头文件：

/** Copyright (c) 2008-2011 Zhang Ming (M. Zhang), zmjerry@163.com** 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*//******************************************************************************                                    cqrd.h** Class template of QR decomposition for complex matrix.** For an m-by-n complex matrix A, the QR decomposition is an m-by-m unitary* matrix Q and an m-by-n upper triangular matrix R so that A = Q*R.** For economy size, denotes p = min(m,n), then Q is m-by-p, and R is n-by-p,* this file provides the economic decomposition format.** The QR decompostion always exists, even if the matrix does not have full* rank, so the constructor will never fail. The Q and R factors can be* retrived via the getQ() and getR() methods. Furthermore, a solve() method* is provided to find the least squares solution of Ax=b or AX=B using the* QR factors.** Zhang Ming, 2010-12, Xi'an Jiaotong University.*****************************************************************************/#ifndef CQRD_H
#define CQRD_H#include <matrix.h>namespace splab
{template <typename Type>class CQRD{public:CQRD();~CQRD();void dec( const Matrix<complex<Type> > &A );bool isFullRank() const;Matrix<complex<Type> > getQ();Matrix<complex<Type> > getR();Vector<complex<Type> > solve( const Vector<complex<Type> > &b );Matrix<complex<Type> > solve( const Matrix<complex<Type> > &B );private:// internal storage of QRMatrix<complex<Type> > QR;// diagonal of RVector<complex<Type> > diagR;// constants for generating Householder vectorVector<Type> betaR;};// class CQRD#include <cqrd-impl.h>}
// namespace splab#endif
// CQRD_H

