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Subvectors provide views on a specific part of a dense or sparse vector. As such, subvectors act as a reference to a specific range within a vector. This reference is valid and can be used in every way any other dense or sparse vector can be used as long as the vector containing the subvector is not resized or entirely destroyed. The subvector also acts as an alias to the vector elements in the specified range: Changes made to the elements (e.g. modifying values, inserting or erasing elements) are immediately visible in the vector and changes made via the vector are immediately visible in the subvector.
Setup of Subvectors
A view on a dense or sparse subvector can be created very conveniently via the subvector() function. It can be included via the header files
#include <blaze/Blaze.h>
// or
#include <blaze/Math.h>
// or
#include <blaze/math/Subvector.h>
and forward declared via the header file
#include <blaze/Forward.h>
The first parameter specifies the offset of the subvector within the underlying dense or sparse vector, the second parameter specifies the size of the subvector. The two parameters can be specified either at compile time or at runtime:
blaze::DynamicVector<double,blaze::rowVector> x;
// ... Resizing and initialization
// Create a subvector from index 4 with a size of 12 (i.e. in the range [4..15]) (compile time arguments)
auto sv1 = subvector<4UL,12UL>( x );
// Create a subvector from index 8 with a size of 16 (i.e. in the range [8..23]) (runtime arguments)
auto sv2 = subvector( x, 8UL, 16UL );
The subvector() function returns an expression representing the subvector view. The type of this expression depends on the given subvector arguments, primarily the type of the vector and the compile time arguments. If the type is required, it can be determined via the decltype specifier:
using VectorType = blaze::DynamicVector<int>;
using SubvectorType = decltype( blaze::subvector<4UL,12UL>( std::declval<VectorType>() ) );
The resulting view can be treated as any other dense or sparse vector, i.e. it can be assigned to, it can be copied from, and it can be used in arithmetic operations. A subvector created from a row vector can be used as any other row vector, a subvector created from a column vector can be used as any other column vector. The view can also be used on both sides of an assignment: The subvector can either be used as an alias to grant write access to a specific subvector of a vector primitive on the left-hand side of an assignment or to grant read-access to a specific subvector of a vector primitive or expression on the right-hand side of an assignment. The following example demonstrates this in detail:
blaze::DynamicVector<double,blaze::rowVector> x;
blaze::CompressedVector<double,blaze::rowVector> y;
blaze::DynamicMatrix<double,blaze::rowMajor> A;
// ... Resizing and initialization
// Create a subvector from index 0 with a size of 10 (i.e. in the range [0..9])
auto sv = subvector( x, 0UL, 10UL );
// Setting the first ten elements of x to the 2nd row of matrix A
sv = row( A, 2UL );
// Setting the second ten elements of x to y
subvector( x, 10UL, 10UL ) = y;
// Setting the 3rd row of A to a subvector of x
row( A, 3UL ) = subvector( x, 3UL, 10UL );
// Setting x to a subvector of the result of the addition between y and the 1st row of A
x = subvector( y + row( A, 1UL ), 2UL, 5UL );
Warning: It is the programmer's responsibility to ensure the subvector does not outlive the viewed vector:
// Creating a subvector on a temporary vector; results in a dangling reference!
auto sv = subvector<1UL,3UL>( DynamicVector<int>{ 1, 2, 3, 4, 5 } );
Element Access
The elements of a subvector can be directly accessed via the subscript operator. The indices to access a subvector are zero-based:
blaze::DynamicVector<double,blaze::rowVector> v;
// ... Resizing and initialization
// Creating an 8-dimensional subvector, starting from index 4
auto sv = subvector( v, 4UL, 8UL );
// Setting the 1st element of the subvector, which corresponds to
// the element at index 5 in vector v
sv[1] = 2.0;
Alternatively, the elements of a subvector can be traversed via iterators. Just as with vectors, in case of non-const subvectors, begin() and end() return an iterator, which allows to manipulate the elements, in case of constant subvectors an iterator to immutable elements is returned:
blaze::DynamicVector<int,blaze::rowVector> v( 256UL );
// ... Resizing and initialization
// Creating a reference to a specific subvector of vector v
auto sv = subvector( v, 16UL, 64UL );
// Traversing the elements via iterators to non-const elements
for( auto it=sv.begin(); it!=sv.end(); ++it ) {
*it = ...; // OK: Write access to the dense subvector value.
... = *it; // OK: Read access to the dense subvector value.
