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BUG: mantis #1373: eigen value/vector calculation now handles repeated eigen values
This commit is contained in:
william
@ -69,5 +69,121 @@ int main()
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Info<< (symm(t7) && t7) - (0.5*(t7 + t7.T()) && t7) << endl;
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Info<< (t7 && symm(t7)) - (t7 && 0.5*(t7 + t7.T())) << endl;
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/*
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// Lots of awkward eigenvector tests ...
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tensor T_rand_real
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(
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0.9999996423721313, 0.3330855667591095, 0.6646450161933899,
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0.9745196104049683, 0.0369445420801640, 0.0846728682518005,
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0.6474838852882385, 0.1617118716239929, 0.2041363865137100
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);
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Debug(T_rand_real);
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vector L_rand_real(eigenValues(T_rand_real));
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Debug(L_rand_real);
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tensor U_rand_real(eigenVectors(T_rand_real));
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Debug(U_rand_real);
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Info << endl << endl;
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tensor T_rand_imag
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(
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0.8668024539947510, 0.1664607226848602, 0.8925783634185791,
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0.9126510620117188, 0.7408077120780945, 0.1499115079641342,
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0.0936608463525772, 0.7615650296211243, 0.8953040242195129
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);
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Debug(T_rand_imag);
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vector L_rand_imag(eigenValues(T_rand_imag));
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Debug(L_rand_imag);
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tensor U_rand_imag(eigenVectors(T_rand_imag));
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Debug(U_rand_imag);
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Info << endl << endl;
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tensor T_rand_symm
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(
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1.9999992847442627, 1.3076051771640778, 1.3121289014816284,
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1.3076051771640778, 0.0738890841603279, 0.2463847398757935,
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1.3121289014816284, 0.2463847398757935, 0.4082727730274200
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);
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Debug(T_rand_symm);
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vector L_rand_symm(eigenValues(T_rand_symm));
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Debug(L_rand_symm);
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tensor U_rand_symm(eigenVectors(T_rand_symm));
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Debug(U_rand_symm);
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Info << endl << endl;
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symmTensor T_rand_Symm
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(
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1.9999992847442627, 1.3076051771640778, 1.3121289014816284,
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0.0738890841603279, 0.2463847398757935,
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0.4082727730274200
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);
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Debug(T_rand_Symm);
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vector L_rand_Symm(eigenValues(T_rand_Symm));
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Debug(L_rand_Symm);
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tensor U_rand_Symm(eigenVectors(T_rand_Symm));
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Debug(U_rand_Symm);
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Info << endl << endl;
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tensor T_rand_diag
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(
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0.8668024539947510, 0, 0,
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0, 0.7408077120780945, 0,
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0, 0, 0.8953040242195129
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);
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Debug(T_rand_diag);
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vector L_rand_diag(eigenValues(T_rand_diag));
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Debug(L_rand_diag);
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tensor U_rand_diag(eigenVectors(T_rand_diag));
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Debug(U_rand_diag);
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Info << endl << endl;
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tensor T_repeated
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(
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0, 1, 1,
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1, 0, 1,
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1, 1, 0
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);
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Debug(T_repeated);
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vector L_repeated(eigenValues(T_repeated));
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Debug(L_repeated);
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tensor U_repeated(eigenVectors(T_repeated));
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Debug(U_repeated);
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Info << endl << endl;
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tensor T_repeated_zero
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(
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1, 1, 1,
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1, 1, 1,
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1, 1, 1
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);
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Debug(T_repeated_zero);
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vector L_repeated_zero(eigenValues(T_repeated_zero));