实现文件：

/** Copyright (c) 2008-2011 Zhang Ming (M. Zhang), zmjerry@163.com** 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*//******************************************************************************                               qrd-impl.h** Implementation for CQRD class.** Zhang Ming, 2010-12, Xi'an Jiaotong University.*****************************************************************************//*** constructor and destructor*/
template<typename Type>
CQRD<Type>::CQRD()
{
}template<typename Type>
CQRD<Type>::~CQRD()
{
}/*** Create a QR factorization for a complex matrix A.*/
template <typename Type>
void CQRD<Type>::dec( const Matrix<complex<Type> > &A )
{int m = A.rows(),n = A.cols(),p = min(m,n);Type absV0, normV, beta;complex<Type> alpha;diagR = Vector<complex<Type> >(p);betaR = Vector<Type>(p);QR = A;// main loop.for( int k=0; k<p; ++k ){// Form k-th Householder vector.normV = 0;for( int i=k; i<m; ++i )normV += norm(QR[i][k]);normV = sqrt(normV);absV0 = abs(QR[k][k]);alpha = -normV * QR[k][k]/absV0;beta  = 1 / ( normV * (normV+absV0) );QR[k][k] -= alpha;// Apply transformation to remaining columns.for( int j=k+1; j<n; ++j ){complex<Type> s = 0;for( int i=k; i<m; ++i )s += conj(QR[i][k]) * QR[i][j];s *= beta;for( int i=k; i<m; ++i )QR[i][j] -= s*QR[i][k];}diagR[k] = alpha;betaR[k] = beta;}//    int m = A.rows(),
//        n = A.cols(),
//        p = min(m,n);
//
//    Type absV0, normV, beta;
//    complex<Type> alpha;
//
//    diagR = Vector<complex<Type> >(p);
//    betaR = Vector<Type>(p);
//  QR = A;
//
//    // main loop.
//    for( int k=0; k<p; ++k )
//    {
//      // Form k-th Householder vector.
//      Vector<complex<Type> > v(m-k);
//      for( int i=k; i<m; ++i )
//          v[i-k] = QR[i][k];
//
//      absV0 = abs(v[0]);
//      normV = norm(v);
//      alpha = -normV * v[0]/absV0;
//      beta  = 1 / ( normV * (normV+absV0) );
//      v[0] -= alpha;
//
//      for( int i=k; i<m; ++i )
//          QR[i][k] = v[i-k];
//
//      // Apply transformation to remaining columns.
//      for( int j=k+1; j<n; ++j )
//      {
//          complex<Type> s = 0;
//          for( int i=k; i<m; ++i )
//              s += conj(v[i-k]) * QR[i][j];
//          for( int i=k; i<m; ++i )
//              QR[i][j] -= beta*s*v[i-k];
//      }
//
//      diagR[k] = alpha;
//      betaR[k] = beta;
//  }
}/*** Flag to denote the matrix is of full rank.*/
template <typename Type>
inline bool CQRD<Type>::isFullRank() const
{for( int j=0; j<diagR.dim(); ++j )if( abs(diagR[j]) == Type(0) )return false;return true;
}/*** Return the upper triangular factorof the QR factorization.*/
template <typename Type>
Matrix<complex<Type> > CQRD<Type>::getQ()
{int m = QR.rows(),p = betaR.dim();Matrix<complex<Type> >Q( m, p );
//  for( int i=0; i<p; ++i )
//      Q[i][i] = 1;for( int k=p-1; k>=0; --k ){Q[k][k] = 1 - betaR[k]*norm(QR[k][k]);for( int i=k+1; i<m; ++i )Q[i][k] = -betaR[k]*QR[i][k]*conj(QR[k][k]);for( int j=k+1; j<p; ++j ){complex<Type> s = 0;for( int i=k; i<m; ++i )s += conj(QR[i][k])*Q[i][j];s *= betaR[k];for( int i=k; i<m; ++i )Q[i][j] -= s * QR[i][k];}}return Q;
}/*** Return the orthogonal factorof the QR factorization.*/
template <typename Type>
Matrix<complex<Type> > CQRD<Type>::getR()
{int n = QR.cols(),p = diagR.dim();Matrix<complex<Type> > R( p, n );for( int i=0; i<p; ++i ){R[i][i] = diagR[i];for( int j=i+1; j<n; ++j )R[i][j] = QR[i][j];}return R;
}/*** Least squares solution of A*x = b, where A and b are complex.* Return x: a n-length vector that minimizes the two norm* of Q*R*X-B. If B is non-conformant, or if QR.isFullRank()* is false, the routinereturns a null (0-length) vector.*/
template <typename Type>
Vector<complex<Type> > CQRD<Type>::solve( const Vector<complex<Type> > &b )
{int m = QR.rows(),n = QR.cols();assert( b.dim() == m );// matrix is rank deficientif( !isFullRank() )return Vector<complex<Type> >();Vector<complex<Type> > x = b;// compute y = Q^H * bfor( int k=0; k<n; ++k ){complex<Type> s = 0;for( int i=k; i<m; ++i )s += conj(QR[i][k])*x[i];s *= betaR[k];for( int i=k; i<m; ++i )x[i] -= s * QR[i][k];}// solve R*x = y;for( int k=n-1; k>=0; --k ){x[k] /= diagR[k];for( int i=0; i<k; ++i )x[i] -= x[k]*QR[i][k];}// return n portion of xVector<complex<Type> > x_(n);for( int i=0; i<n; ++i )x_[i] = x[i];return x_;
}/*** Least squares solution of A*X = B, where A and B are complex.* return X: a matrix that minimizes the two norm of Q*R*X-B.* If B is non-conformant, or if QR.isFullRank() is false, the* routinereturns a null (0) matrix.*/
template <typename Type>
Matrix<complex<Type> > CQRD<Type>::solve( const Matrix<complex<Type> > &B )
{int m = QR.rows();int n = QR.cols();assert( B.rows() == m );// matrix is rank deficientif( !isFullRank() )return Matrix<complex<Type> >(0,0);int nx = B.cols();Matrix<complex<Type> > X = B;// compute Y = Q^H*Bfor( int k=0; k<n; ++k )for( int j=0; j<nx; ++j ){complex<Type> s = 0;for( int i=k; i<m; ++i )s += conj(QR[i][k])*X[i][j];s *= betaR[k];for( int i=k; i<m; ++i )X[i][j] -= s*QR[i][k];}// solve R*X = Y;for( int k=n-1; k>=0; --k ){for( int j=0; j<nx; ++j )X[k][j] /= diagR[k];for( int i=0; i<k; ++i )for( int j=0; j<nx; ++j )X[i][j] -= X[k][j]*QR[i][k];}// return n x nx portion of XMatrix<complex<Type> > X_( n, nx );for( int i=0; i<n; ++i )for( int j=0; j<nx; ++j )X_[i][j] = X[i][j];return X_;
}

测试代码：

/******************************************************************************                               cqrd_test.cpp** CQRD class testing.** Zhang Ming, 2010-12, Xi'an Jiaotong University.*****************************************************************************/#define BOUNDS_CHECK#include <iostream>
#include <iomanip>
#include <cqrd.h>using namespace std;
using namespace splab;typedef double  Type;
const   int     M = 4;
const   int     N = 3;int main()
{Matrix<Type> A(M,N);A[0][0] = 1;    A[0][1] = 2;   A[0][2] = 3;A[1][0] = 1;  A[1][1] = 2;   A[1][2] = 4;A[2][0] = 1;  A[2][1] = 9;   A[2][2] = 7;A[3][0] = 5;  A[3][1] = 6;   A[3][2] = 8;
//  A = trT(A);Matrix<complex<Type> > cA = complexMatrix( A, elemMult(A,A) );CQRD<Type> qr;qr.dec(cA);Matrix<complex<Type> > Q = qr.getQ();Matrix<complex<Type> > R = qr.getR();cout << setiosflags(ios::fixed) << setprecision(3);cout << "The original matrix cA : " << cA << endl;cout << "The orthogonal matrix Q  : " << Q << endl;cout << "The upper triangular matrix R : " << R << endl;cout << "Q^H * Q : " << trMult(Q,Q) << endl;cout << "cA - Q*R : " << cA - Q*R << endl;Vector<Type> b(M);b[0]= 1;   b[1] = 0;  b[2] = 1, b[3] = 2;Vector<complex<Type> > cb = complexVector(b);if( qr.isFullRank() ){Vector<complex<Type> > x = qr.solve(cb);cout << "The constant vector cb : " << cb << endl;cout << "The least squares solution of cA * x = cb : " << x << endl;Matrix<complex<Type> > X = qr.solve( eye( M, complex<Type>(1) ) );cout << "The least squares solution of cA * X = I : " << X << endl;cout << "The cA * X: " << cA*X << endl;}elsecout << " The matrix is rank deficient! " << endl;return 0;
}

运行结果：

The original matrix cA : size: 4 by 3
(1.000,1.000)   (2.000,4.000)   (3.000,9.000)
(1.000,1.000)   (2.000,4.000)   (4.000,16.000)
(1.000,1.000)   (9.000,81.000)  (7.000,49.000)
(5.000,25.000)  (6.000,36.000)  (8.000,64.000)The orthogonal matrix Q  : size: 4 by 3
(-0.055,0.000)  (-0.029,-0.000) (0.370,-0.040)
(-0.055,0.000)  (-0.029,-0.000) (0.913,-0.143)
(-0.055,0.000)  (-0.985,-0.158) (-0.042,-0.000)
(-0.828,-0.552) (0.043,0.039)   (-0.063,-0.030)The upper triangular matrix R : size: 3 by 3
(-18.111,-18.111)       (-25.565,-31.418)       (-42.737,-52.676)
(0.000,0.000)   (-20.071,-77.269)       (-11.956,-45.432)
(0.000,0.000)   (0.000,0.000)   (-0.593,12.784)Q^H * Q : size: 3 by 3
(1.000,0.000)   (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) (1.000,0.000)cA - Q*R : size: 4 by 3
(-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 cb : size: 4 by 1
(1.000,0.000)
(0.000,0.000)
(1.000,0.000)
(2.000,0.000)The least squares solution of cA * x = cb : size: 3 by 1
(-0.002,-0.035)
(-0.002,-0.002)
(0.007,-0.016)The least squares solution of cA * X = I : size: 3 by 4
(-0.007,0.048)  (-0.025,0.120)  (-0.002,0.012)  (0.004,-0.048)
(-0.001,0.017)  (-0.004,0.042)  (0.001,-0.014)  (-0.001,-0.002)
(0.002,-0.029)  (0.008,-0.072)  (0.000,0.003)   (0.003,0.005)The cA * X: size: 4 by 4
(0.143,-0.000)  (0.348,0.017)   (0.016,-0.003)  (0.022,-0.016)
(0.348,-0.017)  (0.858,0.000)   (-0.007,0.001)  (-0.009,0.007)
(0.016,0.003)   (-0.007,-0.001) (1.000,-0.000)  (-0.000,0.000)
(0.022,0.016)   (-0.009,-0.007) (-0.000,-0.000) (0.999,0.000)Process returned 0 (0x0)   execution time : 0.109 s
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