Math Namespace Reference

Contains all definitions in the Math package. More...

Namespaces
	Quadrature
	Namespace for quadrature functions.

Classes
class	AddFunction
struct	AngleInterval
	A contiguous range of angles. More...
struct	AngleSet
	A set of AngleIntervals. More...
struct	BLASInterface
	An interface to BLAS. Activated with the preprocessor flag HAVE_BLAS=1. More...
struct	BlockPrinter
class	BlockTridiagonalMatrix
class	BlockVector
class	CholeskyDecomposition
	Performs the Cholesky decomposition. More...
struct	ClosedInterval
	A closed interval [a,b]. More...
struct	ClosedIntervalSet
class	Complex
	Complex number class (x + iy). More...
class	ComponentVectorFieldFunction
class	Compose_SF_VF_Function
	A scalar field function h(x) = f(g(x)) (g vector-valued) More...
class	Compose_VF_VF_Function
	A vector field function h(x) = f(g(x)) (f,g vector fields) More...
class	ComposeFunction
class	ComposeScalarFieldFunction
	A scalar field h(x) = f(g(x)) More...
class	CompositeInequalityConstraint
class	CompositeVectorFieldFunction
struct	Conditioner_SymmDiag
	A matrix conditioner that does a pre/postmultiply with a diagonal S. More...
class	DiagonalMatrixTemplate
	A templated diagonal matrix, represented by a vector of the entries on the diagonal. More...
class	DiffEqFunction
struct	EquationTerm
class	Gaussian
	Multivariate gaussian N(mu,K) of d dimensions. More...
struct	IndexAndPriority
class	IndexedVectorFieldFunction
	A vector field function f(x) = g(x[xindices])[findices]. More...
class	InequalityConstraint
	A vector field with constraints s.t. ci(x) >= 0. More...
struct	InequalityConstraintAdaptor
	A class that makes it easier to construct "plugin" classes to alter constraint behavior. More...
struct	Interval
	A generic interval {a,b}, where { and } can be either (,[ or ),]. More...
class	InverseFunction
struct	IterativeMethod
struct	JacobiPreconditioner
	Jacobi preconditioning (inverse of diagonal) More...
struct	LAPACKInterface
	An interface to LAPACK. Activated with the preprocessor flag HAVE_CLAPACK=1. More...
struct	LDLDecomposition
	Performs the LDL^t decompositoin of a symmetric matrix A. More...
class	LimitConstraint
class	LinearConstraint
	constraint represented as A*x >= b i.e. constraint value is A*x-b More...
class	LinearFunction
class	LinearScalarFieldFunction
class	LinearVectorFieldFunction
class	LUDecomposition
	Forms the LU decomposition A=PLU. More...
struct	MatrixEquation
class	MatrixEquationPrinter
class	MatrixIterator
	An iterator through MatrixTemplate elements. More...
struct	MatrixPrinter
class	MatrixTemplate
	A matrix over the field T. More...
class	MaximumScalarFieldFunction
	A scalar field max_i xi. More...
class	MinimumScalarFieldFunction
	A scalar field min_i xi. More...
class	MulFunction
class	NormAccumulator
struct	NormScalarFieldFunction
	A scalar field for the L-d norm, where d is passed into the constructor. More...
struct	NormSquaredScalarFieldFunction
	A scalar field ||x||^2. More...
class	NRQRDecomposition
	The QR decomposition as computed by the algorithm in Numerical Recipes in C. More...
struct	NullPreconditioner
	Identity precondtioner. More...
struct	OpenInterval
	An open interval (a,b) More...
struct	OpenIntervalSet
class	PiecewiseLinearFunction
class	PiecewiseLinearPath
struct	QNHessianUpdater
	Maintains the Cholesky decomposition of the hessian H over Quasi-Newton steps. More...
class	QRDecomposition
	Calculates the QR decomposition. More...
class	QuadraticScalarFieldFunction
class	Quaternion
	Complex quaternion class (w + ix + jy + kz). More...
class	RealFunction
	A function from R to R. More...
class	RealFunction2
class	RobustSVD
	Performs a pre/postconditioned singular value decomposition. More...
class	RowEchelon
	Compute reduced row-eschelon form for a matrix A. More...
class	ScalarFieldDirectionalFunction
	A function g(t) that returns f(x+tn) for a scalar field f(x) If ref is true, then x,n are set to refer to the x,n provided in the constructor, rather than to copy. More...
class	ScalarFieldFunction
	A function from R^n to R. More...
class	ScalarFieldProjectionFunction
class	ScaledOffsetConstraint
class	SliceVectorFieldFunction
	A vector field function f(x) = g(x0 | x0[xindices]=x) More...
class	SparseMatrixTemplate_CR
	Row-major, compressed-row sparse matrix. More...
class	SparseMatrixTemplate_RM
	Row-major sparse matrix. More...
struct	SparseVectorAccessor
class	SparseVectorCompressed
class	SparseVectorFunction
	A vector field function with a sparse jacobian. The Jacobian_Sparse and Jacobian_i_Sparse methods must be overridden. More...
class	SparseVectorTemplate
class	SVDecomposition
	Performs the singular value decomposition. More...
class	UniformScaledOffsetConstraint
class	Vector3FieldFunction
class	VectorFieldFunction
	A function from R^n to R^m. More...
class	VectorFieldProjectionFunction
	A function g(x) that returns the i'th component of vector field f(x) More...
class	VectorFunction
	A function from R to R^n, identical to RealFunction except a vector is returned. More...
class	VectorIterator
	An iterator through VectorTemplate elements. More...
struct	VectorPrinter
class	VectorTemplate
	A vector over the field T. More...

Typedefs
typedef DiagonalMatrixTemplate< float >	fDiagonalMatrix
typedef DiagonalMatrixTemplate< double >	dDiagonalMatrix
typedef DiagonalMatrixTemplate< Complex >	cDiagonalMatrix
typedef DiagonalMatrixTemplate< Real >	DiagonalMatrix
typedef VectorTemplate< Real >	Vector
typedef MatrixTemplate< Real >	Matrix
typedef double	Real
typedef class MatrixTemplate< float >	fMatrix
typedef class MatrixTemplate< double >	dMatrix
typedef class MatrixTemplate< Complex >	cMatrix
typedef Real(*	QuadratureFunction) (RealFunction &, Real, Real)
typedef SparseMatrixTemplate_RM< Real >	SparseMatrix
typedef SparseMatrixTemplate_RM< float >	fSparseMatrix_RM
typedef SparseMatrixTemplate_RM< double >	dSparseMatrix_RM
typedef SparseMatrixTemplate_RM< Complex >	cSparseMatrix_RM
typedef SparseMatrixTemplate_CR< float >	fSparseMatrix_CR
typedef SparseMatrixTemplate_CR< double >	dSparseMatrix_CR
typedef SparseMatrixTemplate_CR< Complex >	cSparseMatrix_CR
typedef SparseVectorTemplate< Real >	SparseVector
typedef SparseVectorTemplate< float >	fSparseVector
typedef SparseVectorTemplate< double >	dSparseVector
typedef SparseVectorTemplate< Complex >	cSparseVector
typedef class VectorTemplate< float >	fVector
typedef class VectorTemplate< double >	dVector
typedef class VectorTemplate< Complex >	cVector

Enumerations
enum	ConvergenceResult {   ConvergenceX, ConvergenceF, Divergence, LocalMinimum,   MaxItersReached, ConvergenceError }

