TooN  3.0 About: TooN is a C++ numerics library which is designed to operate efficiently on large numbers of small matrices.  Fossies Dox: TooN-3.0.tar.gz  ("inofficial" and yet experimental doxygen-generated source code documentation)
TooN Documentation

# Introduction

The TooN library is a set of C++14 header files which provide basic numerics facilities:

It provides classes for statically- (known at compile time) and dynamically- (unknown at compile time) sized vectors and matrices and it can delegate advanced functions (like large SVD or multiplication of large matrices) to LAPACK and BLAS (this means you will need libblas and liblapack).

The library makes substantial internal use of templates to achieve run-time speed efficiency whilst retaining a clear programming syntax.

Why use this library?

• Because it supports statically sized vectors and matrices very efficiently.
• Because it provides extensive type safety for statically sized vectors and matrices (you can't attempt to multiply a 3x4 matrix and a 2-vector).
• Because it supports transposition, subscripting and slicing of matrices (to obtain a vector) very efficiently.
• Because it interfaces well to other libraries.
• Because it exploits LAPACK and BLAS (for which optimised versions exist on many platforms).
• Because it is fast, but not at the expense of numerical stability.

# Design philosophy of TooN

• TooN is designed to represent mathematics as closely as possible.
• TooN is a linear algebra library.
• TooN is designed as a linear algebra library and not a generic container and array mathematics library.

- Vectors are not matrices.

• The Vector and Matrix objects are distinct. Vectors and matrices are closely related, but distinct objects which makes things like outer versus inner product clearer, removes ambiguity and special cases and generally makes the code shorter.

TooN generally doesn't allow things which don't make much sense.

• Why would you want to multiply or add Zeros?

A vector is always a Vector and a matrix is always a Matrix

• Both concrete and generic functions take variations on the Vector and Matrix class, no matter where the data comes from. You will never see anything like a BaseVector.

# How to use TooN

This section is arranged as a FAQ. Most answers include code fragments. Assume using namespace TooN;.

## Getting the code and installing

To get the code from git use:

git clone git://github.com/edrosten/TooN.git

The home page for the library with a version of this documentation is at:

http://edwardrosten.com/cvd/toon.html

The code will work as-is and requires no configuration and so should should work on any system. On unix, some more obscure options (such as interfacing with CLAPACK) require configuration. See Manual configuration. On non unix platforms, some use of LAPACK requires configuration.

On a unix system, ./configure && make install  will install TooN to the correct place. Note there is no code to be compiled, but the configure script performs some basic checks. The unit tests can be built and run using make.

## Getting started

To begin, just in include the right file:

@code
#include <TooN/TooN.h>
@endcode

Everything lives in the <code>TooN</code> namespace.

Then, make sure the directory containing TooN is in your compiler's
search path. If you use any decompositions, you will need to link
against LAPACK, BLAS and any required support libraries. On a modern
unix system, linking against LAPACK will do this automatically.


## Comilation errors on Win32

VisualStudio tends to lag behind GCC and CLANG in terms of support for the latest
standards. I (E. Rosten) don't develop on windows so TooN doesn't get regular
testing there. If you find a problem, let me know and ideally, submit a patch!


## How do I create a vector?

Vectors can be statically sized or dynamically sized.

@code
Vector<3> v1;    //Create a static sized vector of size 3
Vector<>  v2(4); //Create a dynamically sized vector of size 4
Vector<Dynamic>  v2(4); //Create a dynamically sized vector of size 4
@endcode



## Can I have a vector with .x, .y, .z members (and so on)?

Yes. You can define new fixed length vectors with named elements of any length. For example:

#include <TooN/TooN.h>
#include <TooN/named_elements.h>
TOON_DECLARE_NAMED_ELEMENT_VECTOR(CMYK, c, m, y, k);
int main()
{
CMYK<> v = TooN::makeVector(1, 2, 3, 4);
std::cout << v << std::endl;
std::cout << " c = " << v.c
<< " m = " << v.m
<< " y = " << v.y
<< " k = " << v.k << endl;
cout << v * TooN::makeVector(1, 2, 3, 4) << endl;
}

Note that the resulting class (CMYK, in this example) is a type of Vector, so it can be used in any place that generically accepts a Vector. See also, sFunctionVector and sGenericCode.

