# Density, Probability, Quantile ('DPQ') Computations

## Documentation for package ‘DPQ’ version 0.6-1

- DESCRIPTION file.
- User guides, package vignettes and other documentation.
- Package NEWS.

## Help Pages

A B C D E F G H I L M N O P Q R S Z misc

DPQ-package | Density, Probability, Quantile ('DPQ') Computations

## -- A --

algdiv | Compute log(gamma(b)/gamma(a+b)) when b >= 8
all_mpfr | Numerical Utilities - Functions, Constants
any_mpfr | Numerical Utilities - Functions, Constants

## -- B --

bd0 | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
bd0C | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
bd0_l1pm | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
bd0_p1l1 | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
bd0_p1l1d | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
bd0_p1l1d1 | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
Bern | Bernoulli Numbers
betaI | (Log) Beta and Ratio of Gammas Approximations
bpser | 'pbeta()' 'bpser' series computation
b_chi | Compute E[chi_nu]/sqrt(nu) useful for t- and chi-Distributions
b_chiAsymp | Compute E[chi_nu]/sqrt(nu) useful for t- and chi-Distributions

## -- C --

chebyshevEval | Chebyshev Polynomial Evaluation
chebyshevPoly | Chebyshev Polynomial Evaluation
chebyshev_nc | Chebyshev Polynomial Evaluation
c_dt | Compute E[chi_nu]/sqrt(nu) useful for t- and chi-Distributions
c_dtAsymp | Compute E[chi_nu]/sqrt(nu) useful for t- and chi-Distributions
c_pt | Compute E[chi_nu]/sqrt(nu) useful for t- and chi-Distributions

## -- D --

dbinom_raw | R's C Mathlib (Rmath) dbinom_raw() Binomial Probability pure R Function
dchisqAsym | Approximations of the (Noncentral) Chi-Squared Density
dgamma.R | Gamma Density Function Alternatives
dhyperBinMolenaar | HyperGeometric (Point) Probabilities via Molenaar's Binomial Approximation
dltgammaInc | TOMS 1006 - Fast and Accurate Generalized Incomplete Gamma Function
dnbinom.mu | Pure R Versions of R's C (Mathlib) dnbinom() Negative Binomial Probabilities
dnbinomR | Pure R Versions of R's C (Mathlib) dnbinom() Negative Binomial Probabilities
dnchisqBessel | Approximations of the (Noncentral) Chi-Squared Density
dnchisqR | Approximations of the (Noncentral) Chi-Squared Density
dnoncentchisq | Approximations of the (Noncentral) Chi-Squared Density
dntJKBf | Non-central t-Distribution Density - Algorithms and Approximations
dntJKBf1 | Non-central t-Distribution Density - Algorithms and Approximations
dpois_raw | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
dpois_simpl | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
dpois_simpl0 | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
DPQ | Density, Probability, Quantile ('DPQ') Computations
dpsifn | Psi Gamma Functions Workhorse from R's API
dtWV | Asymptotic Noncentral t Distribution Density by Viechtbauer

## -- E --

ebd0 | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
ebd0C | Binomial Deviance - Auxiliary Functions for 'dgamma()' Etc
expm1x | Accurate exp(x) - 1 - x (for smallish |x|)
expm1xTser | Accurate exp(x) - 1 - x (for smallish |x|)

## -- F --

f05lchoose | R versions of Simple Formulas for Logarithmic Binomial Coefficients
format01prec | Format Numbers in [0,1] with "Precise" Result
frexp | Base-2 Representation and Multiplication of Numbers

## -- G --

g2 | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
gam1 | Compute 1/Gamma(x+1) - 1 Accurately
gam1d | Compute 1/Gamma(x+1) - 1 Accurately
gamln1 | Compute log( Gamma(x+1) ) Accurately in [-0.2, 1.25]
gamln1. | Compute log( Gamma(x+1) ) Accurately in [-0.2, 1.25]
gammaVer | Gamma Function Versions
gnt | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
G_half | Numerical Utilities - Functions, Constants

## -- H --

h | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
h0 | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
h1 | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
h2 | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
hnt | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
hyper2binomP | Transform Hypergeometric Distribution Parameters to Binomial Probability

## -- I --

Ixpq | Normalized Incomplete Beta Function "Like" 'pbeta()'

