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// Interpolation routines.
//
// Copyright (C) 2020 Juan Marín Noguera
//
// This file is part of Solvned.
//
// Solvned is free software: you can redistribute it and/or modify it under the
// terms of the GNU Lesser General Public License as published by the Free
// Software Foundation, either version 3 of the License, or (at your option) any
// later version.
//
// Solvned is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
// A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
// details.
//
// You should have received a copy of the GNU Lesser General Public License
// along with Solvned. If not, see <https://www.gnu.org/licenses/>.
package mned
// An Interpolator tries to approximate a function based on a finite set of
// known points in it. We'll focus just on interpolation with two points since
// this is precise enough for our purposes, although the interpolator might be
// constructed with some knowledge about the underlying function.
type Interpolator interface {
// Approximate the value of a function in a point `t` given two
// **different** points `p1` and `p2` of the function such that `t`
// is between `p1.Time` and `p2.Time`.
FindValue(p1 *Point, p2 *Point, t float64) []float64
}
// A LinearInterpolator is an interpolator that assumes that there's a straight
// line between the two points given. It can be constructed with `new`.
type LinearInterpolator struct{}
func (li LinearInterpolator) FindValue(
p1 *Point, p2 *Point, t float64,
) []float64 {
// The line between (t1,x1) and (t2,x2) is x1 + (x2-x1)*(t-t1)/(t2-t1).
// If ratio:=(t-t1)/(t2-t1), this is x2*ratio + x1*(1-ratio)
size := len(p1.Value)
result := make([]float64, size)
ratio := (t - p1.Time) / (p2.Time - p1.Time)
for i := 0; i < size; i++ {
result[i] = p1.Value[i]*(1-ratio) + p2.Value[i]*ratio
}
return result
}
// A HermiteInterpolator is an interpolator that takes into account the
// derivative of the solution function in the end points.
type HermiteInterpolator struct {
F func(Point) ([]float64, bool) // The derivative function.
}
func (h HermiteInterpolator) FindValue(
p1 *Point, p2 *Point, t float64,
) []float64 {
diff := p2.Time - p1.Time
off := t - p1.Time
d01, ok := h.F(*p1)
if !ok { // Can't derive: fallback
return LinearInterpolator{}.FindValue(p1, p2, t)
}
d23, ok := h.F(*p2)
if !ok {
return LinearInterpolator{}.FindValue(p1, p2, t)
}
d12 := make([]float64, len(d01))
for i := range d12 {
d12[i] = (p2.Value[i] - p1.Value[i]) / diff
}
d02 := make([]float64, len(d01))
for i := range d02 {
d02[i] = (d12[i] - d01[i]) / diff
}
d13 := make([]float64, len(d01))
for i := range d13 {
d13[i] = (d23[i] - d12[i]) / diff
}
d03 := make([]float64, len(d01))
for i := range d03 {
d03[i] = (d13[i] - d02[i]) / diff
}
result := make([]float64, len(d01))
for i := range result {
result[i] = p1.Value[i] + off*
(d01[i]+off*(d02[i]+(off-diff)*d03[i]))
}
return result
}
// Create a Hermite interpolator suitable for the given IVP.
func HermiteForIVP(ivp *IVP) HermiteInterpolator {
return HermiteInterpolator{F: ivp.Derivative}
}
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