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Fix SelfAdjoingEigenSolver (#2191)
Adjust the relaxation step to use the condition ``` abs(subdiag[i]) <= epsilon * sqrt(abs(diag[i]) + abs(diag[i+1])) ``` for setting the subdiagonal entry to zero. Also adjust Wilkinson shift for small `e = subdiag[end-1]` - I couldn't find a reference for the original, and it was not consistent with the Wilkinson definition. Fixes #2191.
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@ -498,8 +498,6 @@ template<typename MatrixType, typename DiagType, typename SubDiagType>
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EIGEN_DEVICE_FUNC
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ComputationInfo computeFromTridiagonal_impl(DiagType& diag, SubDiagType& subdiag, const Index maxIterations, bool computeEigenvectors, MatrixType& eivec)
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{
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EIGEN_USING_STD(abs);
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ComputationInfo info;
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typedef typename MatrixType::Scalar Scalar;
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@ -510,15 +508,23 @@ ComputationInfo computeFromTridiagonal_impl(DiagType& diag, SubDiagType& subdiag
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typedef typename DiagType::RealScalar RealScalar;
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const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
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const RealScalar precision = RealScalar(2)*NumTraits<RealScalar>::epsilon();
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const RealScalar precision_inv = RealScalar(2)/NumTraits<RealScalar>::epsilon();
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while (end>0)
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{
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for (Index i = start; i<end; ++i)
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if (internal::isMuchSmallerThan(abs(subdiag[i]),(abs(diag[i])+abs(diag[i+1])),precision) || abs(subdiag[i]) <= considerAsZero)
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subdiag[i] = 0;
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for (Index i = start; i<end; ++i) {
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if (numext::abs(subdiag[i]) < considerAsZero) {
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subdiag[i] = RealScalar(0);
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} else {
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// abs(subdiag[i]) <= epsilon * sqrt(abs(diag[i]) + abs(diag[i+1]))
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// Scaled to prevent underflows.
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const RealScalar scaled_subdiag = precision_inv * subdiag[i];
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if (scaled_subdiag * scaled_subdiag <= (numext::abs(diag[i])+numext::abs(diag[i+1]))) {
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subdiag[i] = RealScalar(0);
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}
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}
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}
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// find the largest unreduced block
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// find the largest unreduced block at the end of the matrix.
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while (end>0 && subdiag[end-1]==RealScalar(0))
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{
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end--;
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@ -821,32 +827,38 @@ SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>
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}
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namespace internal {
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// Francis implicit QR step.
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template<int StorageOrder,typename RealScalar, typename Scalar, typename Index>
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EIGEN_DEVICE_FUNC
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static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index start, Index end, Scalar* matrixQ, Index n)
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{
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EIGEN_USING_STD(abs);
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// Wilkinson Shift.
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RealScalar td = (diag[end-1] - diag[end])*RealScalar(0.5);
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RealScalar e = subdiag[end-1];
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// Note that thanks to scaling, e^2 or td^2 cannot overflow, however they can still
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// underflow thus leading to inf/NaN values when using the following commented code:
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// RealScalar e2 = numext::abs2(subdiag[end-1]);
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// RealScalar mu = diag[end] - e2 / (td + (td>0 ? 1 : -1) * sqrt(td*td + e2));
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// RealScalar e2 = numext::abs2(subdiag[end-1]);
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// RealScalar mu = diag[end] - e2 / (td + (td>0 ? 1 : -1) * sqrt(td*td + e2));
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// This explain the following, somewhat more complicated, version:
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RealScalar mu = diag[end];
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if(td==RealScalar(0))
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mu -= abs(e);
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else
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{
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RealScalar e2 = numext::abs2(subdiag[end-1]);
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RealScalar h = numext::hypot(td,e);
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if(e2==RealScalar(0)) mu -= (e / (td + (td>RealScalar(0) ? RealScalar(1) : RealScalar(-1)))) * (e / h);
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else mu -= e2 / (td + (td>RealScalar(0) ? h : -h));
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if(td==RealScalar(0)) {
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mu -= numext::abs(e);
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} else if (e != RealScalar(0)) {
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const RealScalar e2 = numext::abs2(e);
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const RealScalar h = numext::hypot(td,e);
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if(e2 == RealScalar(0)) {
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mu -= e / ((td + (td>RealScalar(0) ? h : -h)) / e);
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} else {
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mu -= e2 / (td + (td>RealScalar(0) ? h : -h));
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}
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}
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RealScalar x = diag[start] - mu;
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RealScalar z = subdiag[start];
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for (Index k = start; k < end; ++k)
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// If z ever becomes zero, the Givens rotation will be the identity and
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// z will stay zero for all future iterations.
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for (Index k = start; k < end && z != RealScalar(0); ++k)
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{
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JacobiRotation<RealScalar> rot;
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rot.makeGivens(x, z);
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@ -859,12 +871,11 @@ static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index sta
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diag[k+1] = rot.s() * sdk + rot.c() * dkp1;
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subdiag[k] = rot.c() * sdk - rot.s() * dkp1;
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if (k > start)
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subdiag[k - 1] = rot.c() * subdiag[k-1] - rot.s() * z;
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// "Chasing the bulge" to return to triangular form.
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x = subdiag[k];
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if (k < end - 1)
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{
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z = -rot.s() * subdiag[k+1];
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