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Constructors

Instantiating and setting up a vector is very easy and intuitive. However, there are a few rules to respect:

  • In case the last template parameter (the transpose flag) is omitted, the vector is per default a column vector.
  • The elements of a StaticVector or HybridVector are default initialized (i.e. built-in data types are initialized to 0, class types are initialized via the default constructor).
  • Newly allocated elements of a DynamicVector or CompressedVector remain uninitialized if they are of built-in type and are default constructed if they are of class type.

Default Construction

using blaze::StaticVector;
using blaze::DynamicVector;
using blaze::CompressedVector;

// All vectors can be default constructed. Whereas the size of StaticVectors is fixed via the second template
// parameter, the initial size of a default constructed DynamicVector or CompressedVector is 0.

StaticVector<int,2UL> v1;                // Instantiation of a 2D integer column vector. All elements are initialized to 0.
StaticVector<long,3UL,columnVector> v2;  // Instantiation of a 3D long integer column vector. Again, all elements are initialized to 0L.

DynamicVector<float> v3;                 // Instantiation of a dynamic single precision column vector of size 0.
DynamicVector<double,rowVector> v4;      // Instantiation of a dynamic double precision row vector of size 0.

CompressedVector<int> v5;                // Instantiation of a compressed integer column vector of size 0.
CompressedVector<double,rowVector> v6;   // Instantiation of a compressed double precision row vector of size 0.

Construction with Specific Size

The DynamicVector, HybridVector and CompressedVector classes offer a constructor that allows to immediately give the vector the required size. Whereas both dense vectors (i.e. DynamicVector and HybridVector) use this information to allocate memory for all vector elements, CompressedVector merely acquires the size but remains empty.

DynamicVector<int,columnVector> v7( 9UL );      // Instantiation of an integer dynamic column vector
                                                // of size 9. The elements are NOT initialized!
HybridVector< complex<float>, 5UL > v8( 2UL );  // Instantiation of a column vector with two single precision
                                                // complex values. The elements are default constructed.
CompressedVector<int,rowVector> v9( 10UL );     // Instantiation of a compressed row vector with size 10.
                                                // Initially, the vector provides no capacity for non-zero elements.

Initialization Constructors

All dense vector classes offer a constructor that allows for a direct, homogeneous initialization of all vector elements. In contrast, for sparse vectors the predicted number of non-zero elements can be specified:

StaticVector<int,3UL,rowVector> v10( 2 );            // Instantiation of a 3D integer row vector.
                                                     // All elements are initialized to 2.
DynamicVector<float> v11( 3UL, 7.0F );               // Instantiation of a dynamic single precision column vector
                                                     // of size 3. All elements are set to 7.0F.
CompressedVector<float,rowVector> v12( 15UL, 3UL );  // Instantiation of a single precision column vector of size 15,
                                                     // which provides enough space for at least 3 non-zero elements.

Array Construction

Alternatively, all dense vector classes offer a constructor for an initialization with a dynamic or static array, or with a std::array. If the vector is initialized from a dynamic array, the constructor expects the actual size of the array as first argument, the array as second argument. In case of a static array or std::array, the fixed size of the array is used:

const unique_ptr<double[]> array1( new double[2] );
// ... Initialization of the dynamic array
blaze::StaticVector<double,2UL> v13( 2UL, array1.get() );

const int array2[4] = { 4, -5, -6, 7 };
blaze::StaticVector<int,4UL> v14( array2 );

const std::array<float,3UL> array3{ 1.1F, 2.2F, 3.3F };
blaze::StaticVector<float,3UL> v15( array3 );

Initializer List Construction

In addition, all dense and sparse vector classes can be directly initialized by means of an initializer list:

blaze::DynamicVector<float> v16{ 1.0F, 2.0F, 3.0F, 4.0F };
blaze::CompressedVector<int> v17{ 0, 2, 0, 0, 5, 0, 7, 0 };

Dynamically sized vectors (such as e.g. HybridVector, DynamicVector or CompressedVector) are sized according to the size of the initializer list and all their elements are (copy) assigned the values of the list. For fixed size vectors (such as e.g. StaticVector) missing values are initialized as default and in case the size of the initializer list exceeds the size of the vector a std::invalid_argument exception is thrown. In case of sparse vectors, only the non-zero elements are used to initialize the vector.

Copy Construction

All dense and sparse vectors can be created as the copy of any other dense or sparse vector with the same transpose flag (i.e. blaze::rowVector or blaze::columnVector).

StaticVector<int,9UL,columnVector> v18( v7 );  // Instantiation of the dense column vector v17
                                               // as copy of the dense column vector v7.
DynamicVector<int,rowVector> v19( v9 );        // Instantiation of the dense row vector v18 as
                                               // copy of the sparse row vector v9.
CompressedVector<int,columnVector> v20( v1 );  // Instantiation of the sparse column vector v19
                                               // as copy of the dense column vector v1.
CompressedVector<float,rowVector> v21( v12 );  // Instantiation of the sparse row vector v20 as
                                               // copy of the row vector v12.

Note that it is not possible to create a StaticVector as a copy of a vector with a different size:

StaticVector<int,5UL,columnVector> v22( v7 );  // Runtime error: Size does not match!
StaticVector<int,4UL,rowVector> v23( v10 );    // Compile time error: Size does not match!

Assignment

There are several types of assignment to dense and sparse vectors: Homogeneous Assignment, Array Assignment, Copy Assignment, and Compound Assignment.

Homogeneous Assignment

Sometimes it may be necessary to assign the same value to all elements of a dense vector. For this purpose, the assignment operator can be used:

blaze::StaticVector<int,3UL> v1;
blaze::DynamicVector<double> v2;

// Setting all integer elements of the StaticVector to 2
v1 = 2;

// Setting all double precision elements of the DynamicVector to 5.0
v2 = 5.0;

Array Assignment

Dense vectors can also be assigned a static array or std::array:

blaze::StaticVector<float,2UL> v1;
blaze::DynamicVector<double,rowVector> v2;

const float array1[2] = { 1.0F, 2.0F };
const std::array<double,5UL> array2{ 2.1, 4.0, -1.7, 8.6, -7.2 };

v1 = array1;
v2 = array2;

Initializer list Assignment

Alternatively, it is possible to directly assign an initializer list to a dense or sparse vector:

blaze::DynamicVector<float> v1;
blaze::CompressedVector<double,rowVector> v2;

v1 = { 1.0F, 2.0F };
v2 = { 2.1, 0.0, -1.7, 0.0, -7.2 };

Dynamically sized vectors (such as e.g. HybridVector, DynamicVector or CompressedVector) are resized according to the size of the initializer list and all their elements are (copy) assigned the values of the list. For fixed size vectors (such as e.g. StaticVector) missing values are reset to their default value and in case the size of the initializer list exceeds the size of the vector a std::invalid_argument exception is thrown. In case of sparse vectors, only the non-zero elements are considered.

