Computing Sparse Jacobians

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@This is cppad-20221105 documentation. Here is a link to its current documentation . Computing Sparse Jacobians
Syntax

n_color = f.sparse_jac_for(

    group_max, x, subset, pattern, coloring, work

)

n_color = f.sparse_jac_rev(

    x, subset, pattern, coloring, work

)

Purpose
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) @]@ 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. In fact, it can be used with a different f and a different subset provided that Jacobian sparsity pattern for f and the sparsity pattern in subset are the same. If any of these values change, use work.clear() to empty this structure.

n_color
The return value n_color has prototype



    size_t n_color

If sparse_jac_for (sparse_jac_rev) is used, n_color is the number of first order forward directions 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. Note that if group_max == 1 , or if we are using sparse_jac_rev, n_color is equal to the number of sweeps.

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: include/cppad/core/sparse_jac.hpp