starpu-max/doc/chapters/advanced-examples.texi

@c -*-texinfo-*-

@c This file is part of the StarPU Handbook.
@c Copyright (C) 2009--2011  Universit@'e de Bordeaux 1
@c Copyright (C) 2010, 2011, 2012, 2013  Centre National de la Recherche Scientifique
@c Copyright (C) 2011 Institut National de Recherche en Informatique et Automatique
@c See the file starpu.texi for copying conditions.

@menu
* Using multiple implementations of a codelet::
* Enabling implementation according to capabilities::
* Task and Worker Profiling::
* Partitioning Data::
* Performance model example::
* Theoretical lower bound on execution time example::
* Insert Task Utility::
* Data reduction::
* Temporary buffers::
* Parallel Tasks::
* Debugging::
* The multiformat interface::
* Using the Driver API::
* Defining a New Scheduling Policy::
* On-GPU rendering::
* Defining a New Data Interface::
* Setting the Data Handles for a Task::
* More examples::               More examples shipped with StarPU
@end menu

@node Using multiple implementations of a codelet
@section Using multiple implementations of a codelet
One may want to write multiple implementations of a codelet for a single type of
device and let StarPU choose which one to run. As an example, we will show how
to use SSE to scale a vector. The codelet can be written as follows:

@cartouche
@smallexample
#include <xmmintrin.h>

void scal_sse_func(void *buffers[], void *cl_arg)
@{
    float *vector = (float *) STARPU_VECTOR_GET_PTR(buffers[0]);
    unsigned int n = STARPU_VECTOR_GET_NX(buffers[0]);
    unsigned int n_iterations = n/4;
    if (n % 4 != 0)
        n_iterations++;

    __m128 *VECTOR = (__m128*) vector;
    __m128 factor __attribute__((aligned(16)));
    factor = _mm_set1_ps(*(float *) cl_arg);

    unsigned int i;
    for (i = 0; i < n_iterations; i++)
        VECTOR[i] = _mm_mul_ps(factor, VECTOR[i]);
@}
@end smallexample
@end cartouche

@cartouche
@smallexample
struct starpu_codelet cl = @{
    .where = STARPU_CPU,
    .cpu_funcs = @{ scal_cpu_func, scal_sse_func, NULL @},
    .nbuffers = 1,
    .modes = @{ STARPU_RW @}
@};
@end smallexample
@end cartouche

Schedulers which are multi-implementation aware (only @code{dmda} and
@code{pheft} for now) will use the performance models of all the
implementations it was given, and pick the one that seems to be the fastest.

@node Enabling implementation according to capabilities
@section Enabling implementation according to capabilities

Some implementations may not run on some devices. For instance, some CUDA
devices do not support double floating point precision, and thus the kernel
execution would just fail; or the device may not have enough shared memory for
the implementation being used. The @code{can_execute} field of the @code{struct
starpu_codelet} structure permits to express this. For instance:

@cartouche
@smallexample
static int can_execute(unsigned workerid, struct starpu_task *task, unsigned nimpl)
@{
  const struct cudaDeviceProp *props;
  if (starpu_worker_get_type(workerid) == STARPU_CPU_WORKER)
    return 1;
  /* Cuda device */
  props = starpu_cuda_get_device_properties(workerid);
  if (props->major >= 2 || props->minor >= 3)
    /* At least compute capability 1.3, supports doubles */
    return 1;
  /* Old card, does not support doubles */
  return 0;
@}

struct starpu_codelet cl = @{
    .where = STARPU_CPU|STARPU_CUDA,
    .can_execute = can_execute,
    .cpu_funcs = @{ cpu_func, NULL @},
    .cuda_funcs = @{ gpu_func, NULL @}
    .nbuffers = 1,
    .modes = @{ STARPU_RW @}
@};
@end smallexample
@end cartouche

This can be essential e.g. when running on a machine which mixes various models
of CUDA devices, to take benefit from the new models without crashing on old models.

Note: the @code{can_execute} function is called by the scheduler each time it
tries to match a task with a worker, and should thus be very fast. The
@code{starpu_cuda_get_device_properties} provides a quick access to CUDA
properties of CUDA devices to achieve such efficiency.

Another example is compiling CUDA code for various compute capabilities,
resulting with two CUDA functions, e.g. @code{scal_gpu_13} for compute capability
1.3, and @code{scal_gpu_20} for compute capability 2.0. Both functions can be
provided to StarPU by using @code{cuda_funcs}, and @code{can_execute} can then be
used to rule out the @code{scal_gpu_20} variant on a CUDA device which
will not be able to execute it:

@cartouche
@smallexample
static int can_execute(unsigned workerid, struct starpu_task *task, unsigned nimpl)
@{
  const struct cudaDeviceProp *props;
  if (starpu_worker_get_type(workerid) == STARPU_CPU_WORKER)
    return 1;
  /* Cuda device */
  if (nimpl == 0)
    /* Trying to execute the 1.3 capability variant, we assume it is ok in all cases.  */
    return 1;
  /* Trying to execute the 2.0 capability variant, check that the card can do it.  */
  props = starpu_cuda_get_device_properties(workerid);
  if (props->major >= 2 || props->minor >= 0)
    /* At least compute capability 2.0, can run it */
    return 1;
  /* Old card, does not support 2.0, will not be able to execute the 2.0 variant.  */
  return 0;
@}

struct starpu_codelet cl = @{
    .where = STARPU_CPU|STARPU_CUDA,
    .can_execute = can_execute,
    .cpu_funcs = @{ cpu_func, NULL @},
    .cuda_funcs = @{ scal_gpu_13, scal_gpu_20, NULL @},
    .nbuffers = 1,
    .modes = @{ STARPU_RW @}
@};
@end smallexample
@end cartouche

Note: the most generic variant should be provided first, as some schedulers are
not able to try the different variants.

@node Task and Worker Profiling
@section Task and Worker Profiling

A full example showing how to use the profiling API is available in
the StarPU sources in the directory @code{examples/profiling/}.