Copy Assignment

For all vector types it is generally possible to assign another vector with the same transpose flag (i.e. blaze::columnVector or blaze::rowVector). Note that in case of StaticVector, the assigned vector is required to have the same size as the StaticVector since the size of a StaticVector cannot be changed!

blaze::StaticVector<int,3UL,columnVector> v1;
blaze::DynamicVector<int,columnVector> v2( 3UL );
blaze::DynamicVector<float,columnVector> v3( 5UL );
blaze::CompressedVector<int,columnVector> v4( 3UL );
blaze::CompressedVector<float,rowVector> v5( 3UL );

// ... Initialization of the vectors

v1 = v2;  // OK: Assignment of a 3D dense column vector to another 3D dense column vector
v1 = v4;  // OK: Assignment of a 3D sparse column vector to a 3D dense column vector
v1 = v3;  // Runtime error: Cannot assign a 5D vector to a 3D static vector
v1 = v5;  // Compilation error: Cannot assign a row vector to a column vector

Compound Assignment

Next to plain assignment, it is also possible to use addition assignment, subtraction assignment, and multiplication assignment. Note however, that in contrast to plain assignment the size and the transpose flag of the vectors has be to equal in order to able to perform a compound assignment.

blaze::StaticVector<int,5UL,columnVector> v1;
blaze::DynamicVector<int,columnVector> v2( 5UL );
blaze::CompressedVector<float,columnVector> v3( 7UL );
blaze::DynamicVector<float,rowVector> v4( 7UL );
blaze::CompressedVector<float,rowVector> v5( 7UL );

// ... Initialization of the vectors

v1 += v2;  // OK: Addition assignment between two column vectors of the same size
v1 += v3;  // Runtime error: No compound assignment between vectors of different size
v1 -= v4;  // Compilation error: No compound assignment between vectors of different transpose flag
v4 *= v5;  // OK: Multiplication assignment between two row vectors of the same size

Element Access

Subscript Operator

The easiest and most intuitive way to access a dense or sparse vector is via the subscript operator. The indices to access a vector are zero-based:

blaze::DynamicVector<int> v1( 5UL );
v1[0] = 1;
v1[1] = 3;
// ...

blaze::CompressedVector<float> v2( 5UL );
v2[2] = 7.3F;
v2[4] = -1.4F;

Whereas using the subscript operator on a dense vector only accesses the already existing element, accessing an element of a sparse vector via the subscript operator potentially inserts the element into the vector and may therefore be more expensive. Consider the following example:

blaze::CompressedVector<int> v1( 10UL );

for( size_t i=0UL; i<v1.size(); ++i ) {
   ... = v1[i];  // Inserts the element at index i
}

Although the compressed vector is only used for read access within the for loop, using the subscript operator temporarily inserts 10 non-zero elements into the vector. Therefore the preferred way to traverse the non-zero elements of a sparse vector is to use iterators.

Iterators

An alternate way to traverse the elements contained in a dense or sparse vector is by means of iterators. For that purpose, all vectors provide the begin(), cbegin(), end(), and cend() members functions. In case of non-const vectors, begin() and end() return an Iterator, which allows a manipulation of the (non-zero) value. In case of a constant vector or in case cbegin() or cend() are used a ConstIterator is returned. Iterators on dense vectors traverse all elements of the vector, including the zero elements. Iterators on sparse vectors only traverse the non-zero elements.

The following two examples demonstrate how to traverse the elements of a dense and sparse vector, respectively:

using blaze::DynamicVector;

DynamicVector<int> v1( 10UL );

// Traversing all elements contained in the vector by Iterator
for( DynamicVector<int>::Iterator it=v1.begin(); it!=v1.end(); ++it ) {
   *it = ...;  // OK: Write access to the value of the element.
   ... = *it;  // OK: Read access to the value of the element.
}

// Traversing all elements contained in the vector by ConstIterator
for( DynamicVector<int>::ConstIterator it=v1.cbegin(); it!=v1.cend(); ++it ) {
   *it = ...;  // Compilation error: Assignment to the value via a ConstIterator is invalid.
   ... = *it;  // OK: Read access to the value of the element.
}

// Traversing the vector elements by means of a range-based for loop
for( int& i : v1 ) {
   i = ...;  // OK: Write access to the value of the element.
   ... = i;  // OK: Read access to the value of the element.
}

using blaze::CompressedVector;

CompressedVector<int> v2( 10UL );

// ... Initialization of the vector

// Traversing the non-zero elements contained in the vector by Iterator
for( CompressedVector<int>::Iterator it=v2.begin(); it!=v2.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 non-zero element.
}

// Traversing the non-zero elements contained in the vector by ConstIterator
for( CompressedVector<int>::ConstIterator it=v2.cbegin(); it!=v2.cend(); ++it ) {
   it->value() = ...;  // Compilation error: Assignment to the value via a ConstIterator 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 non-zero element.
}

Note that begin(), cbegin(), end(), and cend() are also available as free functions:

for( CompressedVector<int>::Iterator it=begin( v2 ); it!=end( v2 ); ++it ) {
   // ...
}

for( CompressedVector<int>::ConstIterator it=cbegin( v2 ); it!=cend( v2 ); ++it ) {
   // ...
}

.data() / data()

Sometimes it is necessary to acquire a pointer to the first element of the underlying array of a dense vector. For that purpose the data() member function or the free data() function can be used:

// Instantiating a dynamic vector with 10 elements
blaze::DynamicVector<int> v( 10UL );
v.data();   // Returns a pointer to the first element of the dynamic vector
data( v );  // Same effect as the member function

Element Insertion

In contrast to dense vectors, that store all elements independent of their value and that offer direct access to all elements, spares vectors only store the non-zero elements contained in the vector. Therefore it is necessary to explicitly add elements to the vector.

Subscript Operator

The first option to add elements to a sparse vector is the subscript operator:

using blaze::CompressedVector;

CompressedVector<int> v1( 3UL );
v1[1] = 2;

In case the element at the given index is not yet contained in the vector, it is automatically inserted. Otherwise the old value is replaced by the new value 2. The operator returns a reference to the sparse vector element.

.set()

An alternative to the subscript operator is the set() function: In case the element is not yet contained in the vector the element is inserted, else the element's value is modified:

// Insert or modify the value at index 3
v1.set( 3, 1 );

.insert()

The insertion of elements can be better controlled via the insert() function. In contrast to the subscript operator and the set() function it emits an exception in case the element is already contained in the vector. In order to check for this case, the find() function can be used:

// In case the element at index 4 is not yet contained in the matrix it is inserted with a value of 6.
if( v1.find( 4 ) == v1.end() )
   v1.insert( 4, 6 );

.append()

Although the insert() function is very flexible, due to performance reasons it is not suited for the setup of large sparse vectors. A very efficient, yet also very low-level way to fill a sparse vector is the append() function. It requires the sparse vector to provide enough capacity to insert a new element. Additionally, the index of the new element must be larger than the index of the previous element. Violating these conditions results in undefined behavior!

v1.reserve( 5 );     // Reserving space for 5 non-zero elements
v1.append( 5, -2 );  // Appending the element -2 at index 5
v1.append( 6, 4 );   // Appending the element 4 at index 6
// ...

Element Removal

.erase()

The erase() member functions can be used to remove elements from a sparse vector. The following example gives an impression of the five different flavors of erase():

using blaze::CompressedVector;

CompressedVector<int> v( 42 );
// ... Initialization of the vector

// Erasing the element at index 21
v.erase( 21 );

// Erasing a single element via iterator
v.erase( v.find( 4 ) );

// Erasing all non-zero elements in the range [7..24]
v.erase( v.lowerBound( 7 ), v.upperBound( 24 ) );

// Erasing all non-zero elements with a value larger than 9 by passing a unary predicate
v.erase( []( int i ){ return i > 9; } );

// Erasing all non-zero elements in the range [30..40] with a value larger than 5
v.erase( v.lowerBound( 30 ), v.upperBound( 40 ), []( int i ){ return i > 5; } );

Element Lookup

A sparse vector only stores the non-zero elements contained in the vector. Therefore, whenever accessing a vector element at a specific index a lookup operation is required. Whereas the subscript operator is performing this lookup automatically, it is also possible to use the find(), lowerBound(), and upperBound() member functions for a manual lookup.

