Merge pull request #3854 from pleroy/ZeroTolerance

Change the zero-free test of the Чебышёв series to take an error estimate

numerics/чебышёв_series.hpp

@@ -103,8 +103,11 @@ class ЧебышёвSeries final {
 
   // Returns true if this polynomial may (but doesn't necessarily) have real
   // roots.  Returns false it is guaranteed not to have real roots.  This is
-  // significantly faster than calling |RealRoots|.
-  bool MayHaveRealRoots() const;
+  // significantly faster than calling |RealRoots|.  If |error_estimate| is
+  // given, false is only returned if the envelope of the series at a distance
+  // of |error_estimate| has no real roots.  This is useful if the series is an
+  // approximation of some function with an L∞ error less than |error_estimate|.
+  bool MayHaveRealRoots(Value error_estimate = Value{}) const;
 
   // Returns the real roots of the polynomial, computed as the eigenvalues of
   // the Frobenius companion matrix.

numerics/чебышёв_series_body.hpp

@@ -298,7 +298,9 @@ UnboundedMatrix<double>
 }
 
 template<typename Value, typename Argument>
-bool ЧебышёвSeries<Value, Argument>::MayHaveRealRoots() const {
+bool ЧебышёвSeries<Value, Argument>::MayHaveRealRoots(
+    Value const error_estimate) const {
+  CHECK_LE(Value{}, error_estimate);
   // TODO(phl): This is a property of the series so we should cache it.
   // This code follow [Boy06], theorem 2.  Note that [Boy06] has another
   // criterion, B₁ and concludes: “There was no detectable difference between
@@ -313,7 +315,10 @@ bool ЧебышёвSeries<Value, Argument>::MayHaveRealRoots() const {
     auto const abs_aⱼ = Abs(helper_.coefficients(j));
     B₀ += abs_aⱼ;
   }
-  return B₀ >= Abs(helper_.coefficients(0));
+  auto const abs_a₀ = Abs(helper_.coefficients(0));
+  // The error may shift the curve vertically.  Note that the following
+  // comparison is valid if the right-hand side is negative.
+  return B₀ >= abs_a₀ - error_estimate;
 }
 
 template<typename Value, typename Argument>

numerics/чебышёв_series_test.cpp

@@ -164,6 +164,8 @@ TEST_F(ЧебышёвSeriesTest, MayHaveRealRoots) {
   // B₀ path.
   ЧебышёвSeries<double, Instant> series1({16, 5, 3, 7}, t_min_, t_max_);
   EXPECT_FALSE(series1.MayHaveRealRoots());
+  // An error estimate causes the result to be more pessimistic.
+  EXPECT_TRUE(series1.MayHaveRealRoots(/*error_estimate*/1.5));
   // We don't know, but it actually doesn't have zeroes.
   ЧебышёвSeries<double, Instant> series2({13, 5, 3, 7}, t_min_, t_max_);
   EXPECT_TRUE(series2.MayHaveRealRoots());