}
// Traversing the elements via iterators to const elements
for( auto it=sv.cbegin(); it!=sv.cend(); ++it ) {
*it = ...; // Compilation error: Assignment to the value via iterator-to-const is invalid.
... = *it; // OK: Read access to the dense subvector value.
}
blaze::CompressedVector<int,blaze::rowVector> v( 256UL );
// ... Resizing and initialization
// Creating a reference to a specific subvector of vector v
auto sv = subvector( v, 16UL, 64UL );
// Traversing the elements via iterators to non-const elements
for( auto it=sv.begin(); it!=sv.end(); ++it ) {
it->value() = ...; // OK: Write access to the value of the non-zero element.
... = it->value(); // OK: Read access to the value of the non-zero element.
it->index() = ...; // Compilation error: The index of a non-zero element cannot be changed.
... = it->index(); // OK: Read access to the index of the sparse element.
}
// Traversing the elements via iterators to const elements
for( auto it=sv.cbegin(); it!=sv.cend(); ++it ) {
it->value() = ...; // Compilation error: Assignment to the value via iterator-to-const is invalid.
... = it->value(); // OK: Read access to the value of the non-zero element.
it->index() = ...; // Compilation error: The index of a non-zero element cannot be changed.
... = it->index(); // OK: Read access to the index of the sparse element.
}
Element Insertion
Inserting/accessing elements in a sparse subvector can be done by several alternative functions. The following example demonstrates all options:
blaze::CompressedVector<double,blaze::rowVector> v( 256UL ); // Non-initialized vector of size 256
auto sv = subvector( v, 10UL, 60UL ); // View on the range [10..69] of v
// The subscript operator provides access to all possible elements of the sparse subvector,
// including the zero elements. In case the subscript operator is used to access an element
// that is currently not stored in the sparse subvector, the element is inserted into the
// subvector.
sv[42] = 2.0;
// The second operation for inserting elements is the set() function. In case the element is
// not contained in the subvector it is inserted into the subvector, if it is already contained
// in the subvector its value is modified.
sv.set( 45UL, -1.2 );
// An alternative for inserting elements into the subvector is the insert() function. However,
// it inserts the element only in case the element is not already contained in the subvector.
sv.insert( 50UL, 3.7 );
// Just as in case of vectors, elements can also be inserted via the append() function. In
// case of subvectors, append() also requires that the appended element's index is strictly
// larger than the currently largest non-zero index of the subvector and that the subvector's
// capacity is large enough to hold the new element. Note however that due to the nature of
// a subvector, which may be an alias to the middle of a sparse vector, the append() function
// does not work as efficiently for a subvector as it does for a vector.
sv.reserve( 10UL );
sv.append( 51UL, -2.1 );
Common Operations
A subvector view can be used like any other dense or sparse vector. This means that with only a few exceptions all Vector Operations and Arithmetic Operations can be used. For instance, the current number of elements can be obtained via the size() function, the current capacity via the capacity() function, and the number of non-zero elements via the nonZeros() function. However, since subvectors are references to a specific range of a vector, several operations are not possible, such as resizing and swapping. The following example shows this by means of a dense subvector view:
blaze::DynamicVector<int,blaze::rowVector> v( 42UL );
// ... Resizing and initialization
// Creating a view on the range [5..15] of vector v
auto sv = subvector( v, 5UL, 10UL );
sv.size(); // Returns the number of elements in the subvector
sv.capacity(); // Returns the capacity of the subvector
sv.nonZeros(); // Returns the number of non-zero elements contained in the subvector
sv.resize( 84UL ); // Compilation error: Cannot resize a subvector of a vector
auto sv2 = subvector( v, 15UL, 10UL );
swap( sv, sv2 ); // Compilation error: Swap operation not allowed
Arithmetic Operations
Both dense and sparse subvectors can be used in all arithmetic operations that any other dense or sparse vector can be used in. The following example gives an impression of the use of dense subvectors within arithmetic operations. All operations (addition, subtraction, multiplication, scaling, ...) can be performed on all possible combinations of dense and sparse subvectors with fitting element types:
blaze::DynamicVector<double,blaze::rowVector> d1, d2, d3;
blaze::CompressedVector<double,blaze::rowVector> s1, s2;
// ... Resizing and initialization
