mth Namespace Reference

All MTH functions are contained in this namespace. More...

Classes
class	Vector
	compile-time (static) vector of size N More...
class	Vector< T, 0 >
	run-time (dynamic) vector More...
class	Vector3
	convenience wrapper over apf::Vector<3> More...
class	Matrix
	compile-time (static) matrix More...
class	Matrix< T, 0, 0 >
	run-time (dynamic) matrix More...
class	Matrix3x3
	convenience wrapper over Matrix<T,3,3> More...
class	Tensor
	run-time (dynamic) tensor More...
class	AD
	forward automatic differentiation variable More...
class	AD< T, 0 >
	forward automatic differentiation variable with dynamic variable array More...

Functions
template<class T >
Vector3< T >	cross (Vector< T, 3 > const &a, Vector< T, 3 > const &b)
	returns vector cross product
template<class T >
Matrix3x3< T >	cross (Vector< T, 3 > const &a)
	returns the cross product matrix for the vector
template<class T , unsigned N>
Vector< T, N >	project (Vector< T, N > const &a, Vector< T, N > const &b)
	returns vector a projected onto vector b
template<class T , unsigned N>
Vector< T, N >	reject (Vector< T, N > const &a, Vector< T, N > const &b)
	vector rejection
template<class T , unsigned M, unsigned N>
void	transpose (Matrix< T, M, N > const &a, Matrix< T, N, M > &b)
	transpose of a static matrix
template<class T >
T	determinant (Matrix< T, 2, 2 > const &a)
	determinant of a 2 by 2 matrix
template<class T >
T	determinant (Matrix< T, 3, 3 > const &a)
	determinant of a 3 by 3 matrix
template<class T >
Matrix< T, 2, 2 >	inverse (Matrix< T, 2, 2 > const &a)
	invert a 2 by 2 matrix
template<class T >
Matrix< T, 3, 3 >	inverse (Matrix< T, 3, 3 > const &a)
	invert a 3 by 3 matrix
template<class T >
T	trace (Tensor< T > const &a)
	trace of a tensor
template<class T >
T	norm (Tensor< T > const &a)
	Frobenius norm of a tensor.
template<class T >
T	determinant (Tensor< T > const &a)
	determinant of a tensor
template<class T >
void	transpose (Tensor< T > const &a, Tensor< T > &r)
	transpose of a tensor
template<class T >
void	inverse (Tensor< T > const &a, Tensor< T > &r)
	inverse of a tensor
template<class T >
Tensor< T >	eye (unsigned d)
	identity tensor
template<class T , unsigned M, unsigned N>
unsigned	decomposeQR (Matrix< T, M, N > const &a, Matrix< T, M, M > &q, Matrix< T, M, N > &r)
	finds the QR decomposition of A More...
template<class T , unsigned M, unsigned N>
void	backsubUT (Matrix< T, M, N > const &r, Vector< T, M > const &b, Vector< T, N > &x)
	solves Rx = b for upper triangular R More...
template<class T , unsigned M, unsigned N>
void	solveFromQR (Matrix< T, M, M > const &q, Matrix< T, M, N > const &r, Vector< T, M > const &b, Vector< T, N > &x)
	solves Ax = b given A's QR factorization More...
template<class T , unsigned M, unsigned N>
bool	solveQR (Matrix< T, M, N > const &a, Vector< T, M > const &b, Vector< T, N > &x)
	solves Ax = b using A's QR factorization More...
template<class T , unsigned M>
bool	eigenQR (Matrix< T, M, M > const &a, Matrix< T, M, M > &l, Matrix< T, M, M > &q, unsigned max_iters)
	computes the eigendecomposition of A More...
template<class T , unsigned int N>
AD< T, N >	operator- (AD< T, N > const &A)
	unary subtraction
template<class T , unsigned int N>
AD< T, N >	operator+ (double L, AD< T, N > const &R)
	binary addition between a double and an AD variable
template<class T , unsigned int N>
AD< T, N >	operator+ (AD< T, N > const &L, double R)
	binary addition between an AD variable and a double
template<class T , class B , unsigned int N>
AD< T, N >	operator+ (AD< T, N > const &L, AD< B, N > const &R)
	binary addition between two AD variables
template<class T , unsigned int N>
AD< T, N >	operator- (double L, AD< T, N > const &R)
	binary subtraction between a double and an AD variable
template<class T , unsigned int N>
AD< T, N >	operator- (AD< T, N > const &L, double R)
	binary subtraction between an AD variable and a double
template<class T , class B , unsigned int N>
AD< T, N >	operator- (AD< T, N > const &L, AD< B, N > const &R)
	binary subtraction between two AD variables
template<class T , unsigned int N>
AD< T, N >	operator* (double L, AD< T, N > const &R)
	binary multiplication between a double and an AD variable
template<class T , unsigned int N>
AD< T, N >	operator* (AD< T, N > const &L, double R)
	binary multiplication between an AD variable and a double
template<class T , class B , unsigned int N>
AD< T, N >	operator* (AD< T, N > const &L, AD< B, N > const &R)
	binary multiplication between two AD variables
template<class T , unsigned int N>
AD< T, N >	operator/ (double L, AD< T, N > const &R)
	binary division between a double and an AD variable
template<class T , unsigned int N>
AD< T, N >	operator/ (AD< T, N > const &L, double R)
	binary division between an AD variable and a double
template<class T , class B , unsigned int N>
AD< T, N >	operator/ (AD< B, N > const &L, AD< T, N > const &R)
	binary division between two AD variables
double	exp (double x)
	wrapper to standard exp function
template<class T , unsigned int N>
AD< T, N >	exp (AD< T, N > const &A)
	exponent of an AD variable
double	log (double A)
	wrapper for stander log function
template<class T , unsigned int N>
AD< T, N >	log (AD< T, N > const &A)
	logarithm of an AD variable
double	pow (double A, double e)
	wrapper to standard pow function
template<class T , unsigned int N>
AD< T, N >	pow (AD< T, N > const &A, const int e)
	AD variable raised to an integer power.
template<class T , unsigned int N>
AD< T, N >	pow (AD< T, N > const &A, const double e)
	AD variable raised to a double power.
template<class T , unsigned int N>
AD< T, N >	pow (const int base, AD< T, N > const &A)
	integer raised to an AD power
template<class T , unsigned int N>
AD< T, N >	pow (const double base, AD< T, N > const &A)
	double raised to an AD power
template<class T , unsigned int N>
AD< T, N >	pow (AD< T, N > const &A, AD< T, N > const &e)
	AD variable raised to an AD variable power.
double	sqrt (double A)
	wrapper for standard sqrt function
template<class T , unsigned int N>
AD< T, N >	sqrt (AD< T, N > const &A)
	square root of an AD variable
double	sin (double A)
	wrapper for standard sin function
double	cos (double A)
	wrapper for standard cos function
template<class T , unsigned int N>
AD< T, N >	sin (AD< T, N > const &A)
	sin of an AD variable
template<class T , unsigned int N>
AD< T, N >	cos (AD< T, N > const &A)
	cos of an AD variable
template<class T , unsigned int N>
AD< T, N >	tan (AD< T, N > const &A)
	tan of an AD variable
template<class T , unsigned int N>
AD< T, N >	abs (AD< T, N > const &A)
	absolute value of an AD variable
template<class T , unsigned int N>
bool	operator< (double L, AD< T, N > const &R)
	double less than an AD variable
template<class T , unsigned int N>
bool	operator< (AD< T, N > const &R, double L)
	AD variable less than a double.
template<class T , unsigned int N>
bool	operator< (AD< T, N > const &R, AD< T, N > const &L)
	AD variable less than an AD variable.
template<class T , unsigned int N>
bool	operator<= (double L, AD< T, N > const &R)
	double less than or equal to an AD variable
template<class T , unsigned int N>
bool	operator<= (AD< T, N > const &R, double L)
	AD variable less than or equal to a double.
template<class T , unsigned int N>
bool	operator<= (AD< T, N > const &R, AD< T, N > const &L)
	AD variable less than or equal to an AD variable.
template<class T , unsigned int N>
bool	operator> (double L, AD< T, N > const &R)
	double greater than an AD variable
template<class T , unsigned int N>
bool	operator> (AD< T, N > const &R, double L)
	AD variable greater than a double.
template<class T , unsigned int N>
bool	operator> (AD< T, N > const &R, AD< T, N > const &L)
	AD variable greater than an AD variable.
template<class T , unsigned int N>
bool	operator>= (double L, AD< T, N > const &R)
	double greater than or equal to an AD variable
template<class T , unsigned int N>
bool	operator>= (AD< T, N > const &R, double L)
	AD variable greater than or equal to a double.
template<class T , unsigned int N>
bool	operator>= (AD< T, N > const &R, AD< T, N > const &L)
	AD variable greater than or equal to an AD variable.