Functions
int	AABBLineSearch (const Vector &x0, const Vector &dx, const Vector &bmin, const Vector &bmax, Real &t)
bool	ClipLine1D (Real q, Real p, Real &umin, Real &umax)
bool	AABBClipLine (const Vector &x0, const Vector &dx, const Vector &bmin, const Vector &bmax, Real &u0, Real &u1)
void	AABBClamp (Vector &x, const Vector &bmin, const Vector &bmax, Real d=Zero)
	Clamps x to be contained in the AABB (bmin,bmax)
Real	AABBMargin (const Vector &x, const Vector &bmin, const Vector &bmax, int &index)
Real	AABBMargin (const Vector &x, const Vector &bmin, const Vector &bmax)
bool	AABBContains (const Vector &x, const Vector &bmin, const Vector &bmax, Real d=Zero)
	Returns true if the AABB (bmin,bmax) contains x.
Real	AngleNormalize (Real a)
	Normalizes an angle to [0,2pi)
Real	AngleDiff (Real a, Real b)
	Returns the closest distance to get to a from b (range [-pi,pi])
Real	AngleCCWDiff (Real a, Real b)
	Returns the CCW distance needed to get to a from b.
Real	AngleInterp (Real a, Real b, Real u)
	Interpolates between rotations a and b (u from 0 to 1)
bool	IsValidAngle (Real a)
void	TransformCosSin_Cos (Real a, Real b, Real &c, Real &d)
void	TransformCosSin_Sin (Real a, Real b, Real &c, Real &d)
	Same as above, but returns c*sin(x+d)
bool	SolveCosSinEquation (Real a, Real b, Real c, Real &t1, Real &t2)
void	SolveCosSinGreater (Real a, Real b, Real c, AngleInterval &i)
	Same as above, but a cos(x) + b sin(x) >= c.
void	Acos_All (Real x, Real &a, Real &b)
	returns all solutions y to cos(y)=x in a,b
void	Asin_All (Real x, Real &a, Real &b)
	returns all solutions y to sin(y)=x in a,b
void	Atan_All (Real x, Real &a, Real &b)
	returns all solutions y to tan(y)=x in a,b
char	ASCIIShade (double x)
void	OutputASCIIShade (ostream &out, double x)
void	OutputASCIIShade (ostream &out, const fVector &x, float scale)
void	OutputASCIIShade (ostream &out, const dVector &x, double scale)
void	OutputASCIIShade (ostream &out, const fMatrix &A, float scale, int indent)
void	OutputASCIIShade (ostream &out, const dMatrix &A, double scale, int indent)
void	OutputASCIIShade (std::ostream &out, double x)
void	OutputASCIIShade (std::ostream &out, const fVector &x, float scale=0)
void	OutputASCIIShade (std::ostream &out, const fMatrix &A, float scale=0, int indent=0)
void	OutputASCIIShade (std::ostream &out, const dVector &x, double scale=0)
void	OutputASCIIShade (std::ostream &out, const dMatrix &A, double scale=0, int indent=0)
template<class T >
bool	UBackSubstitute (const MatrixTemplate< T > &a, const VectorTemplate< T > &b, VectorTemplate< T > &x)
template<class T >
bool	LBackSubstitute (const MatrixTemplate< T > &a, const VectorTemplate< T > &b, VectorTemplate< T > &x)
template<class T >
bool	LtBackSubstitute (const MatrixTemplate< T > &a, const VectorTemplate< T > &b, VectorTemplate< T > &x)
template<class T >
void	U1BackSubstitute (const MatrixTemplate< T > &a, const VectorTemplate< T > &b, VectorTemplate< T > &x)
template<class T >
void	L1BackSubstitute (const MatrixTemplate< T > &a, const VectorTemplate< T > &b, VectorTemplate< T > &x)
template<class T >
void	Lt1BackSubstitute (const MatrixTemplate< T > &a, const VectorTemplate< T > &b, VectorTemplate< T > &x)
template<class T >
bool	UBackSubstitute (const MatrixTemplate< T > &a, const MatrixTemplate< T > &b, MatrixTemplate< T > &x)
template<class T >
bool	LBackSubstitute (const MatrixTemplate< T > &a, const MatrixTemplate< T > &b, MatrixTemplate< T > &x)
template<class T >
bool	LtBackSubstitute (const MatrixTemplate< T > &a, const MatrixTemplate< T > &b, MatrixTemplate< T > &x)
template<class T >
void	U1BackSubstitute (const MatrixTemplate< T > &a, const MatrixTemplate< T > &b, MatrixTemplate< T > &x)
template<class T >
void	L1BackSubstitute (const MatrixTemplate< T > &a, const MatrixTemplate< T > &b, MatrixTemplate< T > &x)
template<class T >
void	Lt1BackSubstitute (const MatrixTemplate< T > &a, const MatrixTemplate< T > &b, MatrixTemplate< T > &x)
	DEFINEBACKSUBSTITUTE (float)
	DEFINEBACKSUBSTITUTE (double)
	DEFINEBACKSUBSTITUTE (Complex)
void	PrintBlockVector (const BlockVector &v, ostream &out, char delim, char bracket)
void	PrintBlockMatrix (const BlockTridiagonalMatrix &m, ostream &out, char delim, char bracket)
ostream &	operator<< (ostream &out, const BlockPrinter &bp)
std::ostream &	operator<< (std::ostream &out, const BlockPrinter &bp)
Real	ParabolicExtremum (Real a, Real b, Real c, Real fa, Real fb, Real fc)
void	BracketMin (Real &a, Real &b, Real &c, Real &fa, Real &fb, Real &fc, RealFunction &func)
	Returns a bracketing triplet given some a,b. More...
ConvergenceResult	ParabolicMinimization (Real ax, Real bx, Real cx, RealFunction &f, int &maxIters, Real tol, Real &xmin)
	Brent's algorithm for parabolic minimization. More...
ConvergenceResult	ParabolicMinimization (Real x, RealFunction &f, int &maxIters, Real tol, Real &xmin)
Real	ParabolicLineMinimization (ScalarFieldFunction &f, const Vector &x, const Vector &n, int maxIters, Real tol)
Real	ParabolicLineMinimization_i (ScalarFieldFunction &f, const Vector &x, int i, int maxIters, Real tol)
	Same as above, but minimizes f(x + t*ei).
std::ostream &	operator<< (std::ostream &out, const Complex &x)
std::istream &	operator>> (std::istream &in, Complex &x)
std::ostream &	operator<< (std::ostream &out, const Quaternion &q)
std::istream &	operator>> (std::istream &in, Quaternion &q)
Real	dot (const Complex &a, const Complex &b)
Complex	operator- (const Complex &a)
Complex	operator+ (const Complex &a, const Complex &b)
Complex	operator- (const Complex &a, const Complex &b)
Complex	operator* (const Complex &a, const Complex &b)
Complex	operator/ (const Complex &a, const Complex &b)
Complex	operator+ (const Complex &a, Real b)
Complex	operator- (const Complex &a, Real b)
Complex	operator* (const Complex &a, Real b)
Complex	operator/ (const Complex &a, Real b)
Real	Abs (const Complex &x)
Complex	Sqr (const Complex &x)
Complex	Sqrt (const Complex &x)
Complex	Exp (const Complex &x)
Complex	Log (const Complex &x)
Complex	Pow (const Complex &x, Real y)
Complex	Pow (const Complex &x, const Complex &y)
Complex	Sin (const Complex &z)
Complex	Cos (const Complex &z)
Complex	Tan (const Complex &z)
Complex	Sinh (const Complex &z)
Complex	Cosh (const Complex &z)
Complex	Tanh (const Complex &z)
Complex	Inv (const Complex &x)
bool	FuzzyEquals (const Complex &a, const Complex &b, Real eps=Epsilon)
bool	FuzzyZero (const Complex &a, Real eps=Epsilon)
Complex	PseudoInv (const Complex &x, Real eps=Epsilon)
bool	FuzzyEquals (const Complex &a, const Complex &b, const Complex &eps)
bool	FuzzyZero (const Complex &a, const Complex &eps)
Complex	PseudoInv (const Complex &x, const Complex &eps)
Real	dot (const Quaternion &a, const Quaternion &b)
Quaternion	operator- (const Quaternion &a)
Quaternion	operator+ (const Quaternion &a, const Quaternion &b)
Quaternion	operator- (const Quaternion &a, const Quaternion &b)
Quaternion	operator* (const Quaternion &a, const Quaternion &b)
Quaternion	operator/ (const Quaternion &a, const Quaternion &b)
Quaternion	operator+ (const Quaternion &a, Real b)
Quaternion	operator- (const Quaternion &a, Real b)
Quaternion	operator* (const Quaternion &a, Real b)
Quaternion	operator/ (const Quaternion &a, Real b)
template<class Matrix , class Preconditioner >
int	CG (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &M, int &max_iter, Real &tol)
Real	RungeKutta4_step (RealFunction2 *f, Real t0, Real h, Real w)
	Runge-Kutta 4 step for 1D ODEs. More...
Real	RungeKutta4 (RealFunction2 *f, Real a, Real b, Real alpha, int n)
	Runge-Kutta 4 method for 1D ODEs. More...
Real	RKF (RealFunction2 *f, Real a, Real b, Real alpha, Real tol, Real hmax, Real hmin)
	Runge-Kutta-Fehlberg method for 1D ODEs. More...
Real	AM2I (RealFunction2 *f, Real h, Real t0, Real t1, Real t2, Real w0, Real w1, Real w2)
	Adams-Moulton 2 step implicit method for 1D ODEs. More...
Real	AM2E (RealFunction2 *f, Real h, Real t0, Real t1, Real w0, Real w1)
	Adams-Bashforth 2 step explicit method for 1D ODEs. More...
Real	AM2_predictor_corrector_step (RealFunction2 *f, Real h, Real t0, Real t1, Real t2, Real w0, Real w1)
	Adams-Moulton predictor-corrector step of order 2 for 1D ODEs.
Real	AM2_predictor_corrector (RealFunction2 *f, Real a, Real b, Real alpha0, Real alpha1, int n)
	Adams-Moulton predictor-corrector of order 2 for 1D ODEs.
void	Euler_step (DiffEqFunction *f, Real t0, Real h, const Vector &w0, Vector &w1)
	Euler step for an ODE system. More...
void	RungeKutta4_step (DiffEqFunction *f, Real t0, Real h, const Vector &w0, Vector &w1)
	Runge-Kutta 4 step for an ODE system. More...
void	Euler (DiffEqFunction *f, Real a, Real b, const Vector &alpha, int n, Vector &wn)
	Solve an ODE system using Euler's method. More...
void	RungeKutta4 (DiffEqFunction *f, Real a, Real b, const Vector &alpha, int n, Vector &wn)
	Solve an ODE system using the Runge Kutta 4 method. More...
void	Div (Vector &x, Real h)
void	Div (Matrix &x, Real h)
Real	dfCenteredDifference (RealFunction &f, Real x, Real h)
void	dfCenteredDifference (VectorFunction &f, Real x, Real h, Vector &df)
Real	dfCenteredDifferenceAdaptive (RealFunction &f, Real x, Real h0, Real tol)
void	dfCenteredDifferenceAdaptive (VectorFunction &f, Real x, Real h0, Real tol, Vector &df)
Real	ddfCenteredDifference (RealFunction &f, Real x, Real h)
void	ddfCenteredDifference (VectorFunction &f, Real x, Real h, Vector &ddf)
void	GradientForwardDifference (ScalarFieldFunction &f, Vector &x, Real h, Vector &g)
	x is provided as a temporary, it's restored to its original value later
void	JacobianForwardDifference (VectorFieldFunction &f, Vector &x, Real h, Matrix &J)
void	HessianForwardDifference (ScalarFieldFunction &f, Vector &x, Real h, Matrix &H)
void	HessianForwardDifference_Grad (ScalarFieldFunction &f, Vector &x, Real h, Matrix &H)
void	GradientCenteredDifference (ScalarFieldFunction &f, Vector &x, Real h, Vector &g)
void	GradientCenteredDifferenceAdaptive (ScalarFieldFunction &f, Vector &x, Real h0, Real tol, Vector &g)
void	JacobianCenteredDifference (VectorFieldFunction &f, Vector &x, Real h, Matrix &J)
void	HessianCenteredDifference (ScalarFieldFunction &f, Vector &x, Real h, Matrix &H)
void	HessianCenteredDifference_Grad (ScalarFieldFunction &f, Vector &x, Real h, Matrix &H)
void	GradientForwardDifference (ScalarFieldFunction &f, Vector &x, const Vector &h, Vector &g)
	specified hi in each direction xi
void	JacobianForwardDifference (VectorFieldFunction &f, Vector &x, const Vector &h, Matrix &J)
void	HessianForwardDifference (ScalarFieldFunction &f, Vector &x, const Vector &h, Matrix &H)
void	HessianForwardDifference_Grad (ScalarFieldFunction &f, Vector &x, const Vector &h, Matrix &H)
void	GradientCenteredDifference (ScalarFieldFunction &f, Vector &x, const Vector &h, Vector &g)
void	JacobianCenteredDifference (VectorFieldFunction &f, Vector &x, const Vector &h, Matrix &J)
void	HessianCenteredDifference (ScalarFieldFunction &f, Vector &x, const Vector &h, Matrix &H)
void	HessianCenteredDifference_Grad (ScalarFieldFunction &f, Vector &x, const Vector &h, Matrix &H)
template<class T >
T *	array_create (int n)
template<class T >
void	array_delete (T *v)
template<class T >
void	array_equal (T *a, const T *b, int n)
template<class T >
T *	array_copy (T *a, int n)
template<class T >
void	array_fill (T *a, const T &c, int n)
template<class T >
void	array_zero (T *a, int n)
template<class T >
void	array_negate (T *x, const T *a, int n)
template<class T >
void	array_add (T *x, const T *a, const T *b, int n)
template<class T >
void	array_sub (T *x, const T *a, const T *b, int n)
template<class T >
void	array_mul (T *x, const T *a, T s, int n)
template<class T >
void	array_div (T *x, const T *a, T s, int n)
template<class T >
void	array_acc (T *x, const T *a, int n)
template<class T >
void	array_dec (T *x, const T *a, int n)
template<class T >
void	array_madd (T *x, const T *a, const T &s, int n)
template<class T >
void	array_axpby (T *z, T a, const T *x, T b, const T *y, int n)
template<class T >
void	array_mul (T *a, const T &s, int n)
template<class T >
void	array_div (T *a, const T &s, int n)
template<class T >
T	array_sum_product (const T *a, const T *b, int n)
template<class T >
T	array_dot (const T *a, const T *b, int n)
template<class T >
T	array_norm_squared (const T *a, int n)
template<class T >
void	gen_array_fill (T *x, int xstride, T c, int n)
template<class T >
void	gen_array_mul (T *x, int xstride, T c, int n)
template<class T >
void	gen_array_div (T *x, int xstride, T c, int n)
template<class T >
void	gen_array_negate (T *x, int xstride, const T *a, int astride, int n)
template<class T >
void	gen_array_equal (T *x, int xstride, const T *a, int astride, int n)
template<class T >
void	gen_array_swap (T *x, int xstride, T *y, int ystride, int n)
template<class T >
void	gen_array_mul (T *x, int xstride, const T *a, int astride, T c, int n)
template<class T >
void	gen_array_div (T *x, int xstride, const T *a, int astride, T c, int n)
template<class T >
void	gen_array_add (T *x, int xstride, const T *a, int astride, const T *b, int bstride, int n)
template<class T >
void	gen_array_sub (T *x, int xstride, const T *a, int astride, const T *b, int bstride, int n)
template<class T >
void	gen_array_acc (T *x, int xstride, const T *a, int astride, int n)
template<class T >
void	gen_array_dec (T *x, int xstride, const T *a, int astride, int n)
template<class T >
void	gen_array_madd (T *x, int xstride, const T *a, int astride, T s, int n)
template<class T >
void	gen_array_axpby (T *z, int zstride, T a, const T *x, int xstride, T b, const T *y, int ystride, int n)
template<class T >
T	gen_array_sum_product (const T *a, int astride, const T *b, int bstride, int n)
template<class T >
T	gen_array_dot (const T *a, int astride, const T *b, int bstride, int n)
template<class T >
T	gen_array_norm_squared (const T *a, int astride, int n)
template<class T >
T **	array2d_create (int m, int n)
template<class T >
void	array2d_delete (T **v)
template<class T >
void	array2d_fill (T **X, const T &c, int m, int n)
template<class T >
void	array2d_identity (T **X, int m, int n)
template<class T >
void	array2d_equal (T **X, T **const A, int m, int n)
template<class T >
T **	array2d_copy (T **A, int m, int n)
template<class T >
void	array2d_transpose (T **X, T **const A, int m, int n)
template<class T >
void	array2d_add (T **x, T **const a, T **const b, int m, int n)
template<class T >
void	array2d_sub (T **x, T **const a, T **const b, int m, int n)
template<class T >
void	array2d_mul (T **x, T **const a, T **const b, int m, int n, int p)
template<class T >
void	array2d_multiply_transposeA (T **x, T **const a, T **const b, int m, int n, int p)
template<class T >
void	array2d_multiply_transposeB (T **x, T **const a, T **const b, int m, int n, int p)
template<class T >
void	array2d_vector_multiply (T *x, T **const a, const T *b, int m, int n)
template<class T >
void	array2d_vector_multiply_transpose (T *x, T **const a, const T *b, int m, int n)
template<class T >
void	array2d_vector_madd (T *x, T **const a, const T *b, int m, int n)
template<class T >
void	array2d_vector_madd_transpose (T *x, T **const a, const T *b, int m, int n)
template<class T >
T	array2d_row_dot (T **const a, int i, const T *v, int m, int n)
template<class T >
T	array2d_col_dot (T **const a, int i, const T *v, int m, int n)
template<class T >
T *	gen_array2d_entry (T *X, int xis, int xjs, int i, int j)
template<class T >
T *	gen_array2d_row (T *X, int xis, int xjs, int i)
template<class T >
T *	gen_array2d_col (T *X, int xis, int xjs, int j)
template<class T >
void	gen_array2d_fill (T *X, int xis, int xjs, const T &c, int m, int n)
template<class T >
void	gen_array2d_identity (T *X, int xis, int xjs, int m, int n)
template<class T >
void	gen_array2d_equal (T *X, int xis, int xjs, const T *A, int ais, int ajs, int m, int n)
template<class T >
void	gen_array2d_transpose (T *X, int xis, int xjs, const T *A, int ais, int ajs, int m, int n)
template<class T >
void	gen_array2d_add (T *X, int xis, int xjs, const T *A, int ais, int ajs, const T *B, int bis, int bjs, int m, int n)
template<class T >
void	gen_array2d_sub (T *X, int xis, int xjs, const T *A, int ais, int ajs, const T *B, int bis, int bjs, int m, int n)
template<class T >
void	gen_array2d_mul (T *X, int xis, int xjs, const T *A, int ais, int ajs, const T &c, int m, int n)
template<class T >
void	gen_array2d_mul (T *X, int xis, int xjs, const T &c, int m, int n)
template<class T >
void	gen_array2d_div (T *X, int xis, int xjs, const T &c, int m, int n)
template<class T >
void	gen_array2d_acc (T *X, int xis, int xjs, const T *A, int ais, int ajs, int m, int n)
template<class T >
void	gen_array2d_dec (T *X, int xis, int xjs, const T *A, int ais, int ajs, int m, int n)
template<class T >
void	gen_array2d_madd (T *X, int xis, int xjs, const T *A, int ais, int ajs, const T &c, int m, int n)
template<class T >
void	gen_array2d_negate (T *X, int xis, int xjs, const T *A, int ais, int ajs, int m, int n)
template<class T >
