# Reduction (MPI)
A reduction combines all elements of a collection into a single value using an associative operator. In sequential C, that is a loop:

```c
double sum = 0.0;
for (int i = 0; i < N; i++)
    sum += arr[i];
```

In a parallel program, each process holds a different slice of the data and computes a partial result. To get the global value, those partial results need to be combined. `MPI_Reduce` collects the per-process values and applies the operator, delivering the result only to a root rank. `MPI_Allreduce` does the same but gives every process the result, which is the right choice when the global value drives the next iteration (a convergence check, for example).

```c
double local = compute();
double global;
MPI_Allreduce(&local, &global, 1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
```

Built-in operators include `MPI_SUM`, `MPI_PROD`, `MPI_MAX`, `MPI_MIN`, `MPI_LAND` (logical and), `MPI_LOR`, `MPI_BAND` (bitwise and), `MPI_BOR`, `MPI_MAXLOC`, and `MPI_MINLOC`. `MPI_MAXLOC` and `MPI_MINLOC` return both the extreme value and the rank of the process that holds it, using paired datatypes like `MPI_DOUBLE_INT`. Custom operators can be defined with `MPI_Op_create`.