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Ceres Solver
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1.13
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Why?
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Installation
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Tutorial
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On Derivatives
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Spivak Notation
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Analytic Derivatives
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Numeric derivatives
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Forward Differences
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Implementation Details
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Central Differences
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Ridders’ Method
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Recommendations
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Automatic Derivatives
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Interfacing with Automatic Differentiation
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Modeling Non-linear Least Squares
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Solving Non-linear Least Squares
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Covariance Estimation
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General Unconstrained Minimization
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FAQS, Tips & Tricks
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Users
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Contributing
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Version History
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Bibliography
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License
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Ceres Solver
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Docs »
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On Derivatives »
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Numeric derivatives
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Numeric derivatives¶
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The other extreme from using analytic derivatives is to use numeric
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derivatives. The key observation here is that the process of
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differentiating a function \(f(x)\) w.r.t \(x\) can be written
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as the limiting process:
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\[Df(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}\]
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Forward Differences¶
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Now of course one cannot perform the limiting operation numerically on
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a computer so we do the next best thing, which is to choose a small
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value of \(h\) and approximate the derivative as
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\[Df(x) \approx \frac{f(x + h) - f(x)}{h}\]
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The above formula is the simplest most basic form of numeric
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differentiation. It is known as the Forward Difference formula.
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So how would one go about constructing a numerically differentiated
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version of Rat43Analytic (Rat43) in
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Ceres Solver. This is done in two steps:
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Define Functor that given the parameter values will evaluate the
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residual for a given \((x,y)\).
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Construct a CostFunction by using
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NumericDiffCostFunction to wrap an instance of
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Rat43CostFunctor.
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struct Rat43CostFunctor {
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Rat43CostFunctor(const double x, const double y) : x_(x), y_(y) {}
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bool operator()(const double* parameters, double* residuals) const {
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const double b1 = parameters[0];
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const double b2 = parameters[1];
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const double b3 = parameters[2];
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const double b4 = parameters[3];
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residuals[0] = b1 * pow(1.0 + exp(b2 - b3 * x_), -1.0 / b4) - y_;
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return true;
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}
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const double x_;
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const double y_;
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}
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CostFunction* cost_function =
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new NumericDiffCostFunction<Rat43CostFunctor, FORWARD, 1, 4>(
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new Rat43CostFunctor(x, y));
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