Template Numerical Library version develop:1f26cbe9a
Vectors tutorial

# Introduction

This tutorial 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, hardware architecture transparent and in some cases even faster then Blas or cuBlas implementation.

# Vectors

Vector is, similar to Array, templated class defined in namespace TNL::Containers having three template parameters:

• Real is type of data to be stored in the vector
• Device is the device where the vector is allocated. Currently it can be either Devices::Host for CPU or Devices::Cuda for GPU supporting CUDA.
• Index is the type to be used for indexing the vector elements.

Vector, unlike Array, requires that the Real type is numeric or a type for which basic algebraic operations are defined. What kind of algebraic operations is 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. We refer to to Arrays tutorial for more details.

## Horizontal operations

By horizontal operations we mean vector expressions where we have one or more vectors as an input and a vector as an output. In TNL, this kind of operations is 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 get even more efficient code compared to Blas and cuBlas. See the following example.

1#include <iostream>
2#include <TNL/Containers/Vector.h>
3#include <TNL/Containers/VectorView.h>
4
5using namespace TNL;
6using namespace TNL::Containers;
7
8template< typename Device >
9void expressions()
10{
11 using RealType = float;
12 using VectorType = Vector< RealType, Device >;
13
14 /****
15 * Create vectors
16 */
17 const int size = 11;
18 VectorType a( size ), b( size ), c( size );
19 a.forAllElements( [] __cuda_callable__ ( int i, RealType& value ) { value = 3.14 * ( i - 5.0 ) / 5.0; } );
20 b = a * a;
21 c = 3 * a + sign( a ) * sin( a );
22 std::cout << "a = " << a << std::endl;
23 std::cout << "sin( a ) = " << sin( a ) << std::endl;
24 std::cout << "abs( sin( a ) ) = " << abs( sin ( a ) ) << std::endl;
25 std::cout << "b = " << b << std::endl;
26 std::cout << "c = " << c << std::endl;
27}
28
29int main( int argc, char* argv[] )
30{
31 /****
32 * Perform test on CPU
33 */
34 std::cout << "Expressions on CPU ..." << std::endl;
35 expressions< Devices::Host >();
36
37 /****
38 * Perform test on GPU
39 */
41 std::cout << "Expressions on GPU ..." << std::endl;
42 expressions< Devices::Cuda >();
43}
44
45
Vector extends Array with algebraic operations.
Definition: Vector.h:40
T endl(T... args)
Namespace for TNL containers.
Definition: Array.h:21
The main TNL namespace.
Definition: AtomicOperations.h:18
__cuda_callable__ T sign(const T &a)
This function represents the signum function.
Definition: Math.h:499
T sin(T... args)

Output is:

Expressions on CPU ...
a = [ -3.14, -2.512, -1.884, -1.256, -0.628, 0, 0.628, 1.256, 1.884, 2.512, 3.14 ]
sin( a ) = [ -0.00159255, -0.588816, -0.951351, -0.950859, -0.587528, 0, 0.587528, 0.950859, 0.951351, 0.588816, 0.00159255 ]
abs( sin( a ) ) = [ 0.00159255, 0.588816, 0.951351, 0.950859, 0.587528, 0, 0.587528, 0.950859, 0.951351, 0.588816, 0.00159255 ]
b = [ 9.8596, 6.31014, 3.54946, 1.57754, 0.394384, 0, 0.394384, 1.57754, 3.54946, 6.31014, 9.8596 ]
c = [ -9.41841, -6.94718, -4.70065, -2.81714, -1.29647, 0, 2.47153, 4.71886, 6.60335, 8.12482, 9.42159 ]
Expressions on GPU ...
a = [ -3.14, -2.512, -1.884, -1.256, -0.628, 0, 0.628, 1.256, 1.884, 2.512, 3.14 ]
sin( a ) = [ -0.00159255, -0.588816, -0.951351, -0.950859, -0.587528, 0, 0.587528, 0.950859, 0.951351, 0.588816, 0.00159255 ]
abs( sin( a ) ) = [ 0.00159255, 0.588816, 0.951351, 0.950859, 0.587528, 0, 0.587528, 0.950859, 0.951351, 0.588816, 0.00159255 ]
b = [ 9.8596, 6.31014, 3.54946, 1.57754, 0.394384, 0, 0.394384, 1.57754, 3.54946, 6.31014, 9.8596 ]
c = [ -9.41841, -6.94718, -4.70065, -2.81714, -1.29647, 0, 2.47153, 4.71886, 6.60335, 8.12482, 9.42159 ]