\subsection sCreateMatrix How do I create a matrix?

Matrices can be statically sized or dynamically sized.

@code
Matrix<3> m;              //A 3x3 matrix (statically sized)
Matrix<3,2>  m;           //A 3x2 matrix (statically sized)
Matrix<>  m(5,6);         //A 5x6 matrix (dynamically sized)
Matrix<3,Dynamic> m(3,6); //A 3x6 matrix with a dynamic number of columns and static number of rows.
Matrix<Dynamic,2> m(3,2); //A 2x3 matrix with a dynamic number of rows and static number of columns.
@endcode

\subsection sFunctionVector How do I write a function taking a vector?

To write a function taking a local copy of a vector:
@code
template<int Size> void func(Vector<Size> v);
@endcode

To write a function taking any type of vector by reference:
@code
template<int Size, typename Precision, typename Base> void func(const Vector<Size, Precision, Base>& v);
@endcode

Slices are strange types. If you want to write a function which
uniformly accepts <code>const</code> whole objects as well as slices,
you need to template on the precision.

Note that constness in C++ is tricky (see \ref sConst). If you write
the function to accept <code> Vector<3, double, B>& </code>, then you
will not be able to pass it slices from <code> const Vector</code>s.
If, however you write it to accept <code> Vector<3, const double, B>&
</code>, then the only way to pass in a <code>Vector<3></code> is to
use the <code>.as_slice()</code> method.

\subsection sConst What is wrong with constness?

In TooN, the behaviour of a Vector or Matrix is controlled by the third
template parameter. With one parameter, it owns the data, with another
parameter, it is a slice. A static sized object uses the variable:
@code
double my_data[3];
@endcode
to hold the data. A slice object uses:
@code
double* my_data;
@endcode
When a Vector is made <code>const</code>, C++ inserts <code>const</code> in
to those types.  The <code>const</code> it inserts is top level, so these
become (respectively):
@code
const double my_data[3];
double * const my_data;
@endcode
Now the types behave very differently. In the first case
<code>my_data[0]</code> is immutable. In the second case,
<code>my_data</code> is immutable, but
<code>my_data[0]</code> is mutable.

Therefore a slice <code>const Vector</code> behaves like an immutable
pointer to mutable data. TooN attempts to make <code>const</code>
objects behave as much like pointers to \e immutable data as possible.

The semantics that TooN tries to enforce can be bypassed with
sufficient steps:
@code
//Make v look immutable
template<class P, class B> void fake_immutable(const Vector<2, P, B>& v)
{
Vector<2, P, B> nonconst_v(v);
nonconst_v[0] = 0; //Effectively mutate v
}

void bar()
{
Vector<3> v;
...
fake_immutable(v.slice<0,2>());
//Now v is mutated
}

@endcode

\subsection sElemOps What elementary operations are supported?

\ref sNoResize.

These operators apply to vectors or matrices and scalars.
The operator is applied to every element with the scalar.
@code
=, /=, *, /
@endcode

Vector and vectors or matrices and matrices:
@code
+, -, +=, -=
@endcode

Dot product:
@code
Vector * Vector
@endcode

Matrix multiply:
@code
Matrix * Matrix
@endcode

Matrix multiplying a column vector:
@code
Matrix * Vector
@endcode

Row vector multiplying a matrix:
@code
Vector * Matrix
@endcode

3x3 Vector cross product:
@code
Vector<3> ^ Vector<3>
@endcode

All the functions listed below return slices. The slices
are simply references to the original data and can be used as lvalues.

