atomic Namespace Reference

Namespace with special functions and derivatives. More...

Classes
struct	AtomicGlobal
	For backwards compatibility with CppAD. More...
struct	AtomicLocal
	User interface to checkpointing using TMBad. More...

Functions
template<class Type >
matrix< Type >	absm (matrix< Type > x)
	Matrix absolute value. More...
template<class Type >
CppAD::vector< Type >	bessel_i_10 (CppAD::vector< Type > x)
	Atomic version of \(besselI(x,\nu)\). Valid parameter range: \(x =(x,\nu) \in \mathbb{R}_+\times\mathbb{R}\). More...
template<class Type >
CppAD::vector< Type >	bessel_k_10 (CppAD::vector< Type > x)
	Atomic version of \(besselK(x,\nu)\). Valid parameter range: \(x =(x,\nu) \in \mathbb{R}_+\times\mathbb{R}\). More...
template<class Type >
CppAD::vector< Type >	D_incpl_gamma_shape (CppAD::vector< Type > x)
	Atomic version of scaled incomplete gamma function differentiated to any order wrt. shape parameter \[ \exp(c) \int_0^{y} \exp(-t) t^{\lambda-1} \log(t)^n \:dt \] where the 4 input parameters are passed as a vector \(x=(y,\lambda,n,c)\). Note that the normalized incomplete gamma function is obtained as the special case \(n=0\) and \(c=-\log \Gamma(\lambda)\). Valid parameter range: \(x \in \mathbb{R}_+\times\mathbb{R}_+\times\mathbb{N}_0\times\mathbb{R}\). More...
template<class Type >
CppAD::vector< Type >	D_lgamma (CppAD::vector< Type > x)
	Atomic version of the n'th order derivative of the log gamma function. \[ \frac{d^n}{d\lambda^n}\log \Gamma(\lambda) \] where the 2 input parameters are passed as a vector \(x=(\lambda,n)\). The special case \(n=0\) gives the log gamma function. More...
template<class Type >
matrix< Type >	expm (matrix< Type > x)
	Matrix exponential. More...
template<class Type >
vector< std::complex< Type > >	fft (vector< std::complex< Type > > xc, bool inverse=false)
	FFT (unscaled) More...
template<class Type >
CppAD::vector< Type >	inv_incpl_gamma (CppAD::vector< Type > x)
	Atomic version of inverse of scaled incomplete gamma function. Given \(z\) find \(y\) such that \[ z = \exp(c) \int_0^{y} \exp(-t) t^{\lambda-1} \:dt \] where the 3 input parameters are passed as a vector \(x=(z,\lambda,c)\). The special case \(c=-\log \Gamma(\lambda)\) gives the inverse normalized incomplete gamma function. Valid parameter range: \(x \in \mathbb{R}_+\times\mathbb{R}_+\times\mathbb{R}\). More...
template<class Type >
CppAD::vector< Type >	invpd (CppAD::vector< Type > x)
	Atomic version of log determinant and inverse of positive definite n-by-n matrix. Calculated by Cholesky decomposition. More...
template<class Type >
CppAD::vector< Type >	logdet (CppAD::vector< Type > x)
	Atomic version of log determinant of positive definite n-by-n matrix. More...
template<class Type >
Type	logdet (matrix< Type > x)
	Log-determinant of positive definite matrix.
template<class Type >
CppAD::vector< Type >	matinv (CppAD::vector< Type > x)
	Atomic version of matrix inversion. Inverts n-by-n matrix by LU-decomposition. More...
template<class Type >
matrix< Type >	matinv (matrix< Type > x)
	Matrix inverse. More...
template<class Type >
matrix< Type >	matinvpd (matrix< Type > x, Type &logdet)
	Matrix inverse and determinant. More...
template<class Type >
CppAD::vector< Type >	matmul (CppAD::vector< Type > x)
	Atomic version of matrix multiply. Multiplies n1-by-n2 matrix with n2-by-n3 matrix. More...
template<class Type >
matrix< Type >	matmul (matrix< Type > x, matrix< Type > y)
	Matrix multiply. More...
template<class Type >
CppAD::vector< Type >	pnorm1 (CppAD::vector< Type > x)
	Atomic version of standard normal distribution function. Derivative is known to be 'dnorm1'. More...
template<class Type >
CppAD::vector< Type >	ppois (CppAD::vector< Type > x)
	Atomic version of poisson cdf \(ppois(n,\lambda)\). Valid parameter range: \(x =(n,\lambda) \in \mathbb{N}_0\times\mathbb{R}_+\). More...
template<class Type >
CppAD::vector< Type >	qnorm1 (CppAD::vector< Type > x)
	Atomic version of standard normal quantile function. Derivative is expressed through 'dnorm1'. More...
template<class Type >
matrix< Type >	sqrtm (matrix< Type > x)
	Matrix square root. More...

Detailed Description

Namespace with special functions and derivatives.

This namespace extends the 'derivatives table' of CppAD.

• R's special math library is extended with derivatives in cases where symbolic derivatives are available. These special functions are often iterative and therefore difficult to implement with AD types. Instead, we code the derivatives based on the double versions available from R. This approach requires fewer code lines, and has the benefit of obtaining the same high accuracy as R's math functions.
• Some matrix operations are extended with derivatives. This greatly reduces the AD memory usage. Furthermore, these atomic operations can be linked to a performance library by setting preprocesor flag EIGEN_USE_BLAS.
• New symbols can be added by advanced users. First option is to code the reverse mode derivatives by hand using the TMB_ATOMIC_VECTOR_FUNCTION macro, see source code for examples. Second option is to generate reverse mode derivatives automatically using the macro REGISTER_ATOMIC.