.find() / find()

The find() function can be used to check whether a specific element is contained in a sparse vector. It specifically searches for the element at the given index. In case the element is found, the function returns an iterator to the element. Otherwise an iterator just past the last non-zero element of the compressed vector (the end() iterator) is returned. Note that the returned iterator is subject to invalidation due to inserting operations via the subscript operator, the set() function or the insert() function!

using blaze::CompressedVector;

CompressedVector<int> a( 42 );
// ... Initialization of the vector

// Searching the element at index 7. In case the element is not
// contained in the vector, the end() iterator is returned.
CompressedVector<int>::Iterator pos( a.find( 7 ) );

if( pos != a.end( 7 ) ) {
   // ...
}

Alternatively, the free function find() can be used to find a specific element in a sparse vector:

find( a, 7 );  // Searching the element at index 7; same effect as the member function

.lowerBound() / lowerBound()

The lowerBound() function returns an iterator to the first element with an index not less then the given index. In combination with the upperBound() function this function can be used to create a pair of iterators specifying a range of indices. Note that the returned iterator is subject to invalidation due to inserting operations via the subscript operator, the set() function or the insert() function!

using blaze::CompressedVector;

CompressedVector<int> a( 42 );
// ... Initialization of the vector

// Searching the lower bound of index 17.
CompressedVector<int>::Iterator pos1( a.lowerBound( 17 ) );

// Searching the upper bound of index 28
CompressedVector<int>::Iterator pos2( a.upperBound( 28 ) );

// Erasing all elements in the specified range
a.erase( pos1, pos2 );

Alternatively, the free function lowerBound() can be used to:

lowerBound( a, 17 );  // Searching the lower bound of index 17; same effect as the member function

.upperBound() / upperBound()

The upperBound() function returns an iterator to the first element with an index greater then the given index. In combination with the lowerBound() function this function can be used to create a pair of iterators specifying a range of indices. Note that the returned iterator is subject to invalidation due to inserting operations via the subscript operator, the set() function or the insert() function!

using blaze::CompressedVector;

CompressedVector<int> a( 42 );
// ... Initialization of the vector

// Searching the lower bound of index 17.
CompressedVector<int>::Iterator pos1( a.lowerBound( 17 ) );

// Searching the upper bound of index 28
CompressedVector<int>::Iterator pos2( a.upperBound( 28 ) );

// Erasing all elements in the specified range
a.erase( pos1, pos2 );

Alternatively, the free function upperBound() can be used to:

upperBound( a, 28 );  // Searching the upper bound of index 28; same effect as the member function

Non-Modifying Operations

.size() / size()

Via the size() member function, the current size of a dense or sparse vector can be retrieved:

// Instantiating a dynamic vector with size 10
blaze::DynamicVector<int> v1( 10UL );
v1.size();  // Returns 10

// Instantiating a compressed vector with size 12 and capacity for 3 non-zero elements
blaze::CompressedVector<double> v2( 12UL, 3UL );
v2.size();  // Returns 12

Alternatively, the free function size() can be used to retrieve the current size of a vector. In contrast to the size()member function, the free function can also be used to retrieve the size of vector expressions:

size( v1 );  // Returns 10, i.e. has the same effect as the member function
size( v2 );  // Returns 12, i.e. has the same effect as the member function

blaze::DynamicMatrix<int> A( 15UL, 12UL );
size( A * v2 );  // Returns 15, i.e. the size of the resulting vector

.capacity() / capacity()

Via the capacity() (member) function the internal capacity of a dense or sparse vector can be retrieved. Note that the capacity of a vector doesn't have to be equal to the size of a vector. In case of a dense vector the capacity will always be greater than or equal to the size of the vector, in case of a sparse vector the capacity may even be less than the size.

v1.capacity();   // Returns at least 10

For symmetry reasons, there is also a free function capacity() available that can be used to retrieve the capacity:

capacity( v1 );  // Returns at least 10, i.e. has the same effect as the member function

Note, however, that it is not possible to retrieve the capacity of a vector expression:

capacity( A * v1 );  // Compilation error!

.nonZeros() / nonZeros()

For both dense and sparse vectors the number of non-zero elements can be determined via the nonZeros() member function. Sparse vectors directly return their number of non-zero elements, dense vectors traverse their elements and count the number of non-zero elements.

v1.nonZeros();  // Returns the number of non-zero elements in the dense vector
v2.nonZeros();  // Returns the number of non-zero elements in the sparse vector

There is also a free function nonZeros() available to retrieve the current number of non-zero elements:

nonZeros( v1 );  // Returns the number of non-zero elements in the dense vector
nonZeros( v2 );  // Returns the number of non-zero elements in the sparse vector

The free nonZeros() function can also be used to retrieve the number of non-zero elements in a vector expression. However, the result is not the exact number of non-zero elements, but may be a rough estimation:

nonZeros( A * v1 );  // Estimates the number of non-zero elements in the vector expression

isEmpty()

The isEmpty() function returns whether the total number of elements of the vector is zero:

blaze::DynamicVector<int> a;  // Create an empty vector
isEmpty( a );                 // Returns true
a.resize( 10 );               // Resize to 10 elements
isEmpty( a );                 // Returns false

isnan()

The isnan() function provides the means to check a dense or sparse vector for non-a-number elements:

blaze::DynamicVector<double> a;
// ... Resizing and initialization
if( isnan( a ) ) { ... }

blaze::CompressedVector<double> a;
// ... Resizing and initialization
if( isnan( a ) ) { ... }

If at least one element of the vector is not-a-number, the function returns true, otherwise it returns false. Please note that this function only works for vectors with floating point elements. The attempt to use it for a vector with a non-floating point element type results in a compile time error.

isinf()

The isinf() function checks the given dense or sparse vector for infinite (inf) elements:

blaze::DynamicVector<double> a;
// ... Resizing and initialization
if( isinf( a ) ) { ... }

blaze::CompressedVector<double> a;
// ... Resizing and initialization
if( isinf( a ) ) { ... }

If at least one element of the vector is infinite, the function returns true, otherwise it returns false.

isfinite()

The isfinite() function checks if all elements of the given dense or sparse vector are finite elements (i.e. normal, subnormal or zero elements, but not infinite or NaN):

blaze::DynamicVector<double> a;
// ... Resizing and initialization
if( isfinite( a ) ) { ... }

blaze::CompressedVector<double> a;
// ... Resizing and initialization
if( isfinite( a ) ) { ... }

If all elements of the vector are finite, the function returns true, otherwise it returns false.

isDefault()

The isDefault() function returns whether the given dense or sparse vector is in default state:

blaze::HybridVector<int,20UL> a;
// ... Resizing and initialization
if( isDefault( a ) ) { ... }

A vector is in default state if it appears to just have been default constructed. All resizable vectors (HybridVector, DynamicVector, or CompressedVector) and CustomVector are in default state if its size is equal to zero. A non-resizable vector (StaticVector, all subvectors, element selections, rows, and columns) is in default state if all its elements are in default state. For instance, in case the vector is instantiated for a built-in integral or floating point data type, the function returns true in case all vector elements are 0 and false in case any vector element is not 0.

isUniform()

In order to check if all vector elements are identical, the isUniform() function can be used:

blaze::DynamicVector<int> a;
// ... Resizing and initialization
if( isUniform( a ) ) { ... }

Note that in case of a sparse vector also the zero elements are taken into account!

isZero()

In order to check if all vector elements are zero, the isZero() function can be used:

blaze::DynamicVector<int> a;
// ... Resizing and initialization
if( isZero( a ) ) { ... }

length() / sqrLength()

In order to calculate the length (magnitude) of a dense or sparse vector, both the length() and sqrLength() function can be used:

blaze::StaticVector<float,3UL,blaze::rowVector> v{ -1.2F, 2.7F, -2.3F };

const float len    = length   ( v );  // Computes the current length of the vector
const float sqrlen = sqrLength( v );  // Computes the square length of the vector

