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Computing Sparse Jacobians
Syntax
`n_sweep = f.sparse_jac_for(
group_max, x, subset, pattern, coloring, work
)
` `n_sweep = f.sparse_jac_rev(
x, subset, pattern, coloring, work
)`
Purpose
We use F : R n → 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 ) In the sparse case, this should be faster and take less memory than Jacobian . We use the notation J i , j ( x ) to denote the partial of F i ( x ) with respect to x j .
SizeVector
The type `SizeVector` is a SimpleVector class with elements of type `size_t`.
BaseVector
The type `BaseVector` is a SimpleVector class with elements of type `size_t`.
sparse_jac_for
This function uses first order forward mode sweeps forward_one to compute multiple columns of the Jacobian at the same time.
sparse_jac_rev
This uses function first order reverse mode sweeps reverse_one 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.
group_max
This argument has prototype `
size_t group_max
` and must be greater than zero. It specifies the maximum number of colors to group during a single forward sweep. If a single color is in a group, a single direction for of first order forward mode forward_one is used for each color. If multiple colors are in a group, the multiple direction for of first order forward mode forward_dir is used with one direction for each color. This uses separate memory for each direction (more memory), but my be significantly faster.
x
This argument has prototype `
const BaseVector& x
` and its size is `n` . It specifies the point at which to evaluate the Jacobian J ( x ) .
subset
This argument has prototype `
sparse_rcv<SizeVector, BaseVector>& 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.
pattern
This argument has prototype `
const sparse_rc<SizeVector>& 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.
coloring
The coloring algorithm determines which rows (reverse) or columns (forward) can be computed during the same sweep. This field has prototype `
const std::string& coloring
` This value only matters when work is empty; i.e., after the `work` constructor or `work.clear()` .
cppad
This uses a general purpose coloring algorithm written for Cppad.
colpack
If colpack_prefix is specified on the cmake command line, you can set `coloring` to `colpack`. This uses a general purpose coloring algorithm that is part of Colpack.
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 `
size_t 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. It is also the number of colors determined by the coloring method mentioned above. 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 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.
Example
The files sparse_jac_for.cpp and sparse_jac_rev.cpp are examples and tests of `sparse_jac_for` and `sparse_jac_rev`. They return `true`, if they succeed, and `false` otherwise.

Input File: cppad/core/sparse_jac.hpp