| 1 | #include <gfx/std.h>
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| 2 | #include <gfx/math/Mat3.h>
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| 3 | #include <gfx/math/Mat4.h>
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| 4 |
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| 5 | // Adapted from VTK source code (see vtkMath.cxx)
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| 6 | // which seems to have been adapted directly from Numerical Recipes in C.
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| 7 |
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| 8 |
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| 9 | #define ROT(a,i,j,k,l) g=a[i][j];h=a[k][l];a[i][j]=g-s*(h+g*tau);a[k][l]=h+s*(g-h*tau);
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| 10 |
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| 11 | #define MAX_ROTATIONS 60
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| 12 |
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| 13 |
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| 14 |
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| 15 | // Description:
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| 16 | // Jacobi iteration for the solution of eigenvectors/eigenvalues of a 3x3
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| 17 | // real symmetric matrix. Square 3x3 matrix a; output eigenvalues in w;
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| 18 | // and output eigenvectors in v. Resulting eigenvalues/vectors are sorted
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| 19 | // in decreasing order; eigenvectors are normalized.
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| 20 | //
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| 21 |
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| 22 | using namespace simplif;
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| 23 |
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| 24 | static
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| 25 | bool internal_jacobi(real a[3][3], real w[3], real v[3][3])
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| 26 | {
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| 27 | int i, j, k, iq, ip;
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| 28 | real tresh, theta, tau, t, sm, s, h, g, c;
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| 29 | real b[3], z[3], tmp;
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| 30 |
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| 31 | // initialize
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| 32 | for (ip=0; ip<3; ip++)
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| 33 | {
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| 34 | for (iq=0; iq<3; iq++) v[ip][iq] = 0.0;
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| 35 | v[ip][ip] = 1.0;
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| 36 | }
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| 37 | for (ip=0; ip<3; ip++)
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| 38 | {
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| 39 | b[ip] = w[ip] = a[ip][ip];
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| 40 | z[ip] = 0.0;
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| 41 | }
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| 42 |
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| 43 | // begin rotation sequence
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| 44 | for (i=0; i<MAX_ROTATIONS; i++)
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| 45 | {
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| 46 | sm = 0.0;
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| 47 | for (ip=0; ip<2; ip++)
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| 48 | {
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| 49 | for (iq=ip+1; iq<3; iq++) sm += fabs(a[ip][iq]);
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| 50 | }
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| 51 | if (sm == 0.0) break;
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| 52 |
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| 53 | if (i < 4) tresh = 0.2*sm/(9);
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| 54 | else tresh = 0.0;
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| 55 |
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| 56 | for (ip=0; ip<2; ip++)
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| 57 | {
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| 58 | for (iq=ip+1; iq<3; iq++)
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| 59 | {
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| 60 | g = 100.0*fabs(a[ip][iq]);
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| 61 | if (i > 4 && (fabs(w[ip])+g) == fabs(w[ip])
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| 62 | && (fabs(w[iq])+g) == fabs(w[iq]))
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| 63 | {
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| 64 | a[ip][iq] = 0.0;
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| 65 | }
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| 66 | else if (fabs(a[ip][iq]) > tresh)
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| 67 | {
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| 68 | h = w[iq] - w[ip];
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| 69 | if ( (fabs(h)+g) == fabs(h)) t = (a[ip][iq]) / h;
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| 70 | else
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| 71 | {
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| 72 | theta = 0.5*h / (a[ip][iq]);
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| 73 | t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta));
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| 74 | if (theta < 0.0) t = -t;
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| 75 | }
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| 76 | c = 1.0 / sqrt(1+t*t);
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| 77 | s = t*c;
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| 78 | tau = s/(1.0+c);
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| 79 | h = t*a[ip][iq];
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| 80 | z[ip] -= h;
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| 81 | z[iq] += h;
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| 82 | w[ip] -= h;
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| 83 | w[iq] += h;
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| 84 | a[ip][iq]=0.0;
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| 85 | for (j=0;j<ip-1;j++)
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| 86 | {
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| 87 | ROT(a,j,ip,j,iq)
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| 88 | }
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| 89 | for (j=ip+1;j<iq-1;j++)
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| 90 | {
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| 91 | ROT(a,ip,j,j,iq)
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| 92 | }
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| 93 | for (j=iq+1; j<3; j++)
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| 94 | {
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| 95 | ROT(a,ip,j,iq,j)
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| 96 | }
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| 97 | for (j=0; j<3; j++)
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| 98 | {
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| 99 | ROT(v,j,ip,j,iq)
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| 100 | }
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| 101 | }
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| 102 | }
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| 103 | }
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| 104 |
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| 105 | for (ip=0; ip<3; ip++)
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| 106 | {
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| 107 | b[ip] += z[ip];
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| 108 | w[ip] = b[ip];
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| 109 | z[ip] = 0.0;
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| 110 | }
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| 111 | }