Note that both functions can only be used for vectors with built-in or complex element type.

trans()

As already mentioned, vectors can be either column vectors (blaze::columnVector) or row vectors (blaze::rowVector). A column vector cannot be assigned to a row vector and vice versa. However, vectors can be transposed via the trans() function:

blaze::DynamicVector<int,blaze::columnVector> v1( 4UL );
blaze::CompressedVector<int,blaze::rowVector> v2( 6UL );

v1 = v2;            // Compilation error: Cannot assign a row vector to a column vector
v1 = trans( v2 );   // OK: Transposing the row vector to a column vector and assigning it to the column vector v1
v2 = trans( v1 );   // OK: Transposing the column vector v1 and assigning it to the row vector v2
v1 += trans( v2 );  // OK: Addition assignment of two column vectors

ctrans()

It is also possible to compute the conjugate transpose of a vector. This operation is available via the ctrans() function:

blaze::CompressedVector< complex<float>, rowVector > v1( 4UL );
blaze::DynamicVector< complex<float>, columnVector > v2( 4UL );

v1 = ctrans( v2 );  // Compute the conjugate transpose vector

Note that the ctrans() function has the same effect as manually applying the conj() and trans() function in any order:

v1 = trans( conj( v2 ) );  // Computing the conjugate transpose vector
v1 = conj( trans( v2 ) );  // Computing the conjugate transpose vector

reverse()

Via the reverse() function is is possible to reverse the elements of a dense or sparse vector. The following examples demonstrates this by means of a dense vector:

blaze::DynamicVector<int> a{ 1, 2, 3, 4, 5 };
blaze::DynamicVector<int> b;

b = reverse( a );  // Results in ( 5 4 3 2 1 )

eval() / evaluate()

The evaluate() function forces an evaluation of the given vector expression and enables an automatic deduction of the correct result type of an operation. The following code example demonstrates its intended use for the multiplication of a dense and a sparse vector:

!#c++
using blaze::DynamicVector;
using blaze::CompressedVector;

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

auto c = evaluate( a * b );

In this scenario, the evaluate() function assists in deducing the exact result type of the operation via the auto keyword. Please note that if evaluate() is used in this way, no temporary vector is created and no copy operation is performed. Instead, the result is directly written to the target vector due to the return value optimization (RVO). However, if evaluate() is used in combination with an explicit target type, a temporary will be created and a copy operation will be performed if the used type differs from the type returned from the function:

!#c++
CompressedVector<double> d( a * b );  // No temporary & no copy operation
DynamicVector<double> e( a * b );     // Temporary & copy operation
d = evaluate( a * b );                // Temporary & copy operation

Sometimes it might be desirable to explicitly evaluate a sub-expression within a larger expression. However, please note that evaluate() is not intended to be used for this purpose. This task is more elegantly and efficiently handled by the eval() function:

blaze::DynamicVector<double> a, b, c, d;

d = a + evaluate( b * c );  // Unnecessary creation of a temporary vector
d = a + eval( b * c );      // No creation of a temporary vector

In contrast to the evaluate() function, eval() can take the complete expression into account and therefore can guarantee the most efficient way to evaluate it (see also Intra-Statement Optimization).

noalias()

The Blaze library is able to reliably detect aliasing during the assignment of vectors. In case the aliasing would lead to an incorrect result, Blaze introduces an intermediate temporary of the appropriate type to break the aliasing. For instance, in the following example Blaze performs an alias detection in both assignments, but only, in the second assignment it detects a problematic aliasing and uses an intermediate temporary in order to be able to compute the correct result:

blaze::DynamicVector<double> x, y;
blaze::DynamicMatrix<double> A;

x = x + y;  // No problematic aliasing of x, no intermediate temporary is required.
x = A * x;  // Problematic aliasing of x; intermediate temporary required!

The detection of aliasing effects, however, takes a small runtime effort. In order to disable the aliasing detection, the noalias() function can be used:

blaze::DynamicVector<double> x, y;
blaze::DynamicMatrix<double> A;

x = noalias( x + y );  // No alias detection performed, no intermediate temporary.
x = noalias( A * x );  // No alias detection performed, no intermediate temporary.
                       // Note that the final result will be incorrect!

Warning: The noalias() operation has the semantics of a cast: The caller is completely responsible and the system trusts the given information. Using noalias() in a situation where an aliasing effect occurs leads to undefined behavior (which can be violated invariants or wrong computation results)!

nosimd()

By default, Blaze attempts to vectorize all operations by means of SSE, AVX, etc. in order to achieve maximum performance. However, via the nosimd() operation it is possible to disable the SIMD evaluation of any operation:

blaze::DynamicVector<double> x, y;
blaze::DynamicMatrix<double> A;

x = nosimd( x + y );  // Disables SIMD for the vector/vector addition
x = nosimd( A * x );  // Disables SIMD for the matrix/vector multiplication

Please note that the main purpose of the nosimd() operation is to enable an easy performance comparison between the vectorized and non-vectorized evaluation. Using the nosimd() operation will likely result in significantly reduced performance!

Modifying Operations

.resize() / .reserve()

The size of a StaticVector is fixed by the second template parameter and a CustomVector cannot be resized. In contrast, the size of DynamicVectors, HybridVectors as well as CompressedVectors can be changed via the resize() function:

using blaze::DynamicVector;
using blaze::CompressedVector;

DynamicVector<int,columnVector> v1;
CompressedVector<int,rowVector> v2( 4 );
v2[1] = -2;
v2[3] = 11;

// Adapting the size of the dynamic and compressed vectors. The (optional) second parameter
// specifies whether the existing elements should be preserved. Per default, the existing
// elements are preserved.
v1.resize( 5UL );         // Resizing vector v1 to 5 elements. Elements of built-in type remain
                          // uninitialized, elements of class type are default constructed.
v1.resize( 3UL, false );  // Resizing vector v1 to 3 elements. The old elements are lost, the
                          // new elements are NOT initialized!
v2.resize( 8UL, true );   // Resizing vector v2 to 8 elements. The old elements are preserved.
v2.resize( 5UL, false );  // Resizing vector v2 to 5 elements. The old elements are lost.

Note that resizing a vector invalidates all existing views (see e.g. Subvectors) on the vector:

using VectorType    = blaze::DynamicVector<int,rowVector>;
using SubvectorType = blaze::Subvector<VectorType>;

VectorType v1( 10UL );                         // Creating a dynamic vector of size 10
SubvectorType sv = subvector( v1, 2UL, 5UL );  // Creating a view on the range [2..6]
v1.resize( 6UL );                              // Resizing the vector invalidates the view

When the internal capacity of a vector is no longer sufficient, the allocation of a larger junk of memory is triggered. In order to avoid frequent reallocations, the reserve() function can be used up front to set the internal capacity:

blaze::DynamicVector<int> v1;
v1.reserve( 100 );
v1.size();      // Returns 0
v1.capacity();  // Returns at least 100

Note that the size of the vector remains unchanged, but only the internal capacity is set according to the specified value!

