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 /* -*-C++-*- */ #ifndef _MONOTCUBICINTERPOLATOR_H #define _MONOTCUBICINTERPOLATOR_H #include #include #include /* MonotCubicInterpolator Copyright (C) 2006 Statoil ASA This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program 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 General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. */ /** Class to represent a one-dimensional function f with single-valued argument x. The function is represented by a table of function values. Interpolation between table values is cubic and monotonicity preserving if input values are monotonous. Outside x_min and x_max, the class will extrapolate using the constant f(x_min) or f(x_max). Extra functionality: - Can return (x_1+x_2)/2 where x_1 and x_2 are such that abs(f(x_1) - f(x_2)) is maximized. This is used to determine where one should calculate a new value for increased accuracy in the current function Monotonicity preserving cubic interpolation algorithm is taken from Fritsch and Carlson, "Monotone piecewise cubic interpolation", SIAM J. Numer. Anal. 17, 238--246, no. 2, $Id$ Algorithm also described here: http://en.wikipedia.org/wiki/Monotone_cubic_interpolation @author Håvard Berland , December 2006 @brief Represents one dimensional function f with single valued argument x that can be interpolated using monotone cubic interpolation */ class MonotCubicInterpolator { public: /** @param datafilename A datafile with the x values and the corresponding f(x) values Accepts a filename as input and parses this file for two-column floating point data, interpreting the data as representing function values x and f(x). Ignores all lines not conforming to \\\\\\ */ MonotCubicInterpolator(const std::string & datafilename) throw (const char*) { if (!read(datafilename)) { throw("Unable to constuct MonotCubicInterpolator from file.") ; } ; } ; /** @param datafilename A datafile with the x values and the corresponding f(x) values Accepts a filename as input and parses this file for two-column floating point data, interpreting the data as representing function values x and f(x). Ignores all lines not conforming to \\\\\\ All commas in the file will be treated as spaces when parsing. */ MonotCubicInterpolator(const char* datafilename) throw (const char*) { if (!read(std::string(datafilename))) { throw("Unable to constuct MonotCubicInterpolator from file.") ; } ; } ; /** @param datafilename data file @param XColumn x values @param fColumn f values Accepts a filename as input, and parses the chosen columns in that file. */ MonotCubicInterpolator(const char* datafilename, int xColumn, int fColumn) throw (const char*) { if (!read(std::string(datafilename),xColumn,fColumn)) { throw("Unable to constuct MonotCubicInterpolator from file.") ; } ; } ; /** @param datafilename data file @param XColumn x values @param fColumn f values Accepts a filename as input, and parses the chosen columns in that file. */ MonotCubicInterpolator(const std::string & datafilename, int xColumn, int fColumn) throw (const char*) { if (!read(datafilename,xColumn,fColumn)) { throw("Unable to constuct MonotCubicInterpolator from file.") ; } ; } ; /** @param x vector of x values @param f vector of corresponding f values Accepts two equal-length vectors as input for constructing the interpolation object. First vector is the x-values, the second vector is the function values */ MonotCubicInterpolator(const std::vector & x , const std::vector & f); /** No input, an empty function object is created. This object must be treated with care until populated. */ MonotCubicInterpolator() {;} ; /** @param datafilename A datafile with the x values and the corresponding f(x) values Accepts a filename as input and parses this file for two-column floating point data, interpreting the data as representing function values x and f(x). returns true on success All commas in file will be treated as spaces when parsing Ignores all lines not conforming to \\\\\\ */ bool read(const std::string & datafilename) { return read(datafilename,1,2) ; } ; /** @param datafilename data file @param XColumn x values @param fColumn f values Accepts a filename as input, and parses the chosen columns in that file. */ bool read(const std::string & datafilename, int xColumn, int fColumn) ; /** @param x x value Returns f(x) for given x (input). Interpolates (monotone cubic or linearly) if