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About the Math Toolkit
abscissa
absolute_gini_coefficient
accuracy
Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values
Graphing, Profiling, and Generating Test Data for Special Functions
Handling functions with large features near an endpoint with tanh-sinh quadrature
Setting the Maximum Interval Halvings and Memory Requirements
Some Miscellaneous Examples of the Normal (Gaussian) Distribution
Using With MPFR or GMP - High-Precision Floating-Point Library
Why use a high-precision library rather than built-in floating-point types?
acosh
acoshf
acoshl
Advancing a floating-point Value by a Specific Representation Distance (ULP) float_advance
Ai
Airy Ai Function
Airy Ai' Function
Airy Bi Function
Airy Bi' Function
airy_ai
airy_ai_prime
airy_ai_zero
airy_bi
airy_bi_prime
airy_bi_zero
al
Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions
arcsine
Arcsine Distribution
area
arguments
asinh
asinhf
asinhl
assert
assert_undefined_type
assoc_laguerre
assoc_laguerref
assoc_laguerrel
assoc_legendre
assoc_legendref
assoc_legendrel
atanh
atanhf
atanhl
Barycentric Rational Interpolation
barycentric_rational
bernoulli
Bernoulli Distribution
Bernoulli Numbers
bernoulli_b2n
Bessel Functions of the First and Second Kinds
Beta
beta
Beta Distribution
betac
betaf
betal
binomial
Binomial Coefficients
Binomial Coin-Flipping Example
Binomial Distribution
Binomial Quiz Example
binomial_coefficient
bisect
Bisection
Bivariate Statistics
bool
Boost.Math Macros
Boost.Math Tuning
BOOST_DEFINE_MATH_CONSTANT
BOOST_FLOAT128_C
BOOST_FLOAT16_C
BOOST_FLOAT32_C
BOOST_FLOAT64_C
BOOST_FLOAT80_C
BOOST_FLOATMAX_C
BOOST_FPU_EXCEPTION_GUARD
BOOST_HAS_LOG1P
BOOST_MATH_ASSERT_UNDEFINED_POLICY
BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS
BOOST_MATH_CONTROL_FP
BOOST_MATH_DECLARE_DISTRIBUTIONS
BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS
BOOST_MATH_DENORM_ERROR_POLICY
BOOST_MATH_DIGITS10_POLICY
BOOST_MATH_DISABLE_FLOAT128
BOOST_MATH_DISCRETE_QUANTILE_POLICY
BOOST_MATH_DOMAIN_ERROR_POLICY
BOOST_MATH_EVALUATION_ERROR_POLICY
BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC
BOOST_MATH_INDETERMINATE_RESULT_ERROR_POLICY
BOOST_MATH_INSTRUMENT_CODE
BOOST_MATH_INSTRUMENT_FPU
BOOST_MATH_INSTRUMENT_VARIABLE
BOOST_MATH_INT_TABLE_TYPE
BOOST_MATH_INT_VALUE_SUFFIX
BOOST_MATH_MAX_POLY_ORDER
BOOST_MATH_MAX_ROOT_ITERATION_POLICY
BOOST_MATH_MAX_SERIES_ITERATION_POLICY
BOOST_MATH_NO_ATOMIC_INT
BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS
BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
BOOST_MATH_NO_REAL_CONCEPT_TESTS
BOOST_MATH_OVERFLOW_ERROR_POLICY
BOOST_MATH_POLE_ERROR_POLICY
BOOST_MATH_POLY_METHOD
BOOST_MATH_PROMOTE_DOUBLE_POLICY
BOOST_MATH_PROMOTE_FLOAT_POLICY
BOOST_MATH_RATIONAL_METHOD
BOOST_MATH_ROUNDING_ERROR_POLICY
BOOST_MATH_SMALL_CONSTANT
BOOST_MATH_STD_USING
