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| In the [[mathematics|mathematical]] subfield of [[numerical analysis]], '''monotone cubic interpolation''' is a variant of [[cubic interpolation]] that preserves [[Monotone function|monotonicity]] of the [[data set]] being interpolated.
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| Monotonicity is preserved by [[linear interpolation]] but not guaranteed by [[cubic interpolation]].
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| ==Monotone cubic Hermite interpolation==
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| [[Image:MonotCubInt.png|thumb|300px|right|Example showing non-monotone cubic interpolation (in red) and monotone cubic interpolation (in blue) of a monotone data set.]]
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| Monotone interpolation can be accomplished using [[cubic Hermite spline]] with the tangents <math>m_i</math> modified to ensure the monotonicity of the resulting Hermite spline.
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| ===Interpolant selection===
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| There are several ways of selecting interpolating tangents for each data point. This section will outline the use of the Fritsch–Carlson method.
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| Let the data points be <math>(x_k,y_k)</math> for <math>k=1,...,n</math>
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| # Compute the slopes of the [[secant line]]s between successive points:<blockquote><math>\Delta_k =\frac{y_{k+1}-y_k}{x_{k+1}-x_k}</math></blockquote> for <math>k=1,\dots,n-1</math>.
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| # Initialize the tangents at every data point as the average of the secants, <blockquote><math>m_k = \frac{\Delta_{k-1}+\Delta_k}{2}</math></blockquote> for <math>k=2,\dots,n-1</math>; these may be updated in further steps. For the endpoints, use one-sided differences: <blockquote><math>m_1 = \Delta_1 \quad \text{and} \quad m_n = \Delta_{n-1}</math></blockquote>
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| # For <math>k=1,\dots,n-1</math>, if <math>\Delta_k = 0</math> (if two successive <math>y_k=y_{k+1}</math> are equal), then set <math>m_k = m_{k+1} = 0,</math> as the spline connecting these points must be flat to preserve monotonicity. Ignore step 4 and 5 for those <math>k</math>.
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| # Let <math>\alpha_k = m_k/\Delta_k</math> and <math>\beta_k = m_{k+1}/\Delta_k</math>. If <math>\alpha_k</math> or <math>\beta_{k-1}</math> are computed to be less than zero, then the input data points are not strictly monotone, and <math>(x_k,y_k)</math> is a local extremum. In such cases, piecewise monotone curves can still be generated by choosing <math>m_{k}=0</math>, although global strict monotonicity is not possible.
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| # To prevent [[overshoot]] and ensure monotonicity, at least one of the following conditions must be met:
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| ##the function <blockquote><math>\phi(\alpha, \beta) = \alpha - \frac{(2 \alpha + \beta - 3)^2}{3(\alpha + \beta - 2)}</math></blockquote> must have a value greater than or equal to zero;
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| ##<math>\alpha + 2\beta - 3 \le 0</math>; or
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| ##<math>2\alpha + \beta - 3 \le 0</math>.
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| If monotonicity must be strict then <math>\phi(\alpha, \beta)</math> must have a value strictly greater than zero.
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| One simple way to satisfy this constraint is to restrict the magnitude of vector <math>(\alpha_k, \beta_k)</math> to a circle of radius 3. That is, if <math>\alpha_k^2 + \beta_k^2 > 9</math>, then set <math>m_k = \tau_k \alpha_k \Delta_k</math> and <math>m_{k+1} = \tau_k \beta_k \Delta_k</math> where <math>\tau_k = \frac{3}{\sqrt{\alpha_k^2 + \beta_k^2}}</math>.
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| Alternatively it is sufficient to restrict <math>\alpha_k \le 3</math> and <math>\beta_k \le 3</math>. To accomplish this if <math>\alpha_k > 3</math>, then set <math>m_k = 3\times \Delta_k</math>. Similarly for <math>\beta</math>.
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| Note that only one pass of the algorithm is required.
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| ===Cubic interpolation===
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| After the preprocessing, evaluation of the interpolated spline is equivalent to [[cubic Hermite spline]], using the data <math>x_k</math>, <math>y_k</math>, and <math>m_k</math> for <math>k=1,...,n</math>.
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| To evaluate at <math>x</math>, find the smallest value larger than <math>x</math>, <math>x_\text{upper}</math>, and the largest value smaller than <math>x</math>, <math>x_\text{lower}</math>, among <math>x_k</math> such that <math>x_\text{lower} \leq x \leq x_\text{upper}</math>. Calculate
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| :<math>h = x_\text{upper}-x_\text{lower}</math> and <math>t = \frac{x - x_\text{lower}}{h}</math>
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| then the interpolant is
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| :<math>f_\text{interpolated}(x) = y_\text{lower} h_{00}(t) + h m_\text{lower} h_{10}(t) + y_\text{upper} h_{01}(t) + h m_\text{upper}h_{11}(t)</math>
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| where <math>h_{ii}</math> are the basis functions for the [[cubic Hermite spline]].
