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| 1 | // -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- | ||
| 2 | // vi: set et ts=4 sw=4 sts=4: | ||
| 3 | // | ||
| 4 | // SPDX-FileCopyrightInfo: Copyright © DuMux Project contributors, see AUTHORS.md in root folder | ||
| 5 | // SPDX-License-Identifier: GPL-3.0-or-later | ||
| 6 | // | ||
| 7 | /*! | ||
| 8 | * \file | ||
| 9 | * \ingroup Core | ||
| 10 | * \brief The common code for all 3rd order polynomial splines. | ||
| 11 | */ | ||
| 12 | #ifndef DUMUX_SPLINE_COMMON__HH | ||
| 13 | #define DUMUX_SPLINE_COMMON__HH | ||
| 14 | |||
| 15 | #include <algorithm> | ||
| 16 | #include <iostream> | ||
| 17 | #include <cassert> | ||
| 18 | |||
| 19 | #include <dune/common/exceptions.hh> | ||
| 20 | #include <dune/common/float_cmp.hh> | ||
| 21 | |||
| 22 | #include "math.hh" | ||
| 23 | |||
| 24 | namespace Dumux { | ||
| 25 | |||
| 26 | /*! | ||
| 27 | * \ingroup Core | ||
| 28 | * \brief The common code for all 3rd order polynomial splines. | ||
| 29 | */ | ||
| 30 | template<class ScalarT, class ImplementationT> | ||
| 31 | class SplineCommon_ | ||
| 32 | { | ||
| 33 | using Scalar = ScalarT; | ||
| 34 | using Implementation = ImplementationT; | ||
| 35 | |||
| 36 | Implementation &asImp_() | ||
| 37 | { return *static_cast<Implementation*>(this); } | ||
| 38 | const Implementation &asImp_() const | ||
| 39 | { return *static_cast<const Implementation*>(this); } | ||
| 40 | |||
| 41 | public: | ||
| 42 | /*! | ||
| 43 | * \brief Return true if the given x is in range [x1, xn]. | ||
| 44 | */ | ||
| 45 | bool applies(Scalar x) const | ||
| 46 | { | ||
| 47 | 123016155 | return x_(0) <= x && x <= x_(numSamples_() - 1); | |
| 48 | } | ||
| 49 | |||
| 50 | /*! | ||
| 51 | * \brief Return the x value of the leftmost sampling point. | ||
| 52 | */ | ||
| 53 | Scalar xMin() const | ||
| 54 | 8 | { return x_(0); } | |
| 55 | |||
| 56 | /*! | ||
| 57 | * \brief Return the x value of the rightmost sampling point. | ||
| 58 | */ | ||
| 59 | Scalar xMax() const | ||
| 60 | 9 | { return x_(numSamples_() - 1); } | |
| 61 | |||
| 62 | /*! | ||
| 63 | * \brief Prints k tuples of the format (x, y, dx/dy, isMonotonic) | ||
| 64 | * to stdout. | ||
| 65 | * | ||
| 66 | * If the spline does not apply for parts of [x0, x1] it is | ||
| 67 | * extrapolated using a straight line. The result can be inspected | ||
| 68 | * using the following commands: | ||
| 69 | * | ||
| 70 | ----------- snip ----------- | ||
| 71 | ./yourProgramm > spline.csv | ||
| 72 | gnuplot | ||
| 73 | |||
| 74 | gnuplot> plot "spline.csv" using 1:2 w l ti "Curve", \ | ||
| 75 | "spline.csv" using 1:3 w l ti "Derivative", \ | ||
| 76 | "spline.csv" using 1:4 w p ti "Monotonic" | ||
| 77 | ----------- snap ----------- | ||
| 78 | */ | ||
| 79 | 2 | void printCSV(Scalar xi0, Scalar xi1, int k) const | |
| 80 | { | ||
| 81 | using std::max; | ||
| 82 | using std::min; | ||
| 83 |
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2 | Scalar x0 = min(xi0, xi1); |
| 84 |
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2 | Scalar x1 = max(xi0, xi1); |
| 85 | 4 | const int n = numSamples_() - 1; | |
| 86 |
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2004 | for (int i = 0; i <= k; ++i) { |
| 87 | 2002 | double x = i*(x1 - x0)/k + x0; | |
| 88 | 2002 | double x_p1 = x + (x1 - x0)/k; | |
| 89 | double y; | ||
| 90 | double dy_dx; | ||