@cartouche
@smallexample
struct starpu_task *task = starpu_task_create();
task->cl = &cl;
task->synchronous = 1;
/* We will destroy the task structure by hand so that we can
 * query the profiling info before the task is destroyed. */
task->destroy = 0;

/* Submit and wait for completion (since synchronous was set to 1) */
starpu_task_submit(task);

/* The task is finished, get profiling information */
struct starpu_task_profiling_info *info = task->profiling_info;

/* How much time did it take before the task started ? */
double delay += starpu_timing_timespec_delay_us(&info->submit_time, &info->start_time);

/* How long was the task execution ? */
double length += starpu_timing_timespec_delay_us(&info->start_time, &info->end_time);

/* We don't need the task structure anymore */
starpu_task_destroy(task);
@end smallexample
@end cartouche

@cartouche
@smallexample
/* Display the occupancy of all workers during the test */
int worker;
for (worker = 0; worker < starpu_worker_get_count(); worker++)
@{
        struct starpu_worker_profiling_info worker_info;
        int ret = starpu_worker_get_profiling_info(worker, &worker_info);
        STARPU_ASSERT(!ret);

        double total_time = starpu_timing_timespec_to_us(&worker_info.total_time);
        double executing_time = starpu_timing_timespec_to_us(&worker_info.executing_time);
        double sleeping_time = starpu_timing_timespec_to_us(&worker_info.sleeping_time);
        double overhead_time = total_time - executing_time - sleeping_time;

        float executing_ratio = 100.0*executing_time/total_time;
        float sleeping_ratio = 100.0*sleeping_time/total_time;
        float overhead_ratio = 100.0 - executing_ratio - sleeping_ratio;

        char workername[128];
        starpu_worker_get_name(worker, workername, 128);
        fprintf(stderr, "Worker %s:\n", workername);
        fprintf(stderr, "\ttotal time: %.2lf ms\n", total_time*1e-3);
        fprintf(stderr, "\texec time: %.2lf ms (%.2f %%)\n", executing_time*1e-3,
                executing_ratio);
        fprintf(stderr, "\tblocked time: %.2lf ms (%.2f %%)\n", sleeping_time*1e-3,
                sleeping_ratio);
        fprintf(stderr, "\toverhead time: %.2lf ms (%.2f %%)\n", overhead_time*1e-3,
                overhead_ratio);
@}
@end smallexample
@end cartouche

@node Partitioning Data
@section Partitioning Data

An existing piece of data can be partitioned in sub parts to be used by different tasks, for instance:

@cartouche
@smallexample
int vector[NX];
starpu_data_handle_t handle;

/* Declare data to StarPU */
starpu_vector_data_register(&handle, 0, (uintptr_t)vector,
                            NX, sizeof(vector[0]));

/* Partition the vector in PARTS sub-vectors */
starpu_data_filter f =
@{
    .filter_func = starpu_vector_filter_block,
    .nchildren = PARTS
@};
starpu_data_partition(handle, &f);
@end smallexample
@end cartouche

The task submission then uses @code{starpu_data_get_sub_data} to retrieve the
sub-handles to be passed as tasks parameters.

@cartouche
@smallexample
/* Submit a task on each sub-vector */
for (i=0; i<starpu_data_get_nb_children(handle); i++) @{
    /* Get subdata number i (there is only 1 dimension) */
    starpu_data_handle_t sub_handle = starpu_data_get_sub_data(handle, 1, i);
    struct starpu_task *task = starpu_task_create();

    task->handles[0] = sub_handle;
    task->cl = &cl;
    task->synchronous = 1;
    task->cl_arg = &factor;
    task->cl_arg_size = sizeof(factor);

    starpu_task_submit(task);
@}
@end smallexample
@end cartouche

Partitioning can be applied several times, see
@code{examples/basic_examples/mult.c} and @code{examples/filters/}.

Wherever the whole piece of data is already available, the partitioning will
be done in-place, i.e. without allocating new buffers but just using pointers
inside the existing copy. This is particularly important to be aware of when
using OpenCL, where the kernel parameters are not pointers, but handles. The
kernel thus needs to be also passed the offset within the OpenCL buffer:

@cartouche
@smallexample
void opencl_func(void *buffers[], void *cl_arg)
@{
    cl_mem vector = (cl_mem) STARPU_VECTOR_GET_DEV_HANDLE(buffers[0]);
    unsigned offset = STARPU_BLOCK_GET_OFFSET(buffers[0]);

    ...
    clSetKernelArg(kernel, 0, sizeof(vector), &vector);
    clSetKernelArg(kernel, 1, sizeof(offset), &offset);
    ...
@}
@end smallexample
@end cartouche

And the kernel has to shift from the pointer passed by the OpenCL driver:

@cartouche
@smallexample
__kernel void opencl_kernel(__global int *vector, unsigned offset)
@{
    block = (__global void *)block + offset;
    ...
@}
@end smallexample
@end cartouche

StarPU provides various interfaces and filters for matrices, vectors, etc.,
but applications can also write their own data interfaces and filters, see
@code{examples/interface} and @code{examples/filters/custom_mf} for an example.

@node Performance model example
@section Performance model example

To achieve good scheduling, StarPU scheduling policies need to be able to
estimate in advance the duration of a task. This is done by giving to codelets
a performance model, by defining a @code{starpu_perfmodel} structure and
providing its address in the @code{model} field of the @code{struct starpu_codelet}
structure. The @code{symbol} and @code{type} fields of @code{starpu_perfmodel}
are mandatory, to give a name to the model, and the type of the model, since
there are several kinds of performance models. For compatibility, make sure to
initialize the whole structure to zero, either by using explicit memset, or by
letting the compiler implicitly do it as examplified below.

@itemize
@item
Measured at runtime (@code{STARPU_HISTORY_BASED} model type). This assumes that for a
given set of data input/output sizes, the performance will always be about the
same. This is very true for regular kernels on GPUs for instance (<0.1% error),
and just a bit less true on CPUs (~=1% error). This also assumes that there are
few different sets of data input/output sizes. StarPU will then keep record of
the average time of previous executions on the various processing units, and use
it as an estimation. History is done per task size, by using a hash of the input
and ouput sizes as an index.
It will also save it in @code{$STARPU_HOME/.starpu/sampling/codelets}
for further executions, and can be observed by using the
@code{starpu_perfmodel_display} command, or drawn by using
the @code{starpu_perfmodel_plot} (@pxref{Performance model calibration}).  The
models are indexed by machine name. To
share the models between machines (e.g. for a homogeneous cluster), use
@code{export STARPU_HOSTNAME=some_global_name}. Measurements are only done
when using a task scheduler which makes use of it, such as 
@code{dmda}. Measurements can also be provided explicitly by the application, by
using the @code{starpu_perfmodel_update_history} function.