BOOST_MATH_TEST_VALUE
BOOST_MATH_UNDERFLOW_ERROR_POLICY
BOOST_MATH_USE_C99
BOOST_MATH_USE_FLOAT128
bracket_and_solve_root
branch
brent_find_minima
by
c
C99 and C++ TR1 C-style Functions
C99 and TR1 C Functions Overview
C99 C Functions
Calculation of the Type of the Result
called
Calling User Defined Error Handlers
cancel
case
Catmull-Rom Splines
catmull_rom
cauchy
Cauchy-Lorentz Distribution
cauchy_distribution
Caveats
cbrt
cbrtf
cbrtl
cdf
Cephes
changesign
Changing the Policy Defaults
Changing the Policy on an Ad Hoc Basis for the Special Functions
Chebyshev Polynomials
chebyshev_clenshaw_recurrence
chebyshev_next
chebyshev_t
chebyshev_transform
chebyshev_t_prime
chebyshev_u
checked_narrowing_cast
chf
Chi Squared Distribution
Chi-Square Test for the Standard Deviation
chi_squared
Comparing Different Compilers
Comparison of Cube Root Finding Algorithms
Comparison of Elliptic Integral Root Finding Algoritghms
Comparison of Nth-root Finding Algorithms
Comparison with C, R, FORTRAN-style Free Functions
Comparisons to Other Open Source Libraries
Compile Time Power of a Runtime Base
Compilers
complement
Complements are supported too - and when to use them
complex
complex_newton
complex_step_derivative
compute
Computing the Fifth Root
comp_ellint_1
comp_ellint_1f
comp_ellint_1l
comp_ellint_2
comp_ellint_2f
comp_ellint_2l
comp_ellint_3
comp_ellint_3f
comp_ellint_3l
Conceptual Requirements for Distribution Types
Conceptual Requirements for Real Number Types
Condition Numbers
confidence intervals on the mean with the Students-t distribution
Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution
conf_hyperg
conf_hypergf
conf_hypergl
conj
constants
constexpr
construction_traits
constructors
Continued Fraction Evaluation
continued_fraction_a
continued_fraction_b
conventions
converge
convergence
copysign
copysignf
copysignl
correlation_coefficient
Cost of Finite-Difference Numerical Differentiation
covariance
Credits and Acknowledgements
Cubic B-spline interpolation
cubic_b_spline
Cyclic Hankel Functions
cylindrical
cylindrospherical
cyl_bessel_i
cyl_bessel_if
cyl_bessel_il
cyl_bessel_i_prime
cyl_bessel_j
cyl_bessel_jl
cyl_bessel_j_prime
cyl_bessel_j_zero
cyl_bessel_k
cyl_bessel_kf
cyl_bessel_kl
cyl_bessel_k_prime
cyl_hankel_1
cyl_hankel_2
cyl_neumann
cyl_neumannl
cyl_neumann_prime
cyl_neumann_zero
D
data
DCDFLIB
default_policy
defined
Defining New Constants
denorm_error_type
Derivative of the Incomplete Beta Function
Derivative of the Incomplete Gamma Function
Derivatives of the Bessel Functions
Digamma
Directory and File Structure
Discrete Probability Distributions
Discrete Quantile Policies
discrete_lanczos_derivative
discrete_quantile_type
distribution
Distribution Algorithms
Distribution Construction Examples
Distributions are Objects
Document Conventions
domain_error_type
double
Double Factorial
double_t
ellint_1
ellint_1f
ellint_1l
