diff --git a/tests/linalg/bidiagonal.rs b/tests/linalg/bidiagonal.rs
index aaee393f..80096733 100644
--- a/tests/linalg/bidiagonal.rs
+++ b/tests/linalg/bidiagonal.rs
@@ -1,6 +1,6 @@
-#![cfg(feature = "proptest-support")]
-
-macro_rules! gen_tests(
+#[cfg(feature = "proptest-support")]
+mod proptest_tests {
+    macro_rules! gen_tests(
     ($module: ident, $scalar: expr) => {
         mod $module {
             #[allow(unused_imports)]
@@ -54,8 +54,9 @@ macro_rules! gen_tests(
     }
 );
 
-gen_tests!(complex, complex_f64());
-gen_tests!(f64, PROPTEST_F64);
+    gen_tests!(complex, complex_f64());
+    gen_tests!(f64, PROPTEST_F64);
+}
 
 #[test]
 fn bidiagonal_identity() {
@@ -74,3 +75,31 @@ fn bidiagonal_identity() {
     let (u, d, v_t) = bidiagonal.unpack();
     assert_eq!(m, &u * d * &v_t);
 }
+
+#[test]
+fn bidiagonal_regression_issue_1313() {
+    let s = 6.123234e-16_f32;
+    let mut m = nalgebra::dmatrix![
+        10.0,   0.0, 0.0,  0.0, -10.0, 0.0, 0.0, 0.0;
+        s,     10.0, 0.0, 10.0,     s, 0.0, 0.0, 0.0;
+        20.0, -20.0, 0.0, 20.0,  20.0, 0.0, 0.0, 0.0;
+    ];
+    m.unscale_mut(m.camax());
+    let bidiagonal = m.clone().bidiagonalize();
+    let (u, d, v_t) = bidiagonal.unpack();
+    let m2 = &u * d * &v_t;
+    assert_relative_eq!(m, m2, epsilon = 1e-6);
+}
+
+#[test]
+fn bidiagonal_regression_issue_1313_minimal() {
+    let s = 6.123234e-17_f32;
+    let m = nalgebra::dmatrix![
+        1.0,   0.0, -1.0;
+        s,     1.0,     s;
+    ];
+    let bidiagonal = m.clone().bidiagonalize();
+    let (u, d, v_t) = bidiagonal.unpack();
+    let m2 = &u * &d * &v_t;
+    assert_relative_eq!(m, m2, epsilon = 1e-6);
+}
diff --git a/tests/linalg/svd.rs b/tests/linalg/svd.rs
index 900901ad..d8b23d02 100644
--- a/tests/linalg/svd.rs
+++ b/tests/linalg/svd.rs
@@ -499,3 +499,17 @@ fn svd_regression_issue_1072() {
         epsilon = 1e-9
     );
 }
+
+#[test]
+// Exercises bug reported in issue #1313 of nalgebra (https://github.com/dimforge/nalgebra/issues/1313)
+fn svd_regression_issue_1313() {
+    let s = 6.123234e-16_f32;
+    let m = nalgebra::dmatrix![
+        10.0,   0.0, 0.0,  0.0, -10.0, 0.0, 0.0, 0.0;
+           s,  10.0, 0.0, 10.0,     s, 0.0, 0.0, 0.0;
+        20.0, -20.0, 0.0, 20.0,  20.0, 0.0, 0.0, 0.0;
+    ];
+    let svd = m.clone().svd(true, true);
+    let m2 = svd.recompose().unwrap();
+    assert_relative_eq!(&m, &m2, epsilon = 1e-5);
+}