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/**
* TODO: description of the method
*/
class CubicInterpolation {
constructor(
private firstX: ISafeNumber,
private lastX: ISafeNumber,
private firstY: ISafeNumber,
private lastY: ISafeNumber,
private firstSlope: ISafeNumber,
private lastSlope: ISafeNumber,
private fun: (x: ISafeNumber) => ISafeNumber
) {}
public forX(x: ISafeNumber): ISafeNumber {
// queries outside the range of points are matched by linear interpolation
if (x.lt(this.firstX)) {
return this.firstY.sub(this.firstSlope.mul(this.firstX.sub(x)));
} else if (x.gt(this.lastX)) {
return this.lastY.add(this.lastSlope.mul(x.sub(this.lastX)));
} else {
// we use the calculated cubic interpolation
return this.fun(x);
}
}
public forY(y: ISafeNumber, _precision: ISafeNumber): ISafeNumber {
// queries outside the range of points are matched by linear interpolation
if (y.lt(this.firstY)) {
return this.firstX.sub(this.firstY.sub(y).div(this.firstSlope));
} else if (y.gt(this.lastY)) {
return this.lastX.add(y.sub(this.lastY).div(this.lastSlope));
} else {
// binary search for X for which Y is within the given precision
let low = this.firstX,
high = this.lastX,
tries = 50;
while (low <= high && tries > 0) {
const midX = low.add(high).div(2);
const midY = this.forX(midX);
if (midY.sub(y).abs().lt(_precision)) {
return midX;
} else if (midY.lt(y)) {
low = midX;
} else {
high = midX;
}
tries--;
}
throw new Error(`X was not found for given Y = ${y.toString()}`);
}
}
}
export const monotoneCubicInterpolation = (
xs: ISafeNumber[],
ys: ISafeNumber[]
): CubicInterpolation => {
const { fun, firstSlope, lastSlope } = createInterpolant(xs, ys);
return new CubicInterpolation(
xs[0],
xs[xs.length - 1],
ys[0],
ys[ys.length - 1],
firstSlope,
lastSlope,
fun
);
};
const createInterpolant = (
xs: ISafeNumber[],
ys: ISafeNumber[]
): {
fun: (x: ISafeNumber) => ISafeNumber;
firstSlope: ISafeNumber;
lastSlope: ISafeNumber;
} => {
// checking the initial conditions
if (xs.length != ys.length) {
throw new Error("The number of xs and ys should be equal");
}
if (xs.length === 0) {
throw new Error("Empty array of xs");
}
if (xs.length === 1) {
return { fun: (_x) => ys[0], firstSlope: SafeZero, lastSlope: SafeZero };
}
// sorting points
const indexes = [...Array(xs.length).keys()];
indexes.sort(function (a, b) {
return xs[a] < xs[b] ? -1 : 1;
});
// monotonicity check
if (!isStrictlyMonotonic(indexes, ys)) {
throw new Error("The given points are not monotonic");
}
// consecutive differences and slopes
const dys = [],
dxs = [],
ms = [];
for (let i = 0; i < xs.length - 1; i++) {
const dx = xs[i + 1].sub(xs[i]),
dy = ys[i + 1].sub(ys[i]);
dxs[i] = dx;
dys[i] = dy;
ms[i] = dy.div(dx);
}
// degree-1 coefficients
const c1s = [ms[0]];
for (let i = 0; i < dxs.length - 1; i++) {
const m = ms[i],
mNext = ms[i + 1];
Iif (m.mul(mNext).lte(SafeZero)) {
c1s.push(SafeZero);
} else {
const dx_ = dxs[i],
dxNext = dxs[i + 1],
common = dx_.add(dxNext);
c1s.push(
common
.mul(3)
.div(common.add(dxNext).div(m).add(common.add(dx_).div(mNext)))
);
}
}
c1s.push(ms[ms.length - 1]);
// degree-2 and degree-3 coefficients
const c2s: ISafeNumber[] = [],
c3s: ISafeNumber[] = [];
for (let i = 0; i < c1s.length - 1; i++) {
const c1 = c1s[i],
m = ms[i],
invDx = createSafeNumber(1).div(dxs[i]),
common = c1
.add(c1s[i + 1])
.sub(m)
.sub(m);
c2s.push(m.sub(c1).sub(common).mul(invDx));
c3s.push(common.mul(invDx).mul(invDx));
}
return {
fun: createInterpolantFunction(xs, ys, c1s, c2s, c3s),
firstSlope: ms[0],
lastSlope: ms[ms.length - 1],
};
};
const isStrictlyMonotonic = (indexes: number[], ys: ISafeNumber[]) => {
Iif (ys.length < 2) {
return true;
}
Iif (indexes.length !== ys.length) {
throw new Error("Different lengths of input arrays");
}
// direction of monotonicity
let direction: "increasing" | "decreasing" | "undefined" = "undefined";
let previous = ys[indexes[0]];
for (let i = 1; i < ys.length; i++) {
const current = ys[indexes[i]];
// check monotonicity
if (current.gt(previous)) {
Iif (direction === "decreasing") {
return false;
}
direction = "increasing";
} else if (current.lt(previous)) {
if (direction === "increasing") {
return false;
}
direction = "decreasing";
} else E{
// equality, no strict monotonicity
return false;
}
previous = current;
}
return true;
};
const createInterpolantFunction = (
xs: ISafeNumber[],
ys: ISafeNumber[],
c1s: ISafeNumber[],
c2s: ISafeNumber[],
c3s: ISafeNumber[]
): ((x: ISafeNumber) => ISafeNumber) => {
return (x: ISafeNumber) => {
// checking whether the argument is from the appropriate range
Iif (x.lt(xs[0]) || x.gt(xs[xs.length - 1])) {
throw new Error(
`The function only handles arguments in the range [${xs[0].toString()},${xs[
xs.length - 1
].toString()}]`
);
}
// the rightmost point in the dataset should give an exact result
if (x.eq(xs[xs.length - 1])) {
return ys[xs.length - 1];
}
// search for the interval x is in, returning the corresponding y if x is one of the original xs
let low = 0,
high = c3s.length - 1;
while (low <= high) {
const mid = Math.floor(0.5 * (low + high));
const middleX = xs[mid];
if (middleX < x) {
low = mid + 1;
} else if (middleX > x) {
high = mid - 1;
} else {
return ys[mid];
}
}
const i = Math.max(0, high);
// interpolate
const diff = x.sub(xs[i]),
diffSq = diff.mul(diff);
return ys[i]
.add(c1s[i].mul(diff))
.add(c2s[i].mul(diffSq))
.add(c3s[i].mul(diff).mul(diffSq));
};
};
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