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ROOT
Reference Guide |
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ROOT
Reference Guide |
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Functions | |
| template<unsigned N> | |
| double | horner (const double(&c)[N], double x) |
| Evaluate the polynomial c[0] + c[1]*x + c[2]*x^2 + ... in Horner form, matching the evaluation order of the CERNLIB Landau routines below. | |
| template<unsigned N> | |
| double | horner_deriv (const double(&c)[N], double x) |
| Derivative of horner(c, x) with respect to x. | |
| void | inc_gamma_c_pullback (double a, double x, double _d_y, double *_d_a, double *_d_x) |
| clad::ValueAndPushforward< double, double > | inc_gamma_c_pushforward (double a, double x, double d_a, double d_x) |
| Pushforward of ROOT::Math::inc_gamma_c, which is 1 - inc_gamma. | |
| double | inc_gamma_da (double a, double x) |
| Derivative of the normalized lower incomplete gamma function P(a, x) with respect to a. | |
| void | inc_gamma_da_pullback (double a, double x, double _d_y, double *_d_a, double *_d_x) |
| Pullback of inc_gamma_da(). | |
| double | inc_gamma_dx (double a, double x) |
| Derivative of the normalized lower incomplete gamma function P(a, x) with respect to x. | |
| void | inc_gamma_dx_pullback (double a, double x, double _d_y, double *_d_a, double *_d_x) |
| Pullback of inc_gamma_dx(), using the closed forms of the second derivatives of P(a, x): | |
| void | inc_gamma_pullback (double a, double x, double _d_y, double *_d_a, double *_d_x) |
| clad::ValueAndPushforward< double, double > | inc_gamma_pushforward (double a, double x, double d_a, double d_x) |
| Pushforward of ROOT::Math::inc_gamma. | |
| double | landau_cdf_dv (double v) |
| Derivative with respect to v of the CERNLIB DISLAN rational approximation of the standardized Landau cumulative distribution, obtained by differentiating each branch of landau_cdf() in ProbFuncMathCore.cxx, whose branch structure and coefficient tables this function mirrors. | |
| void | landau_cdf_pullback (double x, double xi, double x0, double d_out, double *d_x, double *d_xi, double *d_x0) |
| clad::ValueAndPushforward< double, double > | landau_cdf_pushforward (double x, double xi, double x0, double d_x, double d_xi, double d_x0) |
| Pushforward of ROOT::Math::landau_cdf, which is the cumulative distribution function of the standardized Landau density evaluated at (x - x0) / xi. | |
| double | landau_pdf_dv (double v) |
| First derivative of the standardized Landau density p(v) = ROOT::Math::landau_pdf(v) with respect to v, obtained by differentiating each branch of the CERNLIB DENLAN rational approximation. | |
| void | landau_pdf_dv_pullback (double v, double _d_y, double *_d_v) |
| Pullback of landau_pdf_dv(). | |
| void | landau_pdf_pullback (double x, double xi, double x0, double d_out, double *d_x, double *d_xi, double *d_x0) |
| clad::ValueAndPushforward< double, double > | landau_pdf_pushforward (double x, double xi, double x0, double d_x, double d_xi, double d_x0) |
| Pushforward of ROOT::Math::landau_pdf, which is p((x - x0) / xi) / xi in terms of the standardized Landau density p. | |
| template<unsigned N, unsigned M> | |
| double | rational_deriv (const double(&p)[N], const double(&q)[M], double x) |
| Derivative of the rational function horner(p, x) / horner(q, x) with respect to x. | |
| double clad::custom_derivatives::ROOT::Math::horner | ( | const double(&) | c[N], |
| double | x ) |
Evaluate the polynomial c[0] + c[1]*x + c[2]*x^2 + ... in Horner form, matching the evaluation order of the CERNLIB Landau routines below.
Definition at line 209 of file CladDerivator.h.
| double clad::custom_derivatives::ROOT::Math::horner_deriv | ( | const double(&) | c[N], |
| double | x ) |
Derivative of horner(c, x) with respect to x.
Definition at line 219 of file CladDerivator.h.
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inline |
Definition at line 522 of file CladDerivator.h.
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inline |
Pushforward of ROOT::Math::inc_gamma_c, which is 1 - inc_gamma.
Definition at line 900 of file CladDerivator.h.