The following is a small code example.

If e.g. the code is recompiled with other compilation options, or several
variants of the code are used, the symbol string should be changed to reflect
that, in order to recalibrate a new model from zero. The symbol string can even
be constructed dynamically at execution time, as long as this is done before
submitting any task using it.

@cartouche
@smallexample
static struct starpu_perfmodel mult_perf_model = @{
    .type = STARPU_HISTORY_BASED,
    .symbol = "mult_perf_model"
@};

struct starpu_codelet cl = @{
    .where = STARPU_CPU,
    .cpu_funcs = @{ cpu_mult, NULL @},
    .nbuffers = 3,
    .modes = @{ STARPU_R, STARPU_R, STARPU_W @},
    /* for the scheduling policy to be able to use performance models */
    .model = &mult_perf_model
@};
@end smallexample
@end cartouche

@item
Measured at runtime and refined by regression (@code{STARPU_*REGRESSION_BASED}
model type). This still assumes performance regularity, but works
with various data input sizes, by applying regression over observed
execution times. STARPU_REGRESSION_BASED uses an a*n^b regression
form, STARPU_NL_REGRESSION_BASED uses an a*n^b+c (more precise than
STARPU_REGRESSION_BASED, but costs a lot more to compute).

For instance,
@code{tests/perfmodels/regression_based.c} uses a regression-based performance
model for the @code{memset} operation.

Of course, the application has to issue
tasks with varying size so that the regression can be computed. StarPU will not
trust the regression unless there is at least 10% difference between the minimum
and maximum observed input size. It can be useful to set the
@code{STARPU_CALIBRATE} environment variable to @code{1} and run the application
on varying input sizes with @code{STARPU_SCHED} set to @code{eager} scheduler,
so as to feed the performance model for a variety of
inputs. The application can also provide the measurements explictly by using
@code{starpu_perfmodel_update_history}. The @code{starpu_perfmodel_display} and
@code{starpu_perfmodel_plot}
tools can be used to observe how much the performance model is calibrated (@pxref{Performance model calibration}); when
their output look good, @code{STARPU_CALIBRATE} can be reset to @code{0} to let
StarPU use the resulting performance model without recording new measures, and
@code{STARPU_SCHED} can be set to @code{dmda} to benefit from the performance models. If
the data input sizes vary a lot, it is really important to set
@code{STARPU_CALIBRATE} to @code{0}, otherwise StarPU will continue adding the
measures, and result with a very big performance model, which will take time a
lot of time to load and save.

For non-linear regression, since computing it
is quite expensive, it is only done at termination of the application. This
means that the first execution of the application will use only history-based
performance model to perform scheduling, without using regression.

@item
Provided as an estimation from the application itself (@code{STARPU_COMMON} model type and @code{cost_function} field),
see for instance
@code{examples/common/blas_model.h} and @code{examples/common/blas_model.c}.

@item
Provided explicitly by the application (@code{STARPU_PER_ARCH} model type): the
@code{.per_arch[arch][nimpl].cost_function} fields have to be filled with pointers to
functions which return the expected duration of the task in micro-seconds, one
per architecture.

@end itemize

For the @code{STARPU_HISTORY_BASED} and @code{STARPU_*REGRESSION_BASE},
the total size of task data (both input and output) is used as an index by
default. The @code{size_base} field of @code{struct starpu_perfmodel} however
permits the application to override that, when for instance some of the data
do not matter for task cost (e.g. mere reference table), or when using sparse
structures (in which case it is the number of non-zeros which matter), or when
there is some hidden parameter such as the number of iterations, etc. The
@code{examples/pi} examples uses this to include the number of iterations in the
base.

How to use schedulers which can benefit from such performance model is explained
in @ref{Task scheduling policy}.

The same can be done for task power consumption estimation, by setting the
@code{power_model} field the same way as the @code{model} field. Note: for
now, the application has to give to the power consumption performance model
a name which is different from the execution time performance model.

The application can request time estimations from the StarPU performance
models by filling a task structure as usual without actually submitting
it. The data handles can be created by calling @code{starpu_*_data_register}
functions with a @code{NULL} pointer and @code{-1} node and the
desired data sizes, and need to be unregistered as usual. The
@code{starpu_task_expected_length} and @code{starpu_task_expected_power}
functions can then be called to get an estimation of the task cost on a given
arch. @code{starpu_task_footprint} can also be used to get the footprint used
for indexing history-based performance models.
@code{starpu_task_destroy}
needs to be called to destroy the dummy task afterwards. See
@code{tests/perfmodels/regression_based.c} for an example.

@node Theoretical lower bound on execution time example
@section Theoretical lower bound on execution time

For kernels with history-based performance models (and provided that they are completely calibrated), StarPU can very easily provide a theoretical lower
bound for the execution time of a whole set of tasks. See for
instance @code{examples/lu/lu_example.c}: before submitting tasks,
call @code{starpu_bound_start}, and after complete execution, call
@code{starpu_bound_stop}. @code{starpu_bound_print_lp} or
@code{starpu_bound_print_mps} can then be used to output a Linear Programming
problem corresponding to the schedule of your tasks. Run it through
@code{lp_solve} or any other linear programming solver, and that will give you a
lower bound for the total execution time of your tasks. If StarPU was compiled
with the glpk library installed, @code{starpu_bound_compute} can be used to
solve it immediately and get the optimized minimum, in ms. Its @code{integer}
parameter allows to decide whether integer resolution should be computed
and returned too.

The @code{deps} parameter tells StarPU whether to take tasks, implicit data, and tag
dependencies into account. Tags released in a callback or similar
are not taken into account, only tags associated with a task are.
It must be understood that the linear programming
problem size is quadratic with the number of tasks and thus the time to solve it
will be very long, it could be minutes for just a few dozen tasks. You should
probably use @code{lp_solve -timeout 1 test.pl -wmps test.mps} to convert the
problem to MPS format and then use a better solver, @code{glpsol} might be
better than @code{lp_solve} for instance (the @code{--pcost} option may be
useful), but sometimes doesn't manage to converge. @code{cbc} might look
slower, but it is parallel. Be sure to try at least all the @code{-B} options
of @code{lp_solve}. For instance, we often just use
@code{lp_solve -cc -B1 -Bb -Bg -Bp -Bf -Br -BG -Bd -Bs -BB -Bo -Bc -Bi} , and
the @code{-gr} option can also be quite useful.

