Levmar:Levenberg-Marquardt非線性最小二乘算法
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簡介
Gauss-Newton算法是一個古老的處理非線性最小二乘問題的方法。該方法在迭代過程中要求矩陣J(x)滿秩。爲了克服這個困難,Levenberg(1944)提出了一種新的方法,但未受到重視。後來Marquardt(1963)又重新提出,並在理論上進行了控討,得到Levenberg-Marquardt方法,簡稱LM方法。在此基礎上,Fletcher(1971)對其實現策略進行了改進,得到了Levenberg-Marquardt-Fletcher方法(LMF)。再後來,More(1978)將LM方法與信賴域方法結合,建立了帶信賴域的LM方法。
LM算法的產生主要是解決曲線最小二乘擬合問題,現在很多軟件使用LM算法來解決通用的曲線擬合問題。
本文主要介紹GPL開源庫levmar2.6使用Visual Studio在Windows上進行編譯。這個開源庫的官方網站是:http://users.ics.forth.gr/~lourakis/levmar/
2. 編譯levmar
下載源碼levmar-2.6解壓,在其README.txt中對levmar的授權GPL、編譯等進行了說明。在Windows操作系統中,可以使用nmake /f Makefile.vc來編譯levmar和一個示例程序。
從官網介紹可知,levmar有些算法依賴LAPACK庫,一個線性代數計算開源庫。所以如果要使用那些算法,編譯的時候必須包含這個庫。從示例程序的源文件lmdemo.c中可以看出,有些問題的求解是需要LAPACK庫的,相關源碼列出如下:
/* 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;
從上述源碼可知,如果LAPACK庫不可用的時候,示例程序中的問題
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
這些問題的求解功能是不能使用的。從頭文件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 */
從頭文件levmar.h中的代碼可以看出,在#ifdef HAVE_LAPACK和#endif /* HAVE_LAPACK */之間的函數都是不可用的。除此之外的函數是可用的,如基本的dlevmar_der和dlevmar_dif等函數是不依賴LAPACK庫的。如果只使用這幾個函數,則可以不用配置LAPACK庫,編譯levmar就很簡單了。
如果不使用LAPACK庫,可以先在頭文件levmar.h中把#define HAVE_LAPACK 這一行註釋掉:
然後再修改Makefile.vc文件,在Makefile.vc中可以看到如下圖所示一句註釋,即當不使用LAPACK庫是,把那一行註釋掉(前面加#):
這時就可以啓動Visual Studio的編譯器CL來編譯levmar庫了。配置好編譯環境的命令工具從Visual Studio的菜單來啓動:
要編譯32位的levmar庫,可以使用x86的命令工具,要編譯64位的levmar,可以使用x64的命令工具。啓動命令工具後,切換到levmar源碼文件夾,並輸入命令
nmake /f Makefile.vc
如下圖所示:
編譯成功生成levmar.lib和lmdemo.exe說明編譯成功了。
接着在命令窗口中運行lmdemo.exe,測試levmar例子程序。如果lmdemo正常運行,說明levmar已經成功編譯。
自己的程序如果要使用levmar,就可以像使用其他開源庫一樣,設置頭文件路徑及庫levmar.lib的路徑,就可以使用了。