Getting the transpose of a matrix:
@code
Matrix.T()
@endcode

Accessing elements:
@code
Vector[i]     //get element i
Matrix(i,j)   //get element i,j
Matrix[i]     //get row i as a vector
Matrix[i][j]  //get element i,j
@endcode

Turning vectors in to matrices:
@code
Vector.as_row() //vector as a 1xN matrix
Vector.as_col() //vector as a Nx1 matrix
@endcode

Slicing with a start position and size:

@code
Vector.slice<Start, Length>();                         //Static slice
Vector.slice(start, length);                           //Dynamic slice
Matrix.slice<RowStart, ColStart, NumRows, NumCols>();  //Static slice
Matrix.slice(rowstart, colstart, numrows, numcols);    //Dynamic slice
@endcode

Slicing diagonals:
@code
Matrix.diagonal_slice();                               //Get the leading diagonal as a vector.
Vector.as_diagonal();                                  //Represent a Vector as a DiagonalMatrix
@endcode

Like other features of TooN, mixed static/dynamic slicing is allowed.
For example:

@code
Vector.slice<Dynamic, 2>(3, 2);   //Slice starting at index 3, of length 2.
@endcode

\subsection sInitialize How I initialize a vector/matrix?

Vectors and matrices start off uninitialized (filled with random garbage).
They can be easily filled with zeros, or ones (see also TooN::Ones):
@code
Vector<3> v = Zeros;
Matrix<3> m = Zeros
Vector<>  v2 = Zeros(2); //Note in they dynamic case, the size must be specified
Matrix<>  m2 = Zeros(2,2); //Note in they dynamic case, the size must be specified
@endcode

Vectors can be filled with makeVector:
@code
Vector<> v = makeVector(2,3,4,5,6);
@endcode

Matrices can be initialized to the identity matrix:
@code
Matrix<2> m = Idendity;
Matrix<> m2 = Identity(3);
@endcode
Note that you need to specify the size in the dynamic case.

Matrices can be filled from data in row-major order:
@code
Matrix<3> m = Data(1, 2, 3,
4, 5, 6,
7, 8, 9);
@endcode

A less general, but visually more pleasing syntax can also be used:
@code
Vector<5> v;
Fill(v) = 1,2,3,4,5;

Matrix<3,3> m;
Fill(m) = 1, 2, 3,
4, 5, 6,
7, 8, 9;
@endcode
Note that underfilling is a run-time check, since it can not be detected
at compile time.

They can also be initialized with data from another source. See also \ref  sWrap.

\subsection sScalars How do I add a scalar to every element of a vector/matrix?

Addition to every element is not an elementary operation in the same way
as multiplication by a scalar. It is supported throught the ::Ones
object:

@code
Vector<3> a, b;
...
b = a + Ones*3;       // b_i = a_i + 3
a+= Ones * 3;         // a_i <- a_i + 3
@endcode

It is supported the same way on Matrix and slices.

\subsection sNoResize Why does assigning mismatched dynamic vectors fail?

Vectors are not generic containers, and dynamic vectors have been designed
to have the same semantics as static vectors where possible. Therefore
trying to assign a vector of length 2 to a vector of length 3 is an error,

\subsection sSTL How do I store Dynamic vectors in STL containers.

As C++ does not yet support move semantics, you can only safely store
static and resizable Vectors in STL containers.

\subsection sResize How do I resize a dynamic vector/matrix?

Do you really want to? If you do, then you have to declare it:

@code
Vector<Resizable> v;
v.resize(3);
v = makeVector(1, 2, 3);

v = makeVector(1, 2); //resize
v = Ones(5); //resize
v = Zeros; // no resize
@endcode

The policy behind the design of TooN is that it is a linear algebra
library, not a generic container library, so resizable Vectors are only
created on request. They provide fewer guarantees than other vectors, so
errors are likely to be more subtle and harder to track down.  One of the
main purposes is to be able to store Dynamic vectors of various sizes in
STL containers.

Assigning vectors of mismatched sizes will cause an automatic resize. Likewise
assigning from entities like Ones with a size specified will cause a resize.
Assigning from an entities like Ones with no size specified will not cause
a resize.

They can also be resized with an explicit call to .resize().
Resizing is efficient since it is implemented internally with
<code>std::vector</code>.  Note that upon resize, existing data elements
are retained but new data elements are uninitialized.

Currently, resizable matrices are unimplemented.  If you want a resizable
matrix, you may consider using a <code>std::vector</code>, and accessing it
as a TooN object when appropriate. See \ref sWrap. Also, the speed and
complexity of resizable matrices depends on the memory layout, so you may
wish to use column major matrices as opposed to the default row major
layout.

\subsection sDebug What debugging options are there?