Data transfer time can only be taken into account when @code{deps} is set. Only
data transfers inferred from implicit data dependencies between tasks are taken
into account.

Setting @code{deps} to 0 will only take into account the actual computations
on processing units. It however still properly takes into account the varying
performances of kernels and processing units, which is quite more accurate than
just comparing StarPU performances with the fastest of the kernels being used.

The @code{prio} parameter tells StarPU whether to simulate taking into account
the priorities as the StarPU scheduler would, i.e. schedule prioritized
tasks before less prioritized tasks, to check to which extend this results
to a less optimal solution. This increases even more computation time.

Note that for simplicity, all this however doesn't take into account data
transfers, which are assumed to be completely overlapped.

@node Insert Task Utility
@section Insert Task Utility

StarPU provides the wrapper function @code{starpu_insert_task} to ease
the creation and submission of tasks. See the definition of the
functions in @ref{Insert Task}.

Here the implementation of the codelet:

@cartouche
@smallexample
void func_cpu(void *descr[], void *_args)
@{
        int *x0 = (int *)STARPU_VARIABLE_GET_PTR(descr[0]);
        float *x1 = (float *)STARPU_VARIABLE_GET_PTR(descr[1]);
        int ifactor;
        float ffactor;

        starpu_codelet_unpack_args(_args, &ifactor, &ffactor);
        *x0 = *x0 * ifactor;
        *x1 = *x1 * ffactor;
@}

struct starpu_codelet mycodelet = @{
        .where = STARPU_CPU,
        .cpu_funcs = @{ func_cpu, NULL @},
        .nbuffers = 2,
        .modes = @{ STARPU_RW, STARPU_RW @}
@};
@end smallexample
@end cartouche

And the call to the @code{starpu_insert_task} wrapper:

@cartouche
@smallexample
starpu_insert_task(&mycodelet,
                   STARPU_VALUE, &ifactor, sizeof(ifactor),
                   STARPU_VALUE, &ffactor, sizeof(ffactor),
                   STARPU_RW, data_handles[0], STARPU_RW, data_handles[1],
                   0);
@end smallexample
@end cartouche

The call to @code{starpu_insert_task} is equivalent to the following
code:

@cartouche
@smallexample
struct starpu_task *task = starpu_task_create();
task->cl = &mycodelet;
task->handles[0] = data_handles[0];
task->handles[1] = data_handles[1];
char *arg_buffer;
size_t arg_buffer_size;
starpu_codelet_pack_args(&arg_buffer, &arg_buffer_size,
                    STARPU_VALUE, &ifactor, sizeof(ifactor),
                    STARPU_VALUE, &ffactor, sizeof(ffactor),
                    0);
task->cl_arg = arg_buffer;
task->cl_arg_size = arg_buffer_size;
int ret = starpu_task_submit(task);
@end smallexample
@end cartouche

Here a similar call using @code{STARPU_DATA_ARRAY}.

@cartouche
@smallexample
starpu_insert_task(&mycodelet,
                   STARPU_DATA_ARRAY, data_handles, 2,
                   STARPU_VALUE, &ifactor, sizeof(ifactor),
                   STARPU_VALUE, &ffactor, sizeof(ffactor),
                   0);
@end smallexample
@end cartouche

If some part of the task insertion depends on the value of some computation,
the @code{STARPU_DATA_ACQUIRE_CB} macro can be very convenient. For
instance, assuming that the index variable @code{i} was registered as handle
@code{i_handle}:

@cartouche
@smallexample
/* Compute which portion we will work on, e.g. pivot */
starpu_insert_task(&which_index, STARPU_W, i_handle, 0);

/* And submit the corresponding task */
STARPU_DATA_ACQUIRE_CB(i_handle, STARPU_R,
                       starpu_insert_task(&work, STARPU_RW, A_handle[i], 0));
@end smallexample
@end cartouche

The @code{STARPU_DATA_ACQUIRE_CB} macro submits an asynchronous request for
acquiring data @code{i} for the main application, and will execute the code
given as third parameter when it is acquired. In other words, as soon as the
value of @code{i} computed by the @code{which_index} codelet can be read, the
portion of code passed as third parameter of @code{STARPU_DATA_ACQUIRE_CB} will
be executed, and is allowed to read from @code{i} to use it e.g. as an
index. Note that this macro is only avaible when compiling StarPU with
the compiler @code{gcc}.

@node Data reduction
@section Data reduction

In various cases, some piece of data is used to accumulate intermediate
results. For instances, the dot product of a vector, maximum/minimum finding,
the histogram of a photograph, etc. When these results are produced along the
whole machine, it would not be efficient to accumulate them in only one place,
incurring data transmission each and access concurrency.

StarPU provides a @code{STARPU_REDUX} mode, which permits to optimize
that case: it will allocate a buffer on each memory node, and accumulate
intermediate results there. When the data is eventually accessed in the normal
@code{STARPU_R} mode, StarPU will collect the intermediate results in just one
buffer.

For this to work, the user has to use the
@code{starpu_data_set_reduction_methods} to declare how to initialize these
buffers, and how to assemble partial results.

For instance, @code{cg} uses that to optimize its dot product: it first defines
the codelets for initialization and reduction:

@cartouche
@smallexample
struct starpu_codelet bzero_variable_cl =
@{
        .cpu_funcs = @{ bzero_variable_cpu, NULL @},
        .cuda_funcs = @{ bzero_variable_cuda, NULL @},
        .nbuffers = 1,
@}

static void accumulate_variable_cpu(void *descr[], void *cl_arg)
@{
        double *v_dst = (double *)STARPU_VARIABLE_GET_PTR(descr[0]);
        double *v_src = (double *)STARPU_VARIABLE_GET_PTR(descr[1]);
        *v_dst = *v_dst + *v_src;
@}

static void accumulate_variable_cuda(void *descr[], void *cl_arg)
@{
        double *v_dst = (double *)STARPU_VARIABLE_GET_PTR(descr[0]);
        double *v_src = (double *)STARPU_VARIABLE_GET_PTR(descr[1]);
        cublasaxpy(1, (double)1.0, v_src, 1, v_dst, 1);
        cudaStreamSynchronize(starpu_cuda_get_local_stream());
@}

struct starpu_codelet accumulate_variable_cl =
@{
        .cpu_funcs = @{ accumulate_variable_cpu, NULL @},
        .cuda_funcs = @{ accumulate_variable_cuda, NULL @},
        .nbuffers = 1,
@}
@end smallexample
@end cartouche

and attaches them as reduction methods for its dtq handle:

@cartouche
@smallexample
starpu_variable_data_register(&dtq_handle, -1, NULL, sizeof(type));
starpu_data_set_reduction_methods(dtq_handle,
        &accumulate_variable_cl, &bzero_variable_cl);
@end smallexample
@end cartouche

and @code{dtq_handle} can now be used in @code{STARPU_REDUX} mode for the dot products
with partitioned vectors:

@cartouche
@smallexample
for (b = 0; b < nblocks; b++)
    starpu_insert_task(&dot_kernel_cl,
        STARPU_REDUX, dtq_handle,
        STARPU_R, starpu_data_get_sub_data(v1, 1, b),
        STARPU_R, starpu_data_get_sub_data(v2, 1, b),
        0);
@end smallexample
@end cartouche

During registration, we have here provided NULL, i.e. there is no initial value
to be taken into account during reduction. StarPU will thus only take into
account the contributions from the @code{dot_kernel_cl} tasks. Also, it will not
allocate any memory for @code{dtq_handle} before @code{dot_kernel_cl} tasks are
ready to run.

If another dot product has to be performed, one could unregister
@code{dtq_handle}, and re-register it. But one can also use
@code{starpu_data_invalidate_submit(dtq_handle)}, which will clear all data from the handle,
thus resetting it back to the initial @code{register(NULL)} state.

The @code{cg} example also uses reduction for the blocked gemv kernel, leading
to yet more relaxed dependencies and more parallelism.

STARPU_REDUX can also be passed to @code{starpu_mpi_insert_task} in the MPI
case. That will however not produce any MPI communication, but just pass
STARPU_REDUX to the underlying @code{starpu_insert_task}. It is up to the
application to call @code{starpu_mpi_redux_data}, which posts tasks that will
reduce the partial results among MPI nodes into the MPI node which owns the
data. For instance, some hypothetical application which collects partial results
into data @code{res}, then uses it for other computation, before looping again
with a new reduction:

@cartouche
@smallexample
for (i = 0; i < 100; i++) @{
    starpu_mpi_insert_task(MPI_COMM_WORLD, &init_res, STARPU_W, res, 0);
    starpu_mpi_insert_task(MPI_COMM_WORLD, &work, STARPU_RW, A,
               STARPU_R, B, STARPU_REDUX, res, 0);
    starpu_mpi_redux_data(MPI_COMM_WORLD, res);
    starpu_mpi_insert_task(MPI_COMM_WORLD, &work2, STARPU_RW, B, STARPU_R, res, 0);
@}
@end smallexample
@end cartouche

@node Temporary buffers
@section Temporary buffers

There are two kinds of temporary buffers: temporary data which just pass results
from a task to another, and scratch data which are needed only internally by
tasks.

@subsection Temporary data

Data can sometimes be entirely produced by a task, and entirely consumed by
another task, without the need for other parts of the application to access
it. In such case, registration can be done without prior allocation, by using
the special -1 memory node number, and passing a zero pointer. StarPU will
actually allocate memory only when the task creating the content gets scheduled,
and destroy it on unregistration.

In addition to that, it can be tedious for the application to have to unregister
the data, since it will not use its content anyway. The unregistration can be
done lazily by using the @code{starpu_data_unregister_submit(handle)} function,
which will record that no more tasks accessing the handle will be submitted, so
that it can be freed as soon as the last task accessing it is over.

The following code examplifies both points: it registers the temporary
data, submits three tasks accessing it, and records the data for automatic
unregistration.

@cartouche
@smallexample
starpu_vector_data_register(&handle, -1, 0, n, sizeof(float));
starpu_insert_task(&produce_data, STARPU_W, handle, 0);
starpu_insert_task(&compute_data, STARPU_RW, handle, 0);
starpu_insert_task(&summarize_data, STARPU_R, handle, STARPU_W, result_handle, 0);
starpu_data_unregister_submit(handle);
@end smallexample
@end cartouche

@subsection Scratch data

Some kernels sometimes need temporary data to achieve the computations, i.e. a
workspace. The application could allocate it at the start of the codelet
function, and free it at the end, but that would be costly. It could also
allocate one buffer per worker (similarly to @ref{Per-worker library
initialization}), but that would make them systematic and permanent. A more
optimized way is to use the SCRATCH data access mode, as examplified below,
which provides per-worker buffers without content consistency.

@cartouche
@smallexample
starpu_vector_data_register(&workspace, -1, 0, sizeof(float));
for (i = 0; i < N; i++)
    starpu_insert_task(&compute, STARPU_R, input[i],
                       STARPU_SCRATCH, workspace, STARPU_W, output[i], 0);
@end smallexample
@end cartouche

StarPU will make sure that the buffer is allocated before executing the task,
and make this allocation per-worker: for CPU workers, notably, each worker has
its own buffer. This means that each task submitted above will actually have its
own workspace, which will actually be the same for all tasks running one after
the other on the same worker. Also, if for instance GPU memory becomes scarce,
StarPU will notice that it can free such buffers easily, since the content does
not matter.

The @code{examples/pi} example uses scratches for some temporary buffer.

@node Parallel Tasks
@section Parallel Tasks

StarPU can leverage existing parallel computation libraries by the means of
parallel tasks. A parallel task is a task which gets worked on by a set of CPUs
(called a parallel or combined worker) at the same time, by using an existing
parallel CPU implementation of the computation to be achieved. This can also be
useful to improve the load balance between slow CPUs and fast GPUs: since CPUs
work collectively on a single task, the completion time of tasks on CPUs become
comparable to the completion time on GPUs, thus relieving from granularity
discrepancy concerns. Hwloc support needs to be enabled to get good performance,
otherwise StarPU will not know how to better group cores.

Two modes of execution exist to accomodate with existing usages.

@subsection Fork-mode parallel tasks

In the Fork mode, StarPU will call the codelet function on one
of the CPUs of the combined worker. The codelet function can use
@code{starpu_combined_worker_get_size()} to get the number of threads it is
allowed to start to achieve the computation. The CPU binding mask for the whole
set of CPUs is already enforced, so that threads created by the function will
inherit the mask, and thus execute where StarPU expected, the OS being in charge
of choosing how to schedule threads on the corresponding CPUs. The application
can also choose to bind threads by hand, using e.g. sched_getaffinity to know
the CPU binding mask that StarPU chose.