.shrinkToFit()

The internal capacity of vectors with dynamic memory is preserved in order to minimize the number of reallocations. For that reason, the resize() and reserve() functions can lead to memory overhead. The shrinkToFit() member function can be used to minimize the internal capacity:

blaze::DynamicVector<int> v1( 1000UL );  // Create a vector of 1000 integers
v1.resize( 10UL );                       // Resize to 10, but the capacity is preserved
v1.shrinkToFit();                        // Remove the unused capacity

Please note that due to padding the capacity might not be reduced exactly to size(). Please also note that in case a reallocation occurs, all iterators (including end() iterators), all pointers and references to elements of the vector are invalidated.

reset() / clear()

In order to reset all elements of a vector, the reset() function can be used:

// Setup of a single precision column vector, whose elements are initialized with 2.0F.
blaze::DynamicVector<float> v1( 3UL, 2.0F );

// Resetting all elements to 0.0F. Only the elements are reset, the size of the vector is unchanged.
reset( v1 );  // Resetting all elements
v1.size();    // Returns 3: size and capacity remain unchanged

In order to return a vector to its default state (i.e. the state of a default constructed vector), the clear() function can be used:

// Setup of a single precision column vector, whose elements are initialized with -1.0F.
blaze::DynamicVector<float> v1( 5, -1.0F );

// Resetting the entire vector.
clear( v1 );  // Resetting the entire vector
v1.size();    // Returns 0: size is reset, but capacity remains unchanged

Note that resetting or clearing both dense and sparse vectors does not change the capacity of the vectors.

swap()

Via the swap() function it is possible to completely swap the contents of two vectors of the same type:

blaze::DynamicVector<int,blaze::columnVector> v1( 10UL );
blaze::DynamicVector<int,blaze::columnVector> v2( 20UL );

swap( v1, v2 );  // Swapping the contents of v1 and v2

Arithmetic Operations

normalize()

The normalize() function can be used to scale any non-zero vector to a length of 1. In case the vector does not contain a single non-zero element (i.e. is a zero vector), the normalize() function returns a zero vector.

blaze::DynamicVector<float,blaze::columnVector>     v1( 10UL );
blaze::CompressedVector<double,blaze::columnVector> v2( 12UL );

v1 = normalize( v1 );  // Normalizing the dense vector v1
length( v1 );          // Returns 1 (or 0 in case of a zero vector)
v1 = normalize( v2 );  // Assigning v1 the normalized vector v2
length( v1 );          // Returns 1 (or 0 in case of a zero vector)

Note that the normalize() function only works for floating point vectors. The attempt to use it for an integral vector results in a compile time error.

min() / max()

The min() and max() functions can be used for a single vector, multiple vectors, and a vector and a scalar.

Single Vector

If passed a single vector, the functions return the smallest and largest element of the given dense vector or the smallest and largest non-zero element of the given sparse vector, respectively:

blaze::StaticVector<int,4UL,rowVector> a{ -5, 2,  7, -4 };

min( a );  // Returns -5
max( a );  // Returns 7

blaze::CompressedVector<int> b{ 1, 0, 3, 0 };

min( b );  // Returns 1
max( b );  // Returns 3

For more information on the unary min() and max() reduction operations see the Reduction Operations section.

Multiple Vectors

If passed two or more dense vectors, the min() and max() functions compute the componentwise minimum or maximum of the given vectors, respectively:

blaze::StaticVector<int,4UL,rowVector> c{ -5, 1, -7, 4 };
blaze::StaticVector<int,4UL,rowVector> d{ -5, 3,  0, 2 };

min( a, c );     // Results in the vector ( -5, 1, -7, -4 )
max( a, c, d );  // Results in the vector ( -5, 3,  7,  4 )

Please note that sparse vectors can only be used in the unary min() and max() functions. Also note that all forms of the min() and max() functions can be used to compute the smallest and largest element of a vector expression:

min( a + b + c );  // Returns -9, i.e. the smallest value of the resulting vector
max( a - b - c );  // Returns 11, i.e. the largest value of the resulting vector

min( a + c, c - d );  // Results in ( -10 -2 -7 0 )
max( a - c, c + d );  // Results in ( 0 4 14 6 )

Vector and Scalar

If passed a dense vector and a scalar, the min() and max() functions compute the componentwise minimum or maximum between the given vector and a uniform vector represented by the scalar value:

min( a, 0 );  // Results in ( -5, 0, 0, -4 )
min( 0, a );  // Results in ( -5, 0, 0, -4 )
max( a, 0 );  // Results in ( 0, 2, 7, 0 )
max( 0, a );  // Results in ( 0, 2, 7, 0 )

softmax()

The softmax function, also called the normalized exponential function, of a given dense vector can be computed via softmax(). The resulting dense vector consists of real values in the range (0..1], which add up to 1.

blaze::StaticVector<double,7UL,rowVector> x{ 1.0, 2.0, 3.0, 4.0, 1.0, 2.0, 3.0 };
blaze::StaticVector<double,7UL,rowVector> y;

// Evaluating the softmax function
y = softmax( x );     // Results in ( 0.024 0.064 0.175 0.475 0.024 0.064 0.175 )
double s = sum( y );  // Results in 1

abs()

The abs() function can be used to compute the absolute values of each element of a vector. For instance, the following computation

blaze::StaticVector<int,3UL,blaze::rowVector> a{ -1, 2, -3 };
blaze::StaticVector<int,3UL,blaze::rowVector> b( abs( a ) );

results in the vector

    ( 1 )
b = ( 2 )
    ( 3 )

sign()

The sign() function can be used to evaluate the sign of each element of a vector v. For each element i the corresponding result is 1 if v[i] is greater than zero, 0 if v[i] is zero, and -1 if v[i] is less than zero. For instance, the following use of the sign() function

blaze::StaticVector<int,3UL,rowVector> a{ -1, 2, 0 };
blaze::StaticVector<int,3UL,rowVector> b( sign( a ) );

results in the vector

    ( -1 )
b = (  1 )
    (  0 )

floor() / ceil() / trunc() / round()

The floor(), ceil(), trunc(), and round() functions can be used to round down/up each element of a vector, respectively:

blaze::StaticVector<double,3UL,rowVector> a, b;

b = floor( a );  // Rounding down each element of the vector
b = ceil ( a );  // Rounding up each element of the vector
b = trunc( a );  // Truncating each element of the vector
b = round( a );  // Rounding each element of the vector

conj()

The conj() function can be applied on a dense or sparse vector to compute the complex conjugate of each element of the vector:

using blaze::StaticVector;

using cplx = std::complex<double>;

// Creating the vector
//    ( (-2,-1) )
//    ( ( 1, 1) )
StaticVector<cplx,2UL> a{ cplx(-2.0,-1.0), cplx(1.0,1.0) };

// Computing the vector of complex conjugates
//    ( (-2, 1) )
//    ( ( 1,-1) )
StaticVector<cplx,2UL> b;
b = conj( a );

Additionally, vectors can be conjugated in-place via the conjugate() function:

blaze::DynamicVector<cplx> c( 5UL );

conjugate( c );  // In-place conjugate operation.
c = conj( c );   // Same as above

real()

The real() function can be used on a dense or sparse vector to extract the real part of each element of the vector:

using blaze::StaticVector;

using cplx = std::complex<double>;

// Creating the vector
//    ( (-2,-1) )
//    ( ( 1, 1) )
StaticVector<cplx,2UL> a{ cplx(-2.0,-1.0), cplx(1.0,1.0) };

// Extracting the real part of each vector element
//    ( -2 )
//    (  1 )
StaticVector<double,2UL> b;
b = real( a );

imag()

The imag() function can be used on a dense or sparse vector to extract the imaginary part of each element of the vector:

using blaze::StaticVector;

using cplx = std::complex<double>;

// Creating the vector
//    ( (-2,-1) )
//    ( ( 1, 1) )
StaticVector<cplx,2UL> a{ cplx(-2.0,-1.0), cplx(1.0,1.0) };

// Extracting the imaginary part of each vector element
//    ( -1 )
//    (  1 )
StaticVector<double,2UL> b;
b = imag( a );

arg()

The arg() function can be used on a dense or sparse vector to compute the phase angle for each element of the vector:

using blaze::StaticVector;

using cplx = std::complex<double>;