| 91 | 2002 | double mono = 1; | |
| 92 | 3964 | if (!applies(x)) { | |
| 93 |
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40 | if (x < 0) { |
| 94 | 60 | dy_dx = evalDerivative(x_(0)); | |
| 95 | 60 | y = (x - x_(0))*dy_dx + y_(0); | |
| 96 | 20 | mono = (dy_dx>0)?1:-1; | |
| 97 | } | ||
| 98 | 40 | else if (x > x_(n)) { | |
| 99 | 60 | dy_dx = evalDerivative(x_(n)); | |
| 100 | 60 | y = (x - x_(n))*dy_dx + y_(n); | |
| 101 | 20 | mono = (dy_dx>0)?1:-1; | |
| 102 | } | ||
| 103 | else { | ||
| 104 | ✗ | std::cerr << "ooops: " << x << "\n"; | |
| 105 | ✗ | exit(1); | |
| 106 | } | ||
| 107 | } | ||
| 108 | else { | ||
| 109 | 1962 | y = eval(x); | |
| 110 | 1962 | dy_dx = evalDerivative(x); | |
| 111 | 1962 | mono = monotonic(max<Scalar>(x_(0), x), min<Scalar>(x_(n), x_p1)); | |
| 112 | } | ||
| 113 | |||
| 114 | 8008 | std::cout << x << " " << y << " " << dy_dx << " " << mono << "\n"; | |
| 115 | } | ||
| 116 | 2 | } | |
| 117 | |||
| 118 | /*! | ||
| 119 | * \brief Evaluate the spline at a given position. | ||
| 120 | * | ||
| 121 | * \param x The value on the abscissa where the spline ought to be | ||
| 122 | * evaluated | ||
| 123 | * \param extrapolate If this parameter is set to true, the spline | ||
| 124 | * will be extended beyond its range by | ||
| 125 | * straight lines, if false calling extrapolate | ||
| 126 | * for \f$ x \not [x_{min}, x_{max}]\f$ will | ||
| 127 | * cause a failed assertion. | ||
| 128 | */ | ||
| 129 | 23092768 | Scalar eval(Scalar x, bool extrapolate=false) const | |
| 130 | { | ||
| 131 |
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23092768 | assert(extrapolate || applies(x)); |
| 132 | |||
| 133 | // handle extrapolation | ||
| 134 |
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23092768 | if (extrapolate) { |
| 135 | 8 | if (x < xMin()) { | |
| 136 | 6 | Scalar m = evalDerivative_(xMin(), /*segmentIdx=*/0); | |
| 137 | 4 | Scalar y0 = y_(0); | |
| 138 | 4 | return y0 + m*(x - xMin()); | |
| 139 | } | ||
| 140 | 4 | else if (x > xMax()) { | |
| 141 | 8 | Scalar m = evalDerivative_(xMax(), /*segmentIdx=*/numSamples_()-2); | |
| 142 | 6 | Scalar y0 = y_(numSamples_()-1); | |
| 143 | 4 | return y0 + m*(x - xMax()); | |
| 144 | } | ||
| 145 | } | ||
| 146 | |||
| 147 | 46181160 | return eval_(x, segmentIdx_(x)); | |
| 148 | } | ||
| 149 | |||
| 150 | /*! | ||
| 151 | * \brief Evaluate the spline's derivative at a given position. | ||
| 152 | * | ||
| 153 | * \param x The value on the abscissa where the spline's | ||
| 154 | * derivative ought to be evaluated | ||
| 155 | * | ||
| 156 | * \param extrapolate If this parameter is set to true, the spline | ||
| 157 | * will be extended beyond its range by | ||
| 158 | * straight lines, if false calling extrapolate | ||
| 159 | * for \f$ x \not [x_{min}, x_{max}]\f$ will | ||
| 160 | * cause a failed assertion. | ||
| 161 | |||
| 162 | */ | ||
| 163 | 1397488 | Scalar evalDerivative(Scalar x, bool extrapolate=false) const | |
| 164 | { | ||
| 165 |
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1397488 | assert(extrapolate || applies(x)); |
| 166 |
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1397488 | if (extrapolate) { |
| 167 | ✗ | if (Dune::FloatCmp::le(x, xMin())) | |
| 168 | ✗ | return evalDerivative_(xMin(), 0); | |
| 169 | ✗ | else if (Dune::FloatCmp::ge(x, xMax())) | |
| 170 | ✗ | return evalDerivative_(xMax(), numSamples_() - 2); | |