For instance, using OpenMP (full source is available in
@code{examples/openmp/vector_scal.c}):

@cartouche
@smallexample
void scal_cpu_func(void *buffers[], void *_args)
@{
    unsigned i;
    float *factor = _args;
    struct starpu_vector_interface *vector = buffers[0];
    unsigned n = STARPU_VECTOR_GET_NX(vector);
    float *val = (float *)STARPU_VECTOR_GET_PTR(vector);

#pragma omp parallel for num_threads(starpu_combined_worker_get_size())
    for (i = 0; i < n; i++)
        val[i] *= *factor;
@}

static struct starpu_codelet cl =
@{
    .modes = @{ STARPU_RW @},
    .where = STARPU_CPU,
    .type = STARPU_FORKJOIN,
    .max_parallelism = INT_MAX,
    .cpu_funcs = @{scal_cpu_func, NULL@},
    .nbuffers = 1,
@};
@end smallexample
@end cartouche

Other examples include for instance calling a BLAS parallel CPU implementation
(see @code{examples/mult/xgemm.c}).

@subsection SPMD-mode parallel tasks

In the SPMD mode, StarPU will call the codelet function on
each CPU of the combined worker. The codelet function can use
@code{starpu_combined_worker_get_size()} to get the total number of CPUs
involved in the combined worker, and thus the number of calls that are made in
parallel to the function, and @code{starpu_combined_worker_get_rank()} to get
the rank of the current CPU within the combined worker. For instance:

@cartouche
@smallexample
static void func(void *buffers[], void *args)
@{
    unsigned i;
    float *factor = _args;
    struct starpu_vector_interface *vector = buffers[0];
    unsigned n = STARPU_VECTOR_GET_NX(vector);
    float *val = (float *)STARPU_VECTOR_GET_PTR(vector);

    /* Compute slice to compute */
    unsigned m = starpu_combined_worker_get_size();
    unsigned j = starpu_combined_worker_get_rank();
    unsigned slice = (n+m-1)/m;

    for (i = j * slice; i < (j+1) * slice && i < n; i++)
        val[i] *= *factor;
@}

static struct starpu_codelet cl =
@{
    .modes = @{ STARPU_RW @},
    .where = STARP_CPU,
    .type = STARPU_SPMD,
    .max_parallelism = INT_MAX,
    .cpu_funcs = @{ func, NULL @},
    .nbuffers = 1,
@}
@end smallexample
@end cartouche

Of course, this trivial example will not really benefit from parallel task
execution, and was only meant to be simple to understand.  The benefit comes
when the computation to be done is so that threads have to e.g. exchange
intermediate results, or write to the data in a complex but safe way in the same
buffer.

@subsection Parallel tasks performance

To benefit from parallel tasks, a parallel-task-aware StarPU scheduler has to
be used. When exposed to codelets with a Fork or SPMD flag, the @code{pheft}
(parallel-heft) and @code{peager} (parallel eager) schedulers will indeed also
try to execute tasks with several CPUs. It will automatically try the various
available combined worker sizes and thus be able to avoid choosing a large
combined worker if the codelet does not actually scale so much.

@subsection Combined workers

By default, StarPU creates combined workers according to the architecture
structure as detected by hwloc. It means that for each object of the hwloc
topology (NUMA node, socket, cache, ...) a combined worker will be created. If
some nodes of the hierarchy have a big arity (e.g. many cores in a socket
without a hierarchy of shared caches), StarPU will create combined workers of
intermediate sizes. The @code{STARPU_SYNTHESIZE_ARITY_COMBINED_WORKER} variable
permits to tune the maximum arity between levels of combined workers.

The combined workers actually produced can be seen in the output of the
@code{starpu_machine_display} tool (the @code{STARPU_SCHED} environment variable
has to be set to a combined worker-aware scheduler such as @code{pheft} or
@code{peager}).

@subsection Concurrent parallel tasks

Unfortunately, many environments and librairies do not support concurrent
calls.

For instance, most OpenMP implementations (including the main ones) do not
support concurrent @code{pragma omp parallel} statements without nesting them in
another @code{pragma omp parallel} statement, but StarPU does not yet support
creating its CPU workers by using such pragma.

Other parallel libraries are also not safe when being invoked concurrently
from different threads, due to the use of global variables in their sequential
sections for instance.

The solution is then to use only one combined worker at a time.  This can be
done by setting @code{single_combined_worker} to 1 in the @code{starpu_conf}
structure, or setting the @code{STARPU_SINGLE_COMBINED_WORKER} environment
variable to 1. StarPU will then run only one parallel task at a time (but other
CPU and GPU tasks are not affected and can be run concurrently). The parallel
task scheduler will however still however still try varying combined worker
sizes to look for the most efficient ones.

@node Debugging
@section Debugging

StarPU provides several tools to help debugging aplications. Execution traces
can be generated and displayed graphically, see @ref{Generating traces}. Some
gdb helpers are also provided to show the whole StarPU state:

@smallexample
(gdb) source tools/gdbinit
(gdb) help starpu
@end smallexample

The Temanejo task debugger can also be used, see @ref{Task debugger}.

@node The multiformat interface
@section The multiformat interface
It may be interesting to represent the same piece of data using two different
data structures: one that would only be used on CPUs, and one that would only
be used on GPUs. This can be done by using the multiformat interface. StarPU
will be able to convert data from one data structure to the other when needed.
Note that the dmda scheduler is the only one optimized for this interface. The
user must provide StarPU with conversion codelets:

@cartouche
@smallexample
#define NX 1024
struct point array_of_structs[NX];
starpu_data_handle_t handle;