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| 112 |
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| 113 | if ( i >= MAX_ROTATIONS )
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| 114 | {
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| 115 | std::cerr << "WARNING -- jacobi() -- Error computing eigenvalues." << std::endl;
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| 116 | return false;
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| 117 | }
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| 118 |
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| 119 | // sort eigenfunctions
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| 120 | for (j=0; j<3; j++)
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| 121 | {
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| 122 | k = j;
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| 123 | tmp = w[k];
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| 124 | for (i=j; i<3; i++)
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| 125 | {
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| 126 | if (w[i] >= tmp)
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| 127 | {
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| 128 | k = i;
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| 129 | tmp = w[k];
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| 130 | }
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| 131 | }
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| 132 | if (k != j)
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| 133 | {
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| 134 | w[k] = w[j];
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| 135 | w[j] = tmp;
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| 136 | for (i=0; i<3; i++)
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| 137 | {
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| 138 | tmp = v[i][j];
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| 139 | v[i][j] = v[i][k];
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| 140 | v[i][k] = tmp;
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| 141 | }
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| 142 | }
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| 143 | }
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| 144 | // insure eigenvector consistency (i.e., Jacobi can compute vectors that
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| 145 | // are negative of one another (.707,.707,0) and (-.707,-.707,0). This can
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| 146 | // reek havoc in hyperstreamline/other stuff. We will select the most
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| 147 | // positive eigenvector.
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| 148 | int numPos;
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| 149 | for (j=0; j<3; j++)
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| 150 | {
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| 151 | for (numPos=0, i=0; i<3; i++) if ( v[i][j] >= 0.0 ) numPos++;
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| 152 | if ( numPos < 2 ) for(i=0; i<3; i++) v[i][j] *= -1.0;
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| 153 | }
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| 154 |
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| 155 | return true;
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| 156 | }
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| 157 |
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| 158 | static
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| 159 | bool internal_jacobi4(real a[4][4], real w[4], real v[4][4])
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| 160 | {
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| 161 | int i, j, k, iq, ip;
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| 162 | real tresh, theta, tau, t, sm, s, h, g, c;
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| 163 | real b[4], z[4], tmp;
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| 164 |
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| 165 | // initialize
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| 166 | for (ip=0; ip<4; ip++)
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| 167 | {
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| 168 | for (iq=0; iq<4; iq++) v[ip][iq] = 0.0;
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| 169 | v[ip][ip] = 1.0;
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| 170 | }
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| 171 | for (ip=0; ip<4; ip++)
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| 172 | {
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| 173 | b[ip] = w[ip] = a[ip][ip];
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| 174 | z[ip] = 0.0;
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| 175 | }
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| 176 |
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| 177 | // begin rotation sequence
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| 178 | for (i=0; i<MAX_ROTATIONS; i++)
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| 179 | {
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| 180 | sm = 0.0;
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| 181 | for (ip=0; ip<3; ip++)
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| 182 | {
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| 183 | for (iq=ip+1; iq<4; iq++) sm += fabs(a[ip][iq]);
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| 184 | }
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| 185 | if (sm == 0.0) break;
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| 186 |
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| 187 | if (i < 4) tresh = 0.2*sm/(16);
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| 188 | else tresh = 0.0;
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| 189 |
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| 190 | for (ip=0; ip<3; ip++)
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| 191 | {
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| 192 | for (iq=ip+1; iq<4; iq++)
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| 193 | {
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| 194 | g = 100.0*fabs(a[ip][iq]);
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| 195 | if (i > 4 && (fabs(w[ip])+g) == fabs(w[ip])
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| 196 | && (fabs(w[iq])+g) == fabs(w[iq]))
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| 197 | {
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| 198 | a[ip][iq] = 0.0;
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| 199 | }
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| 200 | else if (fabs(a[ip][iq]) > tresh)
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| 201 | {
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| 202 | h = w[iq] - w[ip];
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| 203 | if ( (fabs(h)+g) == fabs(h)) t = (a[ip][iq]) / h;
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| 204 | else
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| 205 | {
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| 206 | theta = 0.5*h / (a[ip][iq]);
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| 207 | t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta));
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| 208 | if (theta < 0.0) t = -t;
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| 209 | }
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| 210 | c = 1.0 / sqrt(1+t*t);
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| 211 | s = t*c;
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| 212 | tau = s/(1.0+c);
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| 213 | h = t*a[ip][iq];
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| 214 | z[ip] -= h;
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| 215 | z[iq] += h;
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| 216 | w[ip] -= h;
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| 217 | w[iq] += h;
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| 218 | a[ip][iq]=0.0;
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| 219 | for (j=0;j<ip-1;j++)
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| 220 | {
