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@(@\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}} }@)@

*n_sweep* = *f*.sparse_jac_for(

*group_max*, *x*, *subset*, *pattern*, *coloring*, *work*

)

*n_sweep* = *f*.sparse_jac_rev(

*x*, *subset*, *pattern*, *coloring*, *work*

)

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
@)@.
The type

*SizeVector*

is a SimpleVector
class with
elements of type
`size_t`

.
The type

*BaseVector*

is a SimpleVector
class with
elements of type
`size_t`

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

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

This object has prototype

ADFun<*Base*> *f*

Note that the Taylor coefficients stored in
*f*

are affected
by this operation; see
uses forward
below.
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.
This argument has prototype

const *BaseVector*& *x*

and its size is
*n*

.
It specifies the point at which to evaluate the Jacobian
@(@
J(x)
@)@.
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.
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.
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()

.
This uses a general purpose coloring algorithm written for Cppad.

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.
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.
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.
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.
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