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

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

- Vectors, matrices and etc
- Matrix decompositions
- Function optimization
- Parameterized matrices (eg transformations)
- linear equations
- Functions (eg automatic differentiation and numerical derivatives)

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.

- 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.

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

.

- Getting the code and installing
- Getting started
- Comilation errors on Win32
- How do I create a vector?
- Can I have a vector with .x, .y, .z members (and so on)?
- sCreateMatrix
- sFunctionVector
- sGenericCode
- sConst
- sElemOps
- sInitialize
- sScalars
- Are there any examples?
- sSTL
- sResize
- sDebug
- sSlices
- sFuncSlices
- sPrecision
- sAutomaticDifferentiation
- sSolveLinear
- sOtherStuff
- sHandyFuncs
- sNoInplace
- sColMajor
- sWrap
- How do I interface to other libraries?
- sCpp11
- How is it implemented

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.

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.

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!

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 See also \ref sPrecision.

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>

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 See also \ref sPrecision. \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 See also \ref sPrecision, \ref sGenericCode and \ref sNoInplace 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. See also \ref sGenericCode \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 See also \ref sFunctionVector \subsection sElemOps What elementary operations are supported? Assignments are performed using <code>=</code>. See also \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 See also \ref sSlices \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, so it fails. See also \ref sResize \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 See also \sFuncSlices \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: @link TooN::LU LU @endlink, @link TooN::SVD SVD @endlink, @link TooN::QR QR@endlink, @link TooN::QR_Lapack LAPACK's QR@endlink and gauss_jordan() For square symmetric matrices there are: @link TooN::SymEigen SymEigen @endlink and @link TooN::Cholesky Cholesky @endlink 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: Look at the @link modules modules @endlink. \subsection sHandyFuncs What handy functions are there (normalize, identity, fill, etc...)? See @link gLinAlg here @endlink. \subsection sAutomaticDifferentiation Does TooN support automatic differentiation? TooN has buildin support for <a href="http://www.fadbad.com/fadbad.html">FADBAD++</a>. Just do: @code #include <functions/fadbad.h> @endcode 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 \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 See also wrapVector() and wrapMatrix(). \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 has been set. See \ref stypeof for more information.

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

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.

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).

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);

}

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 ...

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.

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.