necessary. Extrapolates using the constants f(x_min) or f(x_max) if input x is outside (x_min, x_max) @return f(x) for a given x */ double operator () (double x) const { return evaluate(x) ; } ; /** @param x x value Returns f(x) for given x (input). Interpolates (monotone cubic or linearly) if necessary. Extrapolates using the constants f(x_min) or f(x_max) if input x is outside (x_min, x_max) @return f(x) for a given x */ double evaluate(double x) const throw(const char*); /** @param x x value @param errorestimate_output Returns f(x) and an error estimate for given x (input). Interpolates (linearly) if necessary. Throws an exception if extrapolation would be necessary for evaluation. We do not want to do extrapolation (yet). The error estimate for x1 < x < x2 is (x2 - x1)^2/8 * f''(x) where f''(x) is evaluated using the stencil (1 -2 1) using either (x0, x1, x2) or (x1, x2, x3); Throws an exception if the table contains only two x-values. NOT IMPLEMENTED YET! */ double evaluate(double x, double & errorestimate_output ) const ; /** Minimum x-value, returns both x and f in a pair. @return minimum x value @return f(minimum x value) */ std::pair getMinimumX() const { // Easy since the data is sorted on x: return *data.begin(); } /** Maximum x-value, returns both x and f in a pair. @return maximum x value @return f(maximum x value) */ std::pair getMaximumX() const { // Easy since the data is sorted on x: return *data.rbegin(); } /** Maximum f-value, returns both x and f in a pair. @return x value corresponding to maximum f value @return maximum f value */ std::pair getMaximumF() const throw(const char*) ; /** Minimum f-value, returns both x and f in a pair @return x value corresponding to minimal f value @return minimum f value */ std::pair getMinimumF() const throw(const char*) ; /** Provide a copy of the x-data as a vector Unspecified order, but corresponds to get_fVector. @return x values as a vector */ std::vector get_xVector() const ; /** Provide a copy of tghe function data as a vector Unspecified order, but corresponds to get_xVector @return f values as a vector */ std::vector get_fVector() const ; /** @param factor Scaling constant Scale all the function value data by a constant */ void scaleData(double factor); /** Determines if the current function-value-data is strictly monotone. This is a utility function for outsiders if they want to invert the data for example. @return True if f(x) is strictly monotone, else False */ bool isStrictlyMonotone() { /* Use cached value if it can be trusted */ if (strictlyMonotoneCached) { return strictlyMonotone; } else { computeInternalFunctionData(); return strictlyMonotone; } } /** Determines if the current function-value-data is monotone. @return True if f(x) is monotone, else False */ bool isMonotone() const { if (monotoneCached) { return monotone; } else { computeInternalFunctionData(); return monotone; } } /** Determines if the current function-value-data is strictly increasing. This is a utility function for outsiders if they want to invert the data for example. @return True if f(x) is strictly increasing, else False */ bool isStrictlyIncreasing() { /* Use cached value if it can be trusted */ if (strictlyMonotoneCached) { return (strictlyMonotone && strictlyIncreasing); } else { computeInternalFunctionData(); return (strictlyMonotone && strictlyIncreasing); } } /** Determines if the current function-value-data is monotone and increasing. @return True if f(x) is monotone and increasing, else False */ bool isMonotoneIncreasing() const { if (monotoneCached) { return (monotone && increasing); } else { computeInternalFunctionData(); return (monotone && increasing); } } /** Determines if the current function-value-data is strictly decreasing. This is a utility function for outsiders if they want to invert the data for example. @return True if f(x) is strictly decreasing, else False */ bool isStrictlyDecreasing() { /* Use cached value if it can be trusted */ if (strictlyMonotoneCached) { return (strictlyMonotone && strictlyDecreasing); } else { computeInternalFunctionData(); return (strictlyMonotone && strictlyDecreasing); } } /** Determines if the current function-value-data is monotone and decreasing @return True if f(x) is monotone and decreasing, else False */ bool isMonotoneDecreasing() const { if (monotoneCached) { return (monotone && decreasing); } else { computeInternalFunctionData(); return (monotone && decreasing); } } /** @param newx New x point @param newf New f(x) point Adds a new datapoint to the function. This causes all the derivatives at all points of the functions to be recomputed and then adjusted for monotone cubic interpolation. If this function ever enters a critical part of any code, the locality of the algorithm for monotone adjustment must be exploited. */ void addPair(double newx, double newf) throw(const char*); /** Returns an x-value that is believed to yield the best improvement in global accuracy for the interpolation if computed. Searches for the largest jump in f-values, and returns a x value being the average of the two x-values representing the f-value-jump. @return New x value beleived to yield the best improvement in global accuracy @return Maximal difference */ std::pair getMissingX() const throw(const char*) ; /** Constructs a string containing the data in a table @return a string containing the data in a table */ std::string toString() const; /** @return Number of datapoint pairs in this object */ int getSize() const { return data.size(); } /** Checks if the function curve is flat at the endpoints, chop off endpoint data points if that is the case. The notion of "flat" is determined by the input parameter "epsilon" Values whose difference are less than epsilon are regarded as equal. This is implemented to be able to obtain a strictly monotone curve from a data set that is strictly monotone except at the endpoints. Example: The data points (1,3), (2,3), (3,4), (4,5), (5,5), (6,5) will become (2,3), (3,4), (4,5) Assumes at least 3 datapoints. If less than three, this function is a noop. */ void chopFlatEndpoints(const double); /** Wrapper function for chopFlatEndpoints(const double) providing a default epsilon parameter */ void chopFlatEndpoints() { chopFlatEndpoints(1e-14); } /** If function is monotone, but not strictly monotone, this function will remove datapoints from intervals with zero derivative so that the curve become strictly monotone. Example The data points (1,2), (2,3), (3,4), (4,4), (5,5), (6,6) will become (1,2), (2,3), (3,4), (5,5), (6,6) Assumes at least two datapoints, if one or zero datapoint, this is a noop. */ void shrinkFlatAreas(const double); /** Wrapper function for shrinkFlatAreas(const double) providing a default epsilon parameter */ void shrinkFlatAreas() { shrinkFlatAreas(1e-14); }; private: // Data structure to store x- and f-values std::map data; // Data structure to store x- and d-values mutable std::map ddata; // Storage containers for precomputed interpolation data // std::vector dvalues; // derivatives in Hermite interpolation. // Flag to determine whether the boolean strictlyMonotone can be // trusted. mutable bool strictlyMonotoneCached; mutable bool monotoneCached; /* only monotone, not stricly montone */ mutable bool strictlyMonotone; mutable bool monotone; // if strictlyMonotone is true (and can be trusted), the two next are meaningful mutable bool strictlyDecreasing; mutable bool strictlyIncreasing; mutable bool decreasing; mutable bool increasing; /* Hermite basis functions, t \in [0,1] , notation from: http://en.wikipedia.org/w/index.php?title=Cubic_Hermite_spline&oldid=84495502 */ double H00(double t) const { return 2*t*t*t - 3*t*t + 1; } double H10(double t) const { return t*t*t - 2*t*t + t; } double H01(double t) const { return -2*t*t*t + 3*t*t; } double H11(double t) const { return t*t*t - t*t; } void computeInternalFunctionData() const ; /** Computes initial derivative values using centered (second order) difference for internal datapoints, and one-sided derivative for endpoints The internal datastructure map ddata is populated by this method. */ void computeSimpleDerivatives() const ; /** Adjusts the derivative values (ddata) so that we can guarantee that the resulting piecewise Hermite polymial is monotone. This is done according to the algorithm of Fritsch and Carlsson 1980, see Section 4, especially the two last lines. */ void adjustDerivativesForMonotoneness() const ; /** Checks if the coefficient alpha and beta is in the region that guarantees monotoneness of the derivative values they represent See Fritsch and Carlson 1980, Lemma 2, alternatively Step 5 in Wikipedia's article on Monotone cubic interpolation. */ bool isMonotoneCoeff(double alpha, double beta) const { if ((alpha*alpha + beta*beta) <= 9) { return true; } else { return false; } } }; #endif