# Moments and Distributions of Ratios of Quadratic Forms Using Recursion

## Documentation for package ‘qfratio’ version 1.1.1

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A D G H I K M N P Q R S T

qfratio-package | qfratio: Moments and Distributions of Ratios of Quadratic Forms

## -- A --

a1_pk | Recursion for a_{p,k}
ABDpqr_int_E | Internal C++ functions
ABpq_int_E | Internal C++ functions
ApBDqr_int_Ec | Internal C++ functions
ApBDqr_int_Ed | Internal C++ functions
ApBDqr_int_El | Internal C++ functions
ApBDqr_npi_Ec | Internal C++ functions
ApBDqr_npi_Ed | Internal C++ functions
ApBDqr_npi_El | Internal C++ functions
ApBIqr_int_cEd | Internal C++ functions
ApBIqr_int_nEc | Internal C++ functions
ApBIqr_int_nEd | Internal C++ functions
ApBIqr_int_nEl | Internal C++ functions
ApBIqr_npi_Ec | Internal C++ functions
ApBIqr_npi_Ed | Internal C++ functions
ApBIqr_npi_El | Internal C++ functions
ApBq_int_E | Internal C++ functions
ApBq_npi_Ec | Internal C++ functions
ApBq_npi_Ed | Internal C++ functions
ApBq_npi_El | Internal C++ functions
ApIq_int_cE | Internal C++ functions
ApIq_int_nE | Internal C++ functions
ApIq_npi_cE | Internal C++ functions
ApIq_npi_nEc | Internal C++ functions
ApIq_npi_nEd | Internal C++ functions
ApIq_npi_nEl | Internal C++ functions
Ap_int_E | Internal C++ functions

## -- D --

d1_i | Coefficients in polynomial expansion of generating function-single matrix
d2_1j | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d2_1j_m | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d2_1j_v | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d2_ij | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d2_ij_m | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d2_ij_v | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d2_pj | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d2_pj_m | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d2_pj_v | Coefficients in polynomial expansion of generating function-for ratios with two matrices
d3_ijk | Coefficients in polynomial expansion of generating function-for ratios with three matrices
d3_ijk_m | Coefficients in polynomial expansion of generating function-for ratios with three matrices
d3_ijk_v | Coefficients in polynomial expansion of generating function-for ratios with three matrices
d3_pjk | Coefficients in polynomial expansion of generating function-for ratios with three matrices
d3_pjk_m | Coefficients in polynomial expansion of generating function-for ratios with three matrices
d3_pjk_v | Coefficients in polynomial expansion of generating function-for ratios with three matrices
dqfr | Probability distribution of ratio of quadratic forms
dqfr_A1I1 | Probability distribution of ratio of quadratic forms
dqfr_broda | Probability distribution of ratio of quadratic forms
dqfr_butler | Probability distribution of ratio of quadratic forms
dtil1_i | Coefficients in polynomial expansion of generating function-single matrix
dtil1_i_m | Coefficients in polynomial expansion of generating function-single matrix
dtil1_i_v | Coefficients in polynomial expansion of generating function-single matrix
dtil2_1q_m | Coefficients in polynomial expansion of generating function-for products
dtil2_1q_v | Coefficients in polynomial expansion of generating function-for products
dtil2_pq | Coefficients in polynomial expansion of generating function-for products
dtil2_pq_m | Coefficients in polynomial expansion of generating function-for products
dtil2_pq_v | Coefficients in polynomial expansion of generating function-for products
dtil3_pqr | Coefficients in polynomial expansion of generating function-for products
dtil3_pqr_m | Coefficients in polynomial expansion of generating function-for products
dtil3_pqr_v | Coefficients in polynomial expansion of generating function-for products
d_A1I1_Ed | Internal C++ functions
d_broda_Ed | Internal C++ functions
d_butler_Ed | Internal C++ functions

## -- G --

gen_eig | Get range of ratio of quadratic forms
gsl_wrap | Internal C++ wrappers for GSL

