Levmar:Levenberg-Marquardt非线性最小二乘算法

Abstract. Levmar is GPL native ANSI C implementations of the Levenberg-Marquardt optimization algorithm.The blog focus on the compilation of levmar on Windows with Visual Studio.

Key Words. Levmar, C, LM least squares

1. levmar简介

Gauss-Newton算法是一个古老的处理非线性最小二乘问题的方法。该方法在迭代过程中要求矩阵J(x)满秩。为了克服这个困难，Levenberg(1944)提出了一种新的方法，但未受到重视。后来Marquardt(1963)又重新提出，并在理论上进行了控讨，得到Levenberg-Marquardt方法，简称LM方法。在此基础上，Fletcher(1971)对其实现策略进行了改进，得到了Levenberg-Marquardt-Fletcher方法（LMF）。再后来，More(1978)将LM方法与信赖域方法结合，建立了带信赖域的LM方法。

LM算法的产生主要是解决曲线最小二乘拟合问题，现在很多软件使用LM算法来解决通用的曲线拟合问题。

本文主要介绍GPL开源库levmar2.6使用Visual Studio在Windows上进行编译。这个开源库的官方网站是：http://users.ics.forth.gr/~lourakis/levmar/

2. 编译levmar

下载源码levmar-2.6解压，在其README.txt中对levmar的授权GPL、编译等进行了说明。在Windows操作系统中，可以使用nmake /f Makefile.vc来编译levmar和一个示例程序。

从官网介绍可知，levmar有些算法依赖LAPACK库，一个线性代数计算开源库。所以如果要使用那些算法，编译的时候必须包含这个库。从示例程序的源文件lmdemo.c中可以看出，有些问题的求解是需要LAPACK库的，相关源码列出如下：

/* uncomment the appropriate line below to select a minimization problem */
  problem=
          //0; // Rosenbrock function
          //1; // modified Rosenbrock problem
          //2; // Powell's function
      //3; // Wood's function
          4; // Meyer's (reformulated) problem
          //5; // Osborne's problem
      //6; // helical valley function
#ifdef HAVE_LAPACK
      //7; // Boggs & Tolle's problem 3
      //8; // Hock - Schittkowski problem 28
      //9; // Hock - Schittkowski problem 48
      //10; // Hock - Schittkowski problem 51
#else // no LAPACK
#ifdef _MSC_VER
#pragma message("LAPACK not available, some test problems cannot be used")
#else
#warning LAPACK not available, some test problems cannot be used
#endif // _MSC_VER

#endif /* HAVE_LAPACK */
      //11; // Hock - Schittkowski problem 01
      //12; // Hock - Schittkowski modified problem 21
      //13; // hatfldb problem
      //14; // hatfldc problem
      //15; // equilibrium combustion problem
#ifdef HAVE_LAPACK
      //16; // Hock - Schittkowski modified #1 problem 52
      //17; // Schittkowski modified problem 235
      //18; // Boggs & Tolle modified problem #7
      //19; // Hock - Schittkowski modified #2 problem 52
      //20; // Hock - Schittkowski modified problem #76"
#endif /* HAVE_LAPACK */
                
  switch(problem){
  default: fprintf(stderr, "unknown problem specified (#%d)! Note that some minimization problems require LAPACK.\n", problem);
           exit(1);
    break;

从上述源码可知，如果LAPACK库不可用的时候，示例程序中的问题

l 7 Boggs & Tolle’s problem 3

l 8 Hock - Schittkowski problem 28

l 9 Hock - Schittkowski problem 48

l 10 Hock - Schittkowski problem 51

l 16 Hock - Schittkowskit modified #1 problem 52

l 17 Schittkowski modified problem 235

l 18 Boggs & Tolle modified problem #7

l 19 Hock - Schittkowski modified #2 problem 52

l 20 Hock - Schittkowski modified probem #76

这些问题的求解功能是不能使用的。从头文件levmar.h中要以看出，

#ifdef LM_DBL_PREC
/* double precision LM, with & without Jacobian */
/* unconstrained minimization */
extern int dlevmar_der(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      void (*jacf)(double *p, double *j, int m, int n, void *adata),
      double *p, double *x, int m, int n, int itmax, double *opts,
      double *info, double *work, double *covar, void *adata);

extern int dlevmar_dif(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      double *p, double *x, int m, int n, int itmax, double *opts,
      double *info, double *work, double *covar, void *adata);