By default, everything which is checked at compile time in the static case
is checked at run-time in the dynamic case (with some additions). Checks can
be disabled with various macros. Note that the optimizer will usually
remove run-time checks on static objects if the test passes.

Bounds are not checked by default. Bounds checking can be enabled by
defining the macro \c TOON_CHECK_BOUNDS. None of these macros change the
interface, so debugging code can be freely mixed with optimized code.

The debugging checks can be disabled by defining either of the following macros:
- \c TOON_NDEBUG
- \c NDEBUG

Additionally, individual checks can be disabled with the following macros:
- Static/Dynamic mismatch
- Statically determined functions accept and ignore dynamically specified
sizes. Nevertheless, it is an error if they do not match.
- Disable with \c TOON_NDEBUG_MISMATCH
- Slices
- Disable with \c TOON_NDEBUG_SLICE
- Size checks (for assignment)
- Disable with \c TOON_NDEBUG_SIZE
- overfilling using Fill
- Disable with \c TOON_NDEBUG_FILL
- underfilling using Fill (run-time check)
- Disable with \c TOON_NDEBUG_FILL

Errors are manifested to a call to <code>std::abort()</code>.

TooN does not initialize data in a Vector or Matrix.  For debugging purposes
the following macros can be defined:
- \c TOON_INITIALIZE_QNAN or \c TOON_INITIALIZE_NAN Sets every element of newly defined Vectors or
Matrixs to quiet NaN, if it exists, and 0 otherwise. Your code will not compile
if you have made a Vector or Matrix of a type which cannot be constructed
from a number.
- \c TOON_INITIALIZE_SNAN Sets every element of newly defined Vectors or
Matrixs to signalling NaN, if it exists, and 0 otherwise.
- \c TOON_INITIALIZE_VAL Sets every element of newly defined Vectors or
Matrixs to the expansion of this macro.
- \c TOON_INITIALIZE_RANDOM Fills up newly defined Vectors and Matrixs with
random bytes, to trigger non repeatable behaviour. The random number
generator is automatically seeded with a granularity of 1 second. Your
code will not compile if you have a Vector or Matrix of a non-POD type.

\subsection sSlices What are slices?

Slices are references to data belonging to another vector or matrix. Modifying
the data in a slice modifies the original object. Likewise, if the original
object changes, the change will be reflected in the slice. Slices can be
used as lvalues. For example:

@code
Matrix<3> m = Identity;

m.slice<0,0,2,2>() *= 3; //Multiply the top-left 2x2 submatrix of m by 3.

m[2] /=10; //Divide the third row of M by 10.

m.T()[2] +=2; //Add 2 to every element of the second column of M.

m[1].slice<1,2>() = makeVector(3,4); //Set m_1,1 to 3 and m_1,2 to 4

m[0][0]=6;
@endcode

Slices are usually strange types. See \ref sFunctionVector

\subsection sPrecision Can I have a precision other than double?

Yes!
@code
Vector<3, float> v;          //Static sized vector of floats
Vector<Dynamic, float> v(4); //Dynamic sized vector of floats
Vector<Dynamic, std::complex<double> > v(4); //Dynamic sized vector of complex numbers
@endcode

Likewise for matrix. By default, TooN supports all builtin types
and std::complex. Using custom types requires some work. If the
custom type understands +,-,*,/ with builtin types, then specialize
TooN::IsField on the types.

If the type only understands +,-,*,/ with itself, then specialize
TooN::Field on the type.

Note that this is required so that TooN can follow the C++ promotion
rules. The result of multiplying a <code>Matrix<double></code> by a
<code>Vector<float></code> is a <code>Vector<double></code>.

\subsection sFuncSlices How do I return a slice from a function?

If you are using C++11, returning slices is now easy:
@code
auto sliceof(Vector<4>& v)->decltype (v.slice<1,2>())
{
return v.slice<1,2>();
}
@endcode
end even easier in C++14:
@code
auto sliceof(Vector<4>& v)
{
return v.slice<1,2>();
}
@endcode

If not, some tricks are required.
Each vector has a <code>SliceBase</code> type indicating the type of a slice.

They can be slightly tricky to use:
@code
Vector<2, double, Vector<4>::SliceBase> sliceof(Vector<4>& v)
{
return v.slice<1,2>();
}

template<int S, class P, class B>
Vector<2, P, Vector<S, P, B>::SliceBase> sliceof(Vector<S, P, B>& v)
{
return v.template slice<1,2>();
}

template<int S, class P, class B>
const Vector<2, const P, typename Vector<S, P, B>::ConstSliceBase > foo(const Vector<S, P, B>& v)
{
return v.template slice<1,2>();
}

@endcode

\subsection sSolveLinear How do I invert a matrix / solve linear equations?

You use the decomposition objects (see \ref sDecompos "below"), for example to solve Ax=b:

@code
Matrix<3> A;
A[0]=makeVector(1,2,3);
A[1]=makeVector(3,2,1);
A[2]=makeVector(1,0,1);

Vector<3> b = makeVector (2,3,4);

// solve Ax=b using LU
LU<3> luA(A);
Vector<3> x1 = luA.backsub(b);

// solve Ax=b using SVD
SVD<3> svdA(A);
Vector<3> x2 = svdA.backsub(b);
@endcode

Similarly for the other \ref sDecompos "decomposition objects"

For 2x2 matrices, the TooN::inv function can be used.

\subsection sDecompos  Which decomposisions are there?

For general size matrices (not necessarily square) there are:

For square symmetric matrices there are:

If all you want to do is solve a single Ax=b then you may want gaussian_elimination()

\subsection sOtherStuff What other stuff is there:

\subsection sHandyFuncs What handy functions are there (normalize, identity, fill, etc...)?

\subsection sAutomaticDifferentiation Does TooN support automatic differentiation?

Just do:
@code
@endcode
for available functions and parameterisations.

TooN is type generic and so can work on any reasonable types including AD types
if a small amount of interfacing is performed.
See \sPrecision.

\subsection sNoInplace Why don't functions work in place?

Consider the function:
@code
void func(Vector<3>& v);
@endcode
It can accept a <code>Vector<3></code> by reference, and operate on it
in place. A <code>Vector<3></code> is a type which allocates memory on the
stack. A slice merely references memory, and is a subtly different type. To
write a function taking any kind of vector (including slices) you can write:

@code
template<class Base> void func(Vector<3, double, Base>& v);
@endcode

A slice is a
temporary object, and according to the rules of C++, you can't pass a
temporary to a function as a non-const reference. TooN provides the
<code>.ref()</code> method to escape from this restriction, by returning a
reference as a non-temporary. You would then have to write:
@code
Vector<4> v;
...
func(v.slice<0,3>().ref());
@endcode
to get func to accept the slice.

You may also wish to consider writing functions that do not modify structures in
place. The \c unit function of TooN computes a unit vector given an input
vector. In the following context, the code:
@code
//There is some Vector, which may be a slice, etc called v;
v = unit(v);
@endcode
produces exactly the same compiler output as the hypothetical
<code>Normalize(v)</code> which operates in place (for static vectors). Consult the ChangeLog
entries dated Wed 25 Mar, 2009 20:18:16'' and Wed  1 Apr, 2009 16:48:45''
for further discussion.

\subsection sColMajor Can I have a column major matrix?

Yes!
@code
Matrix<3, 3, double, ColMajor> m;          //3x3 Column major matrix
@endcode

\subsection sWrap I have a pointer to a bunch of data. How do I turn it in to a vector/matrix without copying?

To create a vector use:
@code
double d[]={1,2,3,4};
Vector<4,double,Reference> v1(d);
Vector<Dynamic,double,Reference> v2(d,4);
@endcode
Or, a functional form can be used:
@code
double d[]={1,2,3,4};

wrapVector<4>(d);         //Returns a Vector<4>
wrapVector<4,double>(d);  //Returns a Vector<4>

wrapVector(d,3);          //Return a Vector<Dynamic> of size 3
wrapVector<Double>(d,3);  //Return a Vector<Dynamic> of size 3
@endcode

To crate a matrix use
@code
double d[]={1,2,3,4,5,6};
Matrix<2,3,double,Reference::RowMajor> m1(d);
Matrix<2,3,double,Reference::ColMajor> m2(d);
Matrix<Dynamic, Dynamic, double, Reference::RowMajor> m3(d, 2, 3);
Matrix<Dynamic, 3, double, Reference::RowMajor> m4(d, 2, 3); // note two size arguments are required for semi-dynamic matrices
@endcode

\subsection sGenericCode How do I write generic code?

