Template Numerical Library version\ main:a0a944a8c
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This chapter introduces vectors in TNL. Vector, in addition to Array, offers also basic operations from linear algebra. The reader will mainly learn how to do BLAS level 1 operations in TNL. Thanks to the design of TNL, it is easier to implement, independent of the hardware architecture, and in some cases even faster then BLAS or cuBLAS implementation.
Vector is a class template similar to Array, which is defined in the TNL::Containers namespace and has four template parameters:
Unlike Array, Vector requires that the Real type is numeric or a type for which basic algebraic operations are defined. What kind of algebraic operations are required depends on what vector operations the user will call. Vector is derived from Array so it inherits all its methods. In the same way the Array has its counterpart ArraView, Vector has VectorView which is derived from ArrayView. See Arrays for more details.
In addition to the comparison operators == and != defined for Array and ArrayView, the comparison operators <, <=, >, and >= are available for Vector and VectorView. The comparison follows the lexicographic order and it is performed by an algorithm equivalent to std::lexicographical_compare.
By horizontal operations we mean vector expressions where we have one or more vectors as an input and a vector as an output. Horizontal operations in TNL are performed by the expression templates. It makes algebraic operations with vectors easy to do and very efficient at the same time. In some cases, one gets code even more efficient compared to BLAS and cuBLAS. See the following example.
Output is:
The expression is evaluated on the same device where the vectors are allocated, this is done automatically. One cannot, however, mix vectors from different devices in one expression.
Vector expressions consist of operands, operators, and functions. Operands may be vectors, vector views, or other vector expressions. Operators available for vector expressions are listed in the following table, where v is a result vector and expr, expr1, expr2 are vector expressions:
Expression | Meaning |
---|---|
v = expr1 + expr2 | v[ i ] = expr1[ i ] + expr2[ i ] |
v = expr1 - expr2 | v[ i ] = expr1[ i ] - expr2[ i ] |
v = expr1 * expr2 | v[ i ] = expr1[ i ] * expr2[ i ] |
v = expr1 / expr2 | v[ i ] = expr1[ i ] / expr2[ i ] |
v = expr1 / expr2 | v[ i ] = expr1[ i ] / expr2[ i ] |
v = expr1 % expr2 | v[ i ] = expr1[ i ] % expr2[ i ] |
v = expr1 && expr2 | v[ i ] = expr1[ i ] && expr2[ i ] |
v = expr1 || expr2 | v[ i ] = expr1[ i ] || expr2[ i ] |
v = expr1 & expr2 | v[ i ] = expr1[ i ] & expr2[ i ] |
v = expr1 | expr2 | v[ i ] = expr1[ i ] | expr2[ i ] |
v = +expr1 | v[ i ] = +expr1[ i ] |
v = -expr1 | v[ i ] = -expr1[ i ] |
v = !expr1 | v[ i ] = !expr1[ i ] |
v = ~expr1 | v[ i ] = ~expr1[ i ] |
Additionally, vector expressions may contain any function listed in the following table, where v is a result vector and expr, expr1, expr2 are vector expressions:
Expression | Meaning |
---|---|
v = TNL::equalTo( expr1, expr2 ) | v[ i ] = expr1[ i ] == expr2[ i ] |
v = TNL::notEqualTo( expr1, expr2 ) | v[ i ] = expr1[ i ] != expr2[ i ] |
v = TNL::greater( expr1, expr2 ) | v[ i ] = expr1[ i ] > expr2[ i ] |
v = TNL::greaterEqual( expr1, expr2 ) | v[ i ] = expr1[ i ] >= expr2[ i ] |
v = TNL::less( expr1, expr2 ) | v[ i ] = expr1[ i ] < expr2[ i ] |
v = TNL::lessEqual( expr1, expr2 ) | v[ i ] = expr1[ i ] <= expr2[ i ] |
v = TNL::minimum( expr1, expr2 ) | v[ i ] = min( expr1[ i ], expr2[ i ] ) |
v = TNL::maximum( expr1, expr2 ) | v[ i ] = max( expr1[ i ], expr2[ i ] ) |
v = TNL::abs( expr ) | v[ i ] = abs( expr[ i ] ) |
v = TNL::sin( expr ) | v[ i ] = sin( expr[ i ] ) |