| 171 | } | ||
| 172 | |||
| 173 | 2790430 | return evalDerivative_(x, segmentIdx_(x)); | |
| 174 | } | ||
| 175 | |||
| 176 | /*! | ||
| 177 | * \brief Find the intersections of the spline with a cubic | ||
| 178 | * polynomial in the whole interval, throws | ||
| 179 | * Dune::MathError exception if there is more or less than | ||
| 180 | * one solution. | ||
| 181 | */ | ||
| 182 | Scalar intersect(Scalar a, Scalar b, Scalar c, Scalar d) const | ||
| 183 | { | ||
| 184 | return intersectInterval(xMin(), xMax(), a, b, c, d); | ||
| 185 | } | ||
| 186 | |||
| 187 | /*! | ||
| 188 | * \brief Find the intersections of the spline with a cubic | ||
| 189 | * polynomial in a sub-interval of the spline, throws | ||
| 190 | * Dune::MathError exception if there is more or less than | ||
| 191 | * one solution. | ||
| 192 | */ | ||
| 193 | 55775 | Scalar intersectInterval(Scalar x0, Scalar x1, | |
| 194 | Scalar a, Scalar b, Scalar c, Scalar d) const | ||
| 195 | { | ||
| 196 | ✗ | assert(applies(x0) && applies(x1)); | |
| 197 | |||
| 198 | Scalar tmpSol[3]; | ||
| 199 | int nSol = 0; | ||
| 200 | int iFirst = segmentIdx_(x0); | ||
| 201 | int iLast = segmentIdx_(x1); | ||
| 202 |
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111550 | for (int i = iFirst; i <= iLast; ++i) |
| 203 | { | ||
| 204 | 55775 | nSol += intersectSegment_(tmpSol, i, a, b, c, d, x0, x1); | |
| 205 | |||
| 206 |
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55775 | if (nSol > 1) { |
| 207 | ✗ | DUNE_THROW(Dune::MathError, | |
| 208 | "Spline has more than one intersection"); //<<a<<"x^3 + "<<b<"x^2 + "<<c<"x + "<<d); | ||
| 209 | } | ||
| 210 | } | ||
| 211 | |||
| 212 |
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55775 | if (nSol != 1) |
| 213 | ✗ | DUNE_THROW(Dune::MathError, | |
| 214 | "Spline has no intersection"); //<<a<"x^3 + " <<b<"x^2 + "<<c<"x + "<<d<<"!"); | ||
| 215 | |||
| 216 | 55775 | return tmpSol[0]; | |
| 217 | } | ||
| 218 | |||
| 219 | /*! | ||
| 220 | * \brief Returns 1 if the spline is monotonically increasing, -1 | ||
| 221 | * if the spline is monotonously decreasing and 0 if the | ||
| 222 | * spline is not monotonous in the interval (x0, x1). | ||
| 223 | * | ||
| 224 | * In the corner case where the whole spline is flat, it returns | ||
| 225 | * 2. | ||
| 226 | */ | ||
| 227 | 1963 | int monotonic(Scalar x0, Scalar x1) const | |
| 228 | { | ||
| 229 | ✗ | assert(applies(x0)); | |
| 230 | ✗ | assert(applies(x1)); | |
| 231 |
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3926 | assert(Dune::FloatCmp::ne<Scalar>(x0, x1)); |
| 232 | |||
| 233 | // make sure that x0 is smaller than x1 | ||
| 234 |
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1963 | if (x0 > x1) |
| 235 | { | ||
| 236 | using std::swap; | ||
| 237 | ✗ | swap(x0, x1); | |
| 238 | } | ||
| 239 | |||
| 240 | // corner case where the whole spline is a constant | ||
| 241 | 6870 | if (Dune::FloatCmp::eq<Scalar>(moment_(0), 0) && | |
| 242 |
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1963 | Dune::FloatCmp::eq<Scalar>(moment_(1), 0) && |
| 243 | ✗ | Dune::FloatCmp::eq<Scalar>(y_(0), y_(1))) | |
| 244 | { | ||
| 245 | // actually the is monotonically increasing as well as | ||
| 246 | // monotonously decreasing | ||
| 247 | return 2; | ||
| 248 | } | ||
| 249 | |||
| 250 | |||
| 251 | 1963 | int i = segmentIdx_(x0); | |
