复数矩阵Cholesky分解算法的C++实现
头文件：
/* * Copyright (c) 2008-2011 Zhang Ming (M. Zhang), [email protected] * * 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 or any later version. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions are met: * * 1. Redistributions of source code must retain the above copyright notice, * this list of conditions and the following disclaimer. * * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * 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. A copy of the GNU General Public License is available at: * http://www.fsf.org/licensing/licenses */ /***************************************************************************** * ccholesky.h * * Class template of complex matrix Cholesky factorization. * * For a conjugate symmetric, positive definite matrix A, this class computes * the Cholesky factorization, i.e. it computes a lower triangular matrix L * such that A = L*L^H. If the matrix is not conjugate symmetric or positive * definite, the class computes only a partial decomposition. This can be * tested with the isSpd() flag. * * Zhang Ming, 2010-12, Xi'an Jiaotong University. *****************************************************************************/ #ifndef CCHOLESKY_H #define CCHOLESKY_H #include <matrix.h> namespace splab { template <typename Type> class CCholesky { public: CCholesky(); ~CCholesky(); bool isSpd() const; void dec( const Matrix<Type> &A ); Matrix<Type> getL() const; Vector<Type> solve( const Vector<Type> &b ); Matrix<Type> solve( const Matrix<Type> &B ); private: bool spd; Matrix<Type> L; }; // class CCholesky #include <ccholesky-impl.h> } // namespace splab #endif // CCHOLESKY_H
实现文件：
/* * Copyright (c) 2008-2011 Zhang Ming (M. Zhang), [email protected] * * 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 or any later version. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions are met: * * 1. Redistributions of source code must retain the above copyright notice, * this list of conditions and the following disclaimer. * * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * 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. A copy of the GNU General Public License is available at: * http://www.fsf.org/licensing/licenses */ /***************************************************************************** * ccholesky-impl.h * * Implementation for CCholesky (complex Cholesky) class. * * Zhang Ming, 2010-12, Xi'an Jiaotong University. *****************************************************************************/ /** * constructor and destructor */ template<typename Type> CCholesky<Type>::CCholesky() : spd(true) { } template<typename Type> CCholesky<Type>::~CCholesky() { } /** * return true, if original matrix is symmetric positive-definite. */ template<typename Type> inline bool CCholesky<Type>::isSpd() const { return spd; } /** * Constructs a lower triangular matrix L, such that L*L^H= A. * If A is not symmetric positive-definite (SPD), only a partial * factorization is performed. If isspd() evalutate true then * the factorizaiton was successful. */ template <typename Type> void CCholesky<Type>::dec( const Matrix<Type> &A ) { int m = A.rows(); int n = A.cols(); spd = (m == n); if( !spd ) return; L = Matrix<Type>(A.cols(),A.cols()); // main loop for( int j=0; j<A.rows(); ++j ) { Type d = 0; spd = spd && (imag(A[j][j]) == 0); for( int k=0; k<j; ++k ) { Type s = 0; for( int i=0; i<k; ++i ) s += L[k][i] * conj(L[j][i]); L[j][k] = s = (A[j][k]-s) / L[k][k]; d = d + s*conj(s); spd = spd && (A[k][j] == conj(A[j][k])); } d = A[j][j] - d; spd = spd && ( real(d) > 0 ); L[j][j] = sqrt( real(d) > 0 ? d : 0 ); for( int k=j+1; k<A.rows(); ++k ) L[j][k] = 0; } } /** * return the lower triangular factor, L, such that L*L'=A. */ template<typename Type> inline Matrix<Type> CCholesky<Type>::getL() const { return L; } /** * Solve a linear system A*x = b, using the previously computed * cholesky factorization of A: L*L'. */ template <typename Type> Vector<Type> CCholesky<Type>::solve( const Vector<Type> &b ) { int n = L.rows(); if( b.dim() != n ) return Vector<Type>(); Vector<Type> x = b; // solve L*y = b for( int k=0; k<n; ++k ) { for( int i=0; i<k; ++i ) x[k] -= x[i]*L[k][i]; x[k] /= L[k][k]; } // solve L^H*x = y for( int k=n-1; k>=0; --k ) { for( int i=k+1; i<n; ++i ) x[k] -= x[i]*conj(L[i][k]); x[k] /= L[k][k]; } return x; } /** * Solve a linear system A*X = B, using the previously computed * cholesky factorization of A: L*L'. */ template <typename Type> Matrix<Type> CCholesky<Type>::solve( const Matrix<Type> &B ) { int n = L.rows(); if( B.rows() != n ) return Matrix<Type>(); Matrix<Type> X = B; int nx = B.cols(); // solve L*Y = B for( int j=0; j<nx; ++j ) for( int k=0; k<n; ++k ) { for( int i=0; i<k; ++i ) X[k][j] -= X[i][j]*L[k][i]; X[k][j] /= L[k][k]; } // solve L^H*x = y for( int j=0; j<nx; ++j ) for( int k=n-1; k>=0; --k ) { for( int i=k+1; i<n; ++i ) X[k][j] -= X[i][j]*conj(L[i][k]); X[k][j] /= L[k][k]; } return X; }
测试代码：
/*****************************************************************************
* ccholesky_test.cpp
*
* Comnplex Cholesky class testing.
