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<HTML><HEAD><TITLE>Manpage of LMDER_</TITLE></HEAD><BODY><H1>LMDER_</H1>Section: C Library Functions (3)<BR>Updated: March 8, 2002<BR><A HREF="#index">Index</A><A HREF="index.html">Return to Main Contents</A><HR><A NAME="lbAB"> </A><H2>NAME</H2>lmder_, lmder1_ - minimize the sum of squares of m nonlinear functions, with user supplied Jacobian <A NAME="lbAC"> </A><H2>SYNOPSIS</H2><B>include <<A HREF="file:/usr/include/minpack.h">minpack.h</A>></B><DL COMPACT><DT><B>void lmder1_ ( </B><B>void (*</B><I>fcn</I><B>)</B><B>(int *</B><I>m</I><B>,</B><B>int *</B><I>n</I><B>,</B><B>double *</B><I>x</I><B>,</B><B>double *</B><I>fvec</I><B>,</B><B>double *</B><I>fjac</I><B>,</B><DD><B>int *</B><I>ldfjac</I><B>,</B><B>int *</B><I>iflag</I><B>),</B><DL COMPACT><DT><DD><B>int *</B><I>m</I><B>,</B><B>int * </B><I>n</I><B>,</B><B>double *</B><I>x</I><B>,</B><B>double *</B><I>fvec</I><B>,</B><B>double *</B><I>fjac</I><B>,</B><B>int *</B><I>ldfjac</I><B>,</B><DD><B>double *</B><I>tol</I><B>,</B><B>int *</B><I>info</I><B>,</B><B>int *</B><I>iwa</I><B>,</B><B>double *</B><I>wa</I><B>,</B><B>int *</B><I>lwa</I><B>);</B></DL><DT><B>void lmder_</B><B>( void (*</B><I>fcn</I><B>)(</B><B>int *</B><I>m</I><B>,</B><B>int *</B><I>n</I><B>,</B><B>double *</B><I>x</I><B>,</B><B>double *</B><I>fvec</I><B>,</B><B>double *</B><I>fjac</I><B>,</B><DD><B>int *</B><I>ldfjac</I><B>,</B><B>int *</B><I>iflag</I><B>),</B><DL COMPACT><DT><DD><B>int *</B><I>m</I><B>,</B><B>int *</B><I>n</I><B>,</B><B>double *</B><I>x</I><B>,</B><B>double *</B><I>fvec</I><B>,</B><B>double *</B><I>fjac</I><B>,</B><B>int *</B><I>ldfjac</I><B>,</B><DD><B>double *</B><I>ftol</I><B>,</B><B>double *</B><I>xtol</I><B>,</B><B>double *</B><I>gtol</I><B>,</B><B>int *</B><I>maxfev</I><B>,</B><B>double *</B><I>diag</I><B>,</B><B>int *</B><I>mode</I><B>,</B><DD><B>double *</B><I>factor</I><B>,</B><B>int *</B><I>nprint</I><B>,</B><B>int *</B><I>info</I><B>,</B><DD><B>int *</B><I>nfev</I><B>,</B><B>int *</B><I>njev</I><B>,</B><B>int *</B><I>ipvt</I><B>,</B><DD><B>double *</B><I>qtf</I><B>,</B><B>double *</B><I>wa1</I><B>,</B><B>double *</B><I>wa2</I><B>,</B><B>double *</B><I>wa3</I><B>,</B><B>double *</B><I>wa4</I><B> );</B></DL><DD></DL><A NAME="lbAD"> </A><H2>DESCRIPTION</H2><P><DD>The purpose of <B>lmder_</B> is to minimize the sum of the squares of<I>m</I> nonlinear functions in <I>n</I> variables by a modification ofthe Levenberg-Marquardt algorithm. The user must provide a function which calculates the functions and the Jacobian.<P><B>lmder1_</B> performs the same functionas <B>lmder_</B> but has a simplified calling sequence. <P><B>lmstr</B> and <B>lmstr1</B> also perform the same function but useminimal storage.<BR><A NAME="lbAE"> </A><H3>Language notes</H3>These functions are written in FORTRAN. If calling fromC, keep these points in mind:<DL COMPACT><DT>Name mangling.<DD>With <B>g77</B> version 2.95 or 3.0, all the function names end in anunderscore. This may change with future versions of <B>g77</B>.<DT>Compile with <B>g77</B>.<DD>Even if your program is all C code, you should link with <B>g77</B>so it will pull in the FORTRAN libraries automatically. It's easiestjust to use <B>g77</B> to do all the compiling. (It handles C just fine.)<DT>Call by reference.<DD>All function parameters must be pointers.<DT>Column-major arrays.<DD>Suppose a function returns an array with 5 rows and 3 columns in anarray <I>z</I> and in the call you have declared a leading dimension of7. The FORTRAN and equivalent C references are:<P><PRE> z(1,1) z[0] z(2,1) z[1] z(5,1) z[4] z(1,2) z[7] z(1,3) z[14] z(i,j) z[(i-1) + (j-1)*7]</PRE><BR></DL><A NAME="lbAF"> </A><H3>User-supplied Function</H3><P><I>fcn</I> is the name of the user-supplied subroutine which calculatesthe functions. In FORTRAN, <I>fcn</I> must be declared in an externalstatement in the user calling program, and should be written asfollows:<P><PRE> subroutine fcn(m,n,x,fvec,fjac,ldfjac,iflag) integer m,n,iflag double precision x(n),fvec(m),fjac(ldfjac,n) ---------- if iflag = 1 calculate the functions at x and return this vector in fvec. do not alter fjac. if iflag = 2 calculate the jacobian at x and return this matrix in fjac. do not alter fvec. ---------- return end</PRE><P>In C, <I>fcn</I> should be written as follows:<P><PRE> void fcn(int m, int n, double *x, double *fvec, double *fjac, int *ldfjac, int *iflag) { /* if iflag = 1 calculate the functions at x and return this vector in fvec[0] through fvec[m-1]. do not alter fjac. if iflag = 2 calculate the jacobian at x and return this matrix in fjac. do not alter fvec. */ }</PRE><P>The value of <I>iflag</I> should not be changed by <I>fcn</I> unless theuser wants to terminate execution of <B>lmder_</B> (or <B>lmder1_</B>). Inthis case set <I>iflag</I> to a negative integer.<BR><A NAME="lbAG"> </A><H3>Parameters for both <B>lmder_</B> and <B>lmder1_</B></H3><P><I>m</I> is a positive integer input variable set to the numberof functions.<P><I>n</I> is a positive integer input variable set to the numberof variables. <I>n</I> must not exceed <I>m</I>.<P><I>x</I> is an array of length <I>n</I>. On input <I>x</I> must containan initial estimate of the solution vector. On output <I>x</I>contains the final estimate of the solution vector.<P><I>fvec</I> is an output array of length <I>m</I> which containsthe functions evaluated at the output <I>x</I>.<P><I>fjac</I> is an output <I>m</I> by <I>n</I> array. The upper <I>n</I> by<I>n</I> submatrix of <I>fjac</I> contains an upper triangular matrix<I>r</I> with diagonal elements of nonincreasing magnitude such that<P><BR> t t t<BR> p *(jac *jac)*p = r *r,<P>where <I>p</I> is a permutation matrix and <I>jac</I> is the finalcalculated Jacobian. column <B>j</B> of <I>p</I> is column<I>ipvt</I>(<B>j</B>) (see below) of the identity matrix. The lowertrapezoidal part of <I>fjac</I> contains information generated duringthe computation of <I>r</I>.<P><I>ldfjac</I> is a positive integer input variable not less than<I>m</I> which specifies the leading dimension of the array<I>fjac</I>.<BR><A NAME="lbAH"> </A><H3>Parameters for <B>lmder1_</B></H3><P><I>tol</I> is a nonnegative input variable. Termination occurs whenthe algorithm estimates either that the relative error in the sum ofsquares is at most <I>tol</I> or that the relative error between<I>x</I> and the solution is at most <I>tol</I>.<P><I>info</I> is an integer output variable. if the user hasterminated execution, <I>info</I> is set to the (negative)value of iflag. see description of <I>fcn</I>. otherwise,<I>info</I> is set as follows.<P><BR> <I>info</I> = 0 improper input parameters.<P><BR> <I>info</I> = 1 algorithm estimates that the relative errorin the sum of squares is at most <I>tol</I>.<P><BR> <I>info</I> = 2 algorithm estimates that the relative errorbetween x and the solution is at most <I>tol</I>.<P><BR> <I>info</I> = 3 conditions for <I>info</I> = 1 and <I>info</I> = 2 both hold.<P><BR> <I>info</I> = 4 <I>fvec</I> is orthogonal to the columns of theJacobian to machine precision.<P><BR> <I>info</I> = 5 number of calls to <I>fcn</I> has reached orexceeded 200*(<I>n</I>+1).<P><BR> <I>info</I> = 6 <I>tol</I> is too small. no further reduction inthe sum of squares is possible.<P><BR> <I>info</I> = 7 <I>tol</I> is too small. no further improvement inthe approximate solution x is possible.<P><I>iwa</I> is an integer work array of length <I>n</I>.<P><I>wa</I> is a work array of length <I>lwa</I>.<P><I>lwa</I> is an integer input variable not less than <I>m</I>*<I>n</I> +5*<I>n</I> + <I>m</I> for <B>lmder1_</B>.<BR><A NAME="lbAI"> </A><H3>Parameters for <B>lmder_</B></H3><P><I>ftol</I> is a nonnegative input variable. Terminationoccurs when both the actual and predicted relativereductions in the sum of squares are at most <I>ftol</I>.Therefore, <I>ftol</I> measures the relative error desiredin the sum of squares.<P><I>xtol</I> is a nonnegative input variable. Terminationoccurs when the relative error between two consecutiveiterates is at most <I>xtol</I>. Therefore, <I>xtol</I> measures therelative error desired in the approximate solution.<P><I>gtol</I> is a nonnegative input variable. Terminationoccurs when the cosine of the angle between <I>fvec</I> andany column of the Jacobian is at most <I>gtol</I> in absolutevalue. Therefore, <I>gtol</I> measures the orthogonalitydesired between the function vector and the columnsof the Jacobian.<P><I>maxfev</I> is a positive integer input variable. Terminationoccurs when the number of calls to <I>fcn</I> is at least<I>maxfev</I> by the end of an iteration.<P><I>diag</I> is an array of length <I>n</I>. If <I>mode</I> = 1 (seebelow), <I>diag</I> is internally set. If <I>mode</I> = 2, <I>diag</I>must contain positive entries that serve asmultiplicative scale factors for the variables.<P><I>mode</I> is an integer input variable. If <I>mode</I> = 1, thevariables will be scaled internally. If <I>mode</I> = 2,the scaling is specified by the input <I>diag</I>. Othervalues of mode are equivalent to <I>mode</I> = 1.<P><I>factor</I> is a positive input variable used in determining theinitial step bound. This bound is set to the product of <I>factor</I>and the euclidean norm of <I>diag</I>*<I>x</I> if the latter isnonzero, or else to <I>factor</I> itself. In most cases factor shouldlie in the interval (.1,100.). 100. is a generally recommendedvalue.<P><I>nprint</I> is an integer input variable that enables controlledprinting of iterates if it is positive. In this case, fcn is calledwith <I>iflag</I> = 0 at the beginning of the first iteration andevery <I>nprint</I> iterations thereafter and immediately prior toreturn, with <I>x</I> and <I>fvec</I> available for printing. If<I>nprint</I> is not positive, no special calls of fcn with<I>iflag</I> = 0 are made.<P><I>info</I> is an integer output variable. If the user hasterminated execution, info is set to the (negative)value of iflag. See description of fcn. Otherwise,info is set as follows.<P><BR> <I>info</I> = 0 improper input parameters.<P><BR> <I>info</I> = 1 both actual and predicted relative reductionsin the sum of squares are at most <I>ftol</I>.<P><BR> <I>info</I> = 2 relative error between two consecutive iteratesis at most <I>xtol</I>.<P><BR> <I>info</I> = 3 conditions for <I>info</I> = 1 and <I>info</I> = 2 both hold.<P><BR> <I>info</I> = 4 the cosine of the angle between fvec and anycolumn of the Jacobian is at most gtol in absolute value.<P><BR> <I>info</I> = 5 number of calls to <I>fcn</I> has reached orexceeded maxfev.<P><BR> <I>info</I> = 6 <I>ftol</I> is too small. No further reduction inthe sum of squares is possible.<P><BR> <I>info</I> = 7 <I>xtol</I> is too small. No further improvement inthe approximate solution x is possible.<P><BR> <I>info</I> = 8 <I>gtol</I> is too small. <I>fvec</I> is orthogonal tothe columns of the Jacobian to machine precision.<P><I>nfev</I> is an integer output variable set to the number ofcalls to <I>fcn</I> with <I>iflag</I> = 1.<P><I>njev</I> is an integer output variable set to the number ofcalls to fcn with <I>iflag</I> = 2.<P><I>ipvt</I> is an integer output array of length <I>n</I>. <I>ipvt</I>defines a permutation matrix <I>p</I> such that <I>jac</I>*<I>p</I> =<I>q</I>*<I>r</I>, where <I>jac</I> is the final calculated Jacobian,<I>q</I> is orthogonal (not stored), and <I>r</I> is upper triangularwith diagonal elements of nonincreasing magnitude. Column <B>j</B>of <I>p</I> is column <I>ipvt</I>(<B>j</B>) of the identity matrix.<P><I>qtf</I> is an output array of length <I>n</I> which containsthe first <I>n</I> elements of the vector (<I>q</I> transpose)*<I>fvec</I>.<P><I>wa1</I>, <I>wa2</I>, and <I>wa3</I> are work arrays of length <I>n</I>.<P><I>wa4</I> is a work array of length <I>m</I>.<BR><A NAME="lbAJ"> </A><H2>SEE ALSO</H2><B><A HREF="lmdif_.html">lmdif</A></B>(3),<B><A HREF="lmdif_.html">lmdif1</A></B>(3),<B><A HREF="lmstr_.html">lmstr</A></B>(3),<B><A HREF="lmstr_.html">lmstr1</A></B>(3).<BR><A NAME="lbAK"> </A><H2>AUTHORS</H2>Jorge More', Burt Garbow, and Ken Hillstrom at Argonne National Laboratory.This manual page was written by Jim Van Zandt <<A HREF="mailto:jrv@debian.org">jrv@debian.org</A>>,for the Debian GNU/Linux system (but may be used by others).<P><HR><A NAME="index"> </A><H2>Index</H2><DL><DT><A HREF="#lbAB">NAME</A><DD><DT><A HREF="#lbAC">SYNOPSIS</A><DD><DT><A HREF="#lbAD">DESCRIPTION</A><DD><DL><DT><A HREF="#lbAE">Language notes</A><DD><DT><A HREF="#lbAF">User-supplied Function</A><DD><DT><A HREF="#lbAG">Parameters for both <B>lmder_</B> and <B>lmder1_</B></A><DD><DT><A HREF="#lbAH">Parameters for <B>lmder1_</B></A><DD><DT><A HREF="#lbAI">Parameters for <B>lmder_</B></A><DD></DL><DT><A HREF="#lbAJ">SEE ALSO</A><DD><DT><A HREF="#lbAK">AUTHORS</A><DD></DL><HR>This document was created by<A HREF="index.html">man2html</A>,using the manual pages.<BR>Time: 10:19:50 GMT, April 20, 2007</BODY></HTML>⌨️ 快捷键说明
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