void	gen_array2d_multiply (T *X, int xis, int xjs, const T *A, int ais, int ajs, const T *B, int bis, int bjs, int m, int n, int p)
template<class T >
void	gen_array2d_multiply_transposeA (T *X, int xis, int xjs, const T *A, int ais, int ajs, const T *B, int bis, int bjs, int m, int n, int p)
template<class T >
void	gen_array2d_multiply_transposeB (T *X, int xis, int xjs, const T *A, int ais, int ajs, const T *B, int bis, int bjs, int m, int n, int p)
template<class T >
void	gen_array2d_vector_multiply (T *x, int xs, const T *A, int ais, int ajs, const T *b, int bs, int m, int n)
template<class T >
void	gen_array2d_vector_multiply_transpose (T *x, int xs, const T *A, int ais, int ajs, const T *b, int bs, int m, int n)
template<class T >
void	gen_array2d_vector_madd (T *x, int xs, const T *A, int ais, int ajs, const T *b, int bs, int m, int n)
template<class T >
void	gen_array2d_vector_madd_transpose (T *x, int xs, const T *A, int ais, int ajs, const T *b, int bs, int m, int n)
template<class T >
void	GaussianCondition (const VectorTemplate< T > &mean, const MatrixTemplate< T > &cov, const VectorTemplate< T > &x, const std::vector< int > &elements, VectorTemplate< T > &newmean, MatrixTemplate< T > &newcov)
template<class T >
void	GaussianMarginalize (const VectorTemplate< T > &mean, const MatrixTemplate< T > &cov, const std::vector< int > &elements, VectorTemplate< T > &newmean, MatrixTemplate< T > &newcov)
	marginalizes the distribution N(mean,cov) over the given elements
template<class T >
void	GaussianTransform (const VectorTemplate< T > &mean, const MatrixTemplate< T > &cov, const MatrixTemplate< T > &A, const VectorTemplate< T > &b, VectorTemplate< T > &newmean, MatrixTemplate< T > &newcov)
	transforms the distribution N(mean,cov) with the linear transform Ax+b
template<class T >
void	GaussianTransformJoint (const VectorTemplate< T > &xmean, const MatrixTemplate< T > &xcov, const MatrixTemplate< T > &A, const VectorTemplate< T > &b, VectorTemplate< T > &jointmean, MatrixTemplate< T > &jointcov)
template void	GaussianCondition (const VectorTemplate< float > &mean, const MatrixTemplate< float > &cov, const VectorTemplate< float > &x, const std::vector< int > &elements, VectorTemplate< float > &newmean, MatrixTemplate< float > &newcov)
template void	GaussianMarginalize (const VectorTemplate< float > &mean, const MatrixTemplate< float > &cov, const std::vector< int > &elements, VectorTemplate< float > &newmean, MatrixTemplate< float > &newcov)
template void	GaussianTransform (const VectorTemplate< float > &mean, const MatrixTemplate< float > &cov, const MatrixTemplate< float > &A, const VectorTemplate< float > &b, VectorTemplate< float > &newmean, MatrixTemplate< float > &newcov)
template void	GaussianTransformJoint (const VectorTemplate< float > &mean, const MatrixTemplate< float > &cov, const MatrixTemplate< float > &A, const VectorTemplate< float > &b, VectorTemplate< float > &jointmean, MatrixTemplate< float > &jointcov)
template void	GaussianCondition (const VectorTemplate< double > &mean, const MatrixTemplate< double > &cov, const VectorTemplate< double > &x, const std::vector< int > &elements, VectorTemplate< double > &newmean, MatrixTemplate< double > &newcov)
template void	GaussianMarginalize (const VectorTemplate< double > &mean, const MatrixTemplate< double > &cov, const std::vector< int > &elements, VectorTemplate< double > &newmean, MatrixTemplate< double > &newcov)
template void	GaussianTransform (const VectorTemplate< double > &mean, const MatrixTemplate< double > &cov, const MatrixTemplate< double > &A, const VectorTemplate< double > &b, VectorTemplate< double > &newmean, MatrixTemplate< double > &newcov)
template void	GaussianTransformJoint (const VectorTemplate< double > &mean, const MatrixTemplate< double > &cov, const MatrixTemplate< double > &A, const VectorTemplate< double > &b, VectorTemplate< double > &jointmean, MatrixTemplate< double > &jointcov)
template<class T >
int	OrthonormalBasis (const VectorTemplate< T > *x, VectorTemplate< T > *basis, int n)
template<class T >
int	OrthogonalBasis (const VectorTemplate< T > *x, VectorTemplate< T > *basis, int n)
	Same as above, but does not normalize.
template<class T >
void	Orthogonalize (VectorTemplate< T > &x, const VectorTemplate< T > *basis, int n)
	orthogonalizes a vector w.r.t the orthogonal basis of n vectors
	DEFINEGRAMSCHMIDT (float)
	DEFINEGRAMSCHMIDT (double)
	DEFINEGRAMSCHMIDT (Complex)
template<class T >
T	HouseholderTransform (VectorTemplate< T > &v)
template<class T >
void	HouseholderPreMultiply (T tau, const VectorTemplate< T > &v, MatrixTemplate< T > &A)
template<class T >
void	HouseholderPostMultiply (T tau, const VectorTemplate< T > &v, MatrixTemplate< T > &A)
template<class T >
void	HouseholderApply (T tau, const VectorTemplate< T > &v, VectorTemplate< T > &w)
template<class T >
void	HouseholderHM1 (T tau, MatrixTemplate< T > &A)
	DEFINEHH (float)
	DEFINEHH (double)
template<class T >
void	SetElements (VectorTemplate< T > &A, const std::vector< int > &indices, T fill)
	Sets A[indices] = fill.
template<class T >
void	SetElements (VectorTemplate< T > &A, const std::vector< int > &indices, const VectorTemplate< T > &fill)
	Sets A[indices] = fill. More...
template<class T >
void	GetElements (const VectorTemplate< T > &A, const std::vector< int > &indices, VectorTemplate< T > &B)
	Sets B = A[indices].
template<class T >
void	CopyElements (VectorTemplate< T > &A, const std::vector< int > &aindices, const VectorTemplate< T > &B, const std::vector< int > &bindices)
	Sets A[aindices] = B[bindices].
template<class T >
void	RemoveElements (VectorTemplate< T > &A, const std::vector< int > &indices)
	Removes the indexed elements of A (shrinking A)
template<class T >
void	RemoveElements (SparseArray< T > &A, const std::vector< int > &indices)
	Removes the indexed elements of A (shrinking A)
template<class T >
void	AddElements (VectorTemplate< T > &A, const std::vector< int > &indices, T fill)
	The inverse of RemoveElements. More...
template<class T >
void	SetRows (MatrixTemplate< T > &A, const std::vector< int > &indices, T fill)
	Sets the indexed rows of A to fill.
template<class T >
void	SetElements (MatrixTemplate< T > &A, const std::vector< int > &rows, const std::vector< int > &cols, T fill)
	Sets A[r,c] = fill for all r in rows and all c in columns.
template<class T >
void	SetElements (MatrixTemplate< T > &A, const std::vector< int > &rows, const std::vector< int > &cols, const MatrixTemplate< T > &fill)
	Sets A[r,c] = fill for all r in rows and all c in columns. More...
template<class T >
void	GetElements (const MatrixTemplate< T > &A, const std::vector< int > &rows, const std::vector< int > &cols, MatrixTemplate< T > &B)
	Sets B[i,j] = A[rows[i],cols[j]] for all i and j.
template<class T >
void	CopyElements (MatrixTemplate< T > &A, const std::vector< int > &arows, const std::vector< int > &acols, const MatrixTemplate< T > &B, const std::vector< int > &brows, const std::vector< int > &bcols)
	Sets A[ar,ac] = B[br,bc] for all (ar,br) in arows x brows, (ac,bc) in acols x bcols.
template<class T >
void	RemoveElements (MatrixTemplate< T > &A, const std::vector< int > &rows, const std::vector< int > &cols)
	Removes the indexed rows and columns of A (shrinking A)
template<class T >
void	SetColumns (MatrixTemplate< T > &A, const std::vector< int > &indices, T fill)
	Sets the indexed columns of A to fill.
template<class T >
void	SetRows (MatrixTemplate< T > &A, const std::vector< int > &indices, const VectorTemplate< T > &fill)
	Sets the indexed rows of A to the vector fill.
template<class T >
void	SetColumns (MatrixTemplate< T > &A, const std::vector< int > &indices, const VectorTemplate< T > &fill)
	Sets the indexed columns of A to the vector fill.
template<class T >
void	SetRows (MatrixTemplate< T > &A, const std::vector< int > &indices, const MatrixTemplate< T > &fill)
	Sets the indexed rows of A to the columns of the matrix fill. More...
template<class T >
void	SetColumns (MatrixTemplate< T > &A, const std::vector< int > &indices, const MatrixTemplate< T > &fill)
	Sets the indexed columns of A to the columns of the matrix fill. More...
template<class T >
void	GetRows (const MatrixTemplate< T > &A, const std::vector< int > &indices, MatrixTemplate< T > &B)
	Copies the indexed rows in A into B. More...
template<class T >
void	GetColumns (const MatrixTemplate< T > &A, const std::vector< int > &indices, MatrixTemplate< T > &B)
	Copies the indexed columns in A into B. More...
