jive_helpers Namespace Reference

namespace for all helper functions

Typedefs
typedef Array< double, 4 >	Quadix
typedef Array< IdxVector, 2 >	IdxVectorMatrix

Functions
Vector	funcGrad (const Ref< Function > func, const Vector &args)
	gets the gradient of a multivariate function.
Vector	evalFuncs (const FuncUtils::FuncArray funcs, const Vector &args)
	evaluates an array of multivariate functions.
Matrix	funcHessian (const Ref< Function > func, const Vector &args)
	gets the Hessian of a multivariate function.
Matrix	gradFuncs (const FuncUtils::FuncArray funcs, const Vector &args)
	gets the gradients of an array of multivariate functions.
Matrix	eye (const idx_t dim=3)
	generates an identity matrix of the given dimension.
void	logMat (const Vector &rv, const Matrix &R)
	compute rotational vector from rotation matrix.
void	vec2mat (const Matrix &mat, const Vector &vec)
	rearrange a vector into a matrix.
void	mat2vec (const Vector &vec, const Matrix &mat)
	rearrange a matrix into a vector.
void	mul3 (const Matrix &C, const Matrix &A, const Matrix &B)
	C = A * B for 3 x 3 matrices without heap allocation.
void	mul3 (const Vector &y, const Matrix &A, const Vector &x)
	y = A * x for a 3 x 3 matrix without heap allocation.
void	expVec (const Matrix &Exp, const Vector &v)
	compute the exponential of a axial vector.
void	expVecP (const Matrix &ExpP, const Vector &v, const Vector &vP)
	derivative of jive_helpers::expVec.
double	trace (const Matrix &mat)
	calculate the trace of a matrix.
idx_t	factorial (const idx_t n)
	calculates the factorial of an integer.
idx_t	binom (const idx_t n, const idx_t k)
	calculate the binomial coefficient.
double	matrixNorm2 (const Matrix &mat)
	calculates the 2-norm of a matrix.
Matrix	skew (const Vector &vec)
	construct a skew symetric matrix from a given vector.
void	skew (const Matrix &S, const Vector &vec)
	fill an existing 3 x 3 matrix with the skew symmetric matrix of a vector
Vector	unskew (const Matrix &mat)
	construct a axial vector from a given skew-symmetric matrix
void	setDofValue (jive::util::Constraints &cons, const idx_t idof, const double rval)
	prescribe the value of a dof, i.e. constrain idof = rval.
void	linkDofs (jive::util::Constraints &cons, const idx_t idof, const idx_t jdof, const double coeff)
	link a slave dof to a master dof, i.e. constrain idof = coeff * jdof.

Variables
const Vector	e1 = {1., 0., 0.}
const Vector	e2 = {0., 1., 0.}
const Vector	e3 = {0., 0., 1.}
const Vector	vec0 = {0., 0., 0.}
const double	TINY = __DBL_EPSILON__ * 1e6

Typedef Documentation

IdxVectorMatrix

typedef Array<IdxVector, 2> jive_helpers::IdxVectorMatrix

Quadix

typedef Array<double, 4> jive_helpers::Quadix

Function Documentation

binom()

idx_t jive_helpers::binom	(	const idx_t	n,
		const idx_t	k
	)

Parameters

[in]	n	Parameter n
[in]	k	Parameter k

Returns

binomial coefficient nCk

evalFuncs()

Vector jive_helpers::evalFuncs	(	const FuncUtils::FuncArray	funcs,
		const Vector &	args
	)

Parameters

[in]	funcs	array of functions to evaluate
[in]	args	point to evaluate at

Returns

results of the evaluation

expVec()

void jive_helpers::expVec	(	const Matrix &	Exp,
		const Vector &	v
	)

It computes the matrix exponential of the skew-symmetric matrix of this vector so when \(\Theta=[\theta\times]\) this function will return \(\exp(\Theta)\).

See also

Wikipedia

Parameters

[out]	Exp	3 x 3 matrix exponential
[in]	v	3-component axial rotation vector

expVecP()

void jive_helpers::expVecP	(	const Matrix &	ExpP,
		const Vector &	v,
		const Vector &	vP
	)

See also

jive_helpers::expVec

Differentiation rules

Parameters

[out]	ExpP	Derivative of the 3 x 3 matrix exponential
[in]	v	3-component axial rotation vector
[in]	vP	Derivative of the axial rotation vector

eye()

Matrix jive_helpers::eye

(

const idx_t

dim = 3

)

Parameters

[in]

dim

dimension of the identity matrix (default 3)

Returns

identity matrix

factorial()

idx_t jive_helpers::factorial

(

const idx_t

n

)

Parameters

[in]

n

Integer to calculate the factorial from

Returns

the calculated factorial

funcGrad()

