american.edu/cas/mathstat/People/kalman/pdffiles/mmgautodiff.pdf">Doubly Recursive Multivariate Automatic Differentiation, Mathematics Magazine, vol. 75, no. 3, June 2002. However, in order to avoid performances bottlenecks, the recursive rules are "compiled" once in an unfold form. This class does this recursion unrolling and stores the computation rules as simple loops with pre-computed indirection arrays.
This class maps all derivative computation into single dimension arrays that hold the value and partial derivatives. The class does not hold these arrays, which remains under the responsibility of the caller. For each combination of number of free parameters and derivation order, only one compiler is necessary, and this compiler will be used to perform computations on all arrays provided to it, which can represent hundreds or thousands of different parameters kept together with all theur partial derivatives.
The arrays on which compilers operate contain only the partial derivatives together with the 0th derivative, i.e. the value. The partial derivatives are stored in a compiler-specific order, which can be retrieved using methods {@link #getPartialDerivativeIndex(int) getPartialDerivativeIndex} and {@link #getPartialDerivativeOrders(int)}. The value is guaranteed to be stored as the first element (i.e. the {@link #getPartialDerivativeIndex(int) getPartialDerivativeIndex} method returns0 when called with 0 for all derivation orders and {@link #getPartialDerivativeOrders(int) getPartialDerivativeOrders} returns an array filled with 0 when called with 0 as the index).
Note that the ordering changes with number of parameters and derivation order. For example given 2 parameters x and y, df/dy is stored at index 2 when derivation order is set to 1 (in this case the array has three elements: f, df/dx and df/dy). If derivation order is set to 2, then df/dy will be stored at index 3 (in this case the array has six elements: f, df/dx, df/dxdx, df/dy, df/dxdy and df/dydy).
Given this structure, users can perform some simple operations like adding, subtracting or multiplying constants and negating the elements by themselves, knowing if they want to mutate their array or create a new array. These simple operations are not provided by the compiler. The compiler provides only the more complex operations between several arrays.
This class is mainly used as the engine for scalar variable {@link DerivativeStructure}. It can also be used directly to hold several variables in arrays for more complex data structures. User can for example store a vector of n variables depending on three x, y and z free parameters in one array as follows:
// parameter 0 is x, parameter 1 is y, parameter 2 is z int parameters = 3; DSCompiler compiler = DSCompiler.getCompiler(parameters, order); int size = compiler.getSize(); // pack all elements in a single array double[] array = new double[n * size]; for (int i = 0; i < n; ++i) { // we know value is guaranteed to be the first element array[i * size] = v[i]; // we don't know where first derivatives are stored, so we ask the compiler array[i * size + compiler.getPartialDerivativeIndex(1, 0, 0) = dvOnDx[i][0]; array[i * size + compiler.getPartialDerivativeIndex(0, 1, 0) = dvOnDy[i][0]; array[i * size + compiler.getPartialDerivativeIndex(0, 0, 1) = dvOnDz[i][0]; // we let all higher order derivatives set to 0 } Then in another function, user can perform some operations on all elements stored in the single array, such as a simple product of all variables:
// compute the product of all elements double[] product = new double[size]; prod[0] = 1.0; for (int i = 0; i < n; ++i) { double[] tmp = product.clone(); compiler.multiply(tmp, 0, array, i * size, product, 0); } // value double p = product[0]; // first derivatives double dPdX = product[compiler.getPartialDerivativeIndex(1, 0, 0)]; double dPdY = product[compiler.getPartialDerivativeIndex(0, 1, 0)]; double dPdZ = product[compiler.getPartialDerivativeIndex(0, 0, 1)]; // cross derivatives (assuming order was at least 2) double dPdXdX = product[compiler.getPartialDerivativeIndex(2, 0, 0)]; double dPdXdY = product[compiler.getPartialDerivativeIndex(1, 1, 0)]; double dPdXdZ = product[compiler.getPartialDerivativeIndex(1, 0, 1)]; double dPdYdY = product[compiler.getPartialDerivativeIndex(0, 2, 0)]; double dPdYdZ = product[compiler.getPartialDerivativeIndex(0, 1, 1)]; double dPdZdZ = product[compiler.getPartialDerivativeIndex(0, 0, 2)];
@see DerivativeStructure
@since 3.1