Detailed Description

All MTH functions are contained in this namespace.

Function Documentation

backsubUT()

template<class T , unsigned M, unsigned N>

void mth::backsubUT	(	Matrix< T, M, N > const &	r,
		Vector< T, M > const &	b,
		Vector< T, N > &	x
	)

solves Rx = b for upper triangular R

when M > N, the lower rows are ignored

Parameters

r	the MxN (M >= N) upper triangular input matrix
b	the Mx1 right hand side input vector
x	the Nx1 output solution vector

decomposeQR()

template<class T , unsigned M, unsigned N>

unsigned mth::decomposeQR	(	Matrix< T, M, N > const &	a,
		Matrix< T, M, M > &	q,
		Matrix< T, M, N > &	r
	)

finds the QR decomposition of A

only 3x3 static and dynamic types are explicitly instantiated

Parameters

a	the MxN input matrix (M >= N)
q	the MxM orthogonal output matrix
r	the MxN upper triangular output matrix

Returns

the rank of A

eigenQR()

template<class T , unsigned M>

bool mth::eigenQR	(	Matrix< T, M, M > const &	a,
		Matrix< T, M, M > &	l,
		Matrix< T, M, M > &	q,
		unsigned	max_iters
	)

computes the eigendecomposition of A

this function reduces A to Hessenberg form and then uses a Wilkinson-shifted QR algorithm to identify the eigenvalues and eigenvectors.

Parameters

a	the real, square, symmetric input matrix
l	if successful, the diagonal eigenvalue matrix
q	if successful, the orthogonal eigenvector matrix
max_iters	the number of iterations after which to give up

Returns

true if converged in less than max_iters

solveFromQR()

template<class T , unsigned M, unsigned N>

void mth::solveFromQR	(	Matrix< T, M, M > const &	q,
		Matrix< T, M, N > const &	r,
		Vector< T, M > const &	b,
		Vector< T, N > &	x
	)

solves Ax = b given A's QR factorization

when M > N, the least squares problem is solved. only the dynamic type is explicitly instantiated.

Parameters

q	the MxM orthogonal input matrix
r	the MxN (M >= N) upper triangular input matrix
b	the Mx1 right hand side input vector
x	the Nx1 output solution vector

solveQR()

template<class T , unsigned M, unsigned N>

bool mth::solveQR	(	Matrix< T, M, N > const &	a,
		Vector< T, M > const &	b,
		Vector< T, N > &	x
	)

solves Ax = b using A's QR factorization

when M > N, the least squares problem is solved. only the dynamic type is explicitly instantiated.

Parameters

a	the MxN (M >= N) input matrix
b	the Mx1 right hand side input vector
x	the Nx1 output solution vector