/* box-constrained minimization */
extern int dlevmar_bc_der(
       void (*func)(double *p, double *hx, int m, int n, void *adata),
       void (*jacf)(double *p, double *j, int m, int n, void *adata),  
       double *p, double *x, int m, int n, double *lb, double *ub, double *dscl,
       int itmax, double *opts, double *info, double *work, double *covar, void *adata);

extern int dlevmar_bc_dif(
       void (*func)(double *p, double *hx, int m, int n, void *adata),
       double *p, double *x, int m, int n, double *lb, double *ub, double *dscl,
       int itmax, double *opts, double *info, double *work, double *covar, void *adata);

#ifdef HAVE_LAPACK
/* linear equation constrained minimization */
extern int dlevmar_lec_der(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      void (*jacf)(double *p, double *j, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *A, double *b, int k,
      int itmax, double *opts, double *info, double *work, double *covar, void *adata);

extern int dlevmar_lec_dif(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *A, double *b, int k,
      int itmax, double *opts, double *info, double *work, double *covar, void *adata);

/* box & linear equation constrained minimization */
extern int dlevmar_blec_der(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      void (*jacf)(double *p, double *j, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *lb, double *ub, double *A, double *b, int k, double *wghts,
      int itmax, double *opts, double *info, double *work, double *covar, void *adata);

extern int dlevmar_blec_dif(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *lb, double *ub, double *A, double *b, int k, double *wghts,
      int itmax, double *opts, double *info, double *work, double *covar, void *adata);

/* box, linear equations & inequalities constrained minimization */
extern int dlevmar_bleic_der(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      void (*jacf)(double *p, double *j, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *lb, double *ub,
      double *A, double *b, int k1, double *C, double *d, int k2,
      int itmax, double *opts, double *info, double *work, double *covar, void *adata);

extern int dlevmar_bleic_dif(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *lb, double *ub, 
      double *A, double *b, int k1, double *C, double *d, int k2,
      int itmax, double *opts, double *info, double *work, double *covar, void *adata);

/* box & linear inequality constraints */
extern int dlevmar_blic_der(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      void (*jacf)(double *p, double *j, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *lb, double *ub, double *C, double *d, int k2,
      int itmax, double opts[4], double info[LM_INFO_SZ], double *work, double *covar, void *adata);

extern int dlevmar_blic_dif(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *lb, double *ub, double *C, double *d, int k2,
      int itmax, double opts[5], double info[LM_INFO_SZ], double *work, double *covar, void *adata);

/* linear equation & inequality constraints */
extern int dlevmar_leic_der(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      void (*jacf)(double *p, double *j, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *A, double *b, int k1, double *C, double *d, int k2,
      int itmax, double opts[4], double info[LM_INFO_SZ], double *work, double *covar, void *adata);

extern int dlevmar_leic_dif(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *A, double *b, int k1, double *C, double *d, int k2,
      int itmax, double opts[5], double info[LM_INFO_SZ], double *work, double *covar, void *adata);

/* linear inequality constraints */
extern int dlevmar_lic_der(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      void (*jacf)(double *p, double *j, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *C, double *d, int k2,
      int itmax, double opts[4], double info[LM_INFO_SZ], double *work, double *covar, void *adata);

extern int dlevmar_lic_dif(
      void (*func)(double *p, double *hx, int m, int n, void *adata),
      double *p, double *x, int m, int n, double *C, double *d, int k2,
      int itmax, double opts[5], double info[LM_INFO_SZ], double *work, double *covar, void *adata);
#endif /* HAVE_LAPACK */

#endif /* LM_DBL_PREC */

从头文件levmar.h中的代码可以看出，在#ifdef HAVE_LAPACK和#endif /* HAVE_LAPACK */之间的函数都是不可用的。除此之外的函数是可用的，如基本的dlevmar_der和dlevmar_dif等函数是不依赖LAPACK库的。如果只使用这几个函数，则可以不用配置LAPACK库，编译levmar就很简单了。

如果不使用LAPACK库，可以先在头文件levmar.h中把#define HAVE_LAPACK 这一行注释掉：