ellint_2
ellint_2f
ellint_2l
ellint_3
ellint_3f
ellint_3l
ellint_d
ellint_rc
ellint_rd
ellint_rf
ellint_rg
ellint_rj
Elliptic Integral D - Legendre Form
Elliptic Integral Overview
Elliptic Integrals - Carlson Form
Elliptic Integrals of the First Kind - Legendre Form
Elliptic Integrals of the Second Kind - Legendre Form
Elliptic Integrals of the Third Kind - Legendre Form
epsilon
epsilon_difference
eps_tolerance
equal_ceil
equal_floor
equal_nearest_integer
erf
erfc
erfcf
erfcl
erfc_inv
erff
erfl
erf_inv
Error Function Inverses
Error Functions
Error Handling
Error Handling Example
Error Handling Policies
Error Logs For Error Rate Tables
Error rates for beta
Error rates for beta (incomplete)
Error rates for boost::math::powm1
Error rates for cyl_bessel_i
Error rates for cyl_bessel_i (integer orders)
Error rates for cyl_bessel_i_prime (integer orders)
Error rates for cyl_bessel_j
Error rates for cyl_bessel_j (integer orders)
Error rates for cyl_bessel_j_prime (integer orders)
Error rates for cyl_bessel_k
Error rates for cyl_bessel_k (integer orders)
Error rates for cyl_bessel_k_prime (integer orders)
Error rates for cyl_neumann
Error rates for cyl_neumann (integer orders)
Error rates for cyl_neumann_prime (integer orders)
Error rates for digamma
Error rates for ellint_1
Error rates for ellint_1 (complete)
Error rates for ellint_2
Error rates for ellint_2 (complete)
Error rates for ellint_3
Error rates for ellint_3 (complete)
Error rates for ellint_d
Error rates for ellint_d (complete)
Error rates for ellint_rc
Error rates for ellint_rd
Error rates for ellint_rf
Error rates for ellint_rg
Error rates for ellint_rj
Error rates for erf
Error rates for erfc
Error rates for expint (Ei)
Error rates for expint (En)
Error rates for expm1
Error rates for gamma_p
Error rates for gamma_p_inv
Error rates for gamma_q
Error rates for gamma_q_inv
Error rates for ibeta
Error rates for ibetac
Error rates for ibetac_inv
Error rates for ibeta_inv
Error rates for jacobi_cn
Error rates for jacobi_dn
Error rates for jacobi_sn
Error rates for laguerre(n, m, x)
Error rates for laguerre(n, x)
Error rates for legendre_p
Error rates for legendre_p (associated)
Error rates for legendre_q
Error rates for lgamma
Error rates for log1p
Error rates for non central beta CDF
Error rates for non central beta CDF complement
Error rates for non central chi squared CDF
Error rates for non central chi squared CDF complement
Error rates for non central t CDF
Error rates for non central t CDF complement
Error rates for owens_t
Error rates for polygamma
Error rates for sph_bessel
Error rates for sph_neumann
Error rates for tgamma
Error rates for tgamma (incomplete)
Error rates for tgamma1pm1
Error rates for tgamma_lower
Error rates for trigamma
Error rates for zeta
Estimating Sample Sizes for a Binomial Distribution.
Estimating Sample Sizes for the Negative Binomial.
Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation
eta
evaluate_even_polynomial
evaluate_odd_polynomial
evaluate_polynomial
evaluate_rational
evaluation_condition_number
evaluation_error_type
Exact-Width Floating-Point typedef s
Examples
Examples of Root-Finding (with and without derivatives)
Examples Where Root Finding Goes Wrong
excess_kurtosis
exp2
exp2f
exp2l
expint
expintf
expintl
expint_as_fraction
expint_fraction
expm1
expm1f
expm1l
exponential
Exponential Distribution
Exponential Integral Ei
Exponential Integral En
expression
Calculating confidence intervals on the mean with the Students-t distribution
Finding Zeros of Bessel Functions of the First and Second Kinds
Handling functions with large features near an endpoint with tanh-sinh quadrature
Using With MPFR or GMP - High-Precision Floating-Point Library
Using without expression templates for Boost.Test and others
exp_sinh
Extras/Future Directions
Extreme Value Distribution
extreme_value
e_float
F Distribution
Facets for Floating-Point Infinities and NaNs
Factorial
Falling Factorial
FAQs
fdim
fdimf
fdiml
Find Location (Mean) Example
Find mean and standard deviation example
Find Scale (Standard Deviation) Example
Finding the Cubed Root With and Without Derivatives
Finding the Next Greater Representable Value (float_next)
Finding the Next Representable Value in a Specific Direction (nextafter)
Finding the Next Smaller Representable Value (float_prior)
Finding Zeros of Airy Functions
Finding Zeros of Bessel Functions of the First and Second Kinds
find_alpha
find_degrees_of_freedom
find_location
find_lower_bound_on_p
find_non_centrality
find_scale
find_upper_bound_on_p
finite_difference_derivative
first_four_moments
fisher_f
fisher_f_distribution
float
Floating-Point Classification: Infinities and NaNs
Floating-point Comparison
Floating-Point Constant Macros
Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values
float_advance
float_distance
float_next
float_prior
float_t
float_type
fma
fmaf
fmal
fmax
fmaxf
fmaxl
fmin
fminf
fminl
forwarding_policy
fpclassify
FP_INFINITE
FP_NAN
FP_NORMAL
FP_SUBNORMAL
FP_ZERO
freedom
Frequently Asked Questions FAQ
function
functions
G
Gamma
gamma
Gamma (and Erlang) Distribution
gamma_distribution
gamma_p
gamma_p_derivative
gamma_p_inv
gamma_p_inva
gamma_q
gamma_q_inv
gamma_q_inva
gauss
Gauss-Kronrod Quadrature
Gauss-Legendre quadrature
gauss_kronrod
Generalizing to Compute the nth root
Generic operations common to all distributions are non-member functions
geometric
Geometric Distribution
Geometric Distribution Examples
get
Getting the Best Performance from this Library: Compiler and Compiler Options
get_epsilon
get_from_string
gini_coefficient
Graphing, Profiling, and Generating Test Data for Special Functions
Greatest-width floating-point typedef
GSL
halley_iterate
hamming_distance
Handling functions with large features near an endpoint with tanh-sinh quadrature
hazard
hermite
Hermite Polynomials
hermitef
hermitel
hermite_next
Heuman Lambda Function
heuman_lambda
Hints on using float128 (and __float128)
History and What's New
hoyer_sparsity
hyperexponential
Hyperexponential Distribution
hyperg
hypergeometric
Hypergeometric Distribution
hypergf
hypergl
hypot
hypotf
hypotl
i
ibeta
ibetac
ibetac_inv
ibetac_inva
ibetac_invb
ibeta_derivative
ibeta_inv
ibeta_inva
ibeta_invb
ilogb
ilogbf
ilogbl
Implementation
Implementation and Accuracy
Implementation Notes
Incomplete Beta Function Inverses
Incomplete Beta Functions
Incomplete Gamma Function Inverses
Incomplete Gamma Functions
indeterminate_result_error_type
infinity
integrate
Internal Floating-point Promotion Policies
Interpreting these Results
interval
Introduction
inverse
Inverse Chi Squared Distribution