| 252 | 7852 | if (x_(i) <= x0 && x1 <= x_(i+1)) { | |
| 253 | // the interval to check is completely included in a | ||
| 254 | // single segment | ||
| 255 | 1956 | return monotonic_(i, x0, x1); | |
| 256 | } | ||
| 257 | |||
| 258 | // make sure that the segments which are completely in the | ||
| 259 | // interval [x0, x1] all exhibit the same monotonicity. | ||
| 260 | 7 | int iEnd = segmentIdx_(x1); | |
| 261 | 14 | int r = monotonic_(i, x0, x_(1)); | |
| 262 |
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9 | for (; i < iEnd - 1; ++i) |
| 263 |
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2 | if (r != monotonic_(i, x_(i), x_(i + 1))) |
| 264 | return 0; | ||
| 265 | |||
| 266 | // check for the last segment | ||
| 267 |
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7 | if (x_(iEnd) < x1 && r != monotonic_(iEnd, x_(iEnd), x1)) |
| 268 | 2 | { return 0; } | |
| 269 | |||
| 270 | return r; | ||
| 271 | } | ||
| 272 | |||
| 273 | /*! | ||
| 274 | * \brief Same as monotonic(x0, x1), but with the entire range of the | ||
| 275 | * spline as interval. | ||
| 276 | */ | ||
| 277 | int monotonic() const | ||
| 278 | { return monotonic(xMin(), xMax()); } | ||
| 279 | |||
| 280 | protected: | ||
| 281 | // this is an internal class, so everything is protected! | ||
| 282 | SplineCommon_() = default; | ||
| 283 | |||
| 284 | /*! | ||
| 285 | * \brief Set the sampling point vectors. | ||
| 286 | * | ||
| 287 | * This takes care that the order of the x-values is ascending, | ||
| 288 | * although the input must be ordered already! | ||
| 289 | */ | ||
| 290 | template <class DestVector, class SourceVector> | ||
| 291 | void assignSamplingPoints_(DestVector &destX, | ||
| 292 | DestVector &destY, | ||
| 293 | const SourceVector &srcX, | ||
| 294 | const SourceVector &srcY, | ||
| 295 | int numSamples) | ||
| 296 | { | ||
| 297 | assert(numSamples >= 2); | ||
| 298 | |||
| 299 | // copy sample points, make sure that the first x value is | ||
| 300 | // smaller than the last one | ||
| 301 | bool reverse = (srcX[0] > srcX[numSamples - 1]); | ||
| 302 | |||
| 303 | for (int i = 0; i < numSamples; ++i) { | ||
| 304 | int idx = i; | ||
| 305 | if (reverse) | ||
| 306 | idx = numSamples - i - 1; | ||
| 307 | destX[i] = srcX[idx]; | ||
| 308 | destY[i] = srcY[idx]; | ||
| 309 | } | ||
| 310 | } | ||
| 311 | |||
| 312 | /*! | ||
| 313 | * \brief Set the sampling point vectors. | ||
| 314 | * | ||
| 315 | * This takes care that the order of the x-values is ascending, | ||
| 316 | * although the input must be ordered already! | ||
| 317 | */ | ||
| 318 | template <class DestVector, class ListIterator> | ||
| 319 | void assignFromArrayList_(DestVector &destX, | ||
| 320 | DestVector &destY, | ||
| 321 | const ListIterator &srcBegin, | ||
| 322 | const ListIterator &srcEnd, | ||
| 323 | int numSamples) | ||
| 324 | { | ||
| 325 | assert(numSamples >= 2); | ||
| 326 | |||
| 327 | // find out whether the x values are in reverse order | ||
| 328 | ListIterator it = srcBegin; | ||
| 329 | ++it; | ||
| 330 | bool reverse = ((*srcBegin)[0] > (*it)[0]); | ||
| 331 | --it; | ||
| 332 | |||
| 333 | // loop over all sampling points | ||
| 334 | for (int i = 0; it != srcEnd; ++i, ++it) { | ||
| 335 | int idx = i; | ||
| 336 | if (reverse) | ||
| 337 | idx = numSamples - i - 1; | ||
| 338 | destX[idx] = (*it)[0]; | ||
| 339 | destY[idx] = (*it)[1]; | ||
| 340 | } | ||
| 341 | } | ||