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 expression may contain any common function like the following:

Expression Meaning
v = TNL::min( expr1, expr2 ) v[ i ] = min( expr1[ i ], expr2[ i ] )
v = TNL::max( 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 ] )

Where v is a result vector and expr, expr1 and expr2 are vector expressions. Vector expressions can be combined with vector views (TNL::Containers::VectorView) as well.

## Vertical operations

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.

1#include <iostream>
2#include <TNL/Containers/Vector.h>
3#include <TNL/Containers/VectorView.h>
4
5using namespace TNL;
6using namespace TNL::Containers;
7
8template< typename Device >
9void expressions()
10{
11 using RealType = float;
12 using VectorType = Vector< RealType, Device >;
13 using ViewType = VectorView< RealType, Device >;
14
15 /****
16 * Create vectors
17 */
18 const int size = 11;
19 VectorType a_v( size ), b_v( size ), c_v( size );
20 ViewType a = a_v.getView();
21 ViewType b = b_v.getView();
22 ViewType c = c_v.getView();
23 a.forAllElements( [] __cuda_callable__ ( int i, RealType& value ) { value = i; } );
24 b.forAllElements( [] __cuda_callable__ ( int i, RealType& value ) { value = i - 5.0; } );
25 c = -5;
26
27 std::cout << "a = " << a << std::endl;
28 std::cout << "b = " << b << std::endl;
29 std::cout << "c = " << c << std::endl;
30 auto arg_min_a = argMin( a );
31 auto arg_max_a = argMax( a );
32 auto arg_min_b = argMin( b );
33 auto arg_max_b = argMax( b );
34 std::cout << "min( a ) = " << arg_min_a.second << " at " << arg_min_a.first << std::endl;
35 std::cout << "max( a ) = " << arg_max_a.second << " at " << arg_max_a.first << std::endl;
36 std::cout << "min( b ) = " << arg_min_b.second << " at " << arg_min_b.first << std::endl;
37 std::cout << "max( b ) = " << arg_max_b.second << " at " << arg_max_b.first << std::endl;
38 std::cout << "min( abs( b ) ) = " << min( abs( b ) ) << std::endl;
39 std::cout << "sum( b ) = " << sum( b ) << std::endl;
40 std::cout << "sum( abs( b ) ) = " << sum( abs( b ) ) << std::endl;
41 std::cout << "Scalar product: ( a, b ) = " << ( a, b ) << std::endl;
42 std::cout << "Scalar product: ( a + 3, abs( b ) / 2 ) = " << ( a + 3, abs( b ) / 2 ) << std::endl;
43 if( abs( a + b ) <= abs( a ) + abs( b ) )
44 std::cout << "abs( a + b ) <= abs( a ) + abs( b ) holds" << std::endl;
45}
46
47int main( int argc, char* argv[] )
48{
49 /****
50 * Perform test on CPU
51 */
52 std::cout << "Expressions on CPU ..." << std::endl;
53 expressions< Devices::Host >();
54
55 /****
56 * Perform test on GPU
57 */
58#ifdef HAVE_CUDA
60 std::cout << "Expressions on GPU ..." << std::endl;
61 expressions< Devices::Cuda >();
62#endif
63}
64
65
VectorView extends ArrayView with algebraic operations.
Definition: VectorView.h:30
T min(T... args)

