int main(void){
linear();
newton2();
newton3();
lagrange();
return 0;
}
/**
* \brief Find eigenvalues and vectors of a 3x3 matrix.
* \param symmat 0 if matrix is not symetric, 1 otherwise.
* \param mat pointer toward the matrix.
* \param lambda eigenvalues.
* \param v eigenvectors.
* \return order of eigenvalues (1,2,3) or 0 if failed.
*/
int _MMG5_eigenv(int symmat,double *mat,double lambda[3],double v[3][3]) {
double a11,a12,a13,a21,a22,a23,a31,a32,a33;
double aa,bb,cc,dd,ee,ii,vx1[3],vx2[3],vx3[3],dd1,dd2,dd3;
double maxd,maxm,valm,p[4],w1[3],w2[3],w3[3];
int k,n;
/* default */
memcpy(v,Id,9*sizeof(double));
if ( symmat ) {
lambda[0] = (double)mat[0];
lambda[1] = (double)mat[3];
lambda[2] = (double)mat[5];
maxm = fabs(mat[0]);
for (k=1; k<6; k++) {
valm = fabs(mat[k]);
if ( valm > maxm ) maxm = valm;
}
/* single float accuracy */
if ( maxm < _MG_EPS6 ) return(1);
/* normalize matrix */
dd = 1.0 / maxm;
a11 = mat[0] * dd;
a12 = mat[1] * dd;
a13 = mat[2] * dd;
a22 = mat[3] * dd;
a23 = mat[4] * dd;
a33 = mat[5] * dd;
/* diagonal matrix */
maxd = fabs(a12);
valm = fabs(a13);
if ( valm > maxd ) maxd = valm;
valm = fabs(a23);
if ( valm > maxd ) maxd = valm;
if ( maxd < _MG_EPSD ) return(1);
a21 = a12;
a31 = a13;
a32 = a23;
/* build characteristic polynomial
P(X) = X^3 - trace X^2 + (somme des mineurs)X - det = 0 */
aa = a11*a22;
bb = a23*a32;
cc = a12*a21;
dd = a13*a31;
p[0] = a11*bb + a33*(cc-aa) + a22*dd -2.0*a12*a13*a23;
p[1] = a11*(a22 + a33) + a22*a33 - bb - cc - dd;
p[2] = -a11 - a22 - a33;
p[3] = 1.0;
}
else {
lambda[0] = (double)mat[0];
lambda[1] = (double)mat[4];
lambda[2] = (double)mat[8];
maxm = fabs(mat[0]);
for (k=1; k<9; k++) {
valm = fabs(mat[k]);
if ( valm > maxm ) maxm = valm;
}
if ( maxm < _MG_EPS6 ) return(1);
/* normalize matrix */
dd = 1.0 / maxm;
a11 = mat[0] * dd;
a12 = mat[1] * dd;
a13 = mat[2] * dd;
a21 = mat[3] * dd;
a22 = mat[4] * dd;
a23 = mat[5] * dd;
a31 = mat[6] * dd;
a32 = mat[7] * dd;
a33 = mat[8] * dd;
/* diagonal matrix */
maxd = fabs(a12);
valm = fabs(a13);
if ( valm > maxd ) maxd = valm;
valm = fabs(a23);
if ( valm > maxd ) maxd = valm;
valm = fabs(a21);
if ( valm > maxd ) maxd = valm;
valm = fabs(a31);
if ( valm > maxd ) maxd = valm;
valm = fabs(a32);
if ( valm > maxd ) maxd = valm;
if ( maxd < _MG_EPSD ) return(1);
/* build characteristic polynomial
P(X) = X^3 - trace X^2 + (somme des mineurs)X - det = 0 */
aa = a22*a33 - a23*a32;
bb = a23*a31 - a21*a33;
cc = a21*a32 - a31*a22;
ee = a11*a33 - a13*a31;
ii = a11*a22 - a12*a21;
p[0] = -a11*aa - a12*bb - a13*cc;
p[1] = aa + ee + ii;
p[2] = -a11 - a22 - a33;