Vector jive_helpers::funcGrad	(	const Ref< Function >	func,
		const Vector &	args
	)

Parameters

[in]	func	function of which to evaluate the gradient
[in]	args	point to evaluate the gradient

Returns

gradient

funcHessian()

Matrix jive_helpers::funcHessian	(	const Ref< Function >	func,
		const Vector &	args
	)

Parameters

[in]	func	function of which to evaluate the Hessian
[in]	args	point to evaluate the Hessian

Returns

Hessian

gradFuncs()

Matrix jive_helpers::gradFuncs	(	const FuncUtils::FuncArray	funcs,
		const Vector &	args
	)

Parameters

[in]	funcs	array of functions of which to evaluate the gradient
[in]	args	point to evaluate the gradient

Returns

Matrix of the gradients

linkDofs()

void jive_helpers::linkDofs	(	jive::util::Constraints &	cons,
		const idx_t	idof,
		const idx_t	jdof,
		const double	coeff
	)

Parameters

[in,out]	cons	constraints to modify
[in]	idof	slave dof index
[in]	jdof	master dof index
[in]	coeff	coefficient of the master dof

logMat()

void jive_helpers::logMat	(	const Vector &	rv,
		const Matrix &	R
	)

This function computes the logarithmic map of a rotation matrix as a member of the Lie group SO(3) to the Lie algebra \(\mathfrak{so}(3)\). It returns the axial vector of the rotation matrix.

See also

Wikipedia

Parameters

[out]	rv	3-component rotational vector
[in]	R	3 x 3 rotational matrix

mat2vec()

void jive_helpers::mat2vec	(	const Vector &	vec,
		const Matrix &	mat
	)

Parameters

[out]	vec	Vector populated with the matrix values by row
[in]	mat	Matrix whose entry count matches the vector size

matrixNorm2()

double jive_helpers::matrixNorm2

(

const Matrix &

mat

)

Parameters

[in]

mat

Matrix

Returns

resulting 2-norm

mul3() [1/2]

void jive_helpers::mul3	(	const Matrix &	C,
		const Matrix &	A,
		const Matrix &	B
	)

The product is formed on the stack first, so C may alias A or B (unlike jem::numeric::matmul(C, A, B)); the 3 x 3 sizes are fixed, which makes it faster than the general matmul. Use it for every 3 x 3 product in hot code.

Parameters

[out]	C	3 x 3 result
[in]	A	3 x 3 left factor
[in]	B	3 x 3 right factor

mul3() [2/2]

void jive_helpers::mul3	(	const Vector &	y,
		const Matrix &	A,
		const Vector &	x
	)

x is copied first, so y may alias x. The sums are formed in the same order as in jem::numeric::matmul (starting from zero), so the results are the same.

Parameters

[out]	y	3 vector result
[in]	A	3 x 3 matrix (may be a transposed view)
[in]	x	3 vector

setDofValue()

void jive_helpers::setDofValue	(	jive::util::Constraints &	cons,
		const idx_t	idof,
		const double	rval
	)

Parameters

[in,out]	cons	constraints to modify
[in]	idof	dof index to prescribe
[in]	rval	prescribed value

skew() [1/2]

void jive_helpers::skew	(	const Matrix &	S,
		const Vector &	vec
	)

Does not create a new matrix, so it is the form for hot code.

Parameters

[out]	S	3 x 3 skew symmetric matrix
[in]	vec	Vector

skew() [2/2]

Matrix jive_helpers::skew

(

const Vector &

vec

)

Convenient form for set-up code; it creates a new matrix. In hot code use the form below with a work matrix.

Parameters

[in]

vec

Vector to be changed to a skew-symmetric matrix

Returns

skew symmetric matrix

trace()

double jive_helpers::trace

(

const Matrix &

mat

)

Parameters

[in]

mat

Matrix to get the trace of

Returns

the trace of the given matrix

unskew()

Vector jive_helpers::unskew

(

const Matrix &

mat

)

Parameters

[in]

mat

Skew-symmetric matrix

Returns

axial vector

vec2mat()

void jive_helpers::vec2mat	(	const Matrix &	mat,
		const Vector &	vec
	)

Parameters

[out]	mat	Matrix populated from the vector values by row
[in]	vec	Vector whose size matches the number of matrix entries

Variable Documentation

e1

const Vector jive_helpers::e1 = {1., 0., 0.}

e2

const Vector jive_helpers::e2 = {0., 1., 0.}

e3

const Vector jive_helpers::e3 = {0., 0., 1.}

TINY

const double jive_helpers::TINY = __DBL_EPSILON__ * 1e6

vec0

const Vector jive_helpers::vec0 = {0., 0., 0.}