/*
 * The conversion of a piece of data is itself a task, though it is created,
 * submitted and destroyed by StarPU internals and not by the user. Therefore,
 * we have to define two codelets.
 * Note that for now the conversion from the CPU format to the GPU format has to
 * be executed on the GPU, and the conversion from the GPU to the CPU has to be
 * executed on the CPU.
 */
#ifdef STARPU_USE_OPENCL
void cpu_to_opencl_opencl_func(void *buffers[], void *args);
struct starpu_codelet cpu_to_opencl_cl = @{
    .where = STARPU_OPENCL,
    .opencl_funcs = @{ cpu_to_opencl_opencl_func, NULL @},
    .nbuffers = 1,
    .modes = @{ STARPU_RW @}
@};

void opencl_to_cpu_func(void *buffers[], void *args);
struct starpu_codelet opencl_to_cpu_cl = @{
    .where = STARPU_CPU,
    .cpu_funcs = @{ opencl_to_cpu_func, NULL @},
    .nbuffers = 1,
    .modes = @{ STARPU_RW @}
@};
#endif

struct starpu_multiformat_data_interface_ops format_ops = @{
#ifdef STARPU_USE_OPENCL
    .opencl_elemsize = 2 * sizeof(float),
    .cpu_to_opencl_cl = &cpu_to_opencl_cl,
    .opencl_to_cpu_cl = &opencl_to_cpu_cl,
#endif
    .cpu_elemsize = 2 * sizeof(float),
    ...
@};
starpu_multiformat_data_register(handle, 0, &array_of_structs, NX, &format_ops);
@end smallexample
@end cartouche

Kernels can be written almost as for any other interface. Note that
STARPU_MULTIFORMAT_GET_CPU_PTR shall only be used for CPU kernels. CUDA kernels
must use STARPU_MULTIFORMAT_GET_CUDA_PTR, and OpenCL kernels must use
STARPU_MULTIFORMAT_GET_OPENCL_PTR. STARPU_MULTIFORMAT_GET_NX may be used in any
kind of kernel.
@cartouche
@smallexample
static void
multiformat_scal_cpu_func(void *buffers[], void *args)
@{
    struct point *aos;
    unsigned int n;

    aos = STARPU_MULTIFORMAT_GET_CPU_PTR(buffers[0]);
    n = STARPU_MULTIFORMAT_GET_NX(buffers[0]);
    ...
@}

extern "C" void multiformat_scal_cuda_func(void *buffers[], void *_args)
@{
    unsigned int n;
    struct struct_of_arrays *soa;

    soa = (struct struct_of_arrays *) STARPU_MULTIFORMAT_GET_CUDA_PTR(buffers[0]);
    n = STARPU_MULTIFORMAT_GET_NX(buffers[0]);

    ...
@}
@end smallexample
@end cartouche

A full example may be found in @code{examples/basic_examples/multiformat.c}.

@node Using the Driver API
@section Using the Driver API

@pxref{Running drivers}

@cartouche
@smallexample
int ret;
struct starpu_driver = @{
    .type = STARPU_CUDA_WORKER,
    .id.cuda_id = 0
@};
ret = starpu_driver_init(&d);
if (ret != 0)
    error();
while (some_condition) @{
    ret = starpu_driver_run_once(&d);
    if (ret != 0)
        error();
@}
ret = starpu_driver_deinit(&d);
if (ret != 0)
    error();
@end smallexample
@end cartouche

@node Defining a New Scheduling Policy
@section Defining a New Scheduling Policy

A full example showing how to define a new scheduling policy is available in
the StarPU sources in the directory @code{examples/scheduler/}.

@pxref{Scheduling Policy}

@cartouche
@smallexample
static struct starpu_sched_policy dummy_sched_policy = @{
    .init_sched = init_dummy_sched,
    .deinit_sched = deinit_dummy_sched,
    .add_workers = dummy_sched_add_workers,
    .remove_workers = dummy_sched_remove_workers,
    .push_task = push_task_dummy,
    .push_prio_task = NULL,
    .pop_task = pop_task_dummy,
    .post_exec_hook = NULL,
    .pop_every_task = NULL,
    .policy_name = "dummy",
    .policy_description = "dummy scheduling strategy"
@};
@end smallexample
@end cartouche

@node On-GPU rendering
@section On-GPU rendering

Graphical-oriented applications need to draw the result of their computations,
typically on the very GPU where these happened. Technologies such as OpenGL/CUDA
interoperability permit to let CUDA directly work on the OpenGL buffers, making
them thus immediately ready for drawing, by mapping OpenGL buffer, textures or
renderbuffer objects into CUDA.  CUDA however imposes some technical
constraints: peer memcpy has to be disabled, and the thread that runs OpenGL has
to be the one that runs CUDA computations for that GPU.

To achieve this with StarPU, pass the @code{--disable-cuda-memcpy-peer} option
to @code{./configure} (TODO: make it dynamic), OpenGL/GLUT has to be initialized
first, and the interoperability mode has to
be enabled by using the @code{cuda_opengl_interoperability} field of the
@code{starpu_conf} structure, and the driver loop has to be run by
the application, by using the @code{not_launched_drivers} field of
@code{starpu_conf} to prevent StarPU from running it in a separate thread, and
by using @code{starpu_driver_run} to run the loop. The @code{gl_interop} and
@code{gl_interop_idle} examples shows how it articulates in a simple case, where
rendering is done in task callbacks. The former uses @code{glutMainLoopEvent}
to make GLUT progress from the StarPU driver loop, while the latter uses
@code{glutIdleFunc} to make StarPU progress from the GLUT main loop.

Then, to use an OpenGL buffer as a CUDA data, StarPU simply needs to be given
the CUDA pointer at registration, for instance:

@cartouche
@smallexample
/* Get the CUDA worker id */
for (workerid = 0; workerid < starpu_worker_get_count(); workerid++)
        if (starpu_worker_get_type(workerid) == STARPU_CUDA_WORKER)
                break;

/* Build a CUDA pointer pointing at the OpenGL buffer */
cudaGraphicsResourceGetMappedPointer((void**)&output, &num_bytes, resource);

/* And register it to StarPU */
starpu_vector_data_register(&handle, starpu_worker_get_memory_node(workerid),
                            output, num_bytes / sizeof(float4), sizeof(float4));

/* The handle can now be used as usual */
starpu_insert_task(&cl, STARPU_RW, handle, 0);

/* ... */

/* This gets back data into the OpenGL buffer */
starpu_data_unregister(handle);
@end smallexample
@end cartouche

and display it e.g. in the callback function.

@node Defining a New Data Interface
@section Defining a New Data Interface

Let's define a new data interface to manage complex numbers.

@cartouche
@smallexample
/* interface for complex numbers */
struct starpu_complex_interface
@{
        double *real;
        double *imaginary;
        int nx;
@};
@end smallexample
@end cartouche

Registering such a data to StarPU is easily done using the function
@code{starpu_data_register} (@pxref{Basic Data Management API}). The last
parameter of the function, @code{interface_complex_ops}, will be
described below.

