IrwinHall.cpp File Reference

Irwin-Hall probability density kernels. More...

Namespaces
namespace	gtsam
	Global functions in a separate testing namespace.

Variables
const PiecewisePolynomial< 0, 1 >	gtsam::kernels::IrwinHall0 ({{1.}, {0., 1.}, 1./2.})
	coefficients for the Irwin Hall Probability Density Functions https://oeis.org/A188816
const PiecewisePolynomial< 1, 2 >	gtsam::kernels::IrwinHall1 ({{0., 1., 2., -1.}, {0., 1., 2.}, 1.})
	Order-one Irwin-Hall PDF.
const PiecewisePolynomial< 2, 3 >	gtsam::kernels::IrwinHall2 ({{0., 0., 1./2., -3./2., 6./2., -2./2., 9./2., -6./2., 1./2.}, {0., 1., 2., 3.}, 3./2.})
	Order-two Irwin-Hall PDF.
const PiecewisePolynomial< 3, 4 >	gtsam::kernels::IrwinHall3 ({{0., 0., 0., 1./6., 2./3., -2., 2., -1./2., -22./3., 10., -4., 1./2., 32./3., -8., 2., -1./6.}, {0., 1., 2., 3., 4.}, 2.})
	Order-three Irwin-Hall PDF.
const PiecewisePolynomial< 4, 5 >	gtsam::kernels::IrwinHall4 ({{0., 0., 0., 0., 1./24., -5./24., 5./6., -5./4., 5./6., -1./6., 155./24., -25./2., 35./4., -5./2., 1./4., -655./24., 65./2., -55./4., 5./2., -1./6., 625./24., -125./6., 25./4., -5./6., 1./24.}, {0., 1., 2., 3., 4., 5.}, 5./2.})
	Order-four Irwin-Hall PDF.
const PiecewisePolynomial< 5, 6 >	gtsam::kernels::IrwinHall5 ({{0., 0., 0., 0., 0., 1./120., 1./20., -1./4., 1./2., -1./2., 1./4., -1./24., -79./20., 39./4., -19./2., 9./2., -1., 1./12., 731./20., -231./4., 71./2., -21./2., 3./2., -1./12., -1829./20., 409./4., -89./2., 19./2., -1., 1./24., 324./5., -54., 18., -3., 1./4., -1./120.}, {0., 1., 2., 3., 4., 5., 6.}, 3.})
	Order-five Irwin-Hall PDF.
const PiecewisePolynomial< 6, 7 >	gtsam::kernels::IrwinHall6 ({{0., 0., 0., 0., 0., 0., 1./720., -7./720., 7./120., -7./48., 7./36., -7./48., 7./120., -1./120., 1337./720., -133./24., 329./48., -161./36., 77./48., -7./24., 1./48., -12089./360., 196./3., -1253./24., 196./9., -119./24., 7./12., -1./36., 59591./360., -700./3., 3227./24., -364./9., 161./24., -7./12., 1./48., -208943./720., 7525./24., -6671./48., 1169./36., -203./48., 7./24., -1./120., 117649./720., -16807./120., 2401./48., -343./36., 49./48., -7./120., 1./720.}, {0., 1., 2., 3., 4., 5., 6., 7.}, 7./2.})
	Order-six Irwin-Hall PDF.

Detailed Description

Irwin-Hall probability density kernels.

Author

Brett Downing

Variable Documentation

IrwinHall0

GTSAM_EXPORT const PiecewisePolynomial< 0, 1 > gtsam::kernels::IrwinHall0

(

{{1.}, {0., 1.}, 1./2.}

)

coefficients for the Irwin Hall Probability Density Functions https://oeis.org/A188816

Irwin-Hall kernels for uniform cardinal splines.

The Irwin-Hall distribution is the distribution of a sum of independent unit-uniform variables. Its probability density is a compactly supported piecewise polynomial: the order- \(n\) PDF is a cardinal B-spline of degree \(n\). Integrating it produces the cumulative blending function used to turn consecutive control-point increments on a Lie group on and off smoothly.

The PDFs below follow OEIS A188816 and the CDFs follow OEIS A188668. The CDF suffix is two less than the resulting interpolating spline degree: CDF0 gives linear interpolation, CDF1 gives a quadratic spline, and CDF2—the default used by CumulativeSplineTrajectory—gives a cubic spline. Higher suffixes provide progressively smoother, wider-support kernels. Order-zero Irwin-Hall PDF.