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eigensolver for symmetric Mat3

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This commit is contained in:
Jiri Pittner 2021-11-20 22:14:40 +01:00
parent 4e08955ed5
commit 30f4d29d82
3 changed files with 331 additions and 2 deletions
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32
t.cc
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@ -2295,7 +2295,7 @@ cout <<"non-zero = "<< ((n&1)?p.odd_powers():p.even_powers())<<std::endl;
cout <<"value = "<<value(p,1.23456789)<<" "<<odd_value(p,1.23456789)+even_value(p,1.23456789)<<endl;
}
if(1)
if(0)
{
NRVec<int> v({1,2,3,4});
cout <<v;
@ -2321,5 +2321,35 @@ Polynomial<double> pp({1,2,3,4,5});
cout<<pp;
}
if(1)
{
//prepare random symmetric mat3
int seed;
int f=open("/dev/random",O_RDONLY);
if(sizeof(int)!=read(f,&seed,sizeof(int))) laerror("cannot read /dev/random");
close(f);
srand(seed);
NRMat<double> tmp(3,3);
tmp.randomize(2.);
Mat3<double> mm(tmp);
mm.symmetrize();
NRMat<double> m(&mm[0][0],3,3);
cout <<m<<"3 3\n"<<mm<<endl;
NRVec<double> w(3);
diagonalize(m,w);
cout << w<<m;
Vec3<double> ww;
Mat3<double> vv;
mm.eivec_sym(ww,vv);
cout <<"3\n"<<ww<<"\n3 3\n"<<vv<<endl;
NRVec<double> www(&ww[0],3);
NRMat<double> vvv(&vv[0][0],3,3);
cout<<"eival error = "<<(w-www).norm()<<endl;
cout<<"eivec error = "<<(m.diffabs(vvv)).norm()<<endl; //just ignore signs due to arb. phases (not full check)
}
}

297
vecmat3.cc
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@ -127,6 +127,7 @@ const Vec3<T> Mat3<T>::operator*(const Vec3<T> &rhs) const
//cf. https://en.wikipedia.org/wiki/Euler_angles and NASA paper cited therein
template<typename T>
void euler2rotmat(const T *eul, Mat3<T> &a, const char *type, bool transpose, bool direction, bool reverse)
@ -486,7 +487,303 @@ return s;
#endif
template <typename T>
void Mat3<T>::symmetrize()
{
T tmp=(q[0][1]+q[1][0])/2;
q[0][1]=q[1][0]=tmp;
tmp=(q[0][2]+q[2][0])/2;
q[0][2]=q[2][0]=tmp;
tmp=(q[2][1]+q[1][2])/2;
q[2][1]=q[1][2]=tmp;
}
//eigensolver for 3x3 matrix by Joachim Kopp - analytic formula version,
//might be unstable for ill-conditioned ones, then use other methods
//cf. arxiv physics 0610206v3
//
//// Numerical diagonalization of 3x3 matrcies
//// Copyright (C) 2006 Joachim Kopp
//// ----------------------------------------------------------------------------
//// This library is free software; you can redistribute it and/or
//// modify it under the terms of the GNU Lesser General Public
//// License as published by the Free Software Foundation; either
//// version 2.1 of the License, or (at your option) any later version.
////
//// This library is distributed in the hope that it will be useful,
//// but WITHOUT ANY WARRANTY; without even the implied warranty of
//// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
//// Lesser General Public License for more details.
////
//// You should have received a copy of the GNU Lesser General Public
//// License along with this library; if not, write to the Free Software
//// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
//// ----------------------------------------------------------------------------
//
//numeric_limits not available on some crosscompilers for small MCUs
#ifdef ARM_SOURCE37
#define DBL_EPSILON 1.19209290e-07f
#else
#define DBL_EPSILON std::numeric_limits<T>::epsilon()
#endif
#define M_SQRT3 1.73205080756887729352744634151 // sqrt(3)
#define SQR(x) ((x)*(x)) // x^2
//
template <typename T>
void Mat3<T>::eival_sym(Vec3<T> &w) const
{
T m, c1, c0;
// Determine coefficients of characteristic poynomial. We write
// | a d f |
// A = | d* b e |
// | f* e* c |
T de = q[0][1] * q[1][2]; // d * e
T dd = SQR(q[0][1]); // d^2
T ee = SQR(q[1][2]); // e^2
T ff = SQR(q[0][2]); // f^2
m = q[0][0] + q[1][1] + q[2][2];
c1 = (q[0][0]*q[1][1] + q[0][0]*q[2][2] + q[1][1]*q[2][2]) // a*b + a*c + b*c - d^2 - e^2 - f^2
- (dd + ee + ff);
c0 = q[2][2]*dd + q[0][0]*ee + q[1][1]*ff - q[0][0]*q[1][1]*q[2][2]
- 2.0 * q[0][2]*de; // c*d^2 + a*e^2 + b*f^2 - a*b*c - 2*f*d*e)
T p, sqrt_p, q, c, s, phi;
p = SQR(m) - 3.0*c1;
q = m*(p - (3.0/2.0)*c1) - (27.0/2.0)*c0;
sqrt_p = sqrt(abs(p));
phi = 27.0 * ( 0.25*SQR(c1)*(p - c1) + c0*(q + 27.0/4.0*c0));
phi = (1.0/3.0) * atan2(sqrt(abs(phi)), q);
c = sqrt_p*cos(phi);
s = (1.0/M_SQRT3)*sqrt_p*sin(phi);
w[1] = (1.0/3.0)*(m - c);
w[2] = w[1] + s;
w[0] = w[1] + c;
w[1] -= s;
//sort in ascending order
if(w[0]>w[1]) {T tmp=w[0]; w[0]=w[1]; w[1]=tmp;}
if(w[0]>w[2]) {T tmp=w[0]; w[0]=w[2]; w[2]=tmp;}
if(w[1]>w[2]) {T tmp=w[1]; w[1]=w[2]; w[2]=tmp;}
}
// Calculates the eigenvalues and normalized eigenvectors of a symmetric 3x3
// matrix A using Cardano's method for the eigenvalues and an analytical
// method based on vector cross products for the eigenvectors.