p[3] = 1.0;
}
/* solve polynomial (find roots using newton) */
n = newton3(p,lambda);
if ( n <= 0 ) return(0);
/* compute eigenvectors:
an eigenvalue belong to orthogonal of Im(A-lambda*Id) */
v[0][0] = 1.0; v[0][1] = v[0][2] = 0.0;
v[1][1] = 1.0; v[1][0] = v[1][2] = 0.0;
v[2][2] = 1.0; v[2][0] = v[2][1] = 0.0;
w1[1] = a12; w1[2] = a13;
w2[0] = a21; w2[2] = a23;
w3[0] = a31; w3[1] = a32;
if ( n == 1 ) {
/* vk = crsprd(wi,wj) */
for (k=0; k<3; k++) {
w1[0] = a11 - lambda[k];
w2[1] = a22 - lambda[k];
w3[2] = a33 - lambda[k];
/* cross product vectors in (Im(A-lambda(i) Id) ortho */
vx1[0] = w1[1]*w3[2] - w1[2]*w3[1];
vx1[1] = w1[2]*w3[0] - w1[0]*w3[2];
vx1[2] = w1[0]*w3[1] - w1[1]*w3[0];
dd1 = vx1[0]*vx1[0] + vx1[1]*vx1[1] + vx1[2]*vx1[2];
vx2[0] = w1[1]*w2[2] - w1[2]*w2[1];
vx2[1] = w1[2]*w2[0] - w1[0]*w2[2];
vx2[2] = w1[0]*w2[1] - w1[1]*w2[0];
dd2 = vx2[0]*vx2[0] + vx2[1]*vx2[1] + vx2[2]*vx2[2];
vx3[0] = w2[1]*w3[2] - w2[2]*w3[1];
vx3[1] = w2[2]*w3[0] - w2[0]*w3[2];
vx3[2] = w2[0]*w3[1] - w2[1]*w3[0];
dd3 = vx3[0]*vx3[0] + vx3[1]*vx3[1] + vx3[2]*vx3[2];
/* find vector of max norm */
if ( dd1 > dd2 ) {
if ( dd1 > dd3 ) {
dd1 = 1.0 / sqrt(dd1);
v[k][0] = vx1[0] * dd1;
v[k][1] = vx1[1] * dd1;
v[k][2] = vx1[2] * dd1;
}
else {
dd3 = 1.0 / sqrt(dd3);
v[k][0] = vx3[0] * dd3;
v[k][1] = vx3[1] * dd3;
v[k][2] = vx3[2] * dd3;
}
}
else {
if ( dd2 > dd3 ) {
dd2 = 1.0 / sqrt(dd2);
v[k][0] = vx2[0] * dd2;
v[k][1] = vx2[1] * dd2;
v[k][2] = vx2[2] * dd2;
}
else {
dd3 = 1.0 / sqrt(dd3);
v[k][0] = vx3[0] * dd3;
v[k][1] = vx3[1] * dd3;
v[k][2] = vx3[2] * dd3;
}
}
}
}
/* (vp1,vp2) double, vp3 simple root */
else if ( n == 2 ) {
w1[0] = a11 - lambda[2];
w2[1] = a22 - lambda[2];
w3[2] = a33 - lambda[2];
/* cross product */
vx1[0] = w1[1]*w3[2] - w1[2]*w3[1];
vx1[1] = w1[2]*w3[0] - w1[0]*w3[2];
vx1[2] = w1[0]*w3[1] - w1[1]*w3[0];
dd1 = vx1[0]*vx1[0] + vx1[1]*vx1[1] + vx1[2]*vx1[2];
vx2[0] = w1[1]*w2[2] - w1[2]*w2[1];
vx2[1] = w1[2]*w2[0] - w1[0]*w2[2];
vx2[2] = w1[0]*w2[1] - w1[1]*w2[0];
dd2 = vx2[0]*vx2[0] + vx2[1]*vx2[1] + vx2[2]*vx2[2];
vx3[0] = w2[1]*w3[2] - w2[2]*w3[1];
vx3[1] = w2[2]*w3[0] - w2[0]*w3[2];
vx3[2] = w2[0]*w3[1] - w2[1]*w3[0];
dd3 = vx3[0]*vx3[0] + vx3[1]*vx3[1] + vx3[2]*vx3[2];
/* find vector of max norm */
if ( dd1 > dd2 ) {
if ( dd1 > dd3 ) {
dd1 = 1.0 / sqrt(dd1);
v[2][0] = vx1[0] * dd1;
v[2][1] = vx1[1] * dd1;
v[2][2] = vx1[2] * dd1;
}
else {
dd3 = 1.0 / sqrt(dd3);
v[2][0] = vx3[0] * dd3;
v[2][1] = vx3[1] * dd3;
v[2][2] = vx3[2] * dd3;
}
}
else {
if ( dd2 > dd3 ) {
dd2 = 1.0 / sqrt(dd2);
v[2][0] = vx2[0] * dd2;
