blocks|key|1503178|text|试试Numerical+Recipes+(Levenberg-Marquardt在15.5节中)。它可以在网上找到，我发现他们以一种详细的方式解释了算法(他们有完整的源代码，你可以得到更详细的……)，但却是可以访问的。|type|unstyled|depth|inlineStyleRanges|entityRanges|offset|length|data|1503179|entityMap|0|LINK|mutability|MUTABLE|url|http://apps.nrbook.com/c/index.html^0|2|H|0|0^^$0|@$1|2|3|4|5|6|7|L|8|@]|9|@$A|M|B|N|1|O]]|C|$]]|$1|D|3|-4|5|6|7|P|8|@]|9|@]|C|$]]]|E|$F|$5|G|H|I|C|$J|K]]]]

Try <a href="http://apps.nrbook.com/c/index.html" rel="nofollow noreferrer">Numerical Recipes</a> (Levenberg-Marquardt is in Section 15.5). It's available online, and I find that they explain algorithms in a way that's detailed (they have complete source code, how much more detailed can you get...), yet accessible.

blocks|key|1288461|text|最小化函数就像试图找到曲面上的最低点。想象一下，你走在一个丘陵的表面上，你正试图到达最低点。你会找到下坡的方向，然后一直走到它不再下坡。然后，你会选择一个新的方向，这个方向是下坡的，沿着这个方向走，直到它不再下坡，依此类推。最终(希望)你会到达一个没有方向走下坡路的地步。然后，您将处于(本地)最小值。|type|unstyled|depth|inlineStyleRanges|entityRanges|data|1288462|LM算法和许多其他最小化算法都使用此方案。|1288463|假设被最小化的函数是F，并且我们在迭代中处于x(n)点。我们希望找到下一个迭代x(n%2B1)使得F(x(n%2B1))+<+F(x(n))，即函数值较小。为了选择x(n%2B1)，我们需要两件事，从x(n)开始的方向和步长(在该方向上走多远)。LM算法按如下方式确定这些值：|1288464|首先，计算F在点x(n)处的线性近似。线性函数的下坡方向很容易找到，因此我们使用线性逼近函数来确定下坡方向。接下来，我们需要知道我们可以在这个选择的方向上走多远。如果我们的近似线性函数在x(n)附近的大范围内是F的一个很好的近似，那么我们可以采取相当大的步骤。如果它只是一个非常接近x(n)的良好近似值，那么我们只能迈出非常小的一步。|1288465|这就是LM所做的-计算F在x(n)处的线性近似，从而给出下坡方向，然后它根据线性函数在x(n)处对F的逼近程度来计算要走多大的一步。LM通过在这样确定的方向上基本上走一步，并将对F的线性近似减少到实际函数F的减少程度来计算近似函数的好坏。如果它们很接近，则近似函数很好，我们可以采取更大的步长。如果它们不接近，那么近似函数就不好，我们应该后退一步，采取更小的步骤。|1288466|entityMap^0|0|0|0|0|0^^$0|@$1|2|3|4|5|6|7|L|8|@]|9|@]|A|$]]|$1|B|3|C|5|6|7|M|8|@]|9|@]|A|$]]|$1|D|3|E|5|6|7|N|8|@]|9|@]|A|$]]|$1|F|3|G|5|6|7|O|8|@]|9|@]|A|$]]|$1|H|3|I|5|6|7|P|8|@]|9|@]|A|$]]|$1|J|3|-4|5|6|7|Q|8|@]|9|@]|A|$]]]|K|$]]

Minimizing a function is like trying to find lowest point on a surface. Think of yourself walking on a hilly surface and that you are trying to get to the lowest point. You would find the direction that goes downhill and walk until it doesn't go downhill anymore. Then you would chose a new direction that goes downhill and walk in that direction until it doesn't go downhill anymore, and so on. Eventually (hopefully) you would reach a point where no direction goes downhill anymore. You would then be at a (local) minimum.

The LM algorithm, and many other minimization algorithms, use this scheme.

Suppose that the function being minimized is F and we are at the point x(n) in our iteration. We wish to find the next iterate x(n+1) such that F(x(n+1)) &lt; F(x(n)), i.e. the function value is smaller. In order to chose x(n+1) we need two things, a direction from x(n) and a step size (how far to go in that direction). The LM algorithm determines these values as follows -

First, compute a linear approximation to F at the point x(n). It is easy to find out the downhill direction of a linear function, so we use the linear approximating function to determine the downhill direction. 
Next, we need to know how far we can go in this chosen direction. If our approximating linear function is a good approximation for F for a large area around x(n), then we can take a fairly large step. If it's a good approximation only very close to x(n), then we can take only a very small step. 

This is what LM does - calculates a linear approximation to F at x(n), thus giving the downhill direction, then it figures out how big a step to take based on how well the linear function approximates F at x(n). LM figures out how good the approximating function is by basically taking a step in the direction thus determined and comparing how much the linear approximation to F decreased to the how much the the actual function F decreased. If they are close, the approximating function is good and we can take a little larger step. If they are not close then the approximation function is not good and we should back off and take a smaller step.

blocks|key|1288519|text|我使用these+notes+from+a+course+at+Purdue+University在MATLAB中编写了一个通用的Levenberg-Marquardt曲线拟合算法，该算法计算数值导数，因此可以接受f(x;p)形式的任何函数，其中p是拟合参数的向量。|type|unstyled|depth|inlineStyleRanges|offset|length|style|CODE|entityRanges|data|1288520|entityMap|0|LINK|mutability|MUTABLE|url|http://cobweb.ecn.purdue.edu/~kak/courses-i-teach/ECE661.08/homework/HW5_LM_handout.pdf^0|2Z|6|3F|1|3|1A|0|0^^$0|@$1|2|3|4|5|6|7|N|8|@$9|O|A|P|B|C]|$9|Q|A|R|B|C]]|D|@$9|S|A|T|1|U]]|E|$]]|$1|F|3|-4|5|6|7|V|8|@]|D|@]|E|$]]]|G|$H|$5|I|J|K|E|$L|M]]]]

I used <a href="http://cobweb.ecn.purdue.edu/~kak/courses-i-teach/ECE661.08/homework/HW5_LM_handout.pdf" rel="nofollow noreferrer">these notes from a course at Purdue University</a> to code up a generic Levenberg-Marquardt curve-fitting algorithm in MATLAB that computes numerical derivatives and therefore accepts any function of the form <code>f(x;p)</code> where <code>p</code> is a vector of fitting parameters.

blocks|key|1288565|text|LM算法的基本思想可以在几页内解释-但对于快速和健壮的生产级实现，许多微妙的优化是必要的。最先进的技术仍然是由Moré等人实现的Minpack，由MORé1978+(http://link.springer.com/content/pdf/10.1007/BFb0067700.pdf)和Minpack用户指南(http://www.mcs.anl.gov/~more/ANL8074b.pdf)详细记录。为了研究代码，我的C翻译(https://jugit.fz-juelich.de/mlz/lmfit)可能比原始的Fortran代码更容易访问。|type|unstyled|depth|inlineStyleRanges|entityRanges|offset|length|data|1288566|entityMap|0|LINK|mutability|MUTABLE|url|http://link.springer.com/content/pdf/10.1007/BFb0067700.pdf|1|http://www.mcs.anl.gov/~more/ANL8074b.pdf|2|https://jugit.fz-juelich.de/mlz/lmfit^0|2B|1N|0|4C|15|1|60|11|2|0^^$0|@$1|2|3|4|5|6|7|P|8|@]|9|@$A|Q|B|R|1|S]|$A|T|B|U|1|V]|$A|W|B|X|1|Y]]|C|$]]|$1|D|3|-4|5|6|7|Z|8|@]|9|@]|C|$]]]|E|$F|$5|G|H|I|C|$J|K]]|L|$5|G|H|I|C|$J|M]]|N|$5|G|H|I|C|$J|O]]]]

The basic ideas of the LM algorithm can be explained in a few pages - but for a production-grade implementation that is fast and robust, many subtle optimizations are necessary. State of the art is still the Minpack implementation by Moré et al., documented in detail by Moré 1978 (<a href="http://link.springer.com/content/pdf/10.1007/BFb0067700.pdf" rel="nofollow noreferrer">http://link.springer.com/content/pdf/10.1007/BFb0067700.pdf</a>) and in the Minpack user guide (<a href="http://www.mcs.anl.gov/~more/ANL8074b.pdf" rel="nofollow noreferrer">http://www.mcs.anl.gov/~more/ANL8074b.pdf</a>). To study the code, my C translation (<a href="https://jugit.fz-juelich.de/mlz/lmfit" rel="nofollow noreferrer">https://jugit.fz-juelich.de/mlz/lmfit</a>) is probably more accessible than the original Fortran code.

Im a programmer that wants to learn how the Levenberg–Marquardt curvefitting algorithm works so that i can implement it myself. Is there a good tutorial anywhere that can explain how it works in detail with the reader beeing a programmer and not a mathemagician.

My goal is to implement this algorithm in opencl so that i can have it run hardware accelerated.

How does the Levenberg–Marquardt algorithm work in detail but in an understandable way?

我是一名程序员，想要了解Levenberg-Marquardt曲线拟合算法是如何工作的，这样我就可以自己实现它。在读者是程序员而不是数学家的情况下，有没有好的教程可以详细解释它是如何工作的？我的目标是在opencl中实现这个算法，这样我就可以让它运行硬件加速。

问Levenberg-Marquardt算法是如何以一种可以理解的方式详细工作的？
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