+ If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
+ diagonal and the eigenvector matrix V is orthogonal. That is,
+ the diagonal values of D are the eigenvalues, and
+ V*V' = I, where I is the identity matrix. The columns of V
+ represent the eigenvectors in the sense that A*V = V*D.
+
+
+ If A is not symmetric, then the eigenvalue matrix D is block diagonal
+ with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
+ a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex
+ eigenvalues look like
+
+
+ u + iv . . . . .
+ . u - iv . . . .
+ . . a + ib . . .
+ . . . a - ib . .
+ . . . . x .
+ . . . . . y
+
+ then D looks like
+
+
+ u v . . . .
+ -v u . . . .
+ . . a b . .
+ . . -b a . .
+ . . . . x .
+ . . . . . y
+
+ This keeps V a real matrix in both symmetric and non-symmetric
+ cases, and A*V = V*D.
+
+
+
+
+ The matrix V may be badly
+ conditioned, or even singular, so the validity of the equation
+ A = V*D*inverse(V) depends upon the condition number of V.
+
+
+ (Adapted from JAMA, a Java Matrix Library, developed by jointly
+ by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
+**/
+
+template
+class Eigenvalue
+{
+
+
+ /** Row and column dimension (square matrix). */
+ int n;
+
+ int issymmetric; /* boolean*/
+
+ /** Arrays for internal storage of eigenvalues. */
+
+ TNT::Array1D d; /* real part */
+ TNT::Array1D e; /* img part */
+
+ /** Array for internal storage of eigenvectors. */
+ TNT::Array2D V;
+
+ /** Array for internal storage of nonsymmetric Hessenberg form.
+ @serial internal storage of nonsymmetric Hessenberg form.
+ */
+ TNT::Array2D H;
+
+
+ /** Working storage for nonsymmetric algorithm.
+ @serial working storage for nonsymmetric algorithm.
+ */
+ TNT::Array1D ort;
+
+
+ // Symmetric Householder reduction to tridiagonal form.
+
+ void tred2() {
+
+ // This is derived from the Algol procedures tred2 by
+ // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
+ // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
+ // Fortran subroutine in EISPACK.
+
+ for (int j = 0; j < n; j++) {
+ d[j] = V[n-1][j];
+ }
+
+ // Householder reduction to tridiagonal form.
+
+ for (int i = n-1; i > 0; i--) {
+
+ // Scale to avoid under/overflow.
+
+ Real scale = 0.0;
+ Real h = 0.0;
+ for (int k = 0; k < i; k++) {
+ scale = scale + fabs(d[k]);
+ }
+ if (scale == 0.0) {
+ e[i] = d[i-1];
+ for (int j = 0; j < i; j++) {
+ d[j] = V[i-1][j];
+ V[i][j] = 0.0;
+ V[j][i] = 0.0;
+ }
+ } else {
+
+ // Generate Householder vector.
+
+ for (int k = 0; k < i; k++) {
+ d[k] /= scale;
+ h += d[k] * d[k];
+ }
+ Real f = d[i-1];
+ Real g = sqrt(h);
+ if (f > 0) {
+ g = -g;
+ }
+ e[i] = scale * g;
+ h = h - f * g;
+ d[i-1] = f - g;
+ for (int j = 0; j < i; j++) {
+ e[j] = 0.0;
+ }
+
+ // Apply similarity transformation to remaining columns.
+
+ for (int j = 0; j < i; j++) {
+ f = d[j];
+ V[j][i] = f;
+ g = e[j] + V[j][j] * f;
+ for (int k = j+1; k <= i-1; k++) {
+ g += V[k][j] * d[k];
+ e[k] += V[k][j] * f;
+ }
+ e[j] = g;
+ }
+ f = 0.0;
+ for (int j = 0; j < i; j++) {
+ e[j] /= h;
+ f += e[j] * d[j];
+ }
+ Real hh = f / (h + h);
+ for (int j = 0; j < i; j++) {
+ e[j] -= hh * d[j];
+ }
+ for (int j = 0; j < i; j++) {
+ f = d[j];
+ g = e[j];
+ for (int k = j; k <= i-1; k++) {
+ V[k][j] -= (f * e[k] + g * d[k]);
+ }
+ d[j] = V[i-1][j];
+ V[i][j] = 0.0;
+ }
+ }
+ d[i] = h;
+ }
+
+ // Accumulate transformations.
+
+ for (int i = 0; i < n-1; i++) {
+ V[n-1][i] = V[i][i];
+ V[i][i] = 1.0;
+ Real h = d[i+1];
+ if (h != 0.0) {
+ for (int k = 0; k <= i; k++) {
+ d[k] = V[k][i+1] / h;
+ }
+ for (int j = 0; j <= i; j++) {
+ Real g = 0.0;
+ for (int k = 0; k <= i; k++) {
+ g += V[k][i+1] * V[k][j];
+ }
+ for (int k = 0; k <= i; k++) {
+ V[k][j] -= g * d[k];
+ }
+ }
+ }
+ for (int k = 0; k <= i; k++) {
+ V[k][i+1] = 0.0;
+ }
+ }
+ for (int j = 0; j < n; j++) {
+ d[j] = V[n-1][j];
+ V[n-1][j] = 0.0;
+ }
+ V[n-1][n-1] = 1.0;
+ e[0] = 0.0;
+ }
+
+ // Symmetric tridiagonal QL algorithm.
+
+ void tql2 () {
+
+ // This is derived from the Algol procedures tql2, by
+ // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
+ // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
+ // Fortran subroutine in EISPACK.
+
+ for (int i = 1; i < n; i++) {
+ e[i-1] = e[i];
+ }
+ e[n-1] = 0.0;
+
+ Real f = 0.0;
+ Real tst1 = 0.0;
+ Real eps = pow(2.0,-52.0);
+ for (int l = 0; l < n; l++) {
+
+ // Find small subdiagonal element
+
+ tst1 = max(tst1,fabs(d[l]) + fabs(e[l]));
+ int m = l;
+
+ // Original while-loop from Java code
+ while (m < n) {
+ if (fabs(e[m]) <= eps*tst1) {
+ break;
+ }
+ m++;
+ }
+
+
+ // If m == l, d[l] is an eigenvalue,
+ // otherwise, iterate.
+
+ if (m > l) {
+ int iter = 0;
+ do {
+ iter = iter + 1; // (Could check iteration count here.)
+
+ // Compute implicit shift
+
+ Real g = d[l];
+ Real p = (d[l+1] - g) / (2.0 * e[l]);
+ Real r = hypot(p,1.0);
+ if (p < 0) {
+ r = -r;
+ }
+ d[l] = e[l] / (p + r);
+ d[l+1] = e[l] * (p + r);
+ Real dl1 = d[l+1];
+ Real h = g - d[l];
+ for (int i = l+2; i < n; i++) {
+ d[i] -= h;
+ }
+ f = f + h;
+
+ // Implicit QL transformation.
+
+ p = d[m];
+ Real c = 1.0;
+ Real c2 = c;
+ Real c3 = c;
+ Real el1 = e[l+1];
+ Real s = 0.0;
+ Real s2 = 0.0;
+ for (int i = m-1; i >= l; i--) {
+ c3 = c2;
+ c2 = c;
+ s2 = