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Easy derivative and primitive from an interpolation setup

Aren't primitives, are definite integrals, but it's a nice word.

Function and domain

x f 6 x 3^* 3 - x 5^1 + / :

X 04 0.2 range :

Y X f vectorize :

xo X min eval :

Derivative

x f' x f x diff :

Y' X f' vectorize :

Primitive

x f.p ξ f ξ xo x int :

Y.p X f.p vectorize :

X Y o augment x f 2 1 sys X Y' o augment x f' 2 1 sys X Y.p o augment x f.p 2 1 sys

Work arround for quickly get the answers

Function

x φ X Y x cinterp :

φ can't be vectorized because use vectors inside: Use ranges

Utils

h 105 -^:

N 300 :

k 1 X length range :

Derivative

x φ' x h + φ x φ - h / :

Ψ' k el X k el φ' :

Primitive

x φ.p x xo - N / xo κ x xo - N / * + φ κ 0 N sum * :

Ψ.p k el X k el φ.p :

Exact Y & cinterp φ

Ψ' & exact f'

Ψ.p & exact f.p

X Y o augment x φ 2 1 sys X Ψ' o augment x f' 2 1 sys X Ψ.p o augment x f.p 2 1 sys

Errors

Ψ k el X k el φ :

y Plot X y o 8 red augment :

Y Ψ - Plot Y' Ψ' - Plot Y.p Ψ.p - Plot

No errors expected here: cinterp is through each point

Bigest errors expected at the extremes of the interval

Bigest errors expected as x increase from zero.

Y Ψ - normi max 5.4008 1016 -^*

Y' Ψ' - normi max 0.5368

Y.p Ψ.p - normi max 0.0192

Note: Using Viacheslav Rkadapt modification for using it for integrals: https://en.smath.com/forum/yaf_postsm57304_ODE-Solvers.aspx#post57304

f x a b N Int y Clear x y D f : 12 mat tx 0 a b N x y D Rkadapt : tx 1 col tx 2 col x cinterp 31 line :

Primitives

x f.p x f x xo X max N Int :

Y.p k el X k el f.p :

x φ.p x φ x xo X max N Int :

Ψ.p k el X k el φ.p :

X Y.p o augment x f.p 2 1 sys X Ψ.p o augment x f.p 2 1 sys Y.p Ψ.p - Plot

Y.p Ψ.p - normi max 0.0023

Notice that this is fastest and probably more accurate than int SMath routine.

Alvaro