Inverse Chi-Squared Distribution Bayes Example
Inverse Gamma Distribution
Inverse Gaussian (or Inverse Normal) Distribution
inverse_chi_squared
inverse_gaussian
inverse_gaussian_distribution
iround
isfinite
isinf
isnan
isnormal
Iteration Limits Policies
itrunc
J
Jacobi Elliptic Function cd
Jacobi Elliptic Function cn
Jacobi Elliptic Function cs
Jacobi Elliptic Function dc
Jacobi Elliptic Function dn
Jacobi Elliptic Function ds
Jacobi Elliptic Function nc
Jacobi Elliptic Function nd
Jacobi Elliptic Function ns
Jacobi Elliptic Function sc
Jacobi Elliptic Function sd
Jacobi Elliptic Function sn
Jacobi Elliptic SN, CN and DN
Jacobi Zeta Function
jacobi_cd
jacobi_cn
jacobi_cs
jacobi_dc
jacobi_dn
jacobi_ds
jacobi_elliptic
jacobi_nc
jacobi_nd
jacobi_ns
jacobi_sc
jacobi_sd
jacobi_sn
jacobi_zeta
k
kahan_sum_series
Known Issues, and TODO List
kurtosis
kurtosis_excess
l
l0_pseudo_norm
l1
l1_distance
l1_norm
l2_distance
l2_norm
laguerre
Laguerre (and Associated) Polynomials
laguerref
laguerrel
laguerre_next
Lambert W function
lambert_w0
lambert_w0_prime
lambert_wm1
lambert_wm1_prime
Lanczos approximation
Lanczos Smoothing Derivatives
laplace
Laplace Distribution
ldexp
legendre
Legendre (and Associated) Polynomials
Legendre-Stieltjes Polynomials
legendref
legendrel
legendre_next
legendre_p
legendre_p_prime
legendre_p_zeros
legendre_q
legendre_stieltjes
lgamma
lgammaf
lgammal
Library Comparison with Microsoft Visual C++ version 14.0 on Windows x64
llrint
llrintf
llrintl
llround
llroundf
llroundl
lltrunc
Locating Function Minima using Brent's algorithm
location
Log Gamma
Log Normal Distribution
log1p
log1pf
log1pl
log1p_series
log2
log2f
log2l
logb
logbf
logbl
logistic
Logistic Distribution
lognormal
lognormal_distribution
lookup
lookup_t
lp_distance
lp_norm
lrint
lrintf
lrintl
lround
lroundf
lroundl
ltrunc
m
m2m4_snr_estimator
m2m4_snr_estimator_db
Mathematical Constants
Mathematically Undefined Function Policies
max_factorial
mean
means_and_covariance
median
median_absolute_deviation
message
min
Minimax Approximations and the Remez Algorithm
mode
Modified Bessel Functions of the First and Second Kinds
More complex example - Inverting the Elliptic Integrals
msg
multipolar
n
Naive Monte Carlo Integration
naive_monte_carlo
Namespaces
nan
nanf
nanl
nearbyint
nearbyintf
nearbyintl
Negative Binomial Distribution
Negative Binomial Sales Quota Example.
negative_binomial
newton_raphson_iterate
nextafter
nextafterf
nextafterl
nexttoward
nexttowardf
nexttowardl
Non-Member Properties
Noncentral Beta Distribution
Noncentral Chi-Squared Distribution
Noncentral F Distribution
Noncentral T Distribution
nonfinite_num_get
nonfinite_num_put
non_central_beta
non_central_beta_distribution
non_central_chi_squared
non_central_chi_squared_distribution
non_central_f
non_central_f_distribution
non_central_t
non_central_t_distribution
norm
normal
Normal (Gaussian) Distribution
normalise
normal_distribution
Norms
Numerical Differentiation
O
Obtaining the Size of a Unit In the Last Place - ULP
octonion
Octonion Creation Functions
Octonion Member Functions
Octonion Member Typedefs
Octonion Non-Member Operators
Octonion Specializations
Octonion Value Operations
operator
oracle_snr
oracle_snr_db
overflow_error_type
Overvew of the Jacobi Elliptic Functions
Overview
Owen's T function
owens_t
P
pareto
Pareto Distribution
performance
Performance Overview
Performance Test Applications
Performance Tuning Macros
point
poisson
Poisson Distribution
polar
pole_error_type
Policies
Policy Class Reference
Policy Overview
policy_type
Polygamma
Polynomial and Rational Function Evaluation
Polynomial Method Comparison with Microsoft Visual C++ version 14.0 on Windows x64
Polynomials
powm1
Precision Policies
precision_type