template<class T >
void	RemoveRows (MatrixTemplate< T > &A, const std::vector< int > &indices)
	Removes the rows of A indexed by the sorted list indices.
template<class T >
void	RemoveColumns (MatrixTemplate< T > &A, const std::vector< int > &indices)
	Removes the columns of A indexed by the sorted indices.
template<class T >
void	RemoveRows (SparseMatrixTemplate_RM< T > &A, const std::vector< int > &indices)
	Removes the rows of A indexed by the sorted list indices.
template<class T >
void	RemoveColumns (SparseMatrixTemplate_RM< T > &A, const std::vector< int > &indices)
	Removes the columns of A indexed by the sorted indices.
int	IsNaN (double x)
	Returns nonzero if x is not-a-number (NaN)
int	IsFinite (double x)
	Returns nonzero unless x is infinite or a NaN.
int	IsInf (double x)
	Returns +1 if x is +inf, -1 if x is -inf, and 0 otherwise.
int	IsNaN (float x)
int	IsFinite (float x)
int	IsInf (float x)
template<class type >
void	interpolate (const type &a, const type &b, Real u, type &x)
template<class T >
void	interpolate (const VectorTemplate< T > &a, const VectorTemplate< T > &b, Real u, VectorTemplate< T > &x)
template<class T >
void	interpolate (const MatrixTemplate< T > &a, const MatrixTemplate< T > &b, Real u, MatrixTemplate< T > &x)
template<class T >
bool	LinearlyDependent_Robust (const VectorTemplate< T > &a, const VectorTemplate< T > &b, T &c, bool &cb, T eps=Epsilon)
	Robust determination of linear dependence. More...
template<class T >
bool	LinearlyDependent_Robust (const MatrixTemplate< T > &A, VectorTemplate< T > &c, T eps=Epsilon)
	A matrix version of the function above. More...
double	Abs (double x)
double	Sqr (double x)
double	Sqrt (double x)
double	Exp (double x)
double	Log (double x)
double	Pow (double x, double y)
double	Sin (double x)
double	Cos (double x)
double	Tan (double x)
double	Sinh (double x)
double	Cosh (double x)
double	Tanh (double x)
double	Asin (double x)
double	Acos (double x)
double	Atan (double x)
double	Atan2 (double y, double x)
double	Floor (double x)
double	Ceil (double x)
double	Mod (double x, double y)
float	Abs (float x)
float	Sqr (float x)
float	Sqrt (float x)
float	Exp (float x)
float	Log (float x)
float	Pow (float x, float y)
float	Sin (float x)
float	Cos (float x)
float	Tan (float x)
float	Sinh (float x)
float	Cosh (float x)
float	Tanh (float x)
float	Asin (float x)
float	Acos (float x)
float	Atan (float x)
float	Atan2 (float y, float x)
float	Floor (float x)
float	Ceil (float x)
float	Mod (float x, float y)
double	Inv (double x)
	1/x
double	Sign (double x)
	Returns {-1,0,1} if x is {<0,0,>0}.
double	Clamp (double x, double a, double b)
	Returns a if x < a, b if x > b, otherwise x.
double	Trunc (double x)
double	Frac (double x)
float	Inv (float x)
float	Sign (float x)
float	Clamp (float x, float a, float b)
float	Trunc (float x)
float	Frac (float x)
double	dot (double a, double b)
	Dot product of a scalar as a 1-d vector.
bool	FuzzyEquals (double a, double b, double eps=dEpsilon)
	Returns true if a and b are within +/- eps.
bool	FuzzyZero (double a, double eps=dEpsilon)
	Returns true if a is zero within +/- eps.
double	PseudoInv (double x, double eps=dZero)
	Returns 1/x if x is nonzero, otherwise 0.
float	dot (float a, float b)
bool	FuzzyEquals (float a, float b, float eps=fEpsilon)
bool	FuzzyZero (float a, float eps=fEpsilon)
float	PseudoInv (float x, float eps=fZero)
template<class T >
Real	Delta (T i, T j)
	Kronecker delta.
template<class T >
Real	Delta (T x)
double	DtoR (double f)
	Degree to radian conversion.
double	RtoD (double f)
	Radian to degree conversion.
float	DtoR (float f)
float	RtoD (float f)
template<class T >
void	PrintMatrix (const MatrixTemplate< T > &x, ostream &out, char delim, char bracket, int indent=0)
template<class T >
void	OutputPlusMinus (ostream &out, const MatrixTemplate< T > &x, int indent, T zeroTolerance=Epsilon)
ostream &	operator<< (ostream &out, const MatrixPrinter &mp)
std::ostream &	operator<< (std::ostream &out, const MatrixPrinter &mp)
template<class T >
ostream &	operator<< (ostream &out, const MatrixTemplate< T > &mat)
template<class T >
istream &	operator>> (istream &in, MatrixTemplate< T > &mat)
template ostream &	operator<< (ostream &out, const MatrixTemplate< float > &v)
template ostream &	operator<< (ostream &out, const MatrixTemplate< double > &v)
template ostream &	operator<< (ostream &out, const MatrixTemplate< Complex > &v)
template istream &	operator>> (istream &in, MatrixTemplate< float > &v)
template istream &	operator>> (istream &in, MatrixTemplate< double > &v)
template istream &	operator>> (istream &in, MatrixTemplate< Complex > &v)
template<class T >
bool	IsFinite (const MatrixTemplate< T > &A)
	returns true if all elements of A are finite
template<class T >
bool	HasNaN (const MatrixTemplate< T > &A)
	returns true if any element of A is NaN
template<class T >
int	HasInf (const MatrixTemplate< T > &A)
	returns nonzero if any element of A is infinite
template<class T >
std::ostream &	operator<< (std::ostream &, const MatrixTemplate< T > &)
template<class T >
std::istream &	operator>> (std::istream &, MatrixTemplate< T > &)
template<class T >
T	DotSelf (const T &a)
double	Pow (double a, float b)
float	Pow (float a, double b)
template<class T >
T	Norm (const V< T > &x, Real norm)
template<class T >
T	Norm_L1 (const V< T > &x)
template<class T >
T	Norm_L2 (const V< T > &x)
template<class T >
T	Norm_L2_Safe (const V< T > &x)
template<class T >
T	Norm_LInf (const V< T > &x)
template<class T >
T	Norm_Mahalanobis (const V< T > &x, const M< T > &A)
template<class T >
T	Norm_Weighted (const V< T > &x, Real norm, const V< T > &w)
template<class T >
T	Norm_WeightedL1 (const V< T > &x, const V< T > &w)
template<class T >
T	Norm_WeightedL2 (const V< T > &x, const V< T > &w)
template<class T >
T	Norm_WeightedLInf (const V< T > &x, const V< T > &w)
template<class T >
T	Distance (const V< T > &x, const V< T > &y, Real norm)
template<class T >
T	Distance_L1 (const V< T > &x, const V< T > &y)
template<class T >
T	Distance_L2 (const V< T > &x, const V< T > &y)
template<class T >
T	Distance_L2_Safe (const V< T > &x, const V< T > &y)
template<class T >
T	Distance_LInf (const V< T > &x, const V< T > &y)
template<class T >
T	Distance_Mahalanobis (const V< T > &x, const V< T > &y, const M< T > &A)
template<class T >
T	Distance_Weighted (const V< T > &x, const V< T > &y, Real norm, const V< T > &w)
template<class T >
T	Distance_WeightedL1 (const V< T > &x, const V< T > &y, const V< T > &w)
template<class T >
T	Distance_WeightedL2 (const V< T > &x, const V< T > &y, const V< T > &w)
template<class T >
T	Distance_WeightedLInf (const V< T > &x, const V< T > &y, const V< T > &w)
template<class T >
T	Norm_L1 (const M< T > &A)
template<class T >
T	Norm_LInf (const M< T > &A)
template<class T >
T	Norm_Frobenius (const M< T > &A)
template<class T >
T	Norm_Frobenius_Safe (const M< T > &A)
template<class T >
T	Distance_L1 (const M< T > &A, const M< T > &B)
template<class T >
T	Distance_LInf (const M< T > &A, const M< T > &B)
template<class T >
T	Distance_Frobenius (const M< T > &A, const M< T > &B)
template<class T >
T	Distance_Frobenius_Safe (const M< T > &A, const M< T > &B)
template<>
Complex	Norm_LInf (const V< Complex > &x)
template<>
Complex	Norm_WeightedLInf (const V< Complex > &x, const V< Complex > &w)
template<>
Complex	Distance_LInf (const V< Complex > &x, const V< Complex > &y)
template<>
Complex	Distance_WeightedLInf (const V< Complex > &x, const V< Complex > &y, const V< Complex > &w)
template<>
Complex	Distance_L2_Safe (const V< Complex > &x, const V< Complex > &y)
template<>
Complex	Norm_L1 (const M< Complex > &A)
template<>
Complex	Norm_LInf (const M< Complex > &A)
template<>
Complex	Distance_L1 (const M< Complex > &A, const M< Complex > &B)
template<>
Complex	Distance_LInf (const M< Complex > &A, const M< Complex > &B)
template<>
Complex	Distance_Frobenius_Safe (const M< Complex > &A, const M< Complex > &B)
	DEFINEMETRIC (float)
	DEFINEMETRIC (double)
template Complex	Norm (const V< Complex > &x, Real norm)
template Complex	Norm_L1 (const V< Complex > &x)
template Complex	Norm_L2 (const V< Complex > &x)
template Complex	Norm_L2_Safe (const V< Complex > &x)
template Complex	Norm_Mahalanobis (const V< Complex > &x, const M< Complex > &k)
template Complex	Norm_Weighted (const V< Complex > &x, Real norm, const V< Complex > &w)
template Complex	Norm_WeightedL1 (const V< Complex > &x, const V< Complex > &w)
template Complex	Norm_WeightedL2 (const V< Complex > &x, const V< Complex > &w)