*
* Zhang Ming, 2010-12, Xi'an Jiaotong University.
*****************************************************************************/
#define BOUNDS_CHECK
#include <iostream>
#include <iomanip>
#include <ccholesky.h>
using namespace std;
using namespace splab;
typedef double Type;
const int N = 5;
int main()
{
cout << setiosflags(ios::fixed) << setprecision(3);
Matrix<complex<Type> > A(N,N), L(N,N);
Vector<complex<Type> > b(N);
for( int i=1; i<N+1; ++i )
{
for( int j=1; j<N+1; ++j )
if( i == j )
A(i,i) = complex<Type>(N+i,0);
else
if( i < j )
A(i,j) = complex<Type>(Type(i),cos(Type(i)));
else
A(i,j) = complex<Type>(Type(j),-cos(Type(j)));
b(i) = i*(i+1)/2 + i*(N-i);
}
cout << "The original matrix A : " << A << endl;
CCholesky<complex<Type> > cho;
cho.dec(A);
if( !cho.isSpd() )
cout << "Factorization was not complete." << endl;
else
{
L = cho.getL();
cout << "The lower triangular matrix L is : " << L << endl;
cout << "A - L*L^H is : " << A - L*trH(L) << endl;
Vector<complex<Type> > x = cho.solve(b);
cout << "The constant vector b : " << b << endl;
cout << "The solution of Ax = b : " << x << endl;
cout << "The solution of Ax - b : " << A*x-b << endl;
Matrix<complex<Type> > IA = cho.solve(eye(N,complex<Type>(1,0)));
cout << "The inverse matrix of A : " << IA << endl;
cout << "The product of A*inv(A) : " << A*IA << endl;
}
return 0;
}
运行结果：
The original matrix A : size: 5 by 5 (6.000,0.000) (1.000,0.540) (1.000,0.540) (1.000,0.540) (1.000,0.540) (1.000,-0.540) (7.000,0.000) (2.000,-0.416) (2.000,-0.416) (2.000,-0.416) (1.000,-0.540) (2.000,0.416) (8.000,0.000) (3.000,-0.990) (3.000,-0.990) (1.000,-0.540) (2.000,0.416) (3.000,0.990) (9.000,0.000) (4.000,-0.654) (1.000,-0.540) (2.000,0.416) (3.000,0.990) (4.000,0.654) (10.000,0.000) The lower triangular matrix L is : size: 5 by 5 (2.449,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.408,-0.221) (2.605,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.408,-0.221) (0.685,0.160) (2.700,0.000) (0.000,0.000) (0.000,0.000) (0.408,-0.221) (0.685,0.160) (0.848,0.367) (2.727,0.000) (0.000,0.000) (0.408,-0.221) (0.685,0.160) (0.848,0.367) (0.893,0.240) (2.753,0.000) A - L*L^H is : size: 5 by 5 (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (-0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) (0.000,0.000) The constant vector b : size: 5 by 1 (5.000,0.000) (9.000,0.000) (12.000,0.000) (14.000,0.000) (15.000,0.000) The solution of Ax = b : size: 5 by 1 (0.356,-0.310) (0.562,0.207) (0.746,0.284) (0.843,-0.037) (0.842,-0.214) The solution of Ax - b : size: 5 by 1 (0.000,-0.000) (-0.000,0.000) (-0.000,0.000) (-0.000,0.000) (0.000,0.000) The inverse matrix of A : size: 5 by 5 (0.177,0.000) (-0.018,-0.007) (-0.014,-0.004) (-0.008,-0.006) (-0.005,-0.007) (-0.018,0.007) (0.164,-0.000) (-0.024,0.013) (-0.020,0.003) (-0.017,-0.002) (-0.014,0.004) (-0.024,-0.013) (0.160,-0.000) (-0.032,0.018) (-0.029,0.008) (-0.008,0.006) (-0.020,-0.003) (-0.032,-0.018) (0.150,0.000) (-0.043,0.012) (-0.005,0.007) (-0.017,0.002) (-0.029,-0.008) (-0.043,-0.012) (0.132,0.000) The product of A*inv(A) : size: 5 by 5 (1.000,0.000) (-0.000,-0.000) (-0.000,-0.000) (-0.000,-0.000) (0.000,-0.000) (0.000,-0.000) (1.000,0.000) (-0.000,0.000) (0.000,-0.000) (0.000,-0.000) (0.000,-0.000) (-0.000,-0.000) (1.000,0.000) (-0.000,0.000) (-0.000,-0.000) (0.000,-0.000) (0.000,0.000) (-0.000,-0.000) (1.000,0.000) (0.000,0.000) (0.000,-0.000) (0.000,0.000) (-0.000,0.000) (0.000,0.000) (1.000,-0.000) Process returned 0 (0x0) execution time : 0.125 s Press any key to continue.