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Debug(L_repeated_zero);
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tensor U_repeated_zero(eigenVectors(T_repeated_zero));
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Debug(U_repeated_zero);
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Info << endl << endl;
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tensor T_triple
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(
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2, 0, 0,
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0, 2, 0,
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0, 0, 2
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);
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Debug(T_triple);
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vector L_triple(eigenValues(T_triple));
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Debug(L_triple);
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tensor U_triple(eigenVectors(T_triple));
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Debug(U_triple);
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*/
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return 0;
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}
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@ -89,99 +89,80 @@ namespace Foam
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Foam::vector Foam::eigenValues(const tensor& t)
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{
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scalar i = 0;
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scalar ii = 0;
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scalar iii = 0;
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// The eigenvalues
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scalar i, ii, iii;
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if
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(
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(
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mag(t.xy()) + mag(t.xz()) + mag(t.yx())
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+ mag(t.yz()) + mag(t.zx()) + mag(t.zy())
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)
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< SMALL
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)
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// Coefficients of the characteristic polynmial
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// x^3 + a*x^2 + b*x + c = 0
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scalar a =
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- t.xx() - t.yy() - t.zz();
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scalar b =
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t.xx()*t.yy() + t.xx()*t.zz() + t.yy()*t.zz()
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- t.xy()*t.yx() - t.yz()*t.zy() - t.zx()*t.xz();
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scalar c =
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- t.xx()*t.yy()*t.zz()
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- t.xy()*t.yz()*t.zx() - t.xz()*t.zy()*t.yx()
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+ t.xx()*t.yz()*t.zy() + t.yy()*t.zx()*t.xz() + t.zz()*t.xy()*t.yx();
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// Auxillary variables
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scalar aBy3 = a/3;
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scalar P = (a*a - 3*b)/9; // == -p_wikipedia/3
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scalar PPP = P*P*P;
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scalar Q = (2*a*a*a - 9*a*b + 27*c)/54; // == q_wikipedia/2
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scalar QQ = Q*Q;
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// Three identical roots
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if (mag(P) < SMALL && mag(Q) < SMALL)
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{
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// diagonal matrix
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i = t.xx();
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ii = t.yy();
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iii = t.zz();
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return vector(- aBy3, - aBy3, - aBy3);
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}
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// Two identical roots and one distinct root
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else if (mag(PPP/QQ - 1) < SMALL)
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{
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scalar sqrtP = sqrt(P);
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scalar signQ = sign(Q);
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i = ii = signQ*sqrtP - aBy3;
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iii = - 2*signQ*sqrtP - aBy3;
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}
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// Three distinct roots
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else if (PPP > QQ)
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{
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scalar sqrtP = sqrt(P);
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scalar value = cos(acos(Q/sqrt(PPP))/3);
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scalar delta = sqrt(3 - 3*value*value);
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i = - 2*sqrtP*value - aBy3;
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ii = sqrtP*(value + delta) - aBy3;
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iii = sqrtP*(value - delta) - aBy3;
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}
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// One real root, two imaginary roots
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// based on the above logic, PPP must be less than QQ
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else
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{
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scalar a = -t.xx() - t.yy() - t.zz();
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WarningIn("eigenValues(const tensor&)")
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<< "complex eigenvalues detected for tensor: " << t
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<< endl;
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scalar b = t.xx()*t.yy() + t.xx()*t.zz() + t.yy()*t.zz()
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- t.xy()*t.yx() - t.xz()*t.zx() - t.yz()*t.zy();
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scalar c = - t.xx()*t.yy()*t.zz() - t.xy()*t.yz()*t.zx()