template Complex	Distance (const V< Complex > &x, const V< Complex > &y, Real norm)
template Complex	Distance_L1 (const V< Complex > &x, const V< Complex > &y)
template Complex	Distance_L2 (const V< Complex > &x, const V< Complex > &y)
template Complex	Distance_Mahalanobis (const V< Complex > &x, const V< Complex > &y, const M< Complex > &A)
template Complex	Distance_Weighted (const V< Complex > &x, const V< Complex > &y, Real norm, const V< Complex > &w)
template Complex	Distance_WeightedL1 (const V< Complex > &x, const V< Complex > &y, const V< Complex > &w)
template Complex	Distance_WeightedL2 (const V< Complex > &x, const V< Complex > &y, const V< Complex > &w)
template Complex	Norm_Frobenius (const M< Complex > &A)
template Complex	Norm_Frobenius_Safe (const M< Complex > &A)
template Complex	Distance_Frobenius (const M< Complex > &A, const M< Complex > &B)
template<class T >
T	Norm (const VectorTemplate< T > &x, Real norm)
template<class T >
T	Norm_L1 (const VectorTemplate< T > &x)
template<class T >
T	Norm_L2 (const VectorTemplate< T > &x)
template<class T >
T	Norm_L2_Safe (const VectorTemplate< T > &x)
	Same as above, but robust to over/underflow.
template<class T >
T	Norm_LInf (const VectorTemplate< T > &x)
template<class T >
T	Norm_Mahalanobis (const VectorTemplate< T > &x, const MatrixTemplate< T > &A)
template<class T >
T	Norm_Weighted (const VectorTemplate< T > &x, Real norm, const VectorTemplate< T > &w)
template<class T >
T	Norm_WeightedL1 (const VectorTemplate< T > &x, const VectorTemplate< T > &w)
template<class T >
T	Norm_WeightedL2 (const VectorTemplate< T > &x, const VectorTemplate< T > &w)
template<class T >
T	Norm_WeightedLInf (const VectorTemplate< T > &x, const VectorTemplate< T > &w)
template<class T >
T	Distance (const VectorTemplate< T > &x, const VectorTemplate< T > &y, Real norm)
template<class T >
T	Distance_L1 (const VectorTemplate< T > &x, const VectorTemplate< T > &y)
template<class T >
T	Distance_L2 (const VectorTemplate< T > &x, const VectorTemplate< T > &y)
template<class T >
T	Distance_L2_Safe (const VectorTemplate< T > &x, const VectorTemplate< T > &y)
	Same as above, but robust to over/underflow.
template<class T >
T	Distance_LInf (const VectorTemplate< T > &x, const VectorTemplate< T > &y)
template<class T >
T	Distance_Mahalanobis (const VectorTemplate< T > &x, const VectorTemplate< T > &y, const MatrixTemplate< T > &A)
template<class T >
T	Distance_Weighted (const VectorTemplate< T > &x, const VectorTemplate< T > &y, Real norm, const VectorTemplate< T > &w)
template<class T >
T	Distance_WeightedL1 (const VectorTemplate< T > &x, const VectorTemplate< T > &y, const VectorTemplate< T > &w)
template<class T >
T	Distance_WeightedL2 (const VectorTemplate< T > &x, const VectorTemplate< T > &y, const VectorTemplate< T > &w)
template<class T >
T	Distance_WeightedLInf (const VectorTemplate< T > &x, const VectorTemplate< T > &y, const VectorTemplate< T > &w)
template<class T >
T	Norm_L1 (const MatrixTemplate< T > &A)
template<class T >
T	Norm_LInf (const MatrixTemplate< T > &A)
template<class T >
T	Norm_Frobenius (const MatrixTemplate< T > &A)
template<class T >
T	Norm_Frobenius_Safe (const MatrixTemplate< T > &A)
	Same as above, but robust to over/underflow.
template<class T >
T	Distance_L1 (const MatrixTemplate< T > &A, const MatrixTemplate< T > &B)
template<class T >
T	Distance_LInf (const MatrixTemplate< T > &A, const MatrixTemplate< T > &B)
template<class T >
T	Distance_Frobenius (const MatrixTemplate< T > &A, const MatrixTemplate< T > &B)
template<class T >
T	Distance_Frobenius_Safe (const MatrixTemplate< T > &A, const MatrixTemplate< T > &B)
	Same as above, but robust to over/underflow.
int	quadratic (double a, double b, double c, double &x1, double &x2)
int	cubic (double a, double b, double c, double d, double x[3])
double	pythag (double a, double b)
	Returns c where a^2+b^2=c^2.
double	pythag_leg (double a, double c)
	Returns b where a^2+b^2=c^2.
double	Sinc (double x)
	Computes the sinc function sin(x)/x (stable for small x)
double	Sinc_Dx (double x)
	Computes the derivative of the sinc function.
double	dFactorial (unsigned int n)
	Computes the factorial of n, with double precision.
double	dLogFactorial (unsigned int n)
	Computes the log of the factorial of n.
double	dChoose (unsigned int n, unsigned int k)
	Computes n choose k in double precision.
double	dLogChoose (unsigned int n, unsigned int k)
	Computes the log of n choose k.
double	TaylorCoeff (double x, unsigned int n)
	Computes the taylor series coefficient x^n/n!
double	Gamma (double x)
	Computes the gamma function.
double	LogGamma (double x)
	Computes log gamma(x)
double	GammaInv (double x)
	Computes 1 / gamma(x)
double	Beta (double a, double b)
	Computes the beta function B(a,b)
double	LogBeta (double a, double b)
	Computes log B(a,b)
double	NormalizedIncompleteBeta (double a, double b, double x)
	Computes the normalized incomplete beta function B(a,b,x)
double	Erf (double x)
	Computes the error function.
double	ErfComplimentary (double x)
	Computes the complimentary error function?
int	quadratic (float a, float b, float c, float &x1, float &x2)
int	cubic (float a, float b, float c, float d, float x[3])
float	pythag (float a, float b)
float	pythag_leg (float a, float c)
float	Sinc (float x)
float	Sinc_Dx (float x)
float	fFactorial (unsigned int n)
float	fLogFactorial (unsigned int n)
float	fChoose (unsigned int n, unsigned int k)
float	fLogChoose (unsigned int n, unsigned int k)
float	TaylorCoeff (float x, unsigned int n)
float	Gamma (float x)
float	LogGamma (float x)
float	GammaInv (float x)
float	Beta (float a, float b)
float	LogBeta (float a, float b)
float	NormalizedIncompleteBeta (float a, float b, float x)
float	Erf (float x)
float	ErfComplimentary (float x)
int	FuzzySign (double x, double eps=dEpsilon)
	Returns -1 if x<eps, 0 if |x|<=eps, 1 if x>eps.
bool	OpposingSigns_pos (double a, double b)
bool	OpposingSigns_neg (double a, double b)
	Same as above, but treats 0 as a negative number.
int	FuzzySign (float x, float eps=fEpsilon)
bool	OpposingSigns_pos (float a, float b)
bool	OpposingSigns_neg (float a, float b)
template<class T >
T	IntegerPower (const T x, int i)
	Calculates x^i where i is an integer (in O(log i) time)
unsigned int	Factorial (unsigned int n)
	Returns n! in O(n) time.
unsigned int	FactorialTruncated (unsigned int n, unsigned int k)
	Returns n!/(n-k)! in O(k) time.
unsigned int	Choose (unsigned int n, unsigned int k)
	Returns `n choose k' = n!/k!(n-k)! in O(k) time.
Real	rFactorial (unsigned int n)
Real	LogFactorial (unsigned int n)
Real	rChoose (unsigned int n, unsigned int k)
Real	LogChoose (unsigned int n, unsigned int k)
void	Srand (unsigned long seed)
long int	RandInt ()
	Generates a random int.
long int	RandInt (long int n)
Real	Rand ()
	Generates a random Real uniformly in [0,1].
Real	Rand (Real a, Real b)
	Generates a random Real uniformly in [a,b].
Real	RandGaussian ()
Real	RandGaussian (Real mean, Real stddev)
	Generates a random Real in a Gaussian distribution.
bool	RandBool ()
	Generates a random boolean that is true with probability 0.5.
bool	RandBool (Real p)
	Generates a random boolean that is true with probability p.
std::ostream &	operator<< (std::ostream &out, ConvergenceResult r)
ConvergenceResult	Root_Bisection (RealFunction &f, Real &a, Real &b, Real &x, Real tol)
	Uses bisection to bracket f's root. More...
ConvergenceResult	Root_Secant (RealFunction &f, Real x0, Real &x1, int &maxIters, Real tolx, Real tolf)
	Uses the secant method to search for f's root. More...
ConvergenceResult	Root_SecantBracket (RealFunction &f, Real &a, Real &b, Real &x, int &iters, Real tolx, Real tolf)
ConvergenceResult	Root_Newton (RealFunction &f, Real &x, int &maxIters, Real tolx, Real tolf)
	Performs Newton's method, which uses the derivative of f to search for it's root. More...
ConvergenceResult	Root_NewtonBracket (RealFunction &f, Real a, Real b, Real &x, int &maxIters, Real tolx, Real tolf)
	Same as above, but the interval [a,b] must be known to contain a root.
ConvergenceResult	Root_Newton (ScalarFieldFunction &f, const Vector &x0, Vector &x, int &maxIters, Real tolx, Real tolf)
	Same as Root_Newton(), but takes a scalar field function as input.
ConvergenceResult	Root_Newton (VectorFieldFunction &f, const Vector &x0, Vector &x, int &maxIters, Real tolx, Real tolf)
	Same as Root_Newton(), but takes a vector field function as input.
template<class T >
int	RowEchelonDecompose (MatrixTemplate< T > &A, MatrixTemplate< T > &B, Real zeroTol)