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| ==Example implementation==
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| The following [[JavaScript]] implementation takes a data set and produces a Fritsch-Carlson cubic spline interpolant function:
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| <syntaxhighlight lang="javascript">
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| /* Fritsch-Carlson monotone cubic spline interpolation
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| Usage example:
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| var f = createInterpolant([0, 1, 2, 3], [0, 1, 4, 9]);
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| var message = '';
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| for (var x = 0; x <= 3; x += 0.5) {
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| var xSquared = f(x);
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| message += x + ' squared is about ' + xSquared + '\n';
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| }
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| alert(message);
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| */
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| var createInterpolant = function(xs, ys) {
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| var i, length = xs.length;
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|
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| // Deal with length issues
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| if (length != ys.length) { throw 'Need an equal count of xs and ys.'; }
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| if (length === 0) { return function(x) { return 0; }; }
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| if (length === 1) {
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| // Impl: Precomputing the result prevents problems if ys is mutated later and allows garbage collection of ys
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| // Impl: Unary plus properly converts values to numbers
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| var result = +ys[0];
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| return function(x) { return result; };
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| }
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|
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| // Rearrange xs and ys so that xs is sorted
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| var indexes = [];
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| for (i = 0; i < length; i++) { indexes.push(i); }
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| indexes.sort(function(a, b) { return xs[a] < xs[b] ? -1 : 1; });
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| var oldXs = xs, oldYs = ys;
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| // Impl: Creating new arrays also prevents problems if the input arrays are mutated later
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| xs = []; ys = [];
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| // Impl: Unary plus properly converts values to numbers
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| for (i = 0; i < length; i++) { xs.push(+oldXs[indexes[i]]); ys.push(+oldYs[indexes[i]]); }
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|
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| // Get consecutive differences and slopes
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| var dys = [], dxs = [], ms = [];
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| for (i = 0; i < length - 1; i++) {
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| var dx = xs[i + 1] - xs[i], dy = ys[i + 1] - ys[i];
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| dxs.push(dx); dys.push(dy); ms.push(dy/dx);
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| }
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|
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| // Get degree-1 coefficients
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| var c1s = [ms[0]];
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| for (i = 0; i < dxs.length - 1; i++) {
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| var m = ms[i], mNext = ms[i + 1];
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| if (m*mNext <= 0) {
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| c1s.push(0);
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| } else {
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| var dx = dxs[i], dxNext = dxs[i + 1], common = dx + dxNext;
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| c1s.push(3*common/((common + dxNext)/m + (common + dx)/mNext));
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| }
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| }
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| c1s.push(ms[ms.length - 1]);
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|
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| // Get degree-2 and degree-3 coefficients
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| var c2s = [], c3s = [];
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| for (i = 0; i < c1s.length - 1; i++) {
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| var c1 = c1s[i], m = ms[i], invDx = 1/dxs[i], common = c1 + c1s[i + 1] - m - m;
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| c2s.push((m - c1 - common)*invDx); c3s.push(common*invDx*invDx);
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| }
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|
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| // Return interpolant function
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| return function(x) {
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| // The rightmost point in the dataset should give an exact result
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| var i = xs.length - 1;
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| if (x == xs[i]) { return ys[i]; }
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|
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| // Search for the interval x is in, returning the corresponding y if x is one of the original xs
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| var low = 0, mid, high = c3s.length - 1;
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| while (low <= high) {
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| mid = Math.floor(0.5*(low + high));
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| var xHere = xs[mid];
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| if (xHere < x) { low = mid + 1; }
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| else if (xHere > x) { high = mid - 1; }
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| else { return ys[mid]; }
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| }
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| i = Math.max(0, high);
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|
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| // Interpolate
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| var diff = x - xs[i], diffSq = diff*diff;
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| return ys[i] + c1s[i]*diff + c2s[i]*diffSq + c3s[i]*diff*diffSq;
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| };
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| };
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| </syntaxhighlight>
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| ==References==
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| *{{cite journal
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| | last = Fritsch
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| | first = F. N.
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| | coauthors = Carlson, R. E.
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| | title = Monotone Piecewise Cubic Interpolation
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| | journal = [[SIAM Journal on Numerical Analysis]]
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| | volume = 17
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| | issue = 2
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| | pages = 238–246
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| | publisher = SIAM
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| | date = 1980
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| | doi = 10.1137/0717021
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| }}
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| ==External links==
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| * [[GPL]]v3 licensed [[C++]] implementation: [https://github.com/OPM/opm-core/blob/master/opm/core/utility/MonotCubicInterpolator.cpp MonotCubicInterpolator.cpp] [https://github.com/OPM/opm-core/blob/master/opm/core/utility/MonotCubicInterpolator.hpp MonotCubicInterpolator.hpp]
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| [[Category:Interpolation]]
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| [[Category:Splines]]
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