## -- L --

lbetaI | (Log) Beta and Ratio of Gammas Approximations
lbetaM | (Log) Beta and Ratio of Gammas Approximations
lbetaMM | (Log) Beta and Ratio of Gammas Approximations
lbeta_asy | (Log) Beta and Ratio of Gammas Approximations
lb_chi0 | Compute E[chi_nu]/sqrt(nu) useful for t- and chi-Distributions
lb_chi00 | Compute E[chi_nu]/sqrt(nu) useful for t- and chi-Distributions
lb_chiAsymp | Compute E[chi_nu]/sqrt(nu) useful for t- and chi-Distributions
ldexp | Base-2 Representation and Multiplication of Numbers
lfastchoose | R versions of Simple Formulas for Logarithmic Binomial Coefficients
lgamma1p | Accurate 'log(gamma(a+1))'
lgamma1p. | Accurate 'log(gamma(a+1))'
lgamma1pC | Accurate 'log(gamma(a+1))'
lgamma1p_series | Accurate 'log(gamma(a+1))'
lgammaAsymp | Asymptotic Log Gamma Function
lgammacor | Stirling's Error Function - Auxiliary for Gamma, Beta, etc
lgammaP11 | Log Gamma(p) for Positive 'p' by Pugh's Method (11 Terms)
log1mexp | Compute log(1 - exp(-a)) and log(1 + exp(x)) Numerically Optimally
log1mexpC | Compute log(1 - exp(-a)) and log(1 + exp(x)) Numerically Optimally
log1pexpC | Compute log(1 - exp(-a)) and log(1 + exp(x)) Numerically Optimally
log1pmx | Accurate 'log(1+x) - x' Computation
log1pmxC | Accurate 'log(1+x) - x' Computation
logcf | Continued Fraction Approximation of Log-Related Power Series
logcfR | Continued Fraction Approximation of Log-Related Power Series
logcfR_vec | Continued Fraction Approximation of Log-Related Power Series
logQab_asy | (Log) Beta and Ratio of Gammas Approximations
logr | Numerical Utilities - Functions, Constants
logspace.add | Logspace Arithmetix - Addition and Subtraction
logspace.sub | Logspace Arithmetix - Addition and Subtraction
lssum | Compute Logarithm of a Sum with Signed Large Summands
lsum | Properly Compute the Logarithm of a Sum (of Exponentials)

## -- M --

modf | Numerical Utilities - Functions, Constants
M_cutoff | Numerical Utilities - Functions, Constants
M_LN2 | Numerical Utilities - Functions, Constants
M_minExp | Numerical Utilities - Functions, Constants
M_SQRT2 | Numerical Utilities - Functions, Constants