// Only the diagonal and upper triangular parts of A need to contain meaningful
// values. However, all of A may be used as temporary storage and may hence be
// destroyed.
// ----------------------------------------------------------------------------
// Parameters:
// A: The symmetric input matrix
// Q: Storage buffer for eigenvectors
// w: Storage buffer for eigenvalues
// ----------------------------------------------------------------------------
// Return value:
// 0: Success
// -1: Error
// ----------------------------------------------------------------------------
// Dependencies:
// dsyevc3()
// ----------------------------------------------------------------------------
// Version history:
// v1.1 (12 Mar 2012): Removed access to lower triangualr part of A
// (according to the documentation, only the upper triangular part needs
// to be filled)
// v1.0: First released version
// ----------------------------------------------------------------------------
template <typename T>
void Mat3<T>::eivec_sym(Vec3<T> &w, Mat3 &v) const
{
T norm; // Squared norm or inverse norm of current eigenvector
T n0, n1; // Norm of first and second columns of A
T n0tmp, n1tmp; // "Templates" for the calculation of n0/n1 - saves a few FLOPS
T thresh; // Small number used as threshold for floating point comparisons
T error; // Estimated maximum roundoff error in some steps
T wmax; // The eigenvalue of maximum modulus
T f, t; // Intermediate storage
int i, j; // Loop counters
// Calculate eigenvalues
eival_sym(w);
Mat3<T> A(*this); //scratch copy
wmax = fabs(w[0]);
if ((t=fabs(w[1])) > wmax)
wmax = t;
if ((t=fabs(w[2])) > wmax)
wmax = t;
thresh = SQR(8.0 * DBL_EPSILON * wmax);
// Prepare calculation of eigenvectors
n0tmp = SQR(A[0][1]) + SQR(A[0][2]);
n1tmp = SQR(A[0][1]) + SQR(A[1][2]);
v[0][1] = A[0][1]*A[1][2] - A[0][2]*A[1][1];
v[1][1] = A[0][2]*A[0][1] - A[1][2]*A[0][0];
v[2][1] = SQR(A[0][1]);
// Calculate first eigenvector by the formula
// v[0] = (A - w[0]).e1 x (A - w[0]).e2
A[0][0] -= w[0];
A[1][1] -= w[0];
v[0][0] = v[0][1] + A[0][2]*w[0];
v[1][0] = v[1][1] + A[1][2]*w[0];
v[2][0] = A[0][0]*A[1][1] - v[2][1];
norm = SQR(v[0][0]) + SQR(v[1][0]) + SQR(v[2][0]);
n0 = n0tmp + SQR(A[0][0]);
n1 = n1tmp + SQR(A[1][1]);
error = n0 * n1;
if (n0 <= thresh) // If the first column is zero, then (1,0,0) is an eigenvector
{
v[0][0] = 1.0;
v[1][0] = 0.0;
v[2][0] = 0.0;
}
else if (n1 <= thresh) // If the second column is zero, then (0,1,0) is an eigenvector
{
v[0][0] = 0.0;
v[1][0] = 1.0;
v[2][0] = 0.0;
}
else if (norm < SQR(64.0 * DBL_EPSILON) * error)
{ // If angle between A[0] and A[1] is too small, don't use
t = SQR(A[0][1]); // cross product, but calculate v ~ (1, -A0/A1, 0)
f = -A[0][0] / A[0][1];
if (SQR(A[1][1]) > t)
{
t = SQR(A[1][1]);
f = -A[0][1] / A[1][1];
}
if (SQR(A[1][2]) > t)
f = -A[0][2] / A[1][2];
norm = 1.0/sqrt(1 + SQR(f));
v[0][0] = norm;
v[1][0] = f * norm;
v[2][0] = 0.0;
}
else // This is the standard branch
{
norm = sqrt(1.0 / norm);
for (j=0; j < 3; j++)
v[j][0] = v[j][0] * norm;
}
// Prepare calculation of second eigenvector
t = w[0] - w[1];
if (fabs(t) > 8.0 * DBL_EPSILON * wmax)
{
// For non-degenerate eigenvalue, calculate second eigenvector by the formula
// v[1] = (A - w[1]).e1 x (A - w[1]).e2
A[0][0] += t;
