TooN 2.1
TooN Documentation


The TooN library is a set of C++ 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?

Design philosophy of TooN

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 cvs use:

cvs -z3 co TooN

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

The code will work as-is, and comes with a default configuration, which should work on any system.

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.

On non-unix systems, e.g. Windows and embedded systems, you may wish to configure the library manually. See Manual configuration.

Getting started

To begin, just in include the right file:

        #include <TooN/TooN.h>

Everything lives in the TooN 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 involving TOON_TYPEOF

If you get errors compiling code that uses TooN, look for the macro TOON_TYPEOF in the messages. Most likely the file internal/config.hh is clobbered. Open it and remove all the defines present there.

Also see Manual configuration for more details on configuring TooN, and Functions using LAPACK, if you want to use LAPACK and BLAS. Define the macro in internal/config.hh.

How do I create a vector?

Vectors can be statically sized or dynamically sized.

            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

See also Can I have a precision other than double?.

How do I create a matrix?

Matrices can be statically sized or dynamically sized.

            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.

See also Can I have a precision other than double?.

How do I write a function taking a vector?

To write a function taking a local copy of a vector:

            template<int Size> void func(Vector<Size> v);

To write a function taking any type of vector by reference:

        template<int Size, typename Precision, typename Base> void func(const Vector<Size, Precision, Base>& v);

See also Can I have a precision other than double?, How do I write generic code? and Why don't functions work in place?

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

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

See also How do I write generic code?

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:

             double my_data[3];

to hold the data. A slice object uses:

             double* my_data;

When a Vector is made const, C++ inserts const in to those types. The const it inserts it top level, so these become (respectively):

             const double my_data[3];
             double * const my_data;

Now the types behave very differently. In the first case my_data[0] is immutable. In the second case, my_data is immutable, but my_data[0] is mutable.

Therefore a slice const Vector behaves like an immutable pointer to mutable data. TooN attempts to make const objects behave as much like pointers to immutable data as possible.

The semantics that TooN tries to enforce can be bypassed with sufficient steps:

            //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;
                //Now v is mutated

See also How do I write a function taking a vector?

What elementary operations are supported?

Assignments are performed using =. See also Why does assigning mismatched dynamic vectors fail?.

These operators apply to vectors or matrices and scalars. The operator is applied to every element with the scalar.

        *=, /=, *, / 

Vector and vectors or matrices and matrices:

        +, -, +=, -= 

Dot product:

        Vector * Vector

Matrix multiply:

        Matrix * Matrix

Matrix multiplying a column vector:

        Matrix * Vector

Row vector multiplying a matrix:

        Vector * Matrix

3x3 Vector cross product:

        Vector<3> ^ Vector<3> 

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:


Accessing elements:

        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

Turning vectors in to matrices:

        Vector.as_row() //vector as a 1xN matrix
        Vector.as_col() //vector as a Nx1 matrix

Slicing with a start position and size:

        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

Slicing diagonals:

        Matrix.diagonal_slice();                               //Get the leading diagonal as a vector.
        Vector.as_diagonal();                                  //Represent a Vector as a DiagonalMatrix

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

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

See also What are slices?

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):

            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

Vectors can be filled with makeVector:

            Vector<> v = makeVector(2,3,4,5,6);

Matrices can be initialized to the identity matrix:

            Matrix<2> m = Idendity;
            Matrix<> m2 = Identity(3);

note that you need to specify the size in the dynamic case.

Matrices can be filled from data in row-major order:

            Matrix<3> m = Data(1, 2, 3, 
                               4, 5, 6, 
                               7, 8, 9);

A less general, but visually more pleasing syntax can also be used:

            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;

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

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:

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

It is supported the same way on Matrix and slices.

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, so it fails. See also How do I resize a dynamic vector/matrix?

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.

How do I resize a dynamic vector/matrix?

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

         Vector<Resizable> v;
         v = makeVector(1, 2, 3);

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

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 std::vector. 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 std::vector, and accessing it as a TooN object when appropriate. See I have a pointer to a bunch of data. How do I turn it in to a vector/matrix without copying?. 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.

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 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:

Additionally, individual checks can be disabled with the following macros:

Errors are manifested to a call to std::abort().

TooN does not initialize data in a Vector or Matrix. For debugging purposes the following macros can be defined:

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:

        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

Slices are usually strange types. See How do I write a function taking a vector?

See also

Can I have a precision other than double?


        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

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 Matrix<double> by a Vector<float> is a Vector<double>.

How do I return a slice from a function?

If you are using C++11, returning slices is now easy:

        auto sliceof(Vector<4>& v)->decltype (v.slice<1,2>())
            return v.slice<1,2>();

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

They can be slightly tricky to use:

        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>();

How do I invert a matrix / solve linear equations?

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

    Matrix<3> A;

    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);

Similarly for the other decomposition objects

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

Which decomposisions are there?

For general size matrices (not necessarily square) there are: LU , SVD , QR, LAPACK's QR and gauss_jordan()

For square symmetric matrices there are: SymEigen and Cholesky

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

What other stuff is there:

Look at the modules .

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

See here .

Does TooN support automatic differentiation?

TooN has buildin support for FADBAD++. Just do:

        #include <functions/fadbad.h>

Then create matrices and vectors of FADBAD types. See functions/fadbad.h 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 .

Why don't functions work in place?

Consider the function:

        void func(Vector<3>& v);

It can accept a Vector<3> by reference, and operate on it in place. A Vector<3> 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:

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

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 .ref() method to escape from this restriction, by returning a reference as a non-temporary. You would then have to write:

        Vector<4> v;

to get func to accept the slice.

You may also wish to consider writing functions that do not modify structures in place. The unit function of TooN computes a unit vector given an input vector. In the following context, the code:

        //There is some Vector, which may be a slice, etc called v;
        v = unit(v);

produces exactly the same compiler output as the hypothetical Normalize(v) 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.

Can I have a column major matrix?


        Matrix<3, 3, double, ColMajor> m;          //3x3 Column major matrix

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:

    double d[]={1,2,3,4};
    Vector<4,double,Reference> v1(d);
    Vector<Dynamic,double,Reference> v2(d,4);

Or, a functional form can be used:

    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

To crate a matrix use

    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

See also wrapVector() and wrapMatrix().

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:

    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;

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:

    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());


For issues relating to constness, see and

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 decltype if a C++11 compiler is present and no overriding configuration has been set. See Typeof for more information.

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)

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

Configuration is controlled by internal/config.hh. If this file is empty then the default configuration will be used and TooN will work. There are several options.


TooN needs a mechanism to determine the type of the result of an expression. One of the following macros can be defined to control the behaviour:

Under Win32, the builtin typeof needs to be used. Comment out all the TOON_TYPEOF_ defines to use it.

If no configuration is present and C++11 is detected, then decltype will be used.

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:

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