## -- N --

newton | Simple R level Newton Algorithm, Mostly for Didactical Reasons

## -- O --

okLongDouble | Numerical Utilities - Functions, Constants

## -- P --

p1l1 | Numerically Stable p1l1(t) = (t+1)*log(1+t) - t
p1l1. | Numerically Stable p1l1(t) = (t+1)*log(1+t) - t
p1l1p | Numerically Stable p1l1(t) = (t+1)*log(1+t) - t
p1l1ser | Numerically Stable p1l1(t) = (t+1)*log(1+t) - t
pbetaAS_eq20 | 'pbeta()' Approximations of Abramowitz & Stegun, 26.5.\{20,21\}
pbetaAS_eq21 | 'pbeta()' Approximations of Abramowitz & Stegun, 26.5.\{20,21\}
pbetaNorm2 | 'pbeta()' Approximations of Abramowitz & Stegun, 26.5.\{20,21\}
pbetaRv1 | Pure R Implementation of Old pbeta()
pchisqV | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
pchisqW | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
pchisqW. | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
pchisqW.R | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
pdhyper | Pure R version of R's C level phyper()
phyper1molenaar | Molenaar's Normal Approximations to the Hypergeometric Distribution
phyper2molenaar | Molenaar's Normal Approximations to the Hypergeometric Distribution
phyperAllBin | Compute Hypergeometric Probabilities via Binomial Approximations
phyperAllBinM | Compute Hypergeometric Probabilities via Binomial Approximations
phyperApprAS152 | Normal Approximation to cumulative Hyperbolic Distribution - AS 152
phyperBin.1 | HyperGeometric Distribution via Approximate Binomial Distribution
phyperBin.2 | HyperGeometric Distribution via Approximate Binomial Distribution
phyperBin.3 | HyperGeometric Distribution via Approximate Binomial Distribution
phyperBin.4 | HyperGeometric Distribution via Approximate Binomial Distribution
phyperBinMolenaar | HyperGeometric Distribution via Molenaar's Binomial Approximation
phyperBinMolenaar.1 | HyperGeometric Distribution via Molenaar's Binomial Approximation
phyperBinMolenaar.2 | HyperGeometric Distribution via Molenaar's Binomial Approximation
phyperBinMolenaar.3 | HyperGeometric Distribution via Molenaar's Binomial Approximation
phyperBinMolenaar.4 | HyperGeometric Distribution via Molenaar's Binomial Approximation
phyperIbeta | Pearson's incomplete Beta Approximation to the Hyperbolic Distribution
phyperPeizer | Peizer's Normal Approximation to the Cumulative Hyperbolic
phyperR | R-only version of R's original phyper() algorithm
phyperR2 | Pure R version of R's C level phyper()
phypers | The Four (4) Symmetric 'phyper()' Calls
pl2curves | Plot 2 Noncentral Distribution Curves for Visual Comparison
plRpois | Compute Relative Size of i-th term of Poisson Distribution Series
pnbetaAppr2 | Noncentral Beta Probabilities
pnbetaAppr2v1 | Noncentral Beta Probabilities
pnbetaAS310 | Noncentral Beta Probabilities
pnchi1sq | (Probabilities of Non-Central Chi-squared Distribution for Special Cases
pnchi3sq | (Probabilities of Non-Central Chi-squared Distribution for Special Cases
pnchisq | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqAbdelAty | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqBolKuz | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqIT | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqPatnaik | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqPearson | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqRC | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqSankaran_d | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqT93 | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqT93.a | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqT93.b | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqTerms | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisqV | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnchisq_ss | (Approximate) Probabilities of Non-Central Chi-squared Distribution
pnormAsymp | Asymptotic Approxmation of (Extreme Tail) 'pnorm()'
pnormL_LD10 | Bounds for 1-Phi(.) - Mill's Ratio related Bounds for pnorm()
pnormU_S53 | Bounds for 1-Phi(.) - Mill's Ratio related Bounds for pnorm()
pnt3150 | Non-central t Probability Distribution - Algorithms and Approximations
pnt3150.1 | Non-central t Probability Distribution - Algorithms and Approximations
pntChShP94 | Non-central t Probability Distribution - Algorithms and Approximations
pntChShP94.1 | Non-central t Probability Distribution - Algorithms and Approximations
pntGST23_1 | Non-central t Probability Distribution - Algorithms and Approximations
pntGST23_T6 | Non-central t Probability Distribution - Algorithms and Approximations
pntGST23_T6.1 | Non-central t Probability Distribution - Algorithms and Approximations
pntJW39 | Non-central t Probability Distribution - Algorithms and Approximations
pntJW39.0 | Non-central t Probability Distribution - Algorithms and Approximations
pntLrg | Non-central t Probability Distribution - Algorithms and Approximations
pntP94 | Non-central t Probability Distribution - Algorithms and Approximations
pntP94.1 | Non-central t Probability Distribution - Algorithms and Approximations
pntR | Non-central t Probability Distribution - Algorithms and Approximations
pntR1 | Non-central t Probability Distribution - Algorithms and Approximations
pntVW13 | Non-central t Probability Distribution - Algorithms and Approximations
pow | X to Power of Y - R C API 'R_pow()'
pow1p | Accurate (1+x)^y, notably for small |x|
pow_di | X to Power of Y - R C API 'R_pow()'
ppoisD | Direct Computation of 'ppois()' Poisson Distribution Probabilities
ppoisErr | Direct Computation of 'ppois()' Poisson Distribution Probabilities
pt_Witkovsky_Tab1 | Viktor Witosky's Table_1 pt() Examples

## -- Q --

Qab_terms | (Log) Beta and Ratio of Gammas Approximations
qbeta.R | Compute (Approximate) Quantiles of the Beta Distribution
qbetaAppr | Compute (Approximate) Quantiles of the Beta Distribution
qbetaAppr.1 | Compute (Approximate) Quantiles of the Beta Distribution
qbetaAppr.2 | Compute (Approximate) Quantiles of the Beta Distribution
qbetaAppr.3 | Compute (Approximate) Quantiles of the Beta Distribution
qbetaAppr.4 | Compute (Approximate) Quantiles of the Beta Distribution
qbinomR | Pure R Implementation of R's qbinom() with Tuning Parameters
qchisqAppr | Compute Approximate Quantiles of the Chi-Squared Distribution
qchisqAppr.0 | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qchisqAppr.1 | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qchisqAppr.2 | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qchisqAppr.3 | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qchisqAppr.R | Compute Approximate Quantiles of the Chi-Squared Distribution
qchisqApprCF1 | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qchisqApprCF2 | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qchisqCappr.2 | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qchisqKG | Compute Approximate Quantiles of the Chi-Squared Distribution
qchisqN | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qchisqWH | Compute Approximate Quantiles of the Chi-Squared Distribution
qgamma.R | Compute (Approximate) Quantiles of the Gamma Distribution
qgammaAppr | Compute (Approximate) Quantiles of the Gamma Distribution
qgammaApprKG | Compute (Approximate) Quantiles of the Gamma Distribution
qgammaApprSmallP | Compute (Approximate) Quantiles of the Gamma Distribution
qnbinomR | Pure R Implementation of R's qnbinom() with Tuning Parameters
qnchisqAbdelAty | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qnchisqAppr | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qnchisqBolKuz | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qnchisqPatnaik | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qnchisqPearson | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qnchisqSankaran_d | Compute Approximate Quantiles of Noncentral Chi-Squared Distribution
qnormAppr | Approximations to 'qnorm()', i.e., z_alpha
qnormAsymp | Asymptotic Approximation to Outer Tail of qnorm()
qnormCappr | Approximations to 'qnorm()', i.e., z_alpha
qnormR | Pure R version of R's 'qnorm()' with Diagnostics and Tuning Parameters
qnormR1 | Pure R version of R's 'qnorm()' with Diagnostics and Tuning Parameters
qnormUappr | Approximations to 'qnorm()', i.e., z_alpha
qnormUappr6 | Approximations to 'qnorm()', i.e., z_alpha
qntR | Pure R Implementation of R's qt() / qnt()
qntR1 | Pure R Implementation of R's qt() / qnt()
qpoisR | Pure R Implementation of R's qpois() with Tuning Parameters
qs | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
qtAppr | Compute Approximate Quantiles of the (Non-Central) t-Distribution
qtNappr | Compute Approximate Quantiles of the (Non-Central) t-Distribution
qtR | Pure R Implementation of R's C-level t-Distribution Quantiles 'qt()'
qtR1 | Pure R Implementation of R's C-level t-Distribution Quantiles 'qt()'
qtU | 'uniroot()'-based Computing of t-Distribution Quantiles
qtU1 | 'uniroot()'-based Computing of t-Distribution Quantiles