| 342 | |||
| 343 | /*! | ||
| 344 | * \brief Set the sampling points. | ||
| 345 | * | ||
| 346 | * Here we assume that the elements of the source vector have an | ||
| 347 | * [] operator where v[0] is the x value and v[1] is the y value | ||
| 348 | * if the sampling point. | ||
| 349 | */ | ||
| 350 | template <class DestVector, class ListIterator> | ||
| 351 | void assignFromTupleList_(DestVector &destX, | ||
| 352 | DestVector &destY, | ||
| 353 | ListIterator srcBegin, | ||
| 354 | ListIterator srcEnd, | ||
| 355 | int numSamples) | ||
| 356 | { | ||
| 357 | assert(numSamples >= 2); | ||
| 358 | |||
| 359 | // copy sample points, make sure that the first x value is | ||
| 360 | // smaller than the last one | ||
| 361 | |||
| 362 | // find out whether the x values are in reverse order | ||
| 363 | ListIterator it = srcBegin; | ||
| 364 | ++it; | ||
| 365 | bool reverse = (std::get<0>(*srcBegin) > std::get<0>(*it)); | ||
| 366 | --it; | ||
| 367 | |||
| 368 | // loop over all sampling points | ||
| 369 | for (int i = 0; it != srcEnd; ++i, ++it) { | ||
| 370 | int idx = i; | ||
| 371 | if (reverse) | ||
| 372 | idx = numSamples - i - 1; | ||
| 373 | destX[idx] = std::get<0>(*it); | ||
| 374 | destY[idx] = std::get<1>(*it); | ||
| 375 | } | ||
| 376 | } | ||
| 377 | |||
| 378 | |||
| 379 | /*! | ||
| 380 | * \brief Make the linear system of equations Mx = d which results | ||
| 381 | * in the moments of the periodic spline. | ||
| 382 | * | ||
| 383 | * When solving Mx = d, it should be noted that x[0] is trash and | ||
| 384 | * needs to be set to x[n-1] | ||
| 385 | */ | ||
| 386 | template <class Vector, class Matrix> | ||
| 387 | void makePeriodicSystem_(Matrix &M, Vector &d) | ||
| 388 | { | ||
| 389 | // a periodic spline only makes sense if the first and the | ||
| 390 | // last sampling point have the same y value | ||
| 391 | assert(y_(0) == y_(numSamples_() - 1)); | ||
| 392 | |||
| 393 | makeNaturalSystem(M, d); | ||
| 394 | |||
| 395 | int n = numSamples_() - 1; | ||
| 396 | |||
| 397 | Scalar lambda_n = h_(1)/(h_(n) + h_(1)); | ||
| 398 | Scalar lambda_1 = M[1][2]; | ||
| 399 | Scalar mu_1 = 1 - lambda_1; | ||
| 400 | Scalar mu_n = 1 - lambda_n; | ||
| 401 | |||
| 402 | Scalar d_n = | ||
| 403 | 6 / (h_(n) + h_(1)) | ||
| 404 | * | ||
| 405 | ( (y_(1) - y_(n))/h_(1) - (y_(n) - y_(n-1))/h_(n)); | ||
| 406 | |||
| 407 | // insert lambda_n and mu_1 | ||
| 408 | M[1][n] = mu_1; | ||
| 409 | M[n][1] = lambda_n; | ||
| 410 | M[n][n-1] = mu_n; | ||
| 411 | d[n] = d_n; | ||
| 412 | } | ||
| 413 | |||
| 414 | /*! | ||
| 415 | * \brief Make the linear system of equations Mx = d which results | ||
| 416 | * in the moments of the full spline. | ||
| 417 | */ | ||
| 418 | template <class Vector, class Matrix> | ||
| 419 | 90666 | void makeFullSystem_(Matrix &M, Vector &d, Scalar m0, Scalar m1) | |
| 420 | { | ||
| 421 | 181306 | makeNaturalSystem_(M, d); | |
| 422 | |||
| 423 | 181332 | int n = numSamples_() - 1; | |
| 424 | // first row | ||
| 425 |
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90666 | M[0][1] = 1; |
| 426 | 543926 | d[0] = 6/h_(1) * ( (y_(1) - y_(0))/h_(1) - m0); | |
| 427 | |||
| 428 | // last row | ||
| 429 |
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181320 | M[n][n - 1] = 1; |
| 430 | |||
| 431 | // right hand side | ||
| 432 | 181346 | d[n] = | |
| 433 | 181318 | 6/h_(n) | |
| 434 | 90666 | * | |
