Hi every one,

I have been using the SLEPc library (with fortran) for several years to solve large eigenvalues problems in parallel on distributed memory architectures.

Since I have started to use julia for a while (and really appreciate it!), I was missing this feature. So I’ve started to write a julia wrapper for this library : SlepcWrap.jl.

SLEPc can be considered as an “extension” to the well-known PETSc library; and heavily rely on it. There are several existing julia wrappers for the PETSc library, for instance : PETSc.jl and GridapPETSc.jl. However none of them were exactly fitting my needs so I have copied some parts of both packages to build my own wrapper : PetscWrap.jl. My concern was to have in hand a parallel and petsc-arch-flexible wrapper. Since I am new to the open-source community, I hope that I didn’t cause any trouble copying some code, I am open to change everything necessary to comply with licence issues.

For now, only the PETSc-SLEPc that I needed are wrapped, but don’t hesitate to contribute and/or post an issue to ask for a new feature.

Below, I have copied an example available in the repo (in the readme / docs) computing the eigenvalues of a 1D - Helmholtz problem (using finite-differences). It can be run using multiple cores : `mpirun -n 4 julia helmholtz.jl`

:

In this example, we use the SLEPc to find the eigenvalues of the following Helmholtz equation:

`u'' + \omega^2 u = 0`

associated to Dirichlet boundary conditions on the domain `[0,1]`

. Hence

the theoritical eigenvalues are `\omega = k \pi`

with `k \in \mathbb{Z}^*`

; and the associated

eigenvectors are `u(x) = \sin(k \pi x)`

.

A centered finite difference scheme is used for the spatial discretization.

The equation is written in matrix form `Au = \alpha Bu`

where `\alpha = \omega^2`

.

In this example, PETSc/SLEPc legacy method names are used. For more fancy names, check the next example.

Note that the way we achieve things in the document can be highly improved and the purpose of this example

is only demonstrate some method calls to give an overview.

Start by importing both `PetscWrap`

, for the distributed matrices, and `SlepcWrap`

for the eigenvalues.

```
using PetscWrap
using SlepcWrap
```

Number of mesh points and mesh step

```
n = 21
Δx = 1. / (n - 1)
```

Initialize SLEPc. Either without arguments, calling `SlepcInitialize()`

or using “command-line” arguments.

To do so, either provide the arguments as one string, for instance

`SlepcInitialize("-eps_max_it 100 -eps_tol 1e-5")`

or provide each argument in

separate strings : `PetscInitialize(["-eps_max_it", "100", "-eps_tol", "1e-5")`

.

Here we ask for the five closest eigenvalues to `0`

, using a non-zero pivot for the LU factorization and a

“shift-inverse” process.

```
SlepcInitialize("-eps_target 0 -eps_nev 5 -st_pc_factor_shift_type NONZERO -st_type sinvert")
```

Create the problem matrices, set sizes and apply “command-line” options. Note that we should

set the number of preallocated non-zeros to increase performance.

```
A = MatCreate()
B = MatCreate()
MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, n, n)
MatSetSizes(B, PETSC_DECIDE, PETSC_DECIDE, n, n)
MatSetFromOptions(A)
MatSetFromOptions(B)
MatSetUp(A)
MatSetUp(B)
```

Get rows handled by the local processor

```
A_rstart, A_rend = MatGetOwnershipRange(A)
B_rstart, B_rend = MatGetOwnershipRange(B)
```

Fill matrix A with second order derivative central scheme

```
for i in A_rstart:A_rend
if(i == 1)
A[1, 1:2] = [-2., 1] / Δx^2
elseif (i == n)
A[n, n-1:n] = [1., -2.] / Δx^2
else
A[i, i-1:i+1] = [1., -2., 1.] / Δx^2
end
end
```

Fill matrix B with identity matrix

```
for i in B_rstart:B_rend
B[i,i] = -1.
end
```

Set boundary conditions : u(0) = 0 and u(1) = 0. Only the processor handling the corresponding rows are playing a role here.

```
(A_rstart == 1) && (A[1, 1:2] = [1. 0.] )
(B_rstart == 1) && (B[1, 1] = 0. )
(A_rend == n) && (A[n, n-1:n] = [0. 1.] )
(B_rend == n) && (B[n, n] = 0. )
```

Assemble the matrices

```
MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)
MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)
MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)
MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)
```

Now we set up the eigenvalue solver

```
eps = EPSCreate()
EPSSetOperators(eps, A, B)
EPSSetFromOptions(eps)
EPSSetUp(eps)
```

Then we solve

```
EPSSolve(eps)
```

And finally we can inspect the solution. Let’s first get the number of converged eigenvalues:

```
nconv = EPSGetConverged(eps)
```

Then we can get/display these eigenvalues (more precisely their square root, i.e `\simeq \omega`

)

```
for ieig in 1:nconv
vpr, vpi = EPSGetEigenvalue(eps, ieig)
@show √(vpr), √(vpi)
end
```

We can also play with eigen vectors. First, create two Petsc vectors to allocate memory

```
vecr, veci = MatCreateVecs(A)
```

Then loop over the eigen pairs and retrieve eigenvectors

```
for ieig in 1:nconv
vpr, vpi, vecpr, vecpi = EPSGetEigenpair(eps, ieig, vecr, veci)
# At this point, you can call VecGetArray to obtain a Julia array (see PetscWrap examples).
# If you are on one processor, you can even plot the solution to check that you have a sinus
# solution. On multiple processors, this would require to "gather" the solution on one processor only.
end
```

Finally, let’s free the memory

```
MatDestroy(A)
MatDestroy(B)
EPSDestroy(eps)
```

And call finalize when you’re done

```
SlepcFinalize()
```