// Creating the vector
//    ( (-2,-1) )
//    ( ( 1, 1) )
StaticVector<cplx,2UL> a{ cplx(-2.0,-1.0), cplx(1.0,1.0) };

// Compute the phase angle of each vector element
//    ( -2.67795  )
//    (  0.785398 )
StaticVector<double,2UL> b;
b = arg( a );

sqrt() / invsqrt()

Via the sqrt() and invsqrt() functions the (inverse) square root of each element of a vector can be computed:

blaze::DynamicVector<double> a, b, c;

b = sqrt( a );     // Computes the square root of each element
c = invsqrt( a );  // Computes the inverse square root of each element

Note that in case of sparse vectors only the non-zero elements are taken into account!

cbrt() / invcbrt()

The cbrt() and invcbrt() functions can be used to compute the the (inverse) cubic root of each element of a vector:

blaze::HybridVector<double,3UL> a, b, c;

b = cbrt( a );     // Computes the cubic root of each element
c = invcbrt( a );  // Computes the inverse cubic root of each element

Note that in case of sparse vectors only the non-zero elements are taken into account!

hypot()

The hypot() function can be used to compute the componentwise hypotenous for a pair of dense vectors:

blaze::StaticVector<double,3UL> a, b, c;

c = hypot( a, b );  // Computes the componentwise hypotenuous

clamp()

The clamp() function can be used to restrict all elements of a vector to a specific range:

blaze::DynamicVector<double> a, b

b = clamp( a, -1.0, 1.0 );  // Restrict all elements to the range [-1..1]

Note that in case of sparse vectors only the non-zero elements are taken into account!

pow()

The pow() function can be used to compute the exponential value of each element of a vector. If passed a vector and a numeric exponent, the function computes the exponential value of each element of the vector using the same exponent. If passed a second vector, the function computes the componentwise exponential value:

blaze::StaticVector<double,3UL> a, b, c;

c = pow( a, 1.2 );  // Computes the exponential value of each element
c = pow( a, b );    // Computes the componentwise exponential value

exp() / exp2() / exp10()

exp(), exp2() and exp10() compute the base e/2/10 exponential of each element of a vector, respectively:

blaze::DynamicVector<double> a, b;

b = exp( a );    // Computes the base e exponential of each element
b = exp2( a );   // Computes the base 2 exponential of each element
b = exp10( a );  // Computes the base 10 exponential of each element

Note that in case of sparse vectors only the non-zero elements are taken into account!

log() / log2() / log10() / log1p() / lgamma()

The log(), log2(), log10(), log1p() and lgamma() functions can be used to compute the natural, binary and common logarithm of each element of a vector:

blaze::StaticVector<double,3UL> a, b;

b = log( a );     // Computes the natural logarithm of each element
b = log2( a );    // Computes the binary logarithm of each element
b = log10( a );   // Computes the common logarithm of each element
b = log1p( a );   // Computes the natural logarithm of x+1 of each element
b = lgamma( a );  // Computes the natural logarithm of the absolute value of the gamma function

sin() / cos() / tan() / asin() / acos() / atan()

The following trigonometric functions are available for both dense and sparse vectors:

blaze::DynamicVector<double> a, b;

b = sin( a );  // Computes the sine of each element of the vector
b = cos( a );  // Computes the cosine of each element of the vector
b = tan( a );  // Computes the tangent of each element of the vector

b = asin( a );  // Computes the inverse sine of each element of the vector
b = acos( a );  // Computes the inverse cosine of each element of the vector
b = atan( a );  // Computes the inverse tangent of each element of the vector

Note that in case of sparse vectors only the non-zero elements are taken into account!

sinh() / cosh() / tanh() / asinh() / acosh() / atanh()

The following hyperbolic functions are available for both dense and sparse vectors:

blaze::DynamicVector<double> a, b;

b = sinh( a );  // Computes the hyperbolic sine of each element of the vector
b = cosh( a );  // Computes the hyperbolic cosine of each element of the vector
b = tanh( a );  // Computes the hyperbolic tangent of each element of the vector

b = asinh( a );  // Computes the inverse hyperbolic sine of each element of the vector
b = acosh( a );  // Computes the inverse hyperbolic cosine of each element of the vector
b = atanh( a );  // Computes the inverse hyperbolic tangent of each element of the vector

Note that in case of sparse vectors only the non-zero elements are taken into account!

atan2()

The multi-valued inverse tangent is available for a pair of dense vectors:

blaze::DynamicVector<double> a, b, c;

c = atan2( a, b );  // Computes the componentwise multi-valued inverse tangent

erf() / erfc()

The erf() and erfc() functions compute the (complementary) error function of each element of a vector:

blaze::StaticVector<double,3UL,rowVector> a, b;

b = erf( a );   // Computes the error function of each element
b = erfc( a );  // Computes the complementary error function of each element

Note that in case of sparse vectors only the non-zero elements are taken into account!

map() / forEach()

Via the map() functions it is possible to execute componentwise custom operations on vectors. The unary map() function can be used to apply a custom operation on each element of a dense or sparse vector. For instance, the following example demonstrates a custom square root computation via a lambda:

blaze::DynamicVector<double> a, b;

b = map( a, []( double d ) { return std::sqrt( d ); } );

The N-ary map() functions can be used to apply an operation componentwise to the elements of N dense vectors (where N <= 6). The following example demonstrates the merging of two column vectors of double precision values into a vector of double precision complex numbers:

blaze::DynamicVector<double> real{ 2.1, -4.2,  1.0,  0.6 };
blaze::DynamicVector<double> imag{ 0.3,  1.4,  2.9, -3.4 };

blaze::DynamicVector< complex<double> > cplx;

// Creating the vector
//    ( ( 2.1,  0.3) )
//    ( (-4.2,  1.4) )
//    ( ( 1.0,  2.9) )
//    ( ( 0.6, -3.4) )
cplx = map( real, imag, []( double r, double i ){ return complex<double>( r, i ); } );

Applying the map() function to a column vector and a row vector results in the outer map of the two vectors. The following example demonstrates the outer sum of a column vector and a row vector:

blaze::DynamicVector<int,columnVector> v1{ 2, 5, -1 };
blaze::DynamicVector<int,rowVector> v2{ -1, 3, -2, 4 };

// Results in the matrix
//
//       (  1  5  0  6 )
//   A = (  4  8  3  9 )
//       ( -2  2 -3  3 )
//
blaze::StaticMatrix<int,3UL,4UL> M1 = map( v1, v2, []( int a, int b ){ return a + b; } );

Although the computation can be parallelized it is not vectorized and thus cannot perform at peak performance. However, it is also possible to create vectorized custom operations. See Custom Operations for a detailed overview of the possibilities of custom operations.

Please note that unary custom operations on vectors have been introduced in Blaze 3.0 in form of the forEach() function. With the introduction of binary custom functions, the forEach() function has been renamed to map(). The forEach() function can still be used, but the function might be deprecated in future releases of Blaze.

select()

The select() function performs a componentwise, conditional selection of elements. Given the three dense vectors cond, a, and b, in case an element in the cond vector evaluates to true, the according element of a is selected, in case the cond element evaluates to false, the according element of b is selected. The following example demonstrates the use of the select() function:

blaze::DynamicVector<bool> cond{ true, false, true false };
blaze::DynamicVector<int> a{ 1, -1, 1, -1 };
blaze::DynamicVector<int> b{ -2, 2, -2, 2 };
blaze::DynamicVector<int> c;
// ... Resizing and initialization

c = select( cond, a, b );  // Results in ( 1, 2, 1, 2 )

Reduction Operations

reduce()

The reduce() function performs a total reduction of the elements of the given dense vector or the non-zero elements of the given sparse vector. The following examples demonstrate the total reduction of a dense and sparse vector:

blaze::DynamicVector<double> a;
// ... Resizing and initialization

const double totalsum1 = reduce( a, blaze::Add() );
const double totalsum2 = reduce( a, []( double a, double b ){ return a + b; } );

blaze::CompressedVector<double> a;
// ... Resizing and initialization

const double totalmin1 = reduce( a, blaze::Min() );
const double totalmin2 = reduce( a, []( double a, double b ){ return blaze::min( a, b ); } );

As demonstrated in the examples it is possible to pass any binary callable as custom reduction operation. However, for instance in the case of lambdas the vectorization of the reduction operation is compiler dependent and might not perform at peak performance. However, it is also possible to create vectorized custom operations. See Custom Operations for a detailed overview of the possibilities of custom operations.