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| 221 | ROT(a,j,ip,j,iq)
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| 222 | }
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| 223 | for (j=ip+1;j<iq-1;j++)
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| 224 | {
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| 225 | ROT(a,ip,j,j,iq)
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| 226 | }
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| 227 | for (j=iq+1; j<4; j++)
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| 228 | {
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| 229 | ROT(a,ip,j,iq,j)
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| 230 | }
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| 231 | for (j=0; j<4; j++)
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| 232 | {
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| 233 | ROT(v,j,ip,j,iq)
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| 234 | }
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| 235 | }
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| 236 | }
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| 237 | }
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| 238 |
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| 239 | for (ip=0; ip<4; ip++)
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| 240 | {
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| 241 | b[ip] += z[ip];
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| 242 | w[ip] = b[ip];
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| 243 | z[ip] = 0.0;
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| 244 | }
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| 245 | }
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| 246 |
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| 247 | if ( i >= MAX_ROTATIONS )
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| 248 | {
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| 249 | std::cerr << "WARNING -- jacobi() -- Error computing eigenvalues." << std::endl;
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| 250 | return false;
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| 251 | }
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| 252 |
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| 253 | // sort eigenfunctions
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| 254 | for (j=0; j<4; j++)
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| 255 | {
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| 256 | k = j;
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| 257 | tmp = w[k];
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| 258 | for (i=j; i<4; i++)
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| 259 | {
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| 260 | if (w[i] >= tmp)
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| 261 | {
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| 262 | k = i;
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| 263 | tmp = w[k];
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| 264 | }
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| 265 | }
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| 266 | if (k != j)
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| 267 | {
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| 268 | w[k] = w[j];
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| 269 | w[j] = tmp;
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| 270 | for (i=0; i<4; i++)
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| 271 | {
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| 272 | tmp = v[i][j];
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| 273 | v[i][j] = v[i][k];
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| 274 | v[i][k] = tmp;
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| 275 | }
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| 276 | }
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| 277 | }
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| 278 | // insure eigenvector consistency (i.e., Jacobi can compute vectors that
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| 279 | // are negative of one another (.707,.707,0) and (-.707,-.707,0). This can
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| 280 | // reek havoc in hyperstreamline/other stuff. We will select the most
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| 281 | // positive eigenvector.
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| 282 | int numPos;
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| 283 | for (j=0; j<4; j++)
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| 284 | {
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| 285 | for (numPos=0, i=0; i<4; i++) if ( v[i][j] >= 0.0 ) numPos++;
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| 286 | if ( numPos < 3 ) for(i=0; i<4; i++) v[i][j] *= -1.0;
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| 287 | }
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| 288 |
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| 289 | return true;
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| 290 | }
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| 291 |
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| 292 |
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| 293 | #undef ROT
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| 294 | #undef MAX_ROTATIONS
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| 295 |
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| 296 |
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| 297 | bool jacobi(const Mat3& m, Vec3& eig_vals, Vec3 eig_vecs[3])
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| 298 | {
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| 299 | real a[3][3], w[3], v[3][3];
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| 300 | int i,j;
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| 301 |
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| 302 | for(i=0;i<3;i++) for(j=0;j<3;j++) a[i][j] = m(i,j);
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| 303 |
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| 304 | bool result = internal_jacobi(a, w, v);
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| 305 | if( result )
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| 306 | {
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| 307 | for(i=0;i<3;i++) eig_vals[i] = w[i];
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| 308 |
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| 309 | for(i=0;i<3;i++) for(j=0;j<3;j++) eig_vecs[i][j] = v[j][i];
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| 310 | }
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| 311 |
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| 312 | return result;
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| 313 | }
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| 314 |
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| 315 | bool jacobi(const Mat4& m, Vec4& eig_vals, Vec4 eig_vecs[4])
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| 316 | {
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| 317 | real a[4][4], w[4], v[4][4];
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| 318 | int i,j;
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| 319 |
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| 320 | for(i=0;i<4;i++) for(j=0;j<4;j++) a[i][j] = m(i,j);
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| 321 |
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| 322 | bool result = internal_jacobi4(a, w, v);
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| 323 | if( result )
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| 324 | {
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| 325 | for(i=0;i<4;i++) eig_vals[i] = w[i];
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| 326 |
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| 327 | for(i=0;i<4;i++) for(j=0;j<4;j++) eig_vecs[i][j] = v[j][i];
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| 328 | }
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| 329 |
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| 330 | return result;
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| 331 | }
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