@cartouche
@smallexample
void starpu_complex_data_register(starpu_data_handle_t *handle,
     unsigned home_node, double *real, double *imaginary, int nx)
@{
        struct starpu_complex_interface complex =
        @{
                .real = real,
                .imaginary = imaginary,
                .nx = nx
        @};

        if (interface_complex_ops.interfaceid == STARPU_UNKNOWN_INTERFACE_ID)
        @{
                interface_complex_ops.interfaceid = starpu_data_interface_get_next_id();
        @}

        starpu_data_register(handleptr, home_node, &complex, &interface_complex_ops);
@}
@end smallexample
@end cartouche

Different operations need to be defined for a data interface through
the type @code{struct starpu_data_interface_ops} (@pxref{Defining
Interface}). We only define here the basic operations needed to
run simple applications. The source code for the different functions
can be found in the file
@code{examples/interface/complex_interface.c}.

@cartouche
@smallexample
static struct starpu_data_interface_ops interface_complex_ops =
@{
        .register_data_handle = complex_register_data_handle,
        .allocate_data_on_node = complex_allocate_data_on_node,
        .copy_methods = &complex_copy_methods,
        .get_size = complex_get_size,
        .footprint = complex_footprint,
        .interfaceid = STARPU_UNKNOWN_INTERFACE_ID,
        .interface_size = sizeof(struct starpu_complex_interface),
@};
@end smallexample
@end cartouche

Functions need to be defined to access the different fields of the
complex interface from a StarPU data handle.

@cartouche
@smallexample
double *starpu_complex_get_real(starpu_data_handle_t handle)
@{
        struct starpu_complex_interface *complex_interface =
          (struct starpu_complex_interface *) starpu_data_get_interface_on_node(handle, 0);
        return complex_interface->real;
@}

double *starpu_complex_get_imaginary(starpu_data_handle_t handle);
int starpu_complex_get_nx(starpu_data_handle_t handle);
@end smallexample
@end cartouche

Similar functions need to be defined to access the different fields of the
complex interface from a @code{void *} pointer to be used within codelet
implemetations.

@cartouche
@smallexample
#define STARPU_COMPLEX_GET_REAL(interface)	\
        (((struct starpu_complex_interface *)(interface))->real)
#define STARPU_COMPLEX_GET_IMAGINARY(interface)	\
        (((struct starpu_complex_interface *)(interface))->imaginary)
#define STARPU_COMPLEX_GET_NX(interface)	\
        (((struct starpu_complex_interface *)(interface))->nx)
@end smallexample
@end cartouche

Complex data interfaces can then be registered to StarPU.

@cartouche
@smallexample
double real = 45.0;
double imaginary = 12.0;
starpu_complex_data_register(&handle1, 0, &real, &imaginary, 1);
starpu_insert_task(&cl_display, STARPU_R, handle1, 0);
@end smallexample
@end cartouche

and used by codelets.

@cartouche
@smallexample
void display_complex_codelet(void *descr[], __attribute__ ((unused)) void *_args)
@{
        int nx = STARPU_COMPLEX_GET_NX(descr[0]);
        double *real = STARPU_COMPLEX_GET_REAL(descr[0]);
        double *imaginary = STARPU_COMPLEX_GET_IMAGINARY(descr[0]);
        int i;

        for(i=0 ; i<nx ; i++)
        @{
                fprintf(stderr, "Complex[%d] = %3.2f + %3.2f i\n", i, real[i], imaginary[i]);
        @}
@}
@end smallexample
@end cartouche

The whole code for this complex data interface is available in the
directory @code{examples/interface/}.

@node Setting the Data Handles for a Task
@section Setting the Data Handles for a Task

The number of data a task can manage is fixed by the
@code{STARPU_NMAXBUFS} which has a default value which can be changed
through the configure option @code{--enable-maxbuffers} (see
@ref{--enable-maxbuffers}).

However, it is possible to define tasks managing more data by using
the field @code{dyn_handles} when defining a task and the field
@code{dyn_modes} when defining the corresponding codelet.

@cartouche
@smallexample
struct starpu_codelet dummy_big_cl =
@{
	.cuda_funcs = @{dummy_big_kernel, NULL@},
	.opencl_funcs = @{dummy_big_kernel, NULL@},
	.cpu_funcs = @{dummy_big_kernel, NULL@},
	.nbuffers = STARPU_NMAXBUFS+1
@};

task = starpu_task_create();
task->cl = &dummy_big_cl;
task->dyn_handles = malloc(task->cl->nbuffers * sizeof(starpu_data_handle_t));
for(i=0 ; i<task->cl->nbuffers ; i++)
@{
	task->dyn_handles[i] = handle;
@}
starpu_task_submit(task);
@end smallexample
@end cartouche

@cartouche
@smallexample
starpu_data_handle_t *handles = malloc(dummy_big_cl.nbuffers * sizeof(starpu_data_handle_t));
for(i=0 ; i<dummy_big_cl.nbuffers ; i++)
@{
	handles[i] = handle;
@}
starpu_insert_task(&dummy_big_cl,
        	 STARPU_VALUE, &dummy_big_cl.nbuffers, sizeof(dummy_big_cl.nbuffers),
		 STARPU_DATA_ARRAY, handles, dummy_big_cl.nbuffers,
		 0);
@end smallexample
@end cartouche

The whole code for this complex data interface is available in the
directory @code{examples/basic_examples/dynamic_handles.c}.

@node More examples
@section More examples


More examples are available in the StarPU sources in the @code{examples/}
directory. Simple examples include:

@table @asis
@item @code{incrementer/}:
    Trivial incrementation test.
@item @code{basic_examples/}:
        Simple documented Hello world and vector/scalar product (as
        shown in @ref{Basic Examples}), matrix
        product examples (as shown in @ref{Performance model example}), an example using the blocked matrix data
        interface, an example using the variable data interface, and an example
        using different formats on CPUs and GPUs.
@item @code{matvecmult/}:
    OpenCL example from NVidia, adapted to StarPU.
@item @code{axpy/}:
    AXPY CUBLAS operation adapted to StarPU.
@item @code{fortran/}:
    Example of Fortran bindings.
@end table

More advanced examples include:

@table @asis
@item @code{filters/}:
    Examples using filters, as shown in @ref{Partitioning Data}.
@item @code{lu/}:
    LU matrix factorization, see for instance @code{xlu_implicit.c}
@item @code{cholesky/}:
    Cholesky matrix factorization, see for instance @code{cholesky_implicit.c}.
@end table