## -- R --

rexpm1 | TOMS 708 Approximation REXP(x) of expm1(x) = exp(x) - 1
rlog1 | Accurate 'log(1+x) - x' Computation
r_pois | Compute Relative Size of i-th term of Poisson Distribution Series
r_pois_expr | Compute Relative Size of i-th term of Poisson Distribution Series

## -- S --

scalefactor | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
ss | (Approximate) Probabilities of Non-Central Chi-squared Distribution
ss2 | (Approximate) Probabilities of Non-Central Chi-squared Distribution
ss2. | (Approximate) Probabilities of Non-Central Chi-squared Distribution
stirlerr | Stirling's Error Function - Auxiliary for Gamma, Beta, etc
stirlerrC | Stirling's Error Function - Auxiliary for Gamma, Beta, etc
stirlerr_simpl | Stirling's Error Function - Auxiliary for Gamma, Beta, etc
sW | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities

## -- Z --

z.f | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
z.s | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities
z0 | Wienergerm Approximations to (Non-Central) Chi-squared Probabilities

## -- misc --

.dntJKBch | Non-central t-Distribution Density - Algorithms and Approximations
.dntJKBch1 | Non-central t-Distribution Density - Algorithms and Approximations
.DT_0 | Distribution Utilities "dpq"
.DT_1 | Distribution Utilities "dpq"
.DT_Cexp | Distribution Utilities "dpq"
.DT_CIv | Distribution Utilities "dpq"
.DT_Clog | Distribution Utilities "dpq"
.DT_Cval | Distribution Utilities "dpq"
.DT_exp | Distribution Utilities "dpq"
.DT_Log | Distribution Utilities "dpq"
.DT_log | Distribution Utilities "dpq"
.DT_qIv | Distribution Utilities "dpq"
.DT_val | Distribution Utilities "dpq"
.D_0 | Distribution Utilities "dpq"
.D_1 | Distribution Utilities "dpq"
.D_Clog | Distribution Utilities "dpq"
.D_Cval | Distribution Utilities "dpq"
.D_exp | Distribution Utilities "dpq"
.D_LExp | Distribution Utilities "dpq"
.D_log | Distribution Utilities "dpq"
.D_Lval | Distribution Utilities "dpq"
.D_qIv | Distribution Utilities "dpq"
.D_val | Distribution Utilities "dpq"
.p1l1ser | Numerically Stable p1l1(t) = (t+1)*log(1+t) - t
.pbeta.eq20 | 'pbeta()' Approximations of Abramowitz & Stegun, 26.5.\{20,21\}
.pbeta.eq21 | 'pbeta()' Approximations of Abramowitz & Stegun, 26.5.\{20,21\}
.pbetaSeq20 | 'pbeta()' Approximations of Abramowitz & Stegun, 26.5.\{20,21\}
.pbetaSeq21 | 'pbeta()' Approximations of Abramowitz & Stegun, 26.5.\{20,21\}
.pow | X to Power of Y - R C API 'R_pow()'
.qgammaApprBnd | Compute (Approximate) Quantiles of the Gamma Distribution
.suppHyper | Compute Hypergeometric Probabilities via Binomial Approximations