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- t.xz()*t.yx()*t.zy() + t.xz()*t.yy()*t.zx()
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+ t.xy()*t.yx()*t.zz() + t.xx()*t.yz()*t.zy();
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// If there is a zero root
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if (mag(c) < 1e-100)
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if (mag(P) < SMALL)
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{
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scalar disc = sqr(a) - 4*b;
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if (disc >= -SMALL)
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{
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scalar q = -0.5*sqrt(max(0.0, disc));
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i = 0;
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ii = -0.5*a + q;
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iii = -0.5*a - q;
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}
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else
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{
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FatalErrorIn("eigenValues(const tensor&)")
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<< "zero and complex eigenvalues in tensor: " << t
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<< abort(FatalError);
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}
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i = cbrt(QQ/2);
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}
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else
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{
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scalar Q = (a*a - 3*b)/9;
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scalar R = (2*a*a*a - 9*a*b + 27*c)/54;
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scalar R2 = sqr(R);
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scalar Q3 = pow3(Q);
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// Three different real roots
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if (R2 < Q3)
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{
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scalar sqrtQ = sqrt(Q);
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scalar theta = acos(min(1.0, max(-1.0, R/(Q*sqrtQ))));
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scalar m2SqrtQ = -2*sqrtQ;
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scalar aBy3 = a/3;
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i = m2SqrtQ*cos(theta/3) - aBy3;
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ii = m2SqrtQ*cos((theta + twoPi)/3) - aBy3;
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iii = m2SqrtQ*cos((theta - twoPi)/3) - aBy3;
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}
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else
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{
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scalar A = cbrt(R + sqrt(R2 - Q3));
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// Three equal real roots
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if (A < SMALL)
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{
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scalar root = -a/3;
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return vector(root, root, root);
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}
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else
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{
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// Complex roots
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WarningIn("eigenValues(const tensor&)")
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<< "complex eigenvalues detected for tensor: " << t
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<< endl;
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return vector::zero;
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}
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}
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scalar w = cbrt(- Q - sqrt(QQ - PPP));
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i = w + P/w - aBy3;
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}
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}
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return vector(-VGREAT, i, VGREAT);
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}
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// Sort the eigenvalues into ascending order
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if (i > ii)
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@ -203,24 +184,35 @@ Foam::vector Foam::eigenValues(const tensor& t)
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}
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Foam::vector Foam::eigenVector(const tensor& t, const scalar lambda)
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Foam::vector Foam::eigenVector
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(
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const tensor& t,
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const scalar lambda
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)
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{
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if (mag(lambda) < SMALL)
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{
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return vector::zero;
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}
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// Constantly rotating direction ensures different eigenvectors are
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// generated when called sequentially with a multiple eigenvalue
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static vector direction(0,0,1);
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vector oldDirection(direction);
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scalar temp = direction[2];
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direction[2] = direction[1];
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direction[1] = direction[0];
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direction[0] = temp;
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// Construct the matrix for the eigenvector problem
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// Construct the linear system for this eigenvalue
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tensor A(t - lambda*I);
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// Calculate the sub-determinants of the 3 components
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scalar sd0 = A.yy()*A.zz() - A.yz()*A.zy();