Output is:

Expressions on CPU ...
a = [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
b = [ -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ]
c = [ -5, -5, -5, -5, -5, -5, -5, -5, -5, -5, -5 ]
min( a ) = 0 at 0
max( a ) = 10 at 10
min( b ) = 0 at -5
max( b ) = 10 at 5
min( abs( b ) ) = 0
sum( b ) = 0
sum( abs( b ) ) = 30
Scalar product: ( a, b ) = 110
Scalar product: ( a + 3, abs( b ) / 2 ) = 120
abs( a + b ) <= abs( a ) + abs( b ) holds
Expressions on GPU ...
a = [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
b = [ -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ]
c = [ -5, -5, -5, -5, -5, -5, -5, -5, -5, -5, -5 ]
min( a ) = 0 at 0
max( a ) = 10 at 10
min( b ) = 0 at -5
max( b ) = 10 at 5
min( abs( b ) ) = 0
sum( b ) = 0
sum( abs( b ) ) = 30
Scalar product: ( a, b ) = 110
Scalar product: ( a + 3, abs( b ) / 2 ) = 120
abs( a + b ) <= abs( a ) + abs( b ) holds

The following table shows vertical operations that can be used on vector expressions:

Expression Meaning
v = TNL::min( expr ) v is minimum of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
std::pair( v, i ) = TNL::argMin( expr ) v is minimum of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ], i is index of the smallest element.
v = TNL::max( expr ) v is maximum of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
std::pair( v, i ) = TNL::argMax( expr ) v is maximum of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ], i is index of the largest element.
v = TNL::sum( expr ) v is sum of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
v = TNL::maxNorm( expr ) v is maximal norm of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
v = TNL::l1Norm( expr ) v is l1 norm of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
v = TNL::l2Norm( expr ) v is l2 norm of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
v = TNL::lpNorm( expr, p ) v is 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::logicalAnd( expr ) v is logical AND of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
v = TNL::logicalOr( expr ) v is logical OR of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
v = TNL::binaryAnd( expr ) v is binary AND of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].
v = TNL::binaryOr( expr ) v is binary OR of expr[ 0 ], expr[ 1 ] , .... expr[ n-1 ].

# Static vectors

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 as well and it is given by a template parameter. Static vector is a templated class defined in namespace TNL::Containers having two template parameters:

• Size is the array size.
• Real is type of numbers stored in the array.

The interface of StaticVectors is smillar to Vector. Probably the most important methods are those related with static vector expressions which are handled by expression templates. They make the use of static vectors simpel and efficient at the same time. See the following simple demonstration:

#include <iostream>
#include <TNL/Containers/StaticVector.h>
using namespace TNL;
using namespace TNL::Containers;
int main( int argc, char* argv[] )
{
using Vector = StaticVector< 3, double >;
Vector v1( 1.0 ), v2( 1.0, 2.0, 3.0 ), v3( v1 - v2 / 2.0 );
Vector v4( 0.0 ), v5( 1.0 );
v4 += v2;
v5 *= v3;
std::cout << "v1 = " << v1 << std::endl;
std::cout << "v2 = " << v2 << std::endl;
std::cout << "v3 = " << v3 << std::endl;
std::cout << "v4 = " << v4 << std::endl;
std::cout << "v5 = " << v5 << std::endl;
std::cout << "abs( v3 - 2.0 ) = " << abs( v3 - 2.0 ) << std::endl;
std::cout << "v2 * v2 = " << v2 * v2 << std::endl;
}

The output looks as:

v1 = [ 1, 1, 1 ]
v2 = [ 1, 2, 3 ]
v3 = [ 0.5, 0, -0.5 ]
v4 = [ 1, 2, 3 ]
v5 = [ 0.5, 0, -0.5 ]
abs( v3 - 2.0 ) = [ 1.5, 2, 2.5 ]
v2 * v2 = [ 1, 4, 9 ]

TODO