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 asTaylorSeries.Taylor1
rem
: the interval boundx0
: expansion pointdom
: domain, interval over which the Taylor model is defined / valid
The approximation is satisfied for all ; n
is the order (degree) of the polynomial .
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 asTaylorSeries.Taylor1
rem
: the interval boundx0
: expansion pointdom
: domain, interval over which the Taylor model is defined / valid
The approximation is satisfied for all (); n
is the order (degree) of the polynomial .
TaylorModelN{N,T,S}
Taylor Models with absolute remainder for N
independent variables.
Methods
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(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(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(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).