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ENH: improve analytical eigendecompositions

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- `tensor` and `tensor2D` returns complex eigenvalues/vectors
  - `symmTensor` and `symmTensor2D` returns real eigenvalues/vectors
  - adds new test routines for eigendecompositions
  - improves numerical stability by:
    - using new robust algorithms,
    - reordering the conditional branches in root-type selection
This commit is contained in:
Kutalmis Bercin
2020-02-11 16:22:09 +00:00
committed by Andrew Heather
parent 6a53794e0a
commit 55e7da670c
28 changed files with 1411 additions and 284 deletions
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120
applications/test/SymmTensor2D/Test-SymmTensor2D.C
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@ -476,6 +476,80 @@ void test_global_opers(Type)
}
// Return false if given eigenvalues fail to satisy eigenvalue relations
// Relations: (Beauregard & Fraleigh (1973), ISBN 0-395-14017-X, p. 307)
void test_eigenvalues(const symmTensor2D& T, const vector2D& EVals)
{
{
Info<< "# Product of eigenvalues = det(T):" << nl;
const scalar determinant = det(T);
const scalar EValsProd = EVals.x()*EVals.y();
cmp("# Product of eigenvalues = det(sT):", EValsProd, determinant);
}
{
const scalar trace = tr(T);
scalar EValsSum = 0.0;
for (const auto& val : EVals)
{
EValsSum += val;
}
cmp("# Sum of eigenvalues = trace(sT):", EValsSum, trace);
}
}
// Return false if a given eigenvalue-eigenvector pair
// fails to satisfy the characteristic equation
void test_characteristic_equation
(
const symmTensor2D& T,
const vector2D& EVals,
const tensor2D& EVecs
)
{
Info<< "# Characteristic equation:" << nl;
for (direction dir = 0; dir < pTraits<vector2D>::nComponents; ++dir)
{
Info<< "EVal = " << EVals[dir] << nl
<< "EVec = " << EVecs.row(dir) << nl;
const vector2D leftSide(T & EVecs.row(dir));
const vector2D rightSide(EVals[dir]*EVecs.row(dir));
const vector2D X(leftSide - rightSide);
for (const auto x : X)
{
cmp(" (sT & EVec - EVal*EVec) = 0:", x, 0.0, 1e-8, 1e-6);
}
}
}
// Return false if the eigen functions fail to satisfy relations
void test_eigen_funcs(const symmTensor2D& T)
{
Info<< "# Operand:" << nl
<< " symmTensor2D = " << T << nl;
Info<< "# Return eigenvalues of a given symmTensor2D:" << nl;
const vector2D EVals(eigenValues(T));
Info<< EVals << endl;
test_eigenvalues(T, EVals);
Info<< "# Return eigenvectors of a given symmTensor2D corresponding to"
<< " given eigenvalues:" << nl;
const tensor2D EVecs0(eigenVectors(T, EVals));
Info<< EVecs0 << endl;
test_characteristic_equation(T, EVals, EVecs0);
Info<< "# Return eigenvectors of a given symmTensor2D by computing"
<< " the eigenvalues of the symmTensor2D in the background:" << nl;
const tensor2D EVecs1(eigenVectors(T));
Info<< EVecs1 << endl;
}
// Do compile-time recursion over the given types
template<std::size_t I = 0, typename... Tp>
inline typename std::enable_if<I == sizeof...(Tp), void>::type
@ -521,6 +595,52 @@ int main(int argc, char *argv[])
run_tests(types, typeID);
Info<< nl << " ## Test symmTensor2D eigen functions: ##" << nl;
const label numberOfTests = 10000;
Random rndGen(1234);
for (label i = 0; i < numberOfTests; ++i)
{
const symmTensor2D T(makeRandomContainer(rndGen));
test_eigen_funcs(T);
}
Info<< nl << " ## Test symmTensor2D eigen functions"
<< " with T.xy = VSMALL: ##" << nl;
for (label i = 0; i < numberOfTests; ++i)
{
symmTensor2D T(makeRandomContainer(rndGen));
T.xy() = VSMALL;
test_eigen_funcs(T);
}
Info<< nl << " ## Test symmTensor2D eigen functions"
<< " with T.xx = T.yy: ##" << nl;
for (label i = 0; i < numberOfTests; ++i)
{
symmTensor2D T(makeRandomContainer(rndGen));
T.xx() = T.yy();
test_eigen_funcs(T);
}
{
Info<< nl
<< " ## Test eigen functions by a zero symmTensor2D: ##"<< nl;
const symmTensor2D zeroT(Zero);
test_eigen_funcs(zeroT);
}
{
Info<< nl << " ## Test eigen functions by a stiff tensor2D: ##"<< nl;
const symmTensor2D stiff
(
pow(10.0, 10), pow(10.0, -8),
pow(10.0, 9)
);
test_eigen_funcs(stiff);
}
if (nFail_)
{
Info<< nl << " #### "