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// Copyright 2012 The Closure Library Authors. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS-IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
/**
* @fileoverview A one dimensional cubic spline interpolator with not-a-knot
* boundary conditions.
*
* See http://en.wikipedia.org/wiki/Spline_interpolation.
*
*/
goog.provide('goog.math.interpolator.Spline1');
goog.require('goog.array');
goog.require('goog.math');
goog.require('goog.math.interpolator.Interpolator1');
goog.require('goog.math.tdma');
/**
* A one dimensional cubic spline interpolator with natural boundary conditions.
* @implements {goog.math.interpolator.Interpolator1}
* @constructor
*/
goog.math.interpolator.Spline1 = function() {
/**
* The abscissa of the data points.
* @type {!Array.<number>}
* @private
*/
this.x_ = [];
/**
* The spline interval coefficients.
* Note that, in general, the length of coeffs and x is not the same.
* @type {!Array.<!Array.<number>>}
* @private
*/
this.coeffs_ = [[0, 0, 0, Number.NaN]];
};
/** @override */
goog.math.interpolator.Spline1.prototype.setData = function(x, y) {
goog.asserts.assert(x.length == y.length,
'input arrays to setData should have the same length');
if (x.length > 0) {
this.coeffs_ = this.computeSplineCoeffs_(x, y);
this.x_ = x.slice();
} else {
this.coeffs_ = [[0, 0, 0, Number.NaN]];
this.x_ = [];
}
};
/** @override */
goog.math.interpolator.Spline1.prototype.interpolate = function(x) {
var pos = goog.array.binarySearch(this.x_, x);
if (pos < 0) {
pos = -pos - 2;
}
pos = goog.math.clamp(pos, 0, this.coeffs_.length - 1);
var d = x - this.x_[pos];
var d2 = d * d;
var d3 = d2 * d;
var coeffs = this.coeffs_[pos];
return coeffs[0] * d3 + coeffs[1] * d2 + coeffs[2] * d + coeffs[3];
};
/**
* Solve for the spline coefficients such that the spline precisely interpolates
* the data points.
* @param {Array.<number>} x The abscissa of the spline data points.
* @param {Array.<number>} y The ordinate of the spline data points.
* @return {!Array.<!Array.<number>>} The spline interval coefficients.
* @private
*/
goog.math.interpolator.Spline1.prototype.computeSplineCoeffs_ = function(x, y) {
var nIntervals = x.length - 1;
var dx = new Array(nIntervals);
var delta = new Array(nIntervals);
for (var i = 0; i < nIntervals; ++i) {
dx[i] = x[i + 1] - x[i];
delta[i] = (y[i + 1] - y[i]) / dx[i];
}
// Compute the spline coefficients from the 1st order derivatives.
var coeffs = [];
if (nIntervals == 0) {
// Nearest neighbor interpolation.
coeffs[0] = [0, 0, 0, y[0]];
} else if (nIntervals == 1) {
// Straight line interpolation.
coeffs[0] = [0, 0, delta[0], y[0]];
} else if (nIntervals == 2) {
// Parabola interpolation.
var c3 = 0;
var c2 = (delta[1] - delta[0]) / (dx[0] + dx[1]);
var c1 = delta[0] - c2 * dx[0];
var c0 = y[0];
coeffs[0] = [c3, c2, c1, c0];
} else {
// General Spline interpolation. Compute the 1st order derivatives from
// the Spline equations.
var deriv = this.computeDerivatives(dx, delta);
for (var i = 0; i < nIntervals; ++i) {
var c3 = (deriv[i] - 2 * delta[i] + deriv[i + 1]) / (dx[i] * dx[i]);
var c2 = (3 * delta[i] - 2 * deriv[i] - deriv[i + 1]) / dx[i];
var c1 = deriv[i];
var c0 = y[i];
coeffs[i] = [c3, c2, c1, c0];
}
}
return coeffs;
};
/**
* Computes the derivative at each point of the spline such that
* the curve is C2. It uses not-a-knot boundary conditions.
* @param {Array.<number>} dx The spacing between consecutive data points.
* @param {Array.<number>} slope The slopes between consecutive data points.
* @return {Array.<number>} The Spline derivative at each data point.
* @protected
*/
goog.math.interpolator.Spline1.prototype.computeDerivatives = function(
dx, slope) {
var nIntervals = dx.length;
// Compute the main diagonal of the system of equations.
var mainDiag = new Array(nIntervals + 1);
mainDiag[0] = dx[1];
for (var i = 1; i < nIntervals; ++i) {
mainDiag[i] = 2 * (dx[i] + dx[i - 1]);
}
mainDiag[nIntervals] = dx[nIntervals - 2];
// Compute the sub diagonal of the system of equations.
var subDiag = new Array(nIntervals);
for (var i = 0; i < nIntervals; ++i) {
subDiag[i] = dx[i + 1];
}
subDiag[nIntervals - 1] = dx[nIntervals - 2] + dx[nIntervals - 1];
// Compute the super diagonal of the system of equations.
var supDiag = new Array(nIntervals);
supDiag[0] = dx[0] + dx[1];
for (var i = 1; i < nIntervals; ++i) {
supDiag[i] = dx[i - 1];
}
// Compute the right vector of the system of equations.
var vecRight = new Array(nIntervals + 1);
vecRight[0] = ((dx[0] + 2 * supDiag[0]) * dx[1] * slope[0] +
dx[0] * dx[0] * slope[1]) / supDiag[0];
for (var i = 1; i < nIntervals; ++i) {
vecRight[i] = 3 * (dx[i] * slope[i - 1] + dx[i - 1] * slope[i]);
}
vecRight[nIntervals] = (dx[nIntervals - 1] * dx[nIntervals - 1] *
slope[nIntervals - 2] + (2 * subDiag[nIntervals - 1] +
dx[nIntervals - 1]) * dx[nIntervals - 2] * slope[nIntervals - 1]) /
subDiag[nIntervals - 1];
// Solve the system of equations.
var deriv = goog.math.tdma.solve(
subDiag, mainDiag, supDiag, vecRight);
return deriv;
};
/**
* Note that the inverse of a cubic spline is not a cubic spline in general.
* As a result the inverse implementation is only approximate. In
* particular, it only guarantees the exact inverse at the original input data
* points passed to setData.
* @override
*/
goog.math.interpolator.Spline1.prototype.getInverse = function() {
var interpolator = new goog.math.interpolator.Spline1();
var y = [];
for (var i = 0; i < this.x_.length; i++) {
y[i] = this.interpolate(this.x_[i]);
}
interpolator.setData(y, this.x_);
return interpolator;
};
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