“Monotone cubic interpolation”的意思、由来-开放百科全书

词条

Monotone cubic interpolation

释义

Monotone cubic Hermite interpolation
Interpolant selection Cubic interpolation
Example implementation
References
External links

In the mathematical subfield of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated.

Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation.

Monotone cubic Hermite interpolation

Monotone interpolation can be accomplished using cubic Hermite spline with the tangents modified to ensure the monotonicity of the resulting Hermite spline.

An algorithm is also available for monotone quintic Hermite interpolation.

Interpolant selection

There are several ways of selecting interpolating tangents for each data point. This section will outline the use of the Fritsch–Carlson method.

Let the data points be for

Compute the slopes of the secant lines between successive points:
for .
Initialize the tangents at every data point as the average of the secants,
for ; if and have different sign, set . These may be updated in further steps. For the endpoints, use one-sided differences:
For , if (if two successive are equal), then set as the spline connecting these points must be flat to preserve monotonicity. Ignore step 4 and 5 for those .
Let and . If or are computed to be less than zero, then the input data points are not strictly monotone, and is a local extremum. In such cases, piecewise monotone curves can still be generated by choosing , although global strict monotonicity is not possible.
To prevent overshoot and ensure monotonicity, at least one of the following conditions must be met:
1. the function
  must have a value greater than or equal to zero;
2. ; or
3. .

If monotonicity must be strict then must have a value strictly greater than zero.

One simple way to satisfy this constraint is to restrict the vector to a circle of radius 3. That is, if , then set and where .

Alternatively it is sufficient to restrict and . To accomplish this if , then set . Similarly for .

Note that only one pass of the algorithm is required.

Cubic interpolation

After the preprocessing, evaluation of the interpolated spline is equivalent to cubic Hermite spline, using the data , , and for .

To evaluate at , find the smallest value larger than , , and the largest value smaller than , , among such that . Calculate

and

then the interpolant is

where are the basis functions for the cubic Hermite spline.

Example implementation

The following JavaScript implementation takes a data set and produces a monotone cubic spline interpolant function:

/* Monotone cubic spline interpolation

var f = createInterpolant([0, 1, 2, 3, 4], [0, 1, 4, 9, 16]);

var message = ;

for (var x = 0; x <= 4; x += 0.5) {

var xSquared = f(x);

message += x + ' squared is about ' + xSquared + '\';

}

alert(message);

var createInterpolant = function(xs, ys) {

var i, length = xs.length;

// Deal with length issues

if (length != ys.length) { throw 'Need an equal count of xs and ys.'; }

if (length === 0) { return function(x) { return 0; }; }

if (length === 1) {

// Impl: Precomputing the result prevents problems if ys is mutated later and allows garbage collection of ys

// Impl: Unary plus properly converts values to numbers

var result = +ys[0];

return function(x) { return result; };

}

// Rearrange xs and ys so that xs is sorted

var indexes = [];

for (i = 0; i < length; i++) { indexes.push(i); }

indexes.sort(function(a, b) { return xs[a] < xs[b] ? -1 : 1; });

var oldXs = xs, oldYs = ys;

// Impl: Creating new arrays also prevents problems if the input arrays are mutated later

xs = []; ys = [];

// Impl: Unary plus properly converts values to numbers

for (i = 0; i < length; i++) { xs.push(+oldXs[indexes[i]]); ys.push(+oldYs[indexes[i]]); }

// Get consecutive differences and slopes

var dys = [], dxs = [], ms = [];

for (i = 0; i < length - 1; i++) {

var dx = xs[i + 1] - xs[i], dy = ys[i + 1] - ys[i];

dxs.push(dx); dys.push(dy); ms.push(dy/dx);

}

// Get degree-1 coefficients

var c1s = [ms[0]];

for (i = 0; i < dxs.length - 1; i++) {

var m = ms[i], mNext = ms[i + 1];

if (m*mNext <= 0) {

c1s.push(0);

} else {

var dx_ = dxs[i], dxNext = dxs[i + 1], common = dx_ + dxNext;

c1s.push(3*common/((common + dxNext)/m + (common + dx_)/mNext));

}

c1s.push(ms[ms.length - 1]);

// Get degree-2 and degree-3 coefficients

var c2s = [], c3s = [];

for (i = 0; i < c1s.length - 1; i++) {

var c1 = c1s[i], m_ = ms[i], invDx = 1/dxs[i], common_ = c1 + c1s[i + 1] - m_ - m_;

c2s.push((m_ - c1 - common_)*invDx); c3s.push(common_*invDx*invDx);

}

// Return interpolant function

return function(x) {

// The rightmost point in the dataset should give an exact result

var i = xs.length - 1;

if (x == xs[i]) { return ys[i]; }

// Search for the interval x is in, returning the corresponding y if x is one of the original xs

var low = 0, mid, high = c3s.length - 1;

while (low <= high) {

mid = Math.floor(0.5*(low + high));

var xHere = xs[mid];

if (xHere < x) { low = mid + 1; }

else if (xHere > x) { high = mid - 1; }

else { return ys[mid]; }

}

i = Math.max(0, high);

// Interpolate

var diff = x - xs[i], diffSq = diff*diff;

return ys[i] + c1s[i]*diff + c2s[i]*diffSq + c3s[i]*diff*diffSq;

};

References

{{cite journal

| last = Fritsch
| first = F. N.
|author2=Carlson, R. E.
| title = Monotone Piecewise Cubic Interpolation
| journal = SIAM Journal on Numerical Analysis
| volume = 17
| issue = 2
| pages = 238–246
| publisher = SIAM
| date = 1980
| doi = 10.1137/0717021
}}

{{cite journal

| last = Dougherty
| first = R.L.
|author2=Edelman, A.
|author3=Hyman, J.M.
| title = Positivity-, monotonicity-, or convexity-preserving cubic and quintic Hermite interpolation
| journal = Mathematics of Computation
| volume = 52
| number = 186
| pages = 471–494
| date = April 1989
| doi=10.2307/2008477
}}

External links

GPLv2 licensed C++ implementation: [https://github.com/bska/opm-origins/blob/master/dune-cornerpoint/common/MonotCubicInterpolator.cpp MonotCubicInterpolator.cpp] [https://github.com/bska/opm-origins/blob/master/dune-cornerpoint/common/MonotCubicInterpolator.hpp MonotCubicInterpolator.hpp]

3 : Interpolation|Splines (mathematics)|Articles with example JavaScript code

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