template<class T >
void	ReduceRowEchelon (MatrixTemplate< T > &A, MatrixTemplate< T > &B)
template<class T >
bool	IsRowEchelon (const MatrixTemplate< T > &A)
template<class T >
bool	IsReducedRowEchelon (const MatrixTemplate< T > &A)
template bool	IsRowEchelon< float > (const MatrixTemplate< float > &A)
template bool	IsRowEchelon< double > (const MatrixTemplate< double > &A)
template bool	IsReducedRowEchelon< float > (const MatrixTemplate< float > &A)
template bool	IsReducedRowEchelon< double > (const MatrixTemplate< double > &A)
int	WeightedSample (const vector< Real > &weights)
	Samples an integer with weighted probability. More...
int	WeightedSample (const vector< Real > &weights, Real totWeight)
	Same as above, but the sum of all weights, W, is provided.
int	CumulativeWeightedSample (const std::vector< Real > &partialSumWeights)
void	RandomAllocate (vector< int > &num, size_t total)
	allocates total samples within the buckets in num, roughly evenly More...
void	RandomAllocate (vector< int > &num, size_t total, const vector< Real > &weights)
	allocates total samples within the buckets in num More...
Real	Sample (const Interval &s)
	Uniformly samples the given intervals.
Real	Sample (const ClosedIntervalSet &s)
	Uniformly samples the interval set.
void	SampleCircle (Real r, Real &x, Real &y)
	Uniform distribution on boundary of a circle with radius r.
void	SampleDisk (Real r, Real &x, Real &y)
	Uniform distribution inside a circle with radius r.
void	SampleTriangle (Real &x, Real &y)
	Uniform distribution in the triangle whose vertices are (0,0),(1,0),(0,1)
void	SampleSphere (Real r, Real &x, Real &y, Real &z)
	Uniform distribution inside a sphere with radius r.
void	SampleBall (Real r, Real &x, Real &y, Real &z)
	Uniform distribution on boundary of a sphere with radius r.
void	SampleHyperSphere (Real r, std::vector< Real > &v)
	Uniformly samples from a hyper-sphere of dimension v.size()
void	SampleHyperBall (Real r, std::vector< Real > &v)
	Uniformly samples from a hyper-ball of dimension v.size()
int	WeightedSample (const std::vector< Real > &weights)
	Samples an integer with weighted probability. More...
int	WeightedSample (const std::vector< Real > &weights, Real totWeight)
	Same as above, but the sum of all weights, W, is provided.
void	RandomAllocate (std::vector< int > &num, size_t total)
	Allocates total samples within the buckets in num, roughly evenly. An O(n) algorithm where n is the number of buckets. More...
void	RandomAllocate (std::vector< int > &num, size_t total, const std::vector< Real > &weights)
	Same as above, but with weights. More...
bool	RFuzzyEquals (Real a, Real b, Real rtol)
bool	CheckError (Real a, Real b, Real atol, Real rtol)
bool	CheckError (const Vector &a, const Vector &b, Vector &err, Real atol, Real rtol)
bool	CheckError (const Matrix &a, const Matrix &b, Matrix &err, Real atol, Real rtol)
void	SelfTest ()
void	BasicSelfTest ()
void	TestVectorBasic ()
void	TestVectorOps ()
void	VectorSelfTest ()
void	MatrixSelfTest ()
void	DifferentiationSelfTest ()
void	QuadratureSelfTest ()
void	BlockVectorSelfTest ()
void	BlockMatrixSelfTest ()
void	BLASSelfTest ()
void	TestDifferentiation (VectorFieldFunction &f, Vector &x, Real h, Real eps)
void	DebugVector (const char *name, const Vector &v)
void	DebugMatrix (const char *name, const Matrix &v)
bool	TestDeriv (RealFunction *f, Real t, Real h, Real atol, Real rtol)
bool	TestDeriv (VectorFunction *f, Real t, Real h, Real atol, Real rtol)
bool	TestGradient (ScalarFieldFunction *f, Vector &x, Real h, Real atol, Real rtol)
bool	TestGradients (ScalarFieldFunction *f, Vector &x, Real h, Real atol, Real rtol)
	these versions test the methods Gradient_i, Jacobian_i, etc.
bool	TestDeriv2 (RealFunction *f, Real t, Real h, Real atol, Real rtol)
bool	TestDeriv2 (VectorFunction *f, Real t, Real h, Real atol, Real rtol)
bool	TestHessian (ScalarFieldFunction *f, Vector &x, Real h, Real atol, Real rtol)
bool	TestJacobian (VectorFieldFunction *f, Vector &x, Real h, Real atol, Real rtol)
bool	TestJacobians (VectorFieldFunction *f, Vector &x, Real h, Real atol, Real rtol)
void	LAPACKSelfTest ()
bool	TestHessians (VectorFieldFunction *f, Vector &x, Real h, Real atol, Real rtol)
template<class T >
std::ostream &	operator<< (std::ostream &out, const SparseMatrixTemplate_RM< T > &A)
template<class T >
std::istream &	operator>> (std::istream &in, SparseMatrixTemplate_RM< T > &A)
template ostream &	operator<< (ostream &out, const SparseMatrixTemplate_RM< float > &v)
template ostream &	operator<< (ostream &out, const SparseMatrixTemplate_RM< double > &v)
template ostream &	operator<< (ostream &out, const SparseMatrixTemplate_RM< Complex > &v)
template istream &	operator>> (istream &in, SparseMatrixTemplate_RM< float > &v)
template istream &	operator>> (istream &in, SparseMatrixTemplate_RM< double > &v)
template istream &	operator>> (istream &in, SparseMatrixTemplate_RM< Complex > &v)
template<class T >
bool	IsFinite (const SparseMatrixTemplate_RM< T > &x)
	returns true if all elements of x are finite
template<class T >
std::ostream &	operator<< (std::ostream &, const SparseMatrixTemplate_CR< T > &)
template<class T >
bool	IsFinite (const SparseVectorTemplate< T > &x)
	returns true if all elements of x are finite
template<class T >
bool	HasNaN (const SparseVectorTemplate< T > &x)
	returns true if any element of x is NaN
template<class T >
bool	HasInf (const SparseVectorTemplate< T > &x)
	returns nonzero if any element of x is infininte
template<class T >
void	Stack (const std::vector< VectorTemplate< T > > &vs, VectorTemplate< T > &out)
template<class T >
void	Stack (const VectorTemplate< T > &v1, const VectorTemplate< T > &v2, VectorTemplate< T > &out)
template<class T >
void	HStack (const std::vector< VectorTemplate< T > > &vs, MatrixTemplate< T > &out)
template<class T >
void	VStack (const std::vector< VectorTemplate< T > > &vs, MatrixTemplate< T > &out)
template<class T >
void	HStack (const MatrixTemplate< T > &v1, const MatrixTemplate< T > &v2, MatrixTemplate< T > &out)
template<class T >
void	VStack (const MatrixTemplate< T > &v1, const MatrixTemplate< T > &v2, MatrixTemplate< T > &out)
template<class T >
void	PrintVector (const VectorTemplate< T > &x, ostream &out, char delim, char bracket)
template<class T >
void	OutputPlusMinus (ostream &out, const VectorTemplate< T > &x, T zeroTolerance=Epsilon)
ostream &	operator<< (ostream &out, const VectorPrinter &vp)
std::ostream &	operator<< (std::ostream &out, const VectorPrinter &vp)
template<class T >
ostream &	operator<< (ostream &out, const VectorTemplate< T > &v)
template<class T >
istream &	operator>> (istream &in, VectorTemplate< T > &v)
template ostream &	operator<< (ostream &out, const VectorTemplate< float > &v)
template ostream &	operator<< (ostream &out, const VectorTemplate< double > &v)
template ostream &	operator<< (ostream &out, const VectorTemplate< Complex > &v)
template istream &	operator>> (istream &in, VectorTemplate< float > &v)
template istream &	operator>> (istream &in, VectorTemplate< double > &v)
template istream &	operator>> (istream &in, VectorTemplate< Complex > &v)
template<class T >
std::ostream &	operator<< (std::ostream &, const VectorTemplate< T > &)
template<class T >
std::istream &	operator>> (std::istream &, VectorTemplate< T > &)
template<class T >
bool	FuzzyEquals (const VectorTemplate< T > &a, const VectorTemplate< T > &b, T eps)
template<class T >
bool	FuzzyZero (const VectorTemplate< T > &a, T eps)
template<class T >
T	dot (const VectorTemplate< T > &a, const VectorTemplate< T > &b)
template<class T >
T	norm (const VectorTemplate< T > &a)
template<class T >
bool	IsFinite (const VectorTemplate< T > &x)
	returns true if all elements of x are finite
template<class T >
bool	HasNaN (const VectorTemplate< T > &x)
	returns true if any element of x is NaN
template<class T >
int	HasInf (const VectorTemplate< T > &x)
	returns nonzero if any element of x is infinite
template<class T >
VectorTemplate< T >	operator+ (const VectorTemplate< T > &a, const VectorTemplate< T > &b)
template<class T >
VectorTemplate< T >	operator- (const VectorTemplate< T > &a, const VectorTemplate< T > &b)
template<class T >
VectorTemplate< T >	operator* (const VectorTemplate< T > &a, T c)
template<class T >
VectorTemplate< T >	operator* (T c, const VectorTemplate< T > &a)
template<class T >
VectorTemplate< T >	operator/ (const VectorTemplate< T > &a, T c)
template<class T >
VectorTemplate< T >	operator/ (T c, const VectorTemplate< T > &a)

