# Uses of Classdev.nm.analysis.function.rn2r1.AbstractRealScalarFunction

Packages that use AbstractRealScalarFunction
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.curvefit.interpolation

Modifier and Type
Class
Description
class
LinearInterpolator
Define a univariate function by linearly interpolating between adjacent points.
class
NevilleTable
Neville's algorithm is a polynomial interpolation algorithm.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.differentiation

Modifier and Type
Class
Description
class
Ridders
Ridders' method computes the numerical derivative of a function.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.differentiation.multivariate

Modifier and Type
Class
Description
class
MultivariateFiniteDifference
A partial derivative of a multivariate function is the derivative with respect to one of the variables with the others held constant.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.differentiation.univariate

Modifier and Type
Class
Description
class
DBeta
This is the first order derivative function of the Beta function w.r.t x, $${\partial \over \partial x} \mathrm{B}(x, y)$$.
class
DBetaRegularized
This is the first order derivative function of the Regularized Incomplete Beta function, BetaRegularized, w.r.t the upper limit, x.
class
DErf
This is the first order derivative function of the Error function, Erf.
class
Dfdx
The first derivative is a measure of how a function changes as its input changes.
class
DGamma
This is the first order derivative function of the Gamma function, $${d \mathrm{\Gamma}(x) \over dx}$$.
class
DGaussian
This is the first order derivative function of a Gaussian function, $${d \mathrm{\phi}(x) \over dx}$$.
class
DPolynomial
This is the first order derivative function of a Polynomial, which, again, is a polynomial.
class
FiniteDifference
A finite difference (divided by a small increment) is an approximation of the derivative of a function.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.polynomial

Modifier and Type
Class
Description
class
CauchyPolynomial
The Cauchy's polynomial of a polynomial takes this form:
class
Polynomial
A polynomial is a UnivariateRealFunction that represents a finite length expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents.
class
QuadraticMonomial
A quadratic monomial has this form: x2 + ux + v.
class
ScaledPolynomial
This constructs a scaled polynomial that has neither too big or too small coefficients, hence avoiding overflow or underflow.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.rn2r1

Modifier and Type
Class
Description
class
AbstractBivariateRealFunction
A bivariate real function takes two real arguments and outputs one real value.
class
AbstractTrivariateRealFunction
A trivariate real function takes three real arguments and outputs one real value.
class
QuadraticFunction
A quadratic function takes this form: $$f(x) = \frac{1}{2} \times x'Hx + x'p + c$$.
class
R1Projection
Projection creates a real-valued function RealScalarFunction from a vector-valued function RealVectorFunction by taking only one of its coordinate components in the vector output.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.rn2r1.univariate

Modifier and Type
Class
Description
class
AbstractUnivariateRealFunction
A univariate real function takes one real argument and outputs one real value.
class
ContinuedFraction
A continued fraction representation of a number has this form: $z = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}}$ ai and bi can be functions of x, which in turn makes z a function of x.
class
StepFunction
A step function (or staircase function) is a finite linear combination of indicator functions of intervals.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.special

Modifier and Type
Class
Description
class
Rastrigin
The Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.special.beta

Modifier and Type
Class
Description
class
Beta
The beta function defined as: $B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}= \int_0^1t^{x-1}(1-t)^{y-1}\,dt, x > 0, y > 0$
class
BetaRegularized
The Regularized Incomplete Beta function is defined as: $I_x(p,q) = \frac{B(x;\,p,q)}{B(p,q)} = \frac{1}{B(p,q)} \int_0^x t^{p-1}\,(1-t)^{q-1}\,dt, p > 0, q > 0$
class
BetaRegularizedInverse
The inverse of the Regularized Incomplete Beta function is defined at: $x = I^{-1}_{(p,q)}(u), 0 \le u \le 1$
class
LogBeta
This class represents the log of Beta function log(B(x, y)).
class
MultinomialBetaFunction
A multinomial Beta function is defined as: $\frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^K \alpha_i\right)},\qquad\boldsymbol{\alpha}=(\alpha_1,\cdots,\alpha_K)$
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.special.gamma

Modifier and Type
Class
Description
class
Digamma
The digamma function is defined as the logarithmic derivative of the gamma function.
class
GammaGergoNemes
The Gergo Nemes' algorithm is very simple and quick to compute the Gamma function, if accuracy is not critical.
class
GammaLanczos
Lanczos approximation provides a way to compute the Gamma function such that the accuracy can be made arbitrarily precise.
class
GammaLanczosQuick
Lanczos approximation, computations are done in double.
class
GammaLowerIncomplete
The Lower Incomplete Gamma function is defined as: $\gamma(s,x) = \int_0^x t^{s-1}\,e^{-t}\,{\rm d}t = P(s,x)\Gamma(s)$ P(s,x) is the Regularized Incomplete Gamma P function.
class
GammaRegularizedP
The Regularized Incomplete Gamma P function is defined as: $P(s,x) = \frac{\gamma(s,x)}{\Gamma(s)} = 1 - Q(s,x), s \geq 0, x \geq 0$
class
GammaRegularizedPInverse
The inverse of the Regularized Incomplete Gamma P function is defined as: $x = P^{-1}(s,u), 0 \geq u \geq 1$ When s > 1, we use the asymptotic inversion method. When s <= 1, we use an approximation of P(s,x) together with a higher-order Newton like method. In both cases, the estimated value is then improved using Halley's method, c.f., HalleyRoot.
class
GammaRegularizedQ
The Regularized Incomplete Gamma Q function is defined as: $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}=1-P(s,x), s \geq 0, x \geq 0$ The algorithm used for computing the regularized incomplete Gamma Q function depends on the values of s and x.
class
GammaUpperIncomplete
The Upper Incomplete Gamma function is defined as: $\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t = Q(s,x) \times \Gamma(s)$ The integrand has the same form as the Gamma function, but the lower limit of the integration is a variable.
class
LogGamma
The log-Gamma function, $$\log (\Gamma(z))$$, for positive real numbers, is the log of the Gamma function.
class
Trigamma
The trigamma function is defined as the logarithmic derivative of the digamma function.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.special.gaussian