## -- H --

h2_ij | Coefficients in polynomial expansion of generating function-for ratios with two matrices
h2_ij_m | Coefficients in polynomial expansion of generating function-for ratios with two matrices
h2_ij_v | Coefficients in polynomial expansion of generating function-for ratios with two matrices
h3_ijk | Coefficients in polynomial expansion of generating function-for ratios with three matrices
h3_ijk_m | Coefficients in polynomial expansion of generating function-for ratios with three matrices
h3_ijk_v | Coefficients in polynomial expansion of generating function-for ratios with three matrices
hgs | Calculate hypergeometric series
hgs_1d | Calculate hypergeometric series
hgs_2d | Calculate hypergeometric series
hgs_3d | Calculate hypergeometric series
hhat2_1j | Coefficients in polynomial expansion of generating function-for ratios with two matrices
hhat2_1j_m | Coefficients in polynomial expansion of generating function-for ratios with two matrices
hhat2_1j_v | Coefficients in polynomial expansion of generating function-for ratios with two matrices
hhat2_pj | Coefficients in polynomial expansion of generating function-for ratios with two matrices
hhat2_pj_m | Coefficients in polynomial expansion of generating function-for ratios with two matrices
hhat2_pj_v | Coefficients in polynomial expansion of generating function-for ratios with two matrices
hhat3_pjk | Coefficients in polynomial expansion of generating function-for ratios with three matrices
hhat3_pjk_m | Coefficients in polynomial expansion of generating function-for ratios with three matrices
hhat3_pjk_v | Coefficients in polynomial expansion of generating function-for ratios with three matrices
htil2_1j | Coefficients in polynomial expansion of generating function-for ratios with two matrices
htil2_1j_m | Coefficients in polynomial expansion of generating function-for ratios with two matrices
htil2_1j_v | Coefficients in polynomial expansion of generating function-for ratios with two matrices
htil2_pj | Coefficients in polynomial expansion of generating function-for ratios with two matrices
htil2_pj_m | Coefficients in polynomial expansion of generating function-for ratios with two matrices
htil2_pj_v | Coefficients in polynomial expansion of generating function-for ratios with two matrices
htil3_pjk | Coefficients in polynomial expansion of generating function-for ratios with three matrices
htil3_pjk_m | Coefficients in polynomial expansion of generating function-for ratios with three matrices
htil3_pjk_v | Coefficients in polynomial expansion of generating function-for ratios with three matrices
hyperg_1F1_vec_b | Internal C++ wrappers for GSL
hyperg_2F1_mat_a_vec_c | Internal C++ wrappers for GSL

## -- I --

IpBDqr_gen_Ec | Internal C++ functions
IpBDqr_gen_Ed | Internal C++ functions
IpBDqr_gen_El | Internal C++ functions
iseq | Are these vectors equal?
is_diagonal | Is this matrix diagonal?

## -- K --

KiK | Matrix square root and generalized inverse

## -- M --

methods.qfrm | Methods for qfrm and qfpm objects

## -- N --

new_qfpm | Construct qfrm object
new_qfrm | Construct qfrm object

## -- P --

plot.qfrm | Methods for qfrm and qfpm objects
pqfr | Probability distribution of ratio of quadratic forms
pqfr_A1B1 | Probability distribution of ratio of quadratic forms
pqfr_butler | Probability distribution of ratio of quadratic forms
pqfr_davies | Probability distribution of ratio of quadratic forms
pqfr_imhof | Probability distribution of ratio of quadratic forms
print.qfpm | Methods for qfrm and qfpm objects
print.qfrm | Methods for qfrm and qfpm objects
p_A1B1_Ec | Internal C++ functions
p_A1B1_Ed | Internal C++ functions
p_A1B1_El | Internal C++ functions
p_butler_Ed | Internal C++ functions
p_imhof_Ed | Internal C++ functions

## -- Q --

qfmrm | Moment of multiple ratio of quadratic forms in normal variables
qfmrm_ApBDqr_int | Moment of multiple ratio of quadratic forms in normal variables
qfmrm_ApBDqr_npi | Moment of multiple ratio of quadratic forms in normal variables
qfmrm_ApBIqr_int | Moment of multiple ratio of quadratic forms in normal variables
qfmrm_ApBIqr_npi | Moment of multiple ratio of quadratic forms in normal variables
qfmrm_IpBDqr_gen | Moment of multiple ratio of quadratic forms in normal variables
qfm_Ap_int | Moment of (product of) quadratic forms in normal variables
qfpm | Moment of (product of) quadratic forms in normal variables
qfpm_ABDpqr_int | Moment of (product of) quadratic forms in normal variables
qfpm_ABpq_int | Moment of (product of) quadratic forms in normal variables
qfrm | Moment of ratio of quadratic forms in normal variables
qfrm_ApBq_int | Moment of ratio of quadratic forms in normal variables
qfrm_ApBq_npi | Moment of ratio of quadratic forms in normal variables
qfrm_ApIq_int | Moment of ratio of quadratic forms in normal variables
qfrm_ApIq_npi | Moment of ratio of quadratic forms in normal variables
qfrm_cpp | Internal C++ functions
qqfr | Probability distribution of ratio of quadratic forms

## -- R --

range_qfr | Get range of ratio of quadratic forms
rqfmr | Monte Carlo sampling of ratio/product of quadratic forms
rqfp | Monte Carlo sampling of ratio/product of quadratic forms
rqfpE | Internal C++ functions
rqfr | Monte Carlo sampling of ratio/product of quadratic forms

## -- S --

sum_counterdiag | Summing up counter-diagonal elements
sum_counterdiag3D | Summing up counter-diagonal elements
S_fromUL | Make covariance matrix from eigenstructure

## -- T --

tr | Matrix trace function