diff --git a/.gitignore b/.gitignore
index 73c33da..2515d99 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,6 +1,8 @@
 build*/
 .spack-env/
 
+*.local*
+
 *.log
 *.out
 *.lock?
\ No newline at end of file
diff --git a/arnoldi.c b/arnoldi.c
index 65d4f83..7c4a52a 100644
--- a/arnoldi.c
+++ b/arnoldi.c
@@ -274,26 +274,30 @@ PetscErrorCode ArnoldiIteration(Mat A, Vec b, PetscInt n, PetscInt m, Vec *Q, do
 
         Vec v;
         PetscCall(VecDuplicate(b, &v));
+
+        // v = A q_k
         PetscCall(MatMult(A, Q[k - 1], v));
 
         // Reorthogonalization using modified Gram-Schmidt
-        for (PetscInt j = 0; j < k; j++) { // anche solo 3
-            // PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[Arnoldi] Reorthogonalization %d %d\n", k, j));
-
+        for (PetscInt j = 0; j < k - 1; j++) { // anche solo 3
             PetscScalar h_ij;
+            
+            // h_ij = q_j . v
             PetscCall(VecDot(Q[j], v, &h_ij));
-
-            // PetscCall(MatSetValue(H, j, k - 1, h, INSERT_VALUES));
-            h[j * n + k - 1] = h_ij;
-
+            
+            // v -= h_ij * q_j
             PetscCall(VecAXPY(v, -h_ij, Q[j]));
+            
+            h[j * n + k - 1] = h_ij;
         }
 
-        // Normalize
+        // Normalize:
+
+        // v = v / ||v||
+        // h_ij = ||v||
+
         PetscScalar h_ij;
         PetscCall(VecNorm(v, NORM_2, &h_ij));
-
-        // PetscCall(MatSetValue(H, k, k - 1, h, INSERT_VALUES));
         h[k * n + k - 1] = h_ij;
 
         // Check for convergence
@@ -301,7 +305,7 @@ PetscErrorCode ArnoldiIteration(Mat A, Vec b, PetscInt n, PetscInt m, Vec *Q, do
             PetscCall(VecNormalize(v, NULL));
             Q[k] = v;
         } else {
-            // PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[Arnoldi] Early breakdown"));
+            PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[Arnoldi] Early breakdown"));
             break;
         }
     }
diff --git a/matrices/laplacian/julia-laplaciano.jl b/matrices/laplacian/laplacian.jl
similarity index 87%
rename from matrices/laplacian/julia-laplaciano.jl
rename to matrices/laplacian/laplacian.jl
index e70bead..3de2fd9 100644
--- a/matrices/laplacian/julia-laplaciano.jl
+++ b/matrices/laplacian/laplacian.jl
@@ -1,6 +1,11 @@
 using SparseArrays
 using MAT
 
+if len(ARGS) == 0
+    println("Usage: julia laplacian.jl <N>")
+    exit(1)
+end
+
 N = parse(Int, ARGS[1])
 println("Generating 3D Laplacian for size $N")