mirror of
https://github.com/triqs/dft_tools
synced 2024-11-01 19:53:45 +01:00
a730f093d6
- fold was not correct in e.g. passing an int as init instead of a double (was leading to narrowing in return). - better return type deduction. - there was an error in the doc (order of argument in the lambda !) - add a more complex example (Frobenius norm of matrices).
154 lines
3.8 KiB
ReStructuredText
154 lines
3.8 KiB
ReStructuredText
.. highlight:: c
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.. _arr_map_fold:
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Functional constructs : map & fold
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###########################################
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Two standard functional constructs are provided :
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* *map* that promotes a function acting on the array element to an array function, acting
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element by element.
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* *fold* is the reduction of a function on the array.
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.. _map:
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map
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========================================================
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* **Purpose** :
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map promotes any function into an `array function`, acting term by term.
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* **Synopsis** ::
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template<class F> auto map (F f);
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If `f` is a function, or a function object ::
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T2 f(T1)
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Then map(f) is a function::
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template<ImmutableCuboidArray A> auto map(f) (A const &)
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with :
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* A::value_type == T1
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* The returned type of map(f) models the :ref:`ImmutableCuboidArray` concept
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* with the same domain as A
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* with value_type == T2
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* **Example** :
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.. compileblock::
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#include <triqs/arrays.hpp>
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using namespace triqs;
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int main() {
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// declare and init a matrix
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clef::placeholder<0> i_; clef::placeholder<1> j_;
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arrays::matrix<int> A (2,2); A(i_,j_) << i_ + j_ ;
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// the mapped function
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auto F = arrays::map([](int i) { return i*2.5;});
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std::cout<< "A = " << A << std::endl;
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std::cout<< "F(A) = " << F(A) << std::endl; // oops no computation done
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std::cout<< "F(A) = " << make_matrix(F(A)) << std::endl;
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std::cout<< "3*F(2*A) = " << make_matrix(3*F(2*A)) << std::endl;
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}
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fold
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========================================================
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* **Purpose** :
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fold implements the folding (or reduction) on the array.
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* **Syntax** :
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If `f` is a function, or a function object of synopsis (T, R being 2 types) ::
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R f (R , T)
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then ::
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auto F = fold(f);
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is a callable object which can fold any array of value_type T.
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So, if
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* A is a type which models the :ref:`ImmutableCuboidArray` concept
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(e.g. an array , a matrix, a vector, an expression, ...)
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* A::value_type is T
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then ::
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fold (f) ( A, R init = R() ) = f(f(f(f(init, a(0,0)), a(0,1)),a(0,2)),a(0,3), ....)
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Note that :
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* The order of traversal is the same as foreach.
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* The precise return type of fold is an implementation detail, depending on the precise type of f,
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use auto to keep it.
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* The function f will be inlined if possible, leading to efficient algorithms.
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* fold is implemented using a foreach loop, hence it is efficient.
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* **Example** :
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Many algorithms can be written in form of map/fold.
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The function :ref:`arr_fnt_sum` which returns the sum of all the elements of the array is implemented as ::
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template <class A>
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typename A::value_type sum(A const & a) { return fold ( std::plus<>()) (a); }
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or the Frobenius norm of a matrix,
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.. math::
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\sum_{i=0}^{N-1} \sum_{j=0}^{N-1} | a_{ij} | ^2
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reads :
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.. compileblock::
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#include <triqs/arrays.hpp>
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#include <triqs/arrays/functional/fold.hpp>
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using namespace triqs;
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double frobenius_norm (arrays::matrix<double> const& a) {
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auto l= [](double r, double x) {
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auto ab = std::abs(x);
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return r + ab * ab;
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};
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return std::sqrt(arrays::fold(l)(a,0));
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}
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int main() {
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// declare and init a matrix
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clef::placeholder<0> i_; clef::placeholder<1> j_;
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arrays::matrix<double> A (2,2); A(i_,j_) << i_ + j_/2.0;
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std::cout<< "A = " << A << std::endl;
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std::cout<< "||A|| = " << frobenius_norm(A) << std::endl;
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}
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Note in this example :
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* the simplicity of the code
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* the genericity : it is valid for any dimension of array.
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* internally, the library will rewrite it as a series of for loop, ordered in the TraversalOrder of the array
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and inline the lambda.
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