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Compute Sparse Jacobians Using Subgraphs


We use @(@ F : \B{R}^n \rightarrow \B{R}^m @)@ to denote the function corresponding to f . Here n is the domain size, and m is the range size, or f . The syntax above takes advantage of sparsity when computing the Jacobian @[@ J(x) = F^{(1)} (x) @]@ The first syntax computes the sparsity pattern and the value of the Jacobian at the same time. If one only wants the sparsity pattern, it should be faster to use subgraph_sparsity .

This routine uses a subgraph technique. To be specific, for each dependent variable, it a subgraph of the operation sequence to determine which independent variables affect it. This avoids to overhead of performing set operations that is inherent in other methods for computing sparsity patterns.

The type BaseVector is a SimpleVector class with elements of type Base .

The type SizeVector is a SimpleVector class with elements of type size_t.

The type BoolVector is a SimpleVector class with elements of type bool.

This object has prototype
Note that the Taylor coefficients stored in f are affected by this operation; see uses forward below.

This argument has prototype
It is the value of x at which we are computing the Jacobian.

Uses Forward
After each call to Forward , the object f contains the corresponding Taylor coefficients . After a call to sparse_jac_forward or sparse_jac_rev, the zero order coefficients correspond to
f.Forward(0, x)
All the other forward mode coefficients are unspecified.

This argument has prototype
SizeVectorBaseVector>& subset
Its row size is == m , and its column size is == n . It specifies which elements of the Jacobian are computed. The input elements in its value vector subset.val() do not matter. Upon return it contains the value of the corresponding elements of the Jacobian.

The argument select_domain has prototype
It has size @(@ n @)@ and specifies which independent variables to include.

The argument select_range has prototype
It has size @(@ m @)@ and specifies which components of the range to include in the calculation. A subgraph is built for each dependent variable and the selected set of independent variables.

This argument has prototype
SizeVectorBaseVector>& matrix_out
This input value of matrix_out does not matter. Upon return matrix_out is sparse matrix representation of @(@ F^{(1)} (x) @)@. The matrix has @(@ m @)@ rows, @(@ n @)@ columns. If select_domain[j] is true, select_range[i] is true, and @(@ F_i (x) @)@ depends on @(@ x_j @)@, then the pair @(@ (i, j) @)@ is in matrix_out . For each k = 0 , ...matrix_out.nnz() , let
i = matrix_out.row()[k]
j = matrix_out.col()[k]
v = matrix_out.val()[k]
It follows that the partial of @(@ F_i (x) @)@ with respect to @(@ x_j @)@ is equal to @(@ v @)@.

The files subgraph_jac_rev.cpp and subgraph_hes2jac.cpp are examples and tests using subgraph_jac_rev. They returns true for success and false for failure.
Input File: cppad/core/subgraph_jac_rev.hpp