A[1][1] += t;
v[0][1] = v[0][1] + A[0][2]*w[1];
v[1][1] = v[1][1] + A[1][2]*w[1];
v[2][1] = A[0][0]*A[1][1] - v[2][1];
norm = SQR(v[0][1]) + SQR(v[1][1]) + SQR(v[2][1]);
n0 = n0tmp + SQR(A[0][0]);
n1 = n1tmp + SQR(A[1][1]);
error = n0 * n1;
if (n0 <= thresh) // If the first column is zero, then (1,0,0) is an eigenvector
{
v[0][1] = 1.0;
v[1][1] = 0.0;
v[2][1] = 0.0;
}
else if (n1 <= thresh) // If the second column is zero, then (0,1,0) is an eigenvector
{
v[0][1] = 0.0;
v[1][1] = 1.0;
v[2][1] = 0.0;
}
else if (norm < SQR(64.0 * DBL_EPSILON) * error)
{ // If angle between A[0] and A[1] is too small, don't use
t = SQR(A[0][1]); // cross product, but calculate v ~ (1, -A0/A1, 0)
f = -A[0][0] / A[0][1];
if (SQR(A[1][1]) > t)
{
t = SQR(A[1][1]);
f = -A[0][1] / A[1][1];
}
if (SQR(A[1][2]) > t)
f = -A[0][2] / A[1][2];
norm = 1.0/sqrt(1 + SQR(f));
v[0][1] = norm;
v[1][1] = f * norm;
v[2][1] = 0.0;
}
else
{
norm = sqrt(1.0 / norm);
for (j=0; j < 3; j++)
v[j][1] = v[j][1] * norm;
}
}
else
{
// For degenerate eigenvalue, calculate second eigenvector according to
// v[1] = v[0] x (A - w[1]).e[i]
//
// This would really get to complicated if we could not assume all of A to
// contain meaningful values.
A[1][0] = A[0][1];
A[2][0] = A[0][2];
A[2][1] = A[1][2];
A[0][0] += w[0];
A[1][1] += w[0];
for (i=0; i < 3; i++)
{
A[i][i] -= w[1];
n0 = SQR(A[0][i]) + SQR(A[1][i]) + SQR(A[2][i]);
if (n0 > thresh)
{
v[0][1] = v[1][0]*A[2][i] - v[2][0]*A[1][i];
v[1][1] = v[2][0]*A[0][i] - v[0][0]*A[2][i];
v[2][1] = v[0][0]*A[1][i] - v[1][0]*A[0][i];
norm = SQR(v[0][1]) + SQR(v[1][1]) + SQR(v[2][1]);
if (norm > SQR(256.0 * DBL_EPSILON) * n0) // Accept cross product only if the angle between
{ // the two vectors was not too small
norm = sqrt(1.0 / norm);
for (j=0; j < 3; j++)
v[j][1] = v[j][1] * norm;
break;
}
}
}
if (i == 3) // This means that any vector orthogonal to v[0] is an EV.
{
for (j=0; j < 3; j++)
if (v[j][0] != 0.0) // Find nonzero element of v[0] ...
{ // ... and swap it with the next one
norm = 1.0 / sqrt(SQR(v[j][0]) + SQR(v[(j+1)%3][0]));
v[j][1] = v[(j+1)%3][0] * norm;
v[(j+1)%3][1] = -v[j][0] * norm;
v[(j+2)%3][1] = 0.0;
break;
}
}
}
// Calculate third eigenvector according to
// v[2] = v[0] x v[1]
v[0][2] = v[1][0]*v[2][1] - v[2][0]*v[1][1];
v[1][2] = v[2][0]*v[0][1] - v[0][0]*v[2][1];
v[2][2] = v[0][0]*v[1][1] - v[1][0]*v[0][1];
}
#undef SQR
//end eigensolver for 3x3 matrix
/////////////////////////////////////////////////////////////////////////////////////////////
//force instantization
#define INSTANTIZE(T) \

4
vecmat3.h
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@ -145,7 +145,9 @@ public:
//C-style IO
int fprintf(FILE *f, const char *format) const {int n= ::fprintf(f,format,q[0][0],q[0][1],q[0][2]); n+=::fprintf(f,format,q[1][0],q[1][1],q[1][2]); n+=::fprintf(f,format,q[2][0],q[2][1],q[2][2]); return n;};
int fscanf(FILE *f, const char *format) const {return ::fscanf(f,format,q[0][0],q[0][1],q[0][2]) + ::fscanf(f,format,q[1][0],q[1][1],q[1][2]) + ::fscanf(f,format,q[2][0],q[2][1],q[2][2]);};
void symmetrize(); //average offdiagonal elements
void eival_sym(Vec3<T> &w) const; //only for real symmetric matrix, symmetry is not checked
void eivec_sym(Vec3<T> &w, Mat3 &v) const; //only for real symmetric matrix, symmetry is not checked
};