v[2][1] = vx2[1] * dd2;
v[2][2] = vx2[2] * dd2;
}
else {
dd3 = 1.0 / sqrt(dd3);
v[2][0] = vx3[0] * dd3;
v[2][1] = vx3[1] * dd3;
v[2][2] = vx3[2] * dd3;
}
}
/* compute v1 and v2 in Im(A-vp3*Id) */
dd1 = w1[0]*w1[0] + w1[1]*w1[1] + w1[2]*w1[2];
dd2 = w2[0]*w2[0] + w2[1]*w2[1] + w2[2]*w2[2];
if ( dd1 > dd2 ) {
dd1 = 1.0 / sqrt(dd1);
v[0][0] = w1[0]*dd1;
v[0][1] = w1[1]*dd1;
v[0][2] = w1[2]*dd1;
}
else {
dd2 = 1.0 / sqrt(dd2);
v[0][0] = w2[0]*dd2;
v[0][1] = w2[1]*dd2;
v[0][2] = w2[2]*dd2;
}
/* 3rd vector orthogonal */
v[1][0] = v[2][1]*v[0][2] - v[2][2]*v[0][1];
v[1][1] = v[2][2]*v[0][0] - v[2][0]*v[0][2];
v[1][2] = v[2][0]*v[0][1] - v[2][1]*v[0][0];
dd1 = v[1][0]*v[1][0] + v[1][1]*v[1][1] + v[1][2]*v[1][2];
dd1 = 1.0 / sqrt(dd1);
v[1][0] *= dd1;
v[1][1] *= dd1;
v[1][2] *= dd1;
}
lambda[0] *= maxm;
lambda[1] *= maxm;
lambda[2] *= maxm;
/* check accuracy */
/*-------------------------------------------------------------------
if ( ddebug && symmat ) {
double err,tmpx,tmpy,tmpz;
float m[6];
int i,j;
k = 0;
for (i=0; i<3; i++)
for (j=i; j<3; j++)
m[k++] = lambda[0]*v[i][0]*v[j][0]
+ lambda[1]*v[i][1]*v[j][1]
+ lambda[2]*v[i][2]*v[j][2];
err = fabs(mat[0]-m[0]);
for (i=1; i<6; i++)
if ( fabs(m[i]-mat[i]) > err ) err = fabs(m[i]-mat[i]);
if ( err > 1.e03*maxm ) {
printf("\nProbleme eigenv3: err= %f\n",err*maxm);
printf("mat depart :\n");
printf("%13.6f %13.6f %13.6f\n",mat[0],mat[1],mat[2]);
printf("%13.6f %13.6f %13.6f\n",mat[1],mat[3],mat[4]);
printf("%13.6f %13.6f %13.6f\n",mat[2],mat[4],mat[5]);
printf("mat finale :\n");
printf("%13.6f %13.6f %13.6f\n",m[0],m[1],m[2]);
printf("%13.6f %13.6f %13.6f\n",m[1],m[3],m[4]);
printf("%13.6f %13.6f %13.6f\n",m[2],m[4],m[5]);
printf("lambda : %f %f %f\n",lambda[0],lambda[1],lambda[2]);
printf(" ordre %d\n",n);
printf("\nOrtho:\n");
printf("v1.v2 = %.14f\n",
v[0][0]*v[1][0]+v[0][1]*v[1][1]+ v[0][2]*v[1][2]);
printf("v1.v3 = %.14f\n",
v[0][0]*v[2][0]+v[0][1]*v[2][1]+ v[0][2]*v[2][2]);
printf("v2.v3 = %.14f\n",
v[1][0]*v[2][0]+v[1][1]*v[2][1]+ v[1][2]*v[2][2]);
printf("Consistency\n");
for (i=0; i<3; i++) {
tmpx = v[0][i]*m[0] + v[1][i]*m[1]
+ v[2][i]*m[2] - lambda[i]*v[0][i];
tmpy = v[0][i]*m[1] + v[1][i]*m[3]
+ v[2][i]*m[4] - lambda[i]*v[1][i];
tmpz = v[0][i]*m[2] + v[1][i]*m[4]
+ v[2][i]*m[5] - lambda[i]*v[2][i];
printf(" Av %d - lambda %d *v %d = %f %f %f\n",
i,i,i,tmpx,tmpy,tmpz);
printf("w1 %f %f %f\n",w1[0],w1[1],w1[2]);
printf("w2 %f %f %f\n",w2[0],w2[1],w2[2]);
printf("w3 %f %f %f\n",w3[0],w3[1],w3[2]);
}
exit(1);
}
}
-------------------------------------------------------------------*/
return(n);
}