The constructors for TooN objects are very permissive in that they
accept run-time size arguments for statically sized objects, and then
discard the values, This allows you to easily write generic code which
works for both static and dynamic inputs.

Here is a function which mixes up a vector with a random matrix:
@code
template<int Size, class Precision, class Base> Vector<Size, Precision> mixup(const Vector<Size, Precision, Base>& v)
{
//Create a square matrix, of the same size as v. If v is of dynamic
//size, then Size == Dynamic, and so Matrix will also be dynamic. In
//this case, TooN will use the constructor arguments to select the
//matrix size. If Size is a real size, then TooN will simply ighore
//the constructor values.

Matrix<Size, Size, Precision> m(v.size(), v.size());

//Fill the matrix with random values that sum up to 1.
Precision sum=0;
for(int i=0; i < v.size(); i++)
for(int j=0; j < v.size(); j++)
sum += (m[i][j] = rand());

m/= sum;

return m * v;
}
@endcode

Writing functions which safely accept multiple objects requires assertions
on the sizes since they may be either static or dynamic. TooN's built in
size check will fail at compile time if mismatched static sizes are given,
and at run-time if mismatched dynamic sizes are given:

@code
template<int S1, class B1, int S2, class B2> void func_of_2_vectors(const Vector<S1, double, B1>& v1, const Vector<S2, double, B2>& v2)
{
//Ensure that vectors are the same size
SizeMismatch<S1, S2>::test(v1.num_rows(), v2.num_rows());

}
@endcode

For issues relating to constness, see \sFunctionVector and \sConst

\subsection sCpp11 What about C++ 11 support?

TooN compiles cleanly under C++ 11, but does not require it. It can also
make use of some C++11 features where present. Internally, it will make use
of \c decltype if a C++11 compiler is present and no overriding configuration


## Are there any examples?

Create two vectors and work out their inner (dot), outer and cross products

// Initialise the vectors
Vector<3> a = makeVector(3,5,0);
Vector<3> b = makeVector(4,1,3);
// Now work out the products
double dot = a*b; // Dot product
Matrix<3,3> outer = a.as_col() * b.as_row(); // Outer product
Vector<3> cross = a ^ b; // Cross product
cout << "a:" << endl << a << endl;
cout << "b:" << endl << b << endl;
cout << "Outer:" << endl << outer << endl;
cout << "Cross:" << endl << cross << endl;

Create a vector and a matrix and multiply the two together

// Initialise a vector
Vector<3> v = makeVector(1,2,3);
// Initialise a matrix
Matrix<2,3> M(d);
M[0] = makeVector(2,4,5);
M[1] = makeVector(6,8,9);
// Now perform calculations
Vector<2> v2 = M*v; // OK - answer is a static 2D vector
Vector<> v3 = M*v; // OK - vector is determined to be 2D at runtime
Vector<> v4 = v*M; // Compile error - dimensions of matrix and vector incompatible

## How is it implemented

### Static-sized vectors and matrices

One aspect that makes this library efficient is that when you declare a 3-vector, all you get are 3 doubles - there's no metadata. So sizeof(Vector<3>) is 24. This means that when you write Vector<3> v; the data for v is allocated on the stack and hence new/delete (malloc/free) overhead is avoided. However, for large vectors and matrices, this would be a Bad Thing since Vector<1000000> v; would result in an object of 8 megabytes being allocated on the stack and potentially overflowing it. TooN gets around that problem by having a cutoff at which statically sized vectors are allocated on the heap. This is completely transparent to the programmer, the objects' behaviour is unchanged and you still get the type safety offered by statically sized vectors and matrices. The cutoff size at which the library changes the representation is defined in TooN.h as the const int TooN::Internal::max_bytes_on_stack=1000;.