Derivative of the normalized lower incomplete gamma function P(a, x) with respect to a.
It has no closed form, but inc_gamma_pullback() computes it exactly by differentiating through the algorithm that evaluates P(a, x).
Definition at line 859 of file CladDerivator.h.
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inline |
Pullback of inc_gamma_da().
The mixed second derivative is known in closed form (it is the same as the a-derivative of inc_gamma_dx(), see inc_gamma_dx_pullback()). For d2P/da2 there is no closed form, so it is approximated by a central difference of the exact first derivative.
For a <= h, the lower stencil point leaves the domain (inc_gamma_da() returns zero for non-positive a), so d2P/da2 is unreliable there. This is acceptable because RooFit never differentiates with respect to a, which is data there.
Definition at line 876 of file CladDerivator.h.
Derivative of the normalized lower incomplete gamma function P(a, x) with respect to x.
This is the integrand of P(a, x), i.e. the gamma distribution density: x^(a-1) * exp(-x) / Gamma(a).
Definition at line 835 of file CladDerivator.h.
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inline |
Pullback of inc_gamma_dx(), using the closed forms of the second derivatives of P(a, x):
d2P/dx2 = inc_gamma_dx(a, x) * ((a - 1) / x - 1) d2P/dxda = inc_gamma_dx(a, x) * (log(x) - digamma(a))
Definition at line 847 of file CladDerivator.h.
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Definition at line 406 of file CladDerivator.h.
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Pushforward of ROOT::Math::inc_gamma.
Besides forward-mode differentiation, this enables second derivatives (e.g. clad::hessian): clad differentiates this function in reverse mode, and all derivatives it needs for that are provided by custom pullbacks.
Definition at line 893 of file CladDerivator.h.
Derivative with respect to v of the CERNLIB DISLAN rational approximation of the standardized Landau cumulative distribution, obtained by differentiating each branch of landau_cdf() in ProbFuncMathCore.cxx, whose branch structure and coefficient tables this function mirrors.
Since DISLAN is an approximation of its own, this is not identical to landau_pdf() (they are consistent to about 1e-7 relative).
Definition at line 330 of file CladDerivator.h.
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Definition at line 394 of file CladDerivator.h.
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Pushforward of ROOT::Math::landau_cdf, which is the cumulative distribution function of the standardized Landau density evaluated at (x - x0) / xi.
Its derivative in x is landau_pdf, so unlike for the density itself, all second derivatives are known in closed form.
That identity is exact only for the mathematical Landau distribution: ROOT implements the cdf (CERNLIB DISLAN) and the density (CERNLIB DENLAN) as independent rational approximations that are consistent with each other to about 1e-7 relative. The derivatives returned here therefore differ at that level both from the exact derivative of the implemented cdf and from landau_cdf_pullback(), which differentiates the DISLAN algorithm itself.
Definition at line 966 of file CladDerivator.h.
First derivative of the standardized Landau density p(v) = ROOT::Math::landau_pdf(v) with respect to v, obtained by differentiating each branch of the CERNLIB DENLAN rational approximation.
The branch structure and the coefficient tables mirror landau_pdf() in PdfFuncMathCore.cxx. Used by landau_pdf_pullback() and by the second-derivative helpers further down.
Definition at line 242 of file CladDerivator.h.
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Pullback of landau_pdf_dv().
The second derivative of the standardized Landau density has no closed form, so it is approximated by a central difference of the exact first derivative (see also inc_gamma_da_pullback()).
The CERNLIB DENLAN approximation of the density is piecewise rational, and its first derivative has small jumps at the branch boundaries. Dividing such a jump by the step size would ruin the difference quotient (up to ~50 % error right at v = 1), so when the stencil would straddle a boundary it is shifted sideways to keep both points on the branch that contains v. The off-center evaluation costs one order in h, which is insignificant at this step size.
Definition at line 916 of file CladDerivator.h.
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Definition at line 310 of file CladDerivator.h.
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Pushforward of ROOT::Math::landau_pdf, which is p((x - x0) / xi) / xi in terms of the standardized Landau density p.
Like for inc_gamma_pushforward(), all derivatives that clad needs to differentiate this function in reverse mode (e.g. for clad::hessian) are provided by custom pullbacks.
Definition at line 944 of file CladDerivator.h.