Variables
const char *	MatrixError_IncompatibleDimensions = "Matrices have incompatible dimensions, %d x %d != %d x %d"
const char *	MatrixError_ArgIncompatibleDimensions = "Matrix arguments have incompatible dimensions"
const char *	MatrixError_DestIncompatibleDimensions = "Matrix destination has incompatible dimensions"
const char *	MatrixError_SizeZero = "Matrix has size 0x0"
const char *	MatrixError_NotSquare = "Matrix is not square"
const char *	MatrixError_NotSymmetric = "Matrix is not square"
const char *	MatrixError_InvalidRow = "Matrix row index [%d] out of range"
const char *	MatrixError_InvalidCol = "Matrix column index [%d] out of range"
StandardRNG	rng

Detailed Description

Contains all definitions in the Math package.

Function Documentation

template<class T >

void Math::GaussianCondition	(	const VectorTemplate< T > &	mean,
		const MatrixTemplate< T > &	cov,
		const VectorTemplate< T > &	x,
		const std::vector< int > &	elements,
		VectorTemplate< T > &	newmean,
		MatrixTemplate< T > &	newcov
	)

sets the conditional distribution of N(mean,cov) given elements x(elements[i]) = x(i)

Referenced by GaussianTransformJoint(), Math::Gaussian< T >::generate(), and Statistics::LinearProcessHMM::Posterior().

template<class T >

void Math::GaussianTransformJoint	(	const VectorTemplate< T > &	mean,
		const MatrixTemplate< T > &	cov,
		const MatrixTemplate< T > &	A,
		const VectorTemplate< T > &	b,
		VectorTemplate< T > &	jointmean,
		MatrixTemplate< T > &	jointcov
	)

if y=Ax+b and x is distributed w.r.t. N(mean,cov), returns a gaussian over the vector [x,y]

References GaussianCondition(), GaussianMarginalize(), and GaussianTransform().

template<class T >

VectorTemplate<T> Math::operator+ ( const VectorTemplate< T > &  a, const VectorTemplate< T > &  b  )

inline

The following are potentially very inefficient, depending on the ability of your compiler to optimize the code to eliminate copy constructor calls.

bool Math::TestDeriv	(	RealFunction *	f,
		Real	t,
		Real	h,
		Real	atol,
		Real	rtol
	)

the TestX functions test the methods of the function objects and prints errors if the error e=x1-x2 exceeds atol and rtol*max(|x1|,|x2|)

References dfCenteredDifference(), and TestDeriv().

Referenced by TestDeriv().