Please note that the evaluation order of the reduce() function is unspecified. Thus the behavior is non-deterministic if the given reduction operation is not associative or not commutative. Also, the operation is undefined if the given reduction operation modifies the values.

sum()

The sum() function reduces the elements of the given dense vector or the non-zero elements of the given sparse vector by means of addition:

blaze::DynamicVector<int> a{ 1, 2, 3, 4 };

const int totalsum = sum( a );  // Results in 10

blaze::CompressedVector<int> a{ 1, 2, 3, 4 };

const int totalsum = sum( a );  // Results in 10

Please note that the evaluation order of the sum() function is unspecified.

prod()

The prod() function reduces the elements of the given dense vector or the non-zero elements of the given sparse vector by means of multiplication:

blaze::DynamicVector<int> a{ 1, 2, 3, 4 };

const int totalprod = prod( a );  // Results in 24

blaze::CompressedVector<int> a{ 1, 2, 3, 4 };

const int totalprod = prod( a );  // Results in 24

min()

The unary min() function returns the smallest element of the given dense vector or the smallest non-zero element of the given sparse vector. It can only be used for element types that support the smaller-than relationship. In case the given vector currently has a size of 0, the returned value is the default value (e.g. 0 in case of fundamental data types).

blaze::DynamicVector<int> a{ 1, -2, 3, 0 };

const int totalmin = min( a );  // Results in -2

blaze::CompressedVector<int> a{ 1, 0, 3, 0 };

const int totalmin = min( a );  // Results in 1

Please note that in case the sparse vector is not completely filled, the implicit zero elements are NOT taken into account. In the previous example the compressed vector has only 2 non-zero elements. However, the minimum of the vector is 1.

max()

The unary max() function returns the largest element of the given dense vector or the largest non-zero element of the given sparse vector. It can only be used for element types that support the smaller-than relationship. In case the given vector currently has a size of 0, the returned value is the default value (e.g. 0 in case of fundamental data types).

blaze::DynamicVector<int> a{ 1, -2, 3, 0 };

const int totalmax = max( a );  // Results in 3

blaze::CompressedVector<int> a{ -1, 0, -3, 0 };

const int totalmin = max( a );  // Results in -1

Please note that in case the sparse vector is not completely filled, the implicit zero elements are NOT taken into account. In the previous example the compressed vector has only 2 non-zero elements. However, the maximum of the vector is -1.

argmin()

The argmin() function returns the index of the first smallest element of the given dense vector. This function can only be used for element types that support the smaller-than relationship. In case the given vector currently has a size of 0, the returned index is 0.

blaze::DynamicVector<int> a{ 1, -2, 3, 0 };
const size_t minindex = argmin( a );  // Results in 1

argmax()

The argmax() function returns the index of the first largest element of the given dense vector. This function can only be used for element types that support the smaller-than relationship. In case the given vector currently has a size of 0, the returned index is 0.

blaze::DynamicVector<int> a{ 1, -2, 3, 0 };
const size_t maxindex = argmax( a );  // Results in 2

Norms

norm()

The norm() function computes the L2 norm of the given dense or sparse vector:

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

const double norm1 = norm( a );
const double norm2 = norm( b );

sqrNorm()

The sqrNorm() function computes the squared L2 norm of the given dense or sparse vector:

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

const double norm1 = sqrNorm( a );
const double norm2 = sqrNorm( b );

l1Norm()

The l1Norm() function computes the squared L1 norm of the given dense or sparse vector:

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

const double norm1 = l1Norm( a );
const double norm2 = l1Norm( b );

l2Norm()

The l2Norm() function computes the squared L2 norm of the given dense or sparse vector:

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

const double norm1 = l2Norm( a );
const double norm2 = l2Norm( b );

l3Norm()

The l3Norm() function computes the squared L3 norm of the given dense or sparse vector:

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

const double norm1 = l3Norm( a );
const double norm2 = l3Norm( b );

l4Norm()

The l4Norm() function computes the squared L4 norm of the given dense or sparse vector:

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

const double norm1 = l4Norm( a );
const double norm2 = l4Norm( b );

lpNorm()

The lpNorm() function computes the general Lp norm of the given dense or sparse vector, where the norm is specified by either a compile time or a runtime argument:

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

const double norm1 = lpNorm<2>( a );    // Compile time argument
const double norm2 = lpNorm( b, 2.3 );  // Runtime argument

linfNorm() / maxNorm()

The linfNorm() and maxNorm() functions compute the infinity/maximum norm of the given dense or sparse vector:

blaze::DynamicVector<double> a;
blaze::CompressedVector<double> b;
// ... Resizing and initialization

const double norm1 = linfNorm( a );
const double norm2 = maxNorm( b );

Scalar Expansion

By means of the uniform() function it is possible to expand a scalar value into a dense, uniform vector. By default, the resulting uniform vector is a column vector, but it is possible to specify the transpose flag explicitly:

using blaze::columnVector;

int scalar = 5;

blaze::DynamicVector<int,columnVector> v;
// ... Resizing and initialization

// Expansion of 'scalar' to a 3-dimensional uniform column vector
//
//    ( 5 )
//    ( 5 )
//    ( 5 )
//
v = uniform( 3UL, scalar );
v = uniform<columnVector>( 3UL, scalar );

Vector Expansion

Via the expand() function it is possible to convert a dense or sparse vector into a matrix. A column vector is expanded into a column-major matrix, a row vector is expanded into a row-major matrix. As demonstrated by the following examples, expand() can be used with both runtime and compile time parameters:

blaze::DynamicVector<int,columnVector> a{ 1, 2, 3 };
blaze::CompressedVector<int,rowVector> b{ 1, 0, 3, 0, 5 };

// Expand the dense column vector ( 1 2 3 ) into a dense 3x5 column-major matrix
//
//   ( 1 1 1 1 1 )
//   ( 2 2 2 2 2 )
//   ( 3 3 3 3 3 )
//
expand( a, 5 );  // Runtime parameter
expand<5>( a );  // Compile time parameter

// Expand the sparse row vector ( 1 0 3 0 5 ) into a sparse 3x5 row-major matrix
//
//   ( 1 0 3 0 5 )
//   ( 1 0 3 0 5 )
//   ( 1 0 3 0 5 )
//
expand( b, 3 );  // Runtime parameter
expand<3>( b );  // Compile time parameter

Vector Repetition

Via the repeat() function it is possible to repeat a dense or sparse vector multiple times to represent a larger vector. Repeating a column vector results in a column vector, repeating a row vector results in a row vector. As demonstrated by the following examples, repeat() can be used with both runtime and compile time parameters:

using blaze::columnVector;
using blaze::rowVector;

blaze::DynamicVector<int,columnVector> a1{ 1, 0, -2 };
blaze::CompressedVector<int,rowVector> b1{ 0, -1, 7 };

blaze::DynamicVector<int,columnVector> a2;
blaze::CompressedVector<int,rowVector> b2;