When you apply the subscript operator to a Matrix<3,3> and the function simply returns a vector which points to the the apropriate hunk of memory as a reference (i.e. it basically does no work apart from moving around a pointer). This avoids copying and also allows the resulting vector to be used as an l-value. Similarly the transpose operation applied to a matrix returns a matrix which referes to the same memory but with the opposite layout which also means the transpose can be used as an l-value so M1 = M2.T(); and M1.T() = M2; do exactly the same thing.

Warning: This also means that M = M.T(); does the wrong thing. However, since .T() essentially costs nothing, it should be very rare that you need to do this.

### Dynamic sized vectors and matrices

These are implemented in the obvious way using metadata with the rule that the object that allocated on the heap also deallocates. Other objects may reference the data (e.g. when you subscript a matrix and get a vector).

## Return value optimisation vs Lazy evaluation

When you write v1 = M * v2; a naive implementation will compute M * v2 and store the result in a temporary object. It will then copy this temporary object into v1. A method often advanced to avoid this is to have M * v2 simply return an special object O which contains references to M and v2. When the compiler then resolves v1 = O, the special object computes M*v2 directly into v1. This approach is often called lazy evaluation and the special objects lazy vectors or lazy matrices. Stroustrup (The C++ programming language Chapter 22) refers to them as composition closure objects or compositors.

The killer is this: What if v1 is just another name for v2? i.e. you write something like v = M * v;. In this case the semantics have been broken because the values of v are being overwritten as the computation progresses and then the remainder of the computation is using the new values. In this library v1 in the expression could equally well alias part of M, thus you can't even solve the problem by having a clever check for aliasing between v1 and v2. This aliasing problem means that the only time the compiler can assume it's safe to omit the temporary is when v1 is being constructed (and thus cannot alias anything else) i.e. Vector<3> v1 = M * v2;.

TooN provides this optimisation by providing the compiler with the opportunity to use a return value optimisation. It does this by making M * v2 call a special constructor for Vector<3> with M and v2 as arguments. Since nothing is happening between the construction of the temporary and the copy construction of v1 from the temporary (which is then destroyed), the compiler is permitted to optimise the construction of the return value directly into v1.

Because a naive implemenation of this strategy would result in the vector and matrix classes having a very large number of constructors, these classes are provided with template constructors that take a standard form. The code that does this, declared in the header of class Vector is:

template <class Op>
inline Vector(const Operator<Op>& op)
: Base::template VLayout<Size, Precision> (op)
{
op.eval(*this);
}

### How it all really works

This documentation is generated from a cleaned-up version of the interface, hiding the implementation that allows all of the magic to work. If you want to know more and can understand idioms like:

template<int, typename, int, typename> struct GenericVBase;
template<int, typename> struct VectorAlloc;
struct VBase {
template<int Size, class Precision>
struct VLayout : public GenericVBase<Size, Precision, 1, VectorAlloc<Size, Precision> > {
...
};
};
template <int Size, class Precision, class Base=VBase>
class Vector: public Base::template VLayout<Size, Precision> {
...
};

then take a look at the source code ...

# Manual configuration

## Functions using LAPACK

Some functions use internal implementations for small sizes and may switch over to LAPACK for larger sizes. In all cases, an equivalent method is used in terms of accuracy (eg Gaussian elimination versus LU decomposition). If the following macro is defined:

• TOON_USE_LAPACK then LAPACK will be used for large systems, where optional. The individual functions are:
• TooN::determinant is controlled by TOON_DETERMINANT_LAPACK
• If the macro is undefined as or defined as -1, then LAPACK will never be used. Otherwise it indicated which the size at which LAPACK should be used.

Note that these macros do not affect classes that are currently only wrappers around LAPACK.

### sConfigureCLAPAK

CLAPACK is an automated transliteration of LAPACK into C. The main advantage is that it's more easily portable as it doesn't require nearly as much in terms of the FORTRAN support libraries, though it's slower. It's useful for embedding the LAPACK codes into the program where LAPACK is not a bottleneck.

In normal FORTRAN, the C int maps to Integer (on 32 and 64 bit systems). In CLAPACK, long maps to Integer. Note that CLAPACK cannot be used out of the box with TooN as it requires libf2c which defines main().

To set the FORTRAN integer type use -DTOON_CLAPACK (to make it long int) or -DTOON_FORTRAN_INTEGER=long to set it to an arbitrary type.