v = TNL::cos( expr ) | v[ i ] = cos( expr[ i ] ) |
v = TNL::tan( expr ) | v[ i ] = tan( expr[ i ] ) |
v = TNL::asin( expr ) | v[ i ] = asin( expr[ i ] ) |
v = TNL::acos( expr ) | v[ i ] = acos( expr[ i ] ) |
v = TNL::atan( expr ) | v[ i ] = atan( expr[ i ] ) |
v = TNL::sinh( expr ) | v[ i ] = sinh( expr[ i ] ) |
v = TNL::cosh( expr ) | v[ i ] = cosh( expr[ i ] ) |
v = TNL::tanh( expr ) | v[ i ] = tanh( expr[ i ] ) |
v = TNL::asinh( expr ) | v[ i ] = asinh( expr[ i ] ) |
v = TNL::acosh( expr ) | v[ i ] = acosh( expr[ i ] ) |
v = TNL::atanh( expr ) | v[ i ] = atanh( expr[ i ] ) |
v = TNL::exp( expr ) | v[ i ] = exp( expr[ i ] ) |
v = TNL::log( expr ) | v[ i ] = log( expr[ i ] ) |
v = TNL::log10( expr ) | v[ i ] = log10( expr[ i ] ) |
v = TNL::log2( expr ) | v[ i ] = log2( expr[ i ] ) |
v = TNL::sqrt( expr ) | v[ i ] = sqrt( expr[ i ] ) |
v = TNL::cbrt( expr ) | v[ i ] = cbrt( expr[ i ] ) |
v = TNL::pow( expr ) | v[ i ] = pow( expr[ i ] ) |
v = TNL::floor( expr ) | v[ i ] = floor( expr[ i ] ) |
v = TNL::ceil( expr ) | v[ i ] = ceil( expr[ i ] ) |
v = TNL::sign( expr ) | v[ i ] = sign( expr[ i ] ) |
By vertical operations we mean (parallel) reduction based operations where we have one vector expressions as an input and one value as an output. For example computing scalar product, vector norm or finding minimum or maximum of vector elements is based on reduction. See the following example.
Output is:
The following table shows vertical operations that can be used on vector expressions:
Expression | Meaning |
---|---|
v = TNL::min( expr ) | v is the minimum of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ]. |
auto [ v, i ] = TNL::argMin( expr ) | v is the minimum of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ], i is the index of the smallest element. |
v = TNL::max( expr ) | v is the maximum of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ]. |
auto [ v, i ] = TNL::argMax( expr ) | v is the maximum of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ], i is the index of the largest element. |
v = TNL::sum( expr ) | v is the sum of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ]. |
v = TNL::maxNorm( expr ) | v is the maximum norm of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ]. |
v = TNL::l1Norm( expr ) | v is the l1 norm of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ]. |
v = TNL::l2Norm( expr ) | v is the l2 norm of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ]. |
v = TNL::lpNorm( expr, p ) | v is the lp norm of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ]. |
v = TNL::product( expr ) | v is product of expr[ 0 ], expr[ 1 ], ..., expr[ n-1 ]. |
v = TNL::all( expr ) | v is the result of expr[ 0 ] && expr[ 1 ] && ... && expr[ n-1 ]. |
v = TNL::any( expr ) | v is the result of expr[ 0 ] || expr[ 1 ] || ... || expr[ n-1 ] . |
auto [ v, i ] = TNL::argAny( expr ) | v is the result of expr[ 0 ] || expr[ 1 ] || ... || expr[ n-1 ] , i is the index of the first true element. |
Static vectors are derived from static arrays and so they are allocated on the stack and can be created in CUDA kernels as well. Their size is fixed and given by a template parameter. The StaticVector class template is defined in the TNL::Containers namespace and has two template parameters:
The interface of StaticVector is smilar to Vector. StaticVector also supports expression templates, which make the use of static vectors simple and efficient at the same time, and lexicographic comparison by operators <, <=, > and >=.
Example:
The output looks as:
DistributedVector extends DistributedArray with algebraic operations. The functionality is similar to how Vector extends Array. DistributedVector also supports expression templates and other operations present in Vector.
Example:
The output looks as: