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Numerical differentiation is an entire topic unto itself, a simple google should bring up enough material for you to work with (just the wiki might be sufficient). There are parameters of your problem that I cannot know, so I can only speak broadly here, but there are direct methods of determining the gradient at a given point, i.e. ones that don't require an interpolation. See the wikipedia for the formulae (ranging from the simple <code>f(x+1)-f(x)</code>, which is where <code>h=1</code>, to the higher order ones). Calculating the partial derivatives is then a simple <code>O(NM)</code> loop with an uber easy formula inside (no interpolation required). The specifics can get gritty: <ol> <li>The higher order formulae need to be reduced for the edges, or discarded altogether.</li> <li>Your precise speed requirements might render more complex formulae useless (depending on the platform sometimes the lookup times for higher order formulae make them too slow; again, it depends on the cache etc.). This is easy to test, the formulae are simple; code them and benchmark.</li> <li>The specific implementation is also dependent on your error requirements. The theory provides error bounds, so that will play a role in what formula you need; but again, there's a trade-off with speed requirements. The in turn can be practically lowered if you know specifics about the types of matrices you'll be processing, if such a thing is known.</li> </ol> The implementation can be made even easier (and maybe faster) if you have existing convolution tools, since this method is really just a convolution of the matrix (note; technically it's called a cross-correlation).
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singulars
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1. POGetting a gradient of a bi-variant function
 singulars
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1. PTAnswer
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1. USdavin
plurals
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