Odwracanie macierzy to poważny problem (szczególnie dla studentów). Z tego powodu udostępniamy nasz kod C++ pozwalający na łatwe, szybkie i przyjemne odwrócenie macierzy kwadratowej o różnym wymiarze. Zaimplementowane zostały dwa popularne algorytmy: Gaussian elimination oraz LU decomposition(factorization) . Z oczywistych względów polecamy korzystanie z tego drugiego. Użycie jest bardzo łatwe. Wystarczy dowolną tablicę o rozmiarze N * N (gdzie N jest rozmiarem wiersza lub kolumny) , w postaci wskaźnika przekazać w sposób następujący:
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<typ> X[N * N]; // gdzie <typ> to double/float/int/... sophisticatedAlgebra::Inverter<typ, N>::gaussian(X); //LU //albo sophisticatedAlgebra::Inverter<typ, N>::lu(X); //LU |
Obliczenia zostaną wykonane w miejscu, więc zawartość tablicy X zostanie nadpisana.
Oto nagłówek, zawierający niezbędny kod:
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#ifndef INVERTER_H #define INVERTER_H #include <cmath> namespace sophisticatedAlgebra { template <typename T> T sgn(const T& val) { if (val >= static_cast<T>(0)) { return static_cast<T>(1); } else { return static_cast<T>(-1); } } template <typename T, int N> class Inverter { public: static void gaussian(T* X) { T a[(2 * N) + 1][(2 * N) + 1]; T d = 0; for (int i = 0; i < 2 * N + 1; ++i) { for (int j = 0; j < 2 * N + 1; ++j) { a[i][j] = 0; } } for (int i = 1; i <= N; i++) { for (int j = 1; j <= 2 * N; j++) { if (j == (i + N)) { a[i][j] = 1; } } } for (int i = 1; i <= N; ++i) { for (int j = 1; j <= N; ++j) { a[i][j] = X[j - 1 + (i - 1) * N]; } } for (int i = N; i > 1; --i) { if (a[i - 1][1] < a[i][1]) { for (int j = 1; j <= N * 2; ++j) { d = a[i][j]; a[i][j] = a[i - 1][j]; a[i - 1][j] = d; } } } for (int i = 1; i <= N; ++i) { for (int j = 1; j <= N * 2; ++j) { if (j != i) { d = a[j][i] / a[i][i]; for (int k = 1; k <= N * 2; ++k) { a[j][k] = a[j][k] - (a[i][k] * d); } } } } for (int i = 1; i <= N; ++i) { d = a[i][i]; for (int j = 1; j <= N * 2; ++j) { a[i][j] /= d; } } for (int i = 1; i <= N; ++i) { for (int j = N + 1; j <= N * 2; ++j) { X[j - (N + 1) + (i - 1) * N] = a[i][j]; } } } static void lu(T* X) { T Z[N][N]; T lower[N][N]; T upper[N][N]; T I[N][N]; for (int column = 0; column < N; ++column) { for (int row = 0; row < N; ++row) { Z[column][row] = 0; if (column == row) { I[column][row] = 1; } else { I[column][row] = 0; } } } luDecomposition(X, lower, upper); for (int column = 0; column < N; ++column) { for (int row = 0; row < N; ++row) { Z[row][column] = valZ(lower, upper, I, Z, row, column); } } for (int column = 0; column < N; ++column) { for (int row = N - 1; row >= 0; --row) { X[column + row * N] = inversal(upper, X, Z, row, column); } } } private: static void luDecomposition(T *X, T lower[N][N], T upper[N][N]) { for (int row = 0; row < N; ++row) { for (int column = 0; column < N; ++column) { if (column < row) { lower[column][row] = 0; } else { lower[column][row] = X[row + column * N]; for (int k = 0; k < row; ++k) { lower[column][row] = lower[column][row] - lower[column][k] * upper[k][row]; } } } for (int column = 0; column < N; ++column) { if (column < row) { upper[row][column] = 0; } else if (column == row) { upper[row][column] = 1; } else { upper[row][column] = X[row + column * N] / lower[row][row]; for (int k = 0; k < row; k++) { upper[row][column] = upper[row][column] - ((lower[row][k] * upper[k][column]) / lower[row][row]); } } } } } static T valZ(const T lower[N][N], const T upper[N][N], const T I[N][N], const T Z[N][N], int row, int column) { T sum = 0; for (int i = 0; i < N; i++) { if (i != row) { sum += lower[row][i] * Z[i][column]; } } T result = I[row][column] - sum; result = result / lower[row][row]; return result; } static T inversal(const T upper[N][N], const T* inversed, const T Z[N][N], int row, int column) { T sum = 0; for (int i = 0; i < N; i++) { if (i != row) { sum += upper[row][i] * inversed[column + i * N]; } } T result = Z[row][column] - sum; result = result / upper[row][row]; return result; } }; template <typename T> class Inverter<T, 0> { static void gaussian(T* X) {} static void lu(T* X) {} }; template <typename T> class Inverter<T, 1> { static void gaussian(T* X) { X[0] = static_cast<T>(1) / X[0]; } static void lu(T* X) { gaussian(X); } }; template <typename T> class Inverter<T, 2> { static void gaussian(T* X) { T determinant; T adjoint[4]; adjoint[0] = X[3]; adjoint[1] = -X[1]; adjoint[2] = -X[2]; adjoint[3] = X[0]; determinant = X[0] * adjoint[0] + X[1] * adjoint[2]; determinant = static_cast<T>(1) / determinant; X[0] = adjoint[0] * determinant; X[1] = adjoint[1] * determinant; X[2] = adjoint[2] * determinant; X[3] = adjoint[3] * determinant; } static void lu(T* X) { gaussian(X); } }; template <typename T> class Inverter<T, 3> { static void gaussian(T* X) { T determinant; T adjoint[9]; adjoint[0] = X[4] * X[8] - X[5] * X[7]; adjoint[1] = -X[1] * X[8] + X[2] * X[7]; adjoint[2] = X[1] * X[5] - X[2] * X[4]; adjoint[3] = -X[3] * X[8] + X[5] * X[6]; adjoint[4] = X[0] * X[8] - X[2] * X[6]; adjoint[5] = -X[0] * X[5] + X[2] * X[3]; adjoint[6] = X[3] * X[7] - X[4] * X[6]; adjoint[7] = -X[0] * X[7] + X[1] * X[6]; adjoint[8] = X[0] * X[4] - X[1] * X[3]; determinant = X[0] * adjoint[0] + X[1] * adjoint[3] + X[2] * adjoint[6]; determinant = static_cast<T>(1) / determinant; X[0] = adjoint[0] * determinant; X[1] = adjoint[1] * determinant; X[2] = adjoint[2] * determinant; X[3] = adjoint[3] * determinant; X[4] = adjoint[4] * determinant; X[5] = adjoint[5] * determinant; X[6] = adjoint[6] * determinant; X[7] = adjoint[7] * determinant; X[8] = adjoint[8] * determinant; } static void lu(T* X) { gaussian(X); } }; } #endif |
Miłej laborki 🙂
PS. Jeżeli wersja Gaussian elimination produkuje błędne wyniki, to jest to zwykle błąd numeryczny, na który ta metoda jest bardzo wrażliwa. To jeden z wielu powodów żeby wybrać LU decomposition.