| 435 | 362650 | (m1 - (y_(n) - y_(n - 1))/h_(n)); | |
| 436 | 90666 | } | |
| 437 | |||
| 438 | /*! | ||
| 439 | * \brief Make the linear system of equations Mx = d which results | ||
| 440 | * in the moments of the natural spline. | ||
| 441 | * Stoer2005: Numerische Mathematik 1, p. 111 \cite stoer2005 | ||
| 442 | */ | ||
| 443 | template <class Vector, class Matrix> | ||
| 444 | 52 | void makeNaturalSystem_(Matrix &M, Vector &d) | |
| 445 | { | ||
| 446 | 90686 | M = 0; | |
| 447 | 90686 | d = 0; | |
| 448 | |||
| 449 | 181328 | const int n = numSamples_() - 1; | |
| 450 | |||
| 451 | // second to next to last rows | ||
| 452 |
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90842 | for (int i = 1; i < n; ++i) { |
| 453 | 354 | const Scalar lambda_i = h_(i + 1) / (h_(i) + h_(i+1)); | |
| 454 | 156 | const Scalar mu_i = 1 - lambda_i; | |
| 455 | 156 | const Scalar d_i = | |
| 456 | 288 | 6 / (h_(i) + h_(i+1)) | |
| 457 | 156 | * | |
| 458 | 912 | ( (y_(i+1) - y_(i))/h_(i+1) - (y_(i) - y_(i-1))/h_(i)); | |
| 459 | |||
| 460 | 156 | M[i][i-1] = mu_i; | |
| 461 | 312 | M[i][i] = 2; | |
| 462 | 312 | M[i][i + 1] = lambda_i; | |
| 463 | 468 | d[i] = d_i; | |
| 464 | } | ||
| 465 | |||
| 466 | // first row | ||
| 467 |
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90686 | M[0][0] = 2; |
| 468 | |||
| 469 | // last row | ||
| 470 |
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181350 | M[n][n] = 2; |
| 471 | // right hand side | ||
| 472 |
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90738 | d[0] = 0.0; |
| 473 |
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90768 | d[n] = 0.0; |
| 474 | 52 | } | |
| 475 | |||
| 476 | // evaluate the spline at a given the position and given the | ||
| 477 | // segment index | ||
| 478 | 23092764 | Scalar eval_(Scalar x, int i) const | |
| 479 | { | ||
| 480 | // See: Stoer2005 "Numerische Mathematik 1", 9th edition, | ||
| 481 | // Springer, 2005, p. 109 | ||
| 482 | 46185266 | Scalar h_i1 = h_(i + 1); | |
| 483 | 46185528 | Scalar x_i = x - x_(i); | |
| 484 | 46185528 | Scalar x_i1 = x_(i+1) - x; | |
| 485 | |||
| 486 | 23092764 | Scalar A_i = | |
| 487 | 69278292 | (y_(i+1) - y_(i))/h_i1 | |
| 488 | - | ||
| 489 | 69278292 | h_i1/6*(moment_(i+1) - moment_(i)); | |
| 490 | 69278292 | Scalar B_i = y_(i) - moment_(i)* (h_i1*h_i1) / 6; | |
| 491 | |||
| 492 | return | ||
| 493 | 46185528 | moment_(i)* x_i1*x_i1*x_i1 / (6 * h_i1) | |
| 494 | 23092764 | + | |
| 495 | 46185528 | moment_(i + 1)* x_i*x_i*x_i / (6 * h_i1) | |
| 496 | 23092764 | + | |
| 497 | 23092764 | A_i*x_i | |
| 498 | + | ||
| 499 | 23092764 | B_i; | |
| 500 | } | ||
| 501 | |||
| 502 | // evaluate the derivative of a spline given the actual position | ||
| 503 | // and the segment index | ||
| 504 | 1397492 | Scalar evalDerivative_(Scalar x, int i) const | |
| 505 | { | ||
| 506 | // See: Stoer2005 "Numerische Mathematik 1", 9th edition, | ||
| 507 | // Springer, 2005, p. 109 | ||
| 508 | 2794660 | Scalar h_i1 = h_(i + 1); | |
| 509 | 2794984 | Scalar x_i = x - x_(i); | |
| 510 | 2794984 | Scalar x_i1 = x_(i+1) - x; | |
| 511 | |||
| 512 | 1397492 | Scalar A_i = | |
| 513 | 4192476 | (y_(i+1) - y_(i))/h_i1 | |
| 514 | - | ||
| 515 | 4192476 | h_i1/6*(moment_(i+1) - moment_(i)); | |
| 516 | |||
| 517 | return | ||
| 518 | 2794984 | -moment_(i) * x_i1*x_i1 / (2 * h_i1) | |
| 519 | 1397492 | + | |
| 520 | 2794984 | moment_(i + 1) * x_i*x_i / (2 * h_i1) | |