// ... Resizing and initialization

// Repeating the dense column vector ( 1  0 -2 ) three times results in
//
//   ( 1  0 -2  1  0 -2  1  0 -2 )
//
a2 = repeat( a1, 3UL );
a2 = repeat<3UL>( a1 );

// Repeating the sparse row vector ( 0 -1  7 ) three times results in
//
//   ( 0 -1  7  0 -1  7  0 -1  7 )
//
b2 = repeat( b1, 3UL );
b2 = repeat<3UL>( b1 );

Statistic Operations

mean()

The (arithmetic) mean of a dense or sparse vector can be computed via the mean() function. In case of a sparse vector, both the non-zero and zero elements are taken into account. The following example demonstrates the computation of the mean of a dense vector:

blaze::DynamicVector<int> v{ 1, 4, 3, 6, 7 };

const double m = mean( v );  // Results in 4.2 (i.e. 21/5)

In case the size of the given vector is 0, a std::invalid_argument is thrown.

var()

The variance of a dense or sparse vector can be computed via the var() function. In case of a sparse vector, both the non-zero and zero elements are taken into account. The following example demonstrates the computation of the variance of a dense vector:

blaze::DynamicVector<int> v{ 1, 4, 3, 6, 7 };

const double v = var( v );  // Results in 5.7

In case the size of the given vector is smaller than 2, a std::invalid_argument is thrown.

stddev()

The standard deviation of a dense or sparse vector can be computed via the stddev() function. In case of a sparse vector, both the non-zero and zero elements are taken into account. The following example demonstrates the computation of the standard deviation of a dense vector:

blaze::DynamicVector<int> v{ 1, 4, 3, 6, 7 };

const double s = stddev( v );  // Results in 2.38747

In case the size of the given vector is smaller than 2, a std::invalid_argument is thrown.

Declaration Operations

declzero()

The declzero() operation can be used to explicitly declare any vector or vector expression as zero vector:

blaze::DynamicVector<double> a, b;
// ... Resizing and initialization

b = declzero( a );

Any vector or vector expression that has been declared as zero vector via declzero() will gain all the benefits of a zero vector, which range from reduced runtime checking to a considerable speed-up in computations:

using blaze::DynamicVector;

DynamicVector<double> a, b, c;
// ... Resizing and initialization

isZero( declzero( a ) );  // Will always return true without runtime effort

c = declzero( a ) + b;  // Declare the left operand of the vector addition as a
                        // zero vector, i.e. no addition needs to be performed

Warning: The declzero() operation has the semantics of a cast: The caller is completely responsible and the system trusts the given information. Declaring a non-zero vector or vector expression as zero vector via the declzero() operation leads to undefined behavior (which can be violated invariants or wrong computation results)!

Vector Generators

generate()

The generate() function returns a dense vector filled elementwise via the given custom operation. By default, the returned vector is a column vector, but this setting can be changed via the BLAZE_DEFAULT_TRANSPOSE_FLAG switch (see Default Vector Storage). Alternatively it is possible to specify the transpose flag explicitly.

The following example demonstrates the use of the generate() function:

using blaze::generate;
using blaze::columnVector;
using blaze::rowVector;

// Generates the homogeneous integer vector ( 2, 2, 2, 2, 2 )
blaze::DynamicVector<int,columnVector> a;
a = generate( 5UL, []( size_t index ){ return 2; } );

// Generates the linearly spaced float vector ( 2.1, 3.2, 4.3, 5.4 )
blaze::DynamicVector<float,columnVector> b;
b = generate( 4UL, []( size_t index ){ return 2.1F + 1.1F*index; } );

// Generates the logarithmically spaced double vector ( 1.0, 10.0, 100.0, 1000.0 )
blaze::DynamicVector<double,columnVector> c;
c = generate<columnVector>( 4UL, []( size_t index ){ return blaze::exp10( 1.0 + 1.0*index ); } );

// Generates the vector of integer vectors ( ( 1, 2 ), ( 2, 3 ), ( 3, 4 ), ( 4, 5 ) )
using VT = blaze::StaticVector<int,2UL>;
blaze::StaticVector<VT,4UL,rowVector> d;
d = generate<rowVector>( []( size_t index ) { return evaluate( VT{ 1, 2 } + index ); } );

linspace()

The linspace() function returns a dense vector filled with linearly spaced elements. By default, the returned vector is a column vector, but this setting can be changed via the BLAZE_DEFAULT_TRANSPOSE_FLAG switch (see Default Vector Storage). Alternatively it is possible to specify the transpose flag explicitly.

The following example demonstrates the use of the linspace() function:

using blaze::linspace;
using blaze::columnVector;
using blaze::rowVector;

// Generates the linearly spaced integer vector ( 2, 3, 4, 5, 6 )
blaze::DynamicVector<int,columnVector> a;
a = linspace( 5UL, 2, 6 );

// Generates the linearly spaced integer vector ( 6, 5, 4, 3, 2 )
blaze::DynamicVector<int,columnVector> b;
b = linspace<columnVector>( 5UL, 6, 2 );

// Generates the linearly spaced float vector ( 2.1, 3.2, 4.3, 5.4 )
blaze::DynamicVector<float,rowVector> c;
c = linspace<rowVector>( 4UL, 2.1F, 5.4F );

logspace()

The logspace() function returns a dense vector filled with logarithmically spaced elements. By default, the returned vector is a column vector, but this setting can be changed via the BLAZE_DEFAULT_TRANSPOSE_FLAG switch (see Default Vector Storage). Alternatively it is possible to specify the transpose flag explicitly.

The following example demonstrates the use of the logspace() function:

using blaze::logspace;
using blaze::columnVector;
using blaze::rowVector;

// Generates the logarithmically spaced double vector ( 1, 10, 100, 1000 )
blaze::DynamicVector<int,columnVector> a;
a = logspace( 4UL, 0, 3 );

// Generates the logarithmically spaced double vector ( 1000.0, 100.0, 10.0, 1.0 )
blaze::DynamicVector<double,rowVector> b;
b = logspace<rowVector>( 4UL, 3.0, 0.0 );

uniform()

The uniform() function creates a uniform vector of the given size. By default, the resulting uniform vector is a column vector, but this setting can be changed via the BLAZE_DEFAULT_TRANSPOSE_FLAG switch (see Default Vector Storage). Alternatively it is possible to specify the transpose flag explicitly.

The following example demonstrates the use of the uniform() function:

using blaze::uniform;
using blaze::columnVector;
using blaze::rowVector;

// Creates the uniform column vector ( 1, 1, 1, 1, 1 )
auto u1 = uniform( 5UL, 1 );

// Creates the uniform column vector ( 1.2, 1.2, 1.2 )
auto u2 = uniform<columnVector>( 3UL, 1.2 );

// Creates the uniform row vector ( 5U, 5U, 5U, 5U )
auto u3 = uniform<rowVector>( 4UL, 5U );

zero()

The zero() function creates a zero vector of the given element type and size. By default, the resulting zero vector is a column vector, but this setting can be changed via the BLAZE_DEFAULT_TRANSPOSE_FLAG switch (see Default Vector Storage). Alternatively it is possible to specify the transpose flag explicitly.

The following example demonstrates the use of the zero() function:

using blaze::zero;
using blaze::columnVector;
using blaze::rowVector;

// Creates the zero column vector ( 0, 0, 0, 0, 0 )
auto z1 = zero<int>( 5UL );

// Creates the zero column vector ( 0.0, 0.0, 0.0 )
auto z2 = zero<double,columnVector>( 3UL );

// Creates the zero row vector ( 0U, 0U, 0U, 0U )
auto z3 = zero<unsigned int,rowVector>( 4UL );

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