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https://github.com/triqs/dft_tools
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108 lines
4.1 KiB
ReStructuredText
108 lines
4.1 KiB
ReStructuredText
.. highlight:: c
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Operations : array and matrix/vector algebras
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=======================================================
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Arithmetic operations
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-----------------------------
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Arrays and matrices can be combined in formal algebraic expressions, which models the :ref:`ImmutableCuboidArray` concept.
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This algebraic expressions can therefore be used in assignment array/matrix contructors.
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For example:
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.. compileblock::
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#include <triqs/arrays.hpp>
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using triqs::arrays::array; using triqs::clef::placeholder;
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int main() {
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// init
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placeholder<0> i_; placeholder<1> j_;
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array<long,2> A (2,2), B(2,2);
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A(i_,j_) << i_ + j_ ;
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B(i_,j_) << i_ - j_ ;
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// use expressions
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array<long,2> C = A+ 2*B;
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array<double,2> D = 0.5 * A; // Type promotion is automatic
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std::cout<< "A= "<< A<< std::endl;
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std::cout<< "B= "<< B<< std::endl;
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std::cout<< "C= "<< C<< std::endl;
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std::cout<< "D= "<< D<< std::endl;
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}
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Arrays vs matrices
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----------------------
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Because their multiplication is not the same, arrays and matrices algebras can not be mixed.
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Mixing them in expression would therefore be meaningless and it is therefore not allowed.
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However, you can always use e.g. `matrix_view` from a array of rank 2 :
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.. compileblock::
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#include <triqs/arrays.hpp>
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using namespace triqs::arrays;
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int main() {
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array<long,2> A(2,2); matrix<long,2> M(2,2);
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//M + A; // --> ERROR. Rejected by the compiler.
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M + make_matrix_view(A); //--> OK.
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}
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.. note::
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Making such a view is very cheap, it only copies the index systems. Nevertheless
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this can still cause significant overhead in very intense loops by disturbing
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optimizers.
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Performance
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---------------------------------------------
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The performance of such compact writing is as good as "hand-written" code or even better.
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Indeed, the operations are implemented with the `expression templates` technique.
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The principle is that the result of A+B is **NOT** an array, but a more complex type which stores
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the expression using the naturally recursive structure of templates.
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Expressions models :ref:`ImmutableCuboidArray` concept.
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They behave like an immutable array : they have a domain, they can be evaluated.
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Hence they can used *anywhere* an object modeling this concept is accepted, e.g. :
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* array, matrix contruction
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* operator =, +=, -=, ...
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When an array in assigned (or constructed from) such expression, it fills itself
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by evaluating the expression.
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This technique allows the elimination of temporaries, so that the clear and readable code::
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Z= A + 2*B + C/2;
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is in fact rewritten by the compiler into ::
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for (i,j,...) indices of A, B :
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C(i,j) = A(i,j) + 2* B(i,j) + C(i,j)/2
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instead of making a chain of temporaries (C/2, 2*B, 2*B + C/2...) that "ordinary" object-oriented programming would produce.
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As a result, the produced code is as fast as if you were writing the loop yourself,
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but with several advantages:
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* It is more **compact** and **readable** : you don't have to write the loop, and the indices range are computed automatically.
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* It is much better for **optimization** :
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* What you want is to tell the compiler/library to compute this expression, not *how* to do it optimally on a given machine.
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* For example, since the traversal order of indices is decided at compile time, the library can traverse the data
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in an optimal way, allowing machine-dependent optimization.
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* The library can perform easy optimisations behind the scene when possible, e.g. for vector it can use blas.
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Expressions are lazy....
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---------------------------
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Note that expressions are lazy objects. It does nothing when constructed, it just "record" the mathematical expression ::
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auto e = A + 2*B; // expression, purely formal, no computation is done
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cout<< e <<endl ; // prints the expression
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cout<< e(1,2) <<endl ; // evaluates just at a point
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cout<< e.domain() <<endl ; // just computes its domain
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array<long,2> D(e); // now really makes the computation and store the result in D.
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D = 2*A +B; // reassign D to the evaluation of the expression.
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