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scalar sd1 = A.xx()*A.zz() - A.xz()*A.zx();
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scalar sd2 = A.xx()*A.yy() - A.xy()*A.yx();
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// Determinants of the 2x2 sub-matrices used to find the eigenvectors
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scalar sd0, sd1, sd2;
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scalar magSd0, magSd1, magSd2;
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scalar magSd0 = mag(sd0);
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scalar magSd1 = mag(sd1);
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scalar magSd2 = mag(sd2);
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// Sub-determinants for a unique eivenvalue
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sd0 = A.yy()*A.zz() - A.yz()*A.zy();
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sd1 = A.zz()*A.xx() - A.zx()*A.xz();
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sd2 = A.xx()*A.yy() - A.xy()*A.yx();
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magSd0 = mag(sd0);
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magSd1 = mag(sd1);
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magSd2 = mag(sd2);
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// Evaluate the eigenvector using the largest sub-determinant
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if (magSd0 >= magSd1 && magSd0 >= magSd2 && magSd0 > SMALL)
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@ -231,9 +223,8 @@ Foam::vector Foam::eigenVector(const tensor& t, const scalar lambda)
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(A.yz()*A.zx() - A.zz()*A.yx())/sd0,
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(A.zy()*A.yx() - A.yy()*A.zx())/sd0
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);
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ev /= mag(ev);
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return ev;
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return ev/mag(ev);
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}
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else if (magSd1 >= magSd2 && magSd1 > SMALL)
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{
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@ -243,9 +234,8 @@ Foam::vector Foam::eigenVector(const tensor& t, const scalar lambda)
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1,
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(A.zx()*A.xy() - A.xx()*A.zy())/sd1
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);
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ev /= mag(ev);
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return ev;
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return ev/mag(ev);
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}
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else if (magSd2 > SMALL)
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{
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@ -255,14 +245,55 @@ Foam::vector Foam::eigenVector(const tensor& t, const scalar lambda)
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(A.yx()*A.xz() - A.xx()*A.yz())/sd2,
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1
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);
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ev /= mag(ev);
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return ev;
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return ev/mag(ev);
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}
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else
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// Sub-determinants for a repeated eigenvalue
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sd0 = A.yy()*direction.z() - A.yz()*direction.y();
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sd1 = A.zz()*direction.x() - A.zx()*direction.z();
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sd2 = A.xx()*direction.y() - A.xy()*direction.x();
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magSd0 = mag(sd0);
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magSd1 = mag(sd1);
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magSd2 = mag(sd2);
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// Evaluate the eigenvector using the largest sub-determinant
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if (magSd0 >= magSd1 && magSd0 >= magSd2 && magSd0 > SMALL)
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{
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return vector::zero;
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vector ev
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(
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1,
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(A.yz()*direction.x() - direction.z()*A.yx())/sd0,
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(direction.y()*A.yx() - A.yy()*direction.x())/sd0
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);
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return ev/mag(ev);
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}
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else if (magSd1 >= magSd2 && magSd1 > SMALL)
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{
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vector ev
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(
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(direction.z()*A.zy() - A.zz()*direction.y())/sd1,
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1,
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(A.zx()*direction.y() - direction.x()*A.zy())/sd1
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);
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return ev/mag(ev);
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}
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else if (magSd2 > SMALL)
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{
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vector ev
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(
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(A.xy()*direction.z() - direction.y()*A.xz())/sd2,
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(direction.x()*A.xz() - A.xx()*direction.z())/sd2,
|
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1
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);
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return ev/mag(ev);
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}
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// Triple eigenvalue
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return oldDirection;
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}
