cpp_sparse_jac#

Computing Sparse Jacobians#

Syntax#

work =  cppad_py::sparse_jac_work ()
n_sweep = f.sparse_jac_for ( subset , x , pattern , work )
n_sweep = f.sparse_jac_rev ( subset , x , pattern , work )

Purpose#

We use \(F : \B{R}^n \rightarrow \B{R}^m\) to denote the function corresponding to f . The syntax above takes advantage of sparsity when computing the Jacobian

\[J(x) = F^{(1)} (x)\]

In the sparse case, this should be faster and take less memory than cpp_fun_jacobian. We use the notation \(J_{i,j} (x)\) to denote the partial of \(F_i (x)\) with respect to \(x_j\).

sparse_jac_for#

This function uses first order forward mode sweeps cpp_fun_forward to compute multiple columns of the Jacobian at the same time.

sparse_jac_rev#

This function uses first order reverse mode sweeps cpp_fun_reverse to compute multiple rows of the Jacobian at the same time.

f#

This object has prototype

ADFun <Base> f

Note that the Taylor coefficients stored in f are affected by this operation; see uses_forward below.

subset#

This argument has prototype

sparse_rcv& subset

Its row size is subset.nr () == m , and its column size is subset.nc () == n . It specifies which elements of the Jacobian are computed. The input value of its value vector subset.val () does not matter. Upon return it contains the value of the corresponding elements of the Jacobian. All of the row, column pairs in subset must also appear in pattern ; i.e., they must be possibly non-zero.

x#

This argument has prototype

const vec_double& x

and its size is n . It specifies the point at which to evaluate the Jacobian \(J(x)\).

pattern#

This argument has prototype

const sparse_rc& pattern

Its row size is pattern.nr () == m , and its column size is pattern.nc () == n . It is a sparsity pattern for the Jacobian \(J(x)\). This argument is not used (and need not satisfy any conditions), when work is non-empty.

work#

This argument has prototype

sparse_jac_work& work

We refer to its initial value, and its value after work.clear () , as empty. If it is empty, information is stored in work . This can be used to reduce computation when a future call is for the same object f , the same member function sparse_jac_for or sparse_jac_rev , and the same subset of the Jacobian. If any of these values change, use work.clear () to empty this structure.

n_sweep#

The return value n_sweep has prototype

int n_sweep

If sparse_jac_for ( sparse_jac_rev ) is used, n_sweep is the number of first order forward (reverse) sweeps used to compute the requested Jacobian values. This is proportional to the total computational work, not counting the zero order forward sweep, or combining multiple columns (rows) into a single sweep.

Uses Forward#

After each call to cpp_fun_forward, the object f contains the corresponding Taylor coefficients for all the variables in the operation sequence.. 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.

Example#

sparse_jac_xam.cpp