| 521 | + | ||
| 522 | 1397492 | A_i; | |
| 523 | } | ||
| 524 | |||
| 525 | |||
| 526 | // returns the monotonicality of an interval of a spline segment | ||
| 527 | // | ||
| 528 | // The return value have the following meaning: | ||
| 529 | // | ||
| 530 | // 1: spline is monotonously increasing in the specified interval | ||
| 531 | // 0: spline is not monotonic in the specified interval | ||
| 532 | // -1: spline is monotonously decreasing in the specified interval | ||
| 533 | 1972 | int monotonic_(int i, Scalar x0, Scalar x1) const | |
| 534 | { | ||
| 535 | // shift the interval so that it is consistent with the | ||
| 536 | // definitions by Stoer2005 | ||
| 537 | 3944 | x0 = x0 - x_(i); | |
| 538 | 3944 | x1 = x1 - x_(i); | |
| 539 | |||
| 540 | 3944 | Scalar a = a_(i); | |
| 541 | 3944 | Scalar b = b_(i); | |
| 542 | 3944 | Scalar c = c_(i); | |
| 543 | |||
| 544 | 1972 | Scalar disc = 4*b*b - 12*a*c; | |
| 545 |
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1972 | if (disc < 0) { |
| 546 | // discriminant is smaller than 0, i.e. the segment does | ||
| 547 | // not exhibit any extrema. | ||
| 548 | ✗ | return (x0*(x0*3*a + 2*b) + c > 0) ? 1 : -1; | |
| 549 | } | ||
| 550 | using std::sqrt; | ||
| 551 | 1972 | disc = sqrt(disc); | |
| 552 | 1972 | Scalar xE1 = (-2*b + disc)/(6*a); | |
| 553 | 1972 | Scalar xE2 = (-2*b - disc)/(6*a); | |
| 554 | |||
| 555 |
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3944 | if (Dune::FloatCmp::eq<Scalar>(disc, 0)) { |
| 556 | // saddle point -> no extrema | ||
| 557 | ✗ | if (Dune::FloatCmp::eq<Scalar>(xE1, x0)) | |
| 558 | // make sure that we're not picking the saddle point | ||
| 559 | // to determine whether we're monotonically increasing | ||
| 560 | // or decreasing | ||
| 561 | ✗ | x0 = x1; | |
| 562 | ✗ | return sign(x0*(x0*3*a + 2*b) + c); | |
| 563 | } | ||
| 564 |
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1972 | if ((x0 < xE1 && xE1 < x1) || |
| 565 |
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631 | (x0 < xE2 && xE2 < x1)) |
| 566 | { | ||
| 567 | // there is an extremum in the range (x0, x1) | ||
| 568 | return 0; | ||
| 569 | } | ||
| 570 | // no extremum in range (x0, x1) | ||
| 571 | 1960 | x0 = (x0 + x1)/2; // pick point in the middle of the interval | |
| 572 | // to avoid extrema on the boundaries | ||
| 573 | 1960 | return sign(x0*(x0*3*a + 2*b) + c); | |
| 574 | } | ||
| 575 | |||
| 576 | /*! | ||
| 577 | * \brief Find all the intersections of a segment of the spline | ||
| 578 | * with a cubic polynomial within a specified interval. | ||
| 579 | */ | ||
| 580 | 55775 | int intersectSegment_(Scalar *sol, | |
| 581 | int segIdx, | ||
| 582 | Scalar a, Scalar b, Scalar c, Scalar d, | ||
| 583 | Scalar x0 = -1e100, Scalar x1 = 1e100) const | ||
| 584 | { | ||
| 585 | 55775 | int n = invertCubicPolynomial(sol, | |
| 586 | 55775 | a_(segIdx) - a, | |
| 587 | 111550 | b_(segIdx) - b, | |
| 588 | 55775 | c_(segIdx) - c, | |
| 589 | 111550 | d_(segIdx) - d); | |
| 590 | using std::max; | ||
| 591 | 167325 | x0 = max(x_(segIdx), x0); | |
| 592 | 167325 | x1 = max(x_(segIdx+1), x1); | |
| 593 | |||
| 594 | // filter the intersections outside of the specified interval | ||
| 595 | 55775 | int k = 0; | |
| 596 |
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184596 | for (int j = 0; j < n; ++j) { |
