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 | AD |
| forward automatic differentiation variable More... | |
| class | AD< T, 0 > |
| forward automatic differentiation variable with dynamic variable array More... | |
Functions | |
| template<class T > | |
| Vector< T, 3 > | cross (Vector< T, 3 > const &a, Vector< T, 3 > const &b) |
| returns vector cross product | |
| template<class T > | |
| Matrix< T, 3, 3 > | 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> | |
| Matrix< T, M, N > | transpose (Matrix< T, M, N > const &a) |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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. | |
All MTH functions are contained in this namespace.
| 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
| r | the MxN (M >= N) upper triangular input matrix | |
| b | the Mx1 right hand side input vector | |
| x | the Nx1 output solution vector |
| 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
| a | the MxN input matrix (M >= N) | |
| q | the MxM orthogonal output matrix | |
| r | the MxN upper triangular output matrix |
| 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.
| 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 |
| 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.
| 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 |
| 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.
| a | the MxN (M >= N) input matrix | |
| b | the Mx1 right hand side input vector | |
| x | the Nx1 output solution vector |
1.7.1