blaze::DynamicMatrix<double,blaze::rowMajor> A;
auto sv( subvector( d1, 0UL, 10UL ) ); // View on the range [0..9] of vector d1
sv = d2; // Dense vector initialization of the range [0..9]
subvector( d1, 10UL, 10UL ) = s1; // Sparse vector initialization of the range [10..19]
d3 = sv + d2; // Dense vector/dense vector addition
s2 = s1 + subvector( d1, 10UL, 10UL ); // Sparse vector/dense vector addition
d2 = sv * subvector( d1, 20UL, 10UL ); // Component-wise vector multiplication
subvector( d1, 3UL, 4UL ) *= 2.0; // In-place scaling of the range [3..6]
d2 = subvector( d1, 7UL, 3UL ) * 2.0; // Scaling of the range [7..9]
d2 = 2.0 * subvector( d1, 7UL, 3UL ); // Scaling of the range [7..9]
subvector( d1, 0UL , 10UL ) += d2; // Addition assignment
subvector( d1, 10UL, 10UL ) -= s2; // Subtraction assignment
subvector( d1, 20UL, 10UL ) *= sv; // Multiplication assignment
double scalar = subvector( d1, 5UL, 10UL ) * trans( s1 ); // Scalar/dot/inner product between two vectors
A = trans( s1 ) * subvector( d1, 4UL, 16UL ); // Outer product between two vectors
Aligned Subvectors
Usually subvectors can be defined anywhere within a vector. They may start at any position and may have an arbitrary size (only restricted by the size of the underlying vector). However, in contrast to vectors themselves, which are always properly aligned in memory and therefore can provide maximum performance, this means that subvectors in general have to be considered to be unaligned. This can be made explicit by the blaze::unaligned flag:
using blaze::unaligned;
blaze::DynamicVector<double,blaze::rowVector> x;
// ... Resizing and initialization
// Identical creations of an unaligned subvector in the range [8..23]
auto sv1 = subvector ( x, 8UL, 16UL );
auto sv2 = subvector<unaligned>( x, 8UL, 16UL );
auto sv3 = subvector<8UL,16UL> ( x );
auto sv4 = subvector<unaligned,8UL,16UL>( x );
All of these calls to the subvector() function are identical. Whether the alignment flag is explicitly specified or not, it always returns an unaligned subvector. Whereas this may provide full flexibility in the creation of subvectors, this might result in performance disadvantages in comparison to vector primitives (even in case the specified subvector could be aligned). Whereas vector primitives are guaranteed to be properly aligned and therefore provide maximum performance in all operations, a general view on a vector might not be properly aligned. This may cause a performance penalty on some platforms and/or for some operations.
However, it is also possible to create aligned subvectors. Aligned subvectors are identical to unaligned subvectors in all aspects, except that they may pose additional alignment restrictions and therefore have less flexibility during creation, but don't suffer from performance penalties and provide the same performance as the underlying vector. Aligned subvectors are created by explicitly specifying the blaze::aligned flag:
using blaze::aligned;
// Creating an aligned subvector in the range [8..23]
auto sv1 = subvector<aligned>( x, 8UL, 16UL );
auto sv2 = subvector<aligned,8UL,16UL>( x );
The alignment restrictions refer to system dependent address restrictions for the used element type and the available vectorization mode (SSE, AVX, ...). In order to be properly aligned the first element of the subvector must be aligned. The following source code gives some examples for a double precision dynamic vector, assuming that AVX is available, which packs 4 double values into a SIMD vector:
using blaze::aligned;
blaze::DynamicVector<double,blaze::columnVector> d( 17UL );
// ... Resizing and initialization
// OK: Starts at the beginning, i.e. the first element is aligned
auto dsv1 = subvector<aligned>( d, 0UL, 13UL );
// OK: Start index is a multiple of 4, i.e. the first element is aligned
auto dsv2 = subvector<aligned>( d, 4UL, 7UL );
// OK: The start index is a multiple of 4 and the subvector includes the last element
auto dsv3 = subvector<aligned>( d, 8UL, 9UL );
// Error: Start index is not a multiple of 4, i.e. the first element is not aligned
auto dsv4 = subvector<aligned>( d, 5UL, 8UL );
Note that the discussed alignment restrictions are only valid for aligned dense subvectors. In contrast, aligned sparse subvectors at this time don't pose any additional restrictions. Therefore aligned and unaligned sparse subvectors are truly fully identical. Still, in case the blaze::aligned flag is specified during setup, an aligned subvector is created:
using blaze::aligned;
blaze::CompressedVector<double,blaze::rowVector> x;
// ... Resizing and initialization
// Creating an aligned subvector in the range [8..23]
auto sv1 = subvector<aligned>( x, 8UL, 16UL );
auto sv2 = subvector<aligned,8UL,16UL>( x );
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