| 597 | 257642 | sol[j] += x_(segIdx); // add the offset of the interval. For details see Stoer2005 | |
| 598 |
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55775 | if (x0 <= sol[j] && sol[j] <= x1) { |
| 599 | 55775 | sol[k] = sol[j]; | |
| 600 | 55775 | ++k; | |
| 601 | } | ||
| 602 | } | ||
| 603 | 55775 | return k; | |
| 604 | } | ||
| 605 | |||
| 606 | // find the segment index for a given x coordinate | ||
| 607 | ✗ | int segmentIdx_(Scalar x) const | |
| 608 | { | ||
| 609 | // bisection | ||
| 610 | 24483712 | int iLow = 0; | |
| 611 | 48967424 | int iHigh = numSamples_() - 1; | |
| 612 | |||
| 613 |
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24500559 | while (iLow + 1 < iHigh) { |
| 614 | 12854 | int i = (iLow + iHigh) / 2; | |
| 615 | 25708 | if (x_(i) > x) | |
| 616 | iHigh = i; | ||
| 617 | else | ||
| 618 | iLow = i; | ||
| 619 | } | ||
| 620 | ✗ | return iLow; | |
| 621 | } | ||
| 622 | |||
| 623 | /*! | ||
| 624 | * \brief Returns x[i] - x[i - 1] | ||
| 625 | */ | ||
| 626 | 397 | Scalar h_(int i) const | |
| 627 | { | ||
| 628 | ✗ | assert(x_(i) > x_(i-1)); // the sampling points must be given | |
| 629 | // in ascending order | ||
| 630 | 75065709 | return x_(i) - x_(i - 1); | |
| 631 | } | ||
| 632 | |||
| 633 | /*! | ||
| 634 | * \brief Returns the y coordinate of the i-th sampling point. | ||
| 635 | */ | ||
| 636 | Scalar x_(int i) const | ||
| 637 |
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|
123464334 | { return asImp_().x_(i); } |
| 638 | |||
| 639 | /*! | ||
| 640 | * \brief Returns the y coordinate of the i-th sampling point. | ||
| 641 | */ | ||
| 642 | Scalar y_(int i) const | ||
| 643 |
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272446 | { return asImp_().y_(i); } |
| 644 | |||
| 645 | /*! | ||
| 646 | * \brief Returns the moment (i.e. second derivative) of the | ||
| 647 | * spline at the i-th sampling point. | ||
| 648 | */ | ||
| 649 | Scalar moment_(int i) const | ||
| 650 |
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242588644 | { return asImp_().moment_(i); } |
| 651 | |||
| 652 | // returns the coefficient in front of the x^0 term. In Stoer2005 this | ||
| 653 | // is delta. | ||
| 654 | 57747 | Scalar a_(int i) const | |
| 655 |
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173241 | { return (moment_(i+1) - moment_(i))/(6*h_(i+1)); } |
| 656 | |||
| 657 | // returns the coefficient in front of the x^2 term In Stoer2005 this | ||
| 658 | // is gamma. | ||
| 659 | Scalar b_(int i) const | ||
| 660 | 115494 | { return moment_(i)/2; } | |
| 661 | |||
| 662 | // returns the coefficient in front of the x^1 term. In Stoer2005 this | ||
| 663 | // is beta. | ||
| 664 | 57747 | Scalar c_(int i) const | |
| 665 |
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173241 | { return (y_(i+1) - y_(i))/h_(i + 1) - h_(i+1)/6*(2*moment_(i) + moment_(i+1)); } |
| 666 | |||
| 667 | // returns the coefficient in front of the x^0 term. In Stoer2005 this | ||
| 668 | // is alpha. | ||
| 669 | Scalar d_(int i) const | ||
| 670 | 111550 | { return y_(i); } | |
| 671 | |||
| 672 | /*! | ||
| 673 | * \brief Returns the number of sampling points. | ||
| 674 | */ | ||
| 675 | ✗ | int numSamples_() const | |
| 676 |
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49268839 | { return asImp_().numSamples(); } |
| 677 | }; | ||
| 678 | |||
| 679 | } // end namespace Dumux | ||
| 680 | |||
| 681 | #endif | ||
| 682 |