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@ -281,198 +312,21 @@ Foam::tensor Foam::eigenVectors(const tensor& t)
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}
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// Return eigenvalues in ascending order of absolute values
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Foam::vector Foam::eigenValues(const symmTensor& t)
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{
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scalar i = 0;
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scalar ii = 0;
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scalar iii = 0;
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|
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if
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(
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(
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mag(t.xy()) + mag(t.xz()) + mag(t.xy())
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+ mag(t.yz()) + mag(t.xz()) + mag(t.yz())
|
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)
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< SMALL
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)
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{
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// diagonal matrix
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i = t.xx();
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ii = t.yy();
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iii = t.zz();
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}
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else
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{
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scalar a = -t.xx() - t.yy() - t.zz();
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scalar b = t.xx()*t.yy() + t.xx()*t.zz() + t.yy()*t.zz()
|
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- t.xy()*t.xy() - t.xz()*t.xz() - t.yz()*t.yz();
|
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scalar c = - t.xx()*t.yy()*t.zz() - t.xy()*t.yz()*t.xz()
|
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- t.xz()*t.xy()*t.yz() + t.xz()*t.yy()*t.xz()
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+ t.xy()*t.xy()*t.zz() + t.xx()*t.yz()*t.yz();
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// If there is a zero root
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if (mag(c) < 1e-100)
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{
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scalar disc = sqr(a) - 4*b;
|
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|
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if (disc >= -SMALL)
|
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{
|
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scalar q = -0.5*sqrt(max(0.0, disc));
|
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|
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i = 0;
|
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ii = -0.5*a + q;
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iii = -0.5*a - q;
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}
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else
|
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{
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FatalErrorIn("eigenValues(const tensor&)")
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<< "zero and complex eigenvalues in tensor: " << t
|
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<< abort(FatalError);
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
scalar Q = (a*a - 3*b)/9;
|
||||
scalar R = (2*a*a*a - 9*a*b + 27*c)/54;
|
||||
|
||||
scalar R2 = sqr(R);
|
||||
scalar Q3 = pow3(Q);
|
||||
|
||||
// Three different real roots
|
||||
if (R2 < Q3)
|
||||
{
|
||||
scalar sqrtQ = sqrt(Q);
|
||||
scalar theta = acos(min(1.0, max(-1.0, R/(Q*sqrtQ))));
|
||||
|
||||
scalar m2SqrtQ = -2*sqrtQ;
|
||||
scalar aBy3 = a/3;
|
||||
|
||||
i = m2SqrtQ*cos(theta/3) - aBy3;
|
||||
ii = m2SqrtQ*cos((theta + twoPi)/3) - aBy3;
|
||||
iii = m2SqrtQ*cos((theta - twoPi)/3) - aBy3;
|
||||
}
|
||||
else
|
||||
{
|
||||
scalar A = cbrt(R + sqrt(R2 - Q3));
|
||||
|
||||
// Three equal real roots
|
||||
if (A < SMALL)
|
||||
{
|
||||
scalar root = -a/3;
|
||||
return vector(root, root, root);
|
||||
}
|
||||
else
|
||||
{
|
||||
// Complex roots
|
||||
WarningIn("eigenValues(const symmTensor&)")
|
||||
<< "complex eigenvalues detected for symmTensor: " << t
|
||||
<< endl;
|
||||
|
||||
return vector::zero;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
// Sort the eigenvalues into ascending order
|
||||
if (i > ii)
|
||||
{
|
||||
Swap(i, ii);
|
||||
}
|
||||
|
||||
if (ii > iii)
|
||||
{
|
||||
Swap(ii, iii);
|
||||
}
|
||||
|
||||
if (i > ii)
|
||||
{
|
||||
Swap(i, ii);
|
||||
}
|
||||
|
||||
return vector(i, ii, iii);
|
||||
return eigenValues(tensor(t));
|
||||
}
|
||||
|
||||
|
||||
Foam::vector Foam::eigenVector(const symmTensor& t, const scalar lambda)
|
||||
{
|
||||
if (mag(lambda) < SMALL)
|
||||
{
|
||||
return vector::zero;
|
||||
}
|
||||
|
||||
// Construct the matrix for the eigenvector problem
|
||||
symmTensor A(t - lambda*I);
|
||||
|
||||
// Calculate the sub-determinants of the 3 components
|
||||
scalar sd0 = A.yy()*A.zz() - A.yz()*A.yz();
|
||||
scalar sd1 = A.xx()*A.zz() - A.xz()*A.xz();
|
||||
scalar sd2 = A.xx()*A.yy() - A.xy()*A.xy();
|
||||
|
||||
scalar magSd0 = mag(sd0);
|
||||
scalar magSd1 = mag(sd1);
|
||||
scalar magSd2 = mag(sd2);
|
||||
|
||||
// Evaluate the eigenvector using the largest sub-determinant
|
||||
if (magSd0 >= magSd1 && magSd0 >= magSd2 && magSd0 > SMALL)
|
||||
{
|
||||
vector ev
|
||||
(
|
||||
1,
|
||||
(A.yz()*A.xz() - A.zz()*A.xy())/sd0,
|
||||
(A.yz()*A.xy() - A.yy()*A.xz())/sd0
|
||||
);
|
||||
ev /= mag(ev);
|
||||
|
||||
return ev;
|
||||
}
|
||||
else if (magSd1 >= magSd2 && magSd1 > SMALL)
|
||||
{
|
||||
vector ev
|
||||
(
|
||||
(A.xz()*A.yz() - A.zz()*A.xy())/sd1,
|
||||
1,
|
||||
(A.xz()*A.xy() - A.xx()*A.yz())/sd1
|
||||
);
|
||||
ev /= mag(ev);
|
||||
|
||||
return ev;
|
||||
}
|
||||
else if (magSd2 > SMALL)
|
||||
{
|
||||
vector ev
|
||||
(
|
||||
(A.xy()*A.yz() - A.yy()*A.xz())/sd2,
|
||||
(A.xy()*A.xz() - A.xx()*A.yz())/sd2,
|
||||
1
|
||||
);
|
||||
ev /= mag(ev);
|
||||
|
||||
return ev;
|
||||
}
|
||||
else
|
||||
{
|
||||
return vector::zero;
|
||||
}
|
||||
return eigenVector(tensor(t), lambda);
|
||||
}
|
||||
|
||||
|
||||
Foam::tensor Foam::eigenVectors(const symmTensor& t)
|
||||
{
|
||||
vector evals(eigenValues(t));
|
||||
|
||||
tensor evs
|
||||
(
|
||||
eigenVector(t, evals.x()),
|
||||
eigenVector(t, evals.y()),
|
||||
eigenVector(t, evals.z())
|
||||
);
|
||||
|
||||
return evs;
|
||||
return eigenVectors(tensor(t));
|
||||
}
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user