Function Documentation

bessel_i_10()

template<class Type >

CppAD::vector<Type> atomic::bessel_i_10

(

CppAD::vector< Type >

x

)

Atomic version of \(besselI(x,\nu)\). Valid parameter range: \(x =(x,\nu) \in \mathbb{R}_+\times\mathbb{R}\).

Note

This atomic function does not handle the derivative wrt. \(\nu\).

Parameters

x	Input vector of length 2.

Returns

Vector of length 1.

bessel_k_10()

template<class Type >

CppAD::vector<Type> atomic::bessel_k_10

(

CppAD::vector< Type >

x

)

Atomic version of \(besselK(x,\nu)\). Valid parameter range: \(x =(x,\nu) \in \mathbb{R}_+\times\mathbb{R}\).

Note

This atomic function does not handle the derivative wrt. \(\nu\).

Parameters

x	Input vector of length 2.

Returns

Vector of length 1.

D_incpl_gamma_shape()

template<class Type >

CppAD::vector<Type> atomic::D_incpl_gamma_shape

(

CppAD::vector< Type >

x

)

Atomic version of scaled incomplete gamma function differentiated to any order wrt. shape parameter

\[ \exp(c) \int_0^{y} \exp(-t) t^{\lambda-1} \log(t)^n \:dt \]

where the 4 input parameters are passed as a vector \(x=(y,\lambda,n,c)\). Note that the normalized incomplete gamma function is obtained as the special case \(n=0\) and \(c=-\log \Gamma(\lambda)\). Valid parameter range: \(x \in \mathbb{R}_+\times\mathbb{R}_+\times\mathbb{N}_0\times\mathbb{R}\).

Warning

No check is performed on parameters

Parameters

x	Input vector of length 4.

Returns

Vector of length 1.

D_lgamma()

template<class Type >

CppAD::vector<Type> atomic::D_lgamma

(

CppAD::vector< Type >

x

)

Atomic version of the n'th order derivative of the log gamma function.

\[ \frac{d^n}{d\lambda^n}\log \Gamma(\lambda) \]

where the 2 input parameters are passed as a vector \(x=(\lambda,n)\). The special case \(n=0\) gives the log gamma function.

Parameters

x	Input vector of length 2.

Returns

Vector of length 1.

inv_incpl_gamma()

template<class Type >

CppAD::vector<Type> atomic::inv_incpl_gamma

(

CppAD::vector< Type >

x

)

Atomic version of inverse of scaled incomplete gamma function. Given \(z\) find \(y\) such that

\[ z = \exp(c) \int_0^{y} \exp(-t) t^{\lambda-1} \:dt \]

where the 3 input parameters are passed as a vector \(x=(z,\lambda,c)\). The special case \(c=-\log \Gamma(\lambda)\) gives the inverse normalized incomplete gamma function. Valid parameter range: \(x \in \mathbb{R}_+\times\mathbb{R}_+\times\mathbb{R}\).

Warning

No check is performed on parameters

Parameters

x	Input vector of length 3.

Returns

Vector of length 1.

invpd()

template<class Type >

CppAD::vector<Type> atomic::invpd

(

CppAD::vector< Type >

x

)

Atomic version of log determinant and inverse of positive definite n-by-n matrix. Calculated by Cholesky decomposition.

Parameters

x	Input vector of length n*n.

Returns

Vector of length 1+n*n.

logdet()

template<class Type >

CppAD::vector<Type> atomic::logdet

(

CppAD::vector< Type >

x

)

Atomic version of log determinant of positive definite n-by-n matrix.

Parameters

x	Input vector of length n*n.

Returns

Vector of length 1.

Examples:

laplace.cpp.

matinv()

template<class Type >

CppAD::vector<Type> atomic::matinv

(

CppAD::vector< Type >

x

)

Atomic version of matrix inversion. Inverts n-by-n matrix by LU-decomposition.

Parameters

x	Input vector of length n*n.

Returns

Vector of length n*n.

Examples:

laplace.cpp.

Referenced by newton::jacobian_sparse_plus_lowrank_t< dummy >::llt_solve(), and matinv().

matmul()

template<class Type >

CppAD::vector<Type> atomic::matmul

(

CppAD::vector< Type >

x

)

Atomic version of matrix multiply. Multiplies n1-by-n2 matrix with n2-by-n3 matrix.

Parameters

x	Input vector of length 2+n1*n2+n2*n3 containing the output dimension (length=2), the first matrix (length=n1*n2) and the second matrix (length=n2*n3).

Returns

Vector of length n1*n3 containing result of matrix multiplication.

Referenced by newton::jacobian_sparse_plus_lowrank_t< dummy >::llt_solve(), and matmul().

pnorm1()

template<class Type >

CppAD::vector<Type> atomic::pnorm1

(

CppAD::vector< Type >

x

)

Atomic version of standard normal distribution function. Derivative is known to be 'dnorm1'.

Parameters

x	Input vector of length 1.

Returns

Vector of length 1.

ppois()

template<class Type >

CppAD::vector<Type> atomic::ppois

(

CppAD::vector< Type >

x

)

Atomic version of poisson cdf \(ppois(n,\lambda)\). Valid parameter range: \(x =(n,\lambda) \in \mathbb{N}_0\times\mathbb{R}_+\).

Warning

No check is performed on parameters

Parameters

x	Input vector of length 2.

Returns

Vector of length 1.

qnorm1()

template<class Type >

CppAD::vector<Type> atomic::qnorm1

(

CppAD::vector< Type >

x

)

Atomic version of standard normal quantile function. Derivative is expressed through 'dnorm1'.

Parameters

x	Input vector of length 1.

Returns

Vector of length 1.