward, one of the many reasons I don't like eval and friends).
It might be better to use evalin('caller',...) in this case, so that when you call the new clear from within a function, it deletes something in the function's workspace, not the base one. I think 'base' should only be used from within a GUI that is expected to control variables in the user's workspace, not from a function that could be called anywhere and is expected (by its name in this case) to do something local.
There are reasons why this might be genuinely useful, but in general you should try to avoid the use of clear just as much as the use of eval and friends. clear slows down program execution. It is much easier (both on the user and on the MATLAB JIT) to assign an empty array to a variable to remove its contents from memory (as suggested by rahnema1 in a comment. Your base workspace would not be cluttered with variables if you used function more: write functions, not scripts!
Im trying to make a n-nested loop method in Julia
function fun(n::Int64)
#nloops n i d->1:3 begin\n
#nexprs n j->(print(i_j))\n
end
end
But the #nloops definition is limited to
_nloops(::Int64, ::Symbol, ::Expr, ::Expr...)
and I get the error
_nloops(::Symbol, ::Symbol, ::Expr, ::Expr)
Is there any way to make this work? Any help greatly appreciated
EDIT:
What I ended up doing was using the combinations method
For my problem, I needed to get all k-combinations of indices to pull values from an array, so the loops would had to look like
for i_1 in 1:100
for i_2 in i_1:100
...
for i_k in i_[k-1]:100
The number of loops needs to be a compile-time constant – a numeric literal, in fact: the code generated for the function body cannot depend on a function argument. Julia's generated functions won't help either since n is just a plain value and not part of the type of any argument. Your best bet for having the number of nested loops depend on a runtime value like n is to use recursion.
In julia-0.4 and above, you can now do this:
function fun(n::Int)
for I in CartesianRange(ntuple(d->1:3, n))
#show I
end
end
In most cases you don't need the Base.Cartesian macros anymore (although there are still some exceptions). It's worth noting that, just as described in StefanKarpinski's answer, this loop will not be "type stable" because n is not a compile-time constant; if performance matters, you can use the "function barrier technique." See http://julialang.org/blog/2016/02/iteration for more information about all topics related to these matters.
I'm trying to evaluate some matrix multiplications in MuPAD. The output is using sigmas as placeholders for the matrix elements since they are long expressions (I assume that's the reason). Is there a way to get MuPAD to display the individual matrix elements as (in my case) the exponential functions that they really are, regardless of the length of the expression?
Below is an example of a case where MuPAD is using sigmas instead of the actual exponential functions. I would like to be able to see what the individual matrix elements of TotT^4 really are.
The commands I executed in the MuPAD interface that lead up to TotT^4 are:
T1 := matrix([[exp((J+B/2)/T),exp(-(J+B/6)/T)],[exp((-J+B/6)/T),exp((J-B/2)/T)]])
T2 := matrix([[exp((J1+B/2)/T),exp(-(J1+B/6)/T)],[exp((-J1+B/6)/T),exp((J1-B/2)/T)]])
T1d := linalg::transpose(T1)
TotT := T1d*T2
The class of your variable can be obtain via type(totT): Dom::Matrix. You may want to look at the many methods of this class in the documentation. As far as I can tell, this issue has something to do with the pretty printing of the class's print method. Other classes exhibit this same substitution, so it may be a function of the overloaded print. I was not able to change the behavior by adjusting setPrintMaxSize, PRETTYPRINT, TEXTWIDTH, or any of the optional arguments to print. You might still try yourself as there are many permutations.
I also tried using the expand function. expand(TotT,IgnoreAnalyticConstraints) nearly works though it could have undesirable effects in some cases if things were expanded too much. Calling simplify does get rid go the substitutions, but it also changes the nature of some of the entries by simplifying. It is probably also not a general solution to this issue.
One way that does work, but is ugly, is to use the expr2text method, which returns a result as a string:
expr2text(TotT)
which returns
"matrix([[exp((B/6 - J)/T)*exp((B/6 - J1)/T) + exp((B/2 + J)/T)*exp((B/2 + J1)/T), ...
exp(-(B/2 - J1)/T)*exp((B/6 - J)/T) + exp((B/2 + J)/T)*exp(-(B/6 + J1)/T)], ...