prime
Prime Numbers
promote_args
promote_double_type
promote_float_type
quad
quadratic_roots
quantile
quaternion
Quaternion Creation Functions
Quaternion Member Functions
Quaternion Member Typedefs
Quaternion Non-Member Operators
Quaternion Specializations
Quaternion Value Operations
r
Random Variates and Distribution Parameters
range
Rational Method Comparison with Microsoft Visual C++ version 14.0 on Windows x64
Ratios of Gamma Functions
rayleigh
Rayleigh Distribution
Reference
References
refinement
Relative Error
Relative Error and Testing
relative_difference
remainder
remainderf
remainderl
Remez Method
remquo
remquof
remquol
Representation Distance Between Two floating-point Values (ULP) float_distance
resize
return_x
return_y
Riemann Zeta Function
riemann_zeta
riemann_zetaf
riemann_zetal
rint
rintf
rintl
Rising Factorial
Rmath
Root Finding With Derivatives: Newton-Raphson, Halley & Schröder
Root Finding Without Derivatives
Root-finding using Boost.Multiprecision
round
roundf
Rounding Functions
rounding_error_type
roundl
sample_absolute_gini_coefficient
sample_gini_coefficient
sample_variance
scalar_type
scalbln
scalblnf
scalblnl
scalbn
scalbnf
scalbnl
scale
schroder_iterate
semipolar
series
Series Evaluation
Setting Polices at Namespace Scope
Setting Policies at Namespace or Translation Unit Scope
Setting Policies for Distributions on an Ad Hoc Basis
Setting the Maximum Interval Halvings and Memory Requirements
set_zero
shape
sign
Sign Manipulation Functions
Signal Statistics
signbit
sinc_pi
sinhc_pi
sinh_sinh
Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview
size
Skew Normal Distribution
skewness
small
Some Miscellaneous Examples of the Normal (Gaussian) Distribution
spherical
Spherical Bessel Functions of the First and Second Kinds
Spherical Hankel Functions
Spherical Harmonics
spherical_harmonic
spherical_harmonic_i
spherical_harmonic_r
sph_bessel
sph_bessell
sph_bessel_prime
sph_hankel_1
sph_hankel_2
sph_legendre
sph_legendref
sph_legendrel
sph_neumann
sph_neumannl
sph_neumann_prime
sqrt1pm1
standard_deviation
step
Students t Distribution
students_t
subtraction
Summary
summation_condition_number
sum_series
sup
support
Supported/Tested Compilers
sup_distance
sup_norm
Synopsis
t
Tables of the power function of the chi 2 test.
Tangent Numbers
tangent_t2n
tanh_sinh
Template Class octonion
Template Class quaternion
Termination Condition Functors
terms
test
Test Program
Testing
Testing a sample mean for difference from a "true" mean
tgamma
tgamma1pm1
tgammaf
tgammal
tgamma_delta_ratio
tgamma_lower
tgamma_ratio
Thread Safety
To Do
tol
toms748_solve
total_variation
TR1 C Functions Quick Reference
Trading Accuracy for Performance
trapezoidal
Trapezoidal Quadrature
triangular
Triangular Distribution
triangular_distribution
Trigamma
trunc
Truncation Functions
truncf
truncl
typeid
ulp
unchecked_bernoulli_b2n
unchecked_factorial
underflow_error_type
uniform
Uniform Distribution
Univariate Statistics
unreal
update_target_error
upper_incomplete_gamma_fract
Use in non-template code
Use in template code
Use With User-Defined Types
user_denorm_error
user_domain_error
user_evaluation_error
user_indeterminate_result_error
user_overflow_error
user_pole_error
user_rounding_error
user_underflow_error
Using Boost.Math with High-Precision Floating-Point Libraries
Using Boost.Multiprecision
Using C++11 Lambda's
Using e_float Library
Using Macros to Change the Policy Defaults
Using NTL Library
Using with GCC's __float128 datatype
Using With MPFR or GMP - High-Precision Floating-Point Library
Using without expression templates for Boost.Test and others
W
weibull
Weibull Distribution
weights
Why use a high-precision library rather than built-in floating-point types?