Modifier and Type
Class
Description
class
CumulativeNormalHastings
Hastings algorithm is faster but less accurate way to compute the cumulative standard Normal.
class
CumulativeNormalInverse
The inverse of the cumulative standard Normal distribution function is defined as: $N^{-1}(u) /] class  CumulativeNormalMarsaglia Marsaglia is about 3 times slower but is more accurate to compute the cumulative standard Normal. class  Erf The Error function is defined as: \[ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt$
class
Erfc
This complementary Error function is defined as: $\operatorname{erfc}(x) = 1-\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt$
class
ErfInverse
The inverse of the Error function is defined as: $\operatorname{erf}^{-1}(x)$
class
Gaussian
The Gaussian function is defined as: $f(x) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }$
• ## Uses of AbstractRealScalarFunction in dev.nm.solver.multivariate.constrained.convex.sdp.socp.qp.problem

Modifier and Type
Class
Description
class
QPProblemOnlyEqualityConstraints
A quadratic programming problem with only equality constraints can be converted into a equivalent quadratic programming problem without constraints, hence a mere quadratic function.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.evt.evd.univariate.fitting.acer

Modifier and Type
Class
Description
class
ACERFunction
The ACER (Average Conditional Exceedance Rate) function $$\epsilon_k(\eta)$$ approximates the probability $\epsilon_k(\eta) = Pr(X_k > \eta | X_1 \le \eta, X_2 \le \eta, ..., X_{k-1} \le \eta)$ for a sequence of stochastic process observations $$X_i$$ with a k-step memory.
class
ACERInverseFunction
The inverse of the ACER function.
class
ACERLogFunction
The ACER function in log scale (base e), i.e., $$log(\epsilon_k(\eta))$$.
class
ACERReturnLevel
Given an ACER function, compute the return level $$\eta$$ for a given return period $$R$$.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.evt.function

Modifier and Type
Class
Description
class
ReturnLevel
Given a GEV distribution of a random variable $$X$$, the return level $$\eta$$ is the value that is expected to be exceeded on average once every interval of time $$T$$, with a probability of $$1 / T$$.
class
ReturnPeriod
The return period $$R$$ of a level $$\eta$$ for a random variable $$X$$ is the mean number of trials that must be done for $$X$$ to exceed $$\eta$$.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.stochasticprocess.univariate.filtration

Modifier and Type
Class
Description
class
Bt
This is a FiltrationFunction that returns $$B(t_i)$$, the Brownian motion value at the i-th time point.
class
F_Sum_BtDt
This represents a function of this integral $I = \int_{0}^{1} B(t)dt$
class
F_Sum_tBtDt
This represents a function of this integral $\int_{0}^{1} (t - 0.5) * B(t) dt$
class
FiltrationFunction
A filtration function, parameterized by a fixed filtration, is a function of time, $$f(\mathfrak{F_{t_i}})$$.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.timeseries.linear.univariate

Modifier and Type
Class
Description
class
AutoCorrelationFunction
This is the auto-correlation function of a univariate time series {xt}.
class
AutoCovarianceFunction
This is the auto-covariance function of a univariate time series {xt}.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.timeseries.linear.univariate.sample

Modifier and Type
Class
Description
class
SampleAutoCorrelation
This is the sample Auto-Correlation Function (ACF) for a univariate data set.
class
SampleAutoCovariance
This is the sample Auto-Covariance Function (ACVF) for a univariate data set.
class
SamplePartialAutoCorrelation
This is the sample partial Auto-Correlation Function (PACF) for a univariate data set.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.timeseries.linear.univariate.stationaryprocess.arma

Modifier and Type
Class
Description
class
AutoCorrelation
Compute the Auto-Correlation Function (ACF) for an AutoRegressive Moving Average (ARMA) model, assuming that EXt = 0.
class
AutoCovariance
Computes the Auto-CoVariance Function (ACVF) for an AutoRegressive Moving Average (ARMA) model by recursion.
• ## Uses of AbstractRealScalarFunction in tech.nmfin.portfoliooptimization.lai2010.ceta

Modifier and Type
Class
Description
class
Ceta
The function C(η) to be maximized (Eq.
• ## Uses of AbstractRealScalarFunction in tech.nmfin.portfoliooptimization.lai2010.ceta.maximizer

Modifier and Type
Class
Description
static class
CetaMaximizer.NegCetaFunction