[exp(-(B/2 - J)/T)*exp((B/6 - J1)/T) + exp((B/2 + J1)/T)*exp(-(B/6 + J)/T), ...
exp(-(B/2 - J)/T)*exp(-(B/2 - J1)/T) + exp(-(B/6 + J)/T)*exp(-(B/6 + J1)/T)]])"
I think that this question would be a good one to ask over at Matlab Central or by filing a service request if you have a license with support.
Suppose one needs to select the real solutions after solving some equation.
Is this the correct and optimal way to do it, or is there a better one?
restart;
mu := 3.986*10^5; T:= 8*60*60:
eq := T = 2*Pi*sqrt(a^3/mu):
sol := solve(eq,a);
select(x->type(x,'realcons'),[sol]);
I could not find real as type. So I used realcons. At first I did this:
select(x->not(type(x,'complex')),[sol]);
which did not work, since in Maple 5 is considered complex! So ended up with no solutions.
type(5,'complex');
(* true *)
Also I could not find an isreal() type of function. (unless I missed one)
Is there a better way to do this that one should use?
update:
To answer the comment below about 5 not supposed to be complex in maple.
restart;
type(5,complex);
true
type(5,'complex');
true
interface(version);
Standard Worksheet Interface, Maple 18.00, Windows 7, February
From help
The type(x, complex) function returns true if x is an expression of the form
a + I b, where a (if present) and b (if present) are finite and of type realcons.
Your solutions sol are all of type complex(numeric). You can select only the real ones with type,numeric, ie.
restart;
mu := 3.986*10^5: T:= 8*60*60:
eq := T = 2*Pi*sqrt(a^3/mu):
sol := solve(eq,a);
20307.39319, -10153.69659 + 17586.71839 I, -10153.69659 - 17586.71839 I
select( type, [sol], numeric );
[20307.39319]
By using the multiple argument calling form of the select command we here can avoid using a custom operator as the first argument. You won't notice it for your small example, but it should be more efficient to do so. Other commands such as map perform similarly, to avoid having to make an additional function call for each individual test.
The types numeric and complex(numeric) cover real and complex integers, rationals, and floats.
The types realcons and complex(realcons) includes the previous, but also allow for an application of evalf done during the test. So Int(sin(x),x=1..3) and Pi and sqrt(2) are all of type realcons since following an application of evalf they become floats of type numeric.
The above is about types. There are also properties to consider. Types are properties, but not necessarily vice versa. There is a real property, but no real type. The is command can test for a property, and while it is often used for mixed numeric-symbolic tests under assumptions (on the symbols) it can also be used in tests like yours.
select( is, [sol], real );
[20307.39319]
It is less efficient to use is for your example. If you know that you have a collection of (possibly non-real) floats then type,numeric should be an efficient test.
And, just to muddy the waters... there is a type nonreal.
remove( type, [sol], nonreal );
[20307.39319]
The one possibility is to restrict the domain before the calculation takes place.
Here is an explanation on the Maplesoft website regarding restricting the domain:
4 Basic Computation
UPD: Basically, according to this and that, 5 is NOT considered complex in Maple, so there might be some bug/error/mistake (try checking what may be wrong there).
For instance, try putting complex without quotes.
Your way seems very logical according to this.
UPD2: According to the Maplesoft Website, all the type checks are done with type() function, so there is rather no isreal() function.
How to simplify a given boolean expression with many variables (>10) so that the number of occurrences of each variable is minimized?
In my scenario, the value of a variable has to be considered ephemeral, that is, has to recomputed for each access (while still being static of course). I therefor need to minimize the number of times a variable has to be evaluated before trying to solve the function.
Consider the function
f(A,B,C,D,E,F) = (ABC)+(ABCD)+(ABEF)
Recursively using the distributive and absorption law one comes up with
f'(A,B,C,E,F) = AB(C+(EF))
I'm now wondering if there is an algorithm or method to solve this task in minimal runtime.
Using only Quine-McCluskey in the example above gives
f'(A,B,C,E,F) = (ABEF) + (ABC)
which is not optimal for my case. Is it save to assume that simplifying with QM first and then use algebra like above to reduce further is optimal?
I usually use Wolfram Alpha for this sort of thing.
Try Logic Friday 1
It features multi-level design of boolean circuits.
For your example, input and output look as follows:
You can use an online boolean expression calculator like https://www.dcode.fr/boolean-expressions-calculator
You can refer to Any good boolean expression simplifiers out there? it will definitely help.