# TaylorModel.jl API

## Types

RTaylorModel1{T,S}

Relative Taylor model in 1 variable, providing a rigurous polynomial approximation given by a Taylor polynomial pol (around x0) and a relative remainder rem for a function f(x) in one variable, valid in the interval dom. This corresponds to definition 2.3.2 of Mioara Joldes' thesis.

Fields:

• pol: polynomial approximation, represented as TaylorSeries.Taylor1

• rem: the interval bound

• x0 : expansion point

• dom: domain, interval over which the Taylor model is defined / valid

The approximation $f(x) = p(x) + \delta (x - x_0)^{n+1}$ is satisfied for all $x\in \mathcal{D}$; n is the order (degree) of the polynomial $p(x)=\sum_{i=0}^n p_i (x - x_0)^i$.

TaylorModel1{T,S}

Absolute Taylor model in 1 variable, providing a rigurous polynomial approximation given by a Taylor polynomial pol (around x0) and an absolute remainder rem for a function f(x) in one variable, valid in the interval dom. This corresponds to definition 2.1.3 of Mioara Joldes' thesis.

Fields:

• pol: polynomial approximation, represented as TaylorSeries.Taylor1

• rem: the interval bound

• x0 : expansion point

• dom: domain, interval over which the Taylor model is defined / valid

The approximation $f(x) = p(x) + \Delta$ is satisfied for all $x\in \mathcal{D}$ ($0\in \Delta$); n is the order (degree) of the polynomial $p(x)=\sum_{i=0}^n p_i (x - x_0)^i$.

TaylorModelN{N,T,S}

Taylor Models with absolute remainder for N independent variables.

## Methods

bound_remainder  - Function

bound_remainder(::Type{TaylorModel1}f::Function, polf::Taylor1, polfI::Taylor1, x0::Interval, I::Interval)

Bound the absolute remainder of the polynomial approximation of f given by the Taylor polynomial polf around x0 on the interval I. It requires the interval extension polfI of the polynomial that approximates f for the whole interval I, in order to compute the Lagrange remainder.

If polfI[end] has a definite sign, then it is monotonic in the intervals [I.lo, x0] and [x0.hi, I.hi], which is exploited; otherwise, it is used to compute the Lagrange remainder. This corresponds to Prop 2.2.1 in Mioara Joldes PhD thesis (pp 52).

bound_remainder(::Type{RTaylorModel1}, f::Function, polf::Taylor1, polfI::Taylor1, x0::Interval, I::Interval)

Bound the relative remainder of the polynomial approximation of f given by the Taylor polynomial polf around x0 on the interval I. It requires an the interval extension polfI of a polynomial that approximates f for the whole interval I, in order to compute the Lagrange remainder.

If polfI[end] has a definite sign, then it is monotonic in the interval I, which is exploited; otherwise, the last coefficients bounds the relative remainder. This corresponds to Prop 2.3.7 in Mioara Joldes' PhD thesis (pp 67).

fp_rpa  - Function
fp_rpa(tm::TaylorModel1{Interval{T},T})
fp_rpa(tm::RTaylorModel1{Interval{T},T})

Convert a tm TaylorModel1 to a TaylorModel1 whose polynomial coefficients are Float64. The accumulated error is added to the remainder. The mid point of the expansion interval is preferentially rounded down if it is not an exactly representable value.

rpa  - Function

rpa(g::Function, tmf::TaylorModel1) rpa(g::Function, tmf::RTaylorModel1) rpa(g::Function, tmf::TaylorModelN)

Rigurous polynomial approximation (RPA) for the function g using the Taylor Model with absolute/relative remainder tmf. The bound is computed exploiting monotonicity if possible, otherwise, it uses Lagrange bound.

integrate  - Function
integrate(a, [x])

Return the integral of a::Taylor1. The constant of integration (0-th order coefficient) is set to x, which is zero if ommitted.

integrate(a, r)

Integrate the a::HomogeneousPolynomial with respect to the r-th variable. The returned HomogeneousPolynomial has no added constant of integration. If the order of a corresponds to get_order(), a zero HomogeneousPolynomial of 0-th order is returned.

integrate(a, r, [x0])

Integrate the a::TaylorN series with respect to the r-th variable, where x0 the integration constant and must be independent of the r-th variable; if x0 is ommitted, it is taken as zero.

integrate(a, c0)

Integrates the one-variable Taylor Model (TaylorModel1 or RTaylorModel1) with respect to the independent variable; c0 is the interval representing the integration constant; if omitted it is considered as the zero interval.

integrate(fT, which)

Integrates a TaylorModelN with respect to which variable. The returned TaylorModelN corresponds to the Taylor Model of the definite integral ∫f(x) - ∫f(expansion_point).