diff --git a/cuda/test/matrix/dense_kernels.cpp b/cuda/test/matrix/dense_kernels.cpp
index a45274cdb9b..7d292a8b5fa 100644
--- a/cuda/test/matrix/dense_kernels.cpp
+++ b/cuda/test/matrix/dense_kernels.cpp
@@ -226,7 +226,21 @@ TEST_F(Dense, MultipleVectorCudaComputeConjDotIsEquivalentToRef)
 }
 
 
-TEST_F(Dense, CudaComputeNorm2IsEquivalentToRef)
+TEST_F(Dense, SingleVectorCudaComputeNorm2IsEquivalentToRef)
+{
+    set_up_vector_data(1);
+    auto norm_size = gko::dim<2>{1, x->get_size()[1]};
+    auto norm_expected = NormVector::create(this->ref, norm_size);
+    auto dnorm = NormVector::create(this->cuda, norm_size);
+
+    x->compute_norm2(norm_expected.get());
+    dx->compute_norm2(dnorm.get());
+
+    GKO_ASSERT_MTX_NEAR(norm_expected, dnorm, 1e-14);
+}
+
+
+TEST_F(Dense, MultipleVectorCudaComputeNorm2IsEquivalentToRef)
 {
     set_up_vector_data(20);
     auto norm_size = gko::dim<2>{1, x->get_size()[1]};
diff --git a/dpcpp/matrix/dense_kernels.dp.cpp b/dpcpp/matrix/dense_kernels.dp.cpp
index 59511d12b0a..872e4be7580 100644
--- a/dpcpp/matrix/dense_kernels.dp.cpp
+++ b/dpcpp/matrix/dense_kernels.dp.cpp
@@ -209,6 +209,7 @@ void compute_conj_dot_dispatch(std::shared_ptr<const DefaultExecutor> exec,
                                const matrix::Dense<ValueType>* y,
                                matrix::Dense<ValueType>* result)
 {
+    // TODO Add onemkl for single column ?
     compute_conj_dot(exec, x, y, result);
 }
 
@@ -221,6 +222,7 @@ void compute_norm2_dispatch(std::shared_ptr<const DefaultExecutor> exec,
                             const matrix::Dense<ValueType>* x,
                             matrix::Dense<remove_complex<ValueType>>* result)
 {
+    // TODO Add onemkl for single column ?
     compute_norm2(exec, x, result);
 }
 
diff --git a/dpcpp/test/matrix/dense_kernels.cpp b/dpcpp/test/matrix/dense_kernels.cpp
index a426c88116f..56ef8b5532e 100644
--- a/dpcpp/test/matrix/dense_kernels.cpp
+++ b/dpcpp/test/matrix/dense_kernels.cpp
@@ -230,7 +230,21 @@ TEST_F(Dense, MultipleVectorDpcppComputeConjDotIsEquivalentToRef)
 }
 
 
-TEST_F(Dense, DpcppComputeNorm2IsEquivalentToRef)
+TEST_F(Dense, SingleVectorDpcppComputeNorm2IsEquivalentToRef)
+{
+    set_up_vector_data(1);
+    auto norm_size = gko::dim<2>{1, x->get_size()[1]};
+    auto norm_expected = NormVector::create(this->ref, norm_size);
+    auto dnorm = NormVector::create(this->dpcpp, norm_size);
+
+    x->compute_norm2(norm_expected.get());
+    dx->compute_norm2(dnorm.get());
+
+    GKO_ASSERT_MTX_NEAR(norm_expected, dnorm, r<vtype>::value);
+}
+
+
+TEST_F(Dense, MultipleVectorDpcppComputeNorm2IsEquivalentToRef)
 {
     set_up_vector_data(20);
     auto norm_size = gko::dim<2>{1, x->get_size()[1]};
diff --git a/hip/test/matrix/dense_kernels.hip.cpp b/hip/test/matrix/dense_kernels.hip.cpp
index 894b5fcc408..a6ae2df195b 100644
--- a/hip/test/matrix/dense_kernels.hip.cpp
+++ b/hip/test/matrix/dense_kernels.hip.cpp
@@ -222,7 +222,21 @@ TEST_F(Dense, MultipleVectorHipComputeConjDotIsEquivalentToRef)
 }
 
 
-TEST_F(Dense, HipComputeNorm2IsEquivalentToRef)
+TEST_F(Dense, SingleHipComputeNorm2IsEquivalentToRef)
+{
+    set_up_vector_data(1);
+    auto norm_size = gko::dim<2>{1, x->get_size()[1]};
+    auto norm_expected = NormVector::create(this->ref, norm_size);
+    auto dnorm = NormVector::create(this->hip, norm_size);
+
+    x->compute_norm2(norm_expected.get());
+    dx->compute_norm2(dnorm.get());
+
+    GKO_ASSERT_MTX_NEAR(norm_expected, dnorm, 1e-14);
+}
+
+
+TEST_F(Dense, MultipleHipComputeNorm2IsEquivalentToRef)
 {
     set_up_vector_data(20);
     auto norm_size = gko::dim<2>{1, x->get_size()[1]};
diff --git a/omp/test/matrix/dense_kernels.cpp b/omp/test/matrix/dense_kernels.cpp
index 9d8db3975b9..7b8c4ac365b 100644
--- a/omp/test/matrix/dense_kernels.cpp
+++ b/omp/test/matrix/dense_kernels.cpp
@@ -239,7 +239,21 @@ TEST_F(Dense, MultipleVectorOmpComputeConjDotIsEquivalentToRef)
 }
 
 
-TEST_F(Dense, ComputesNorm2IsEquivalentToRef)
+TEST_F(Dense, SingleVectorComputesNorm2IsEquivalentToRef)
+{
+    set_up_vector_data(1);
+    auto norm_size = gko::dim<2>{1, x->get_size()[1]};
+    auto norm_expected = NormVector::create(this->ref, norm_size);
+    auto dnorm = NormVector::create(this->omp, norm_size);
+
+    x->compute_norm2(norm_expected.get());
+    dx->compute_norm2(dnorm.get());
+
+    GKO_ASSERT_MTX_NEAR(norm_expected, dnorm, 1e-14);
+}
+
+
+TEST_F(Dense, MultipleVectorComputesNorm2IsEquivalentToRef)
 {
     set_up_vector_data(20);
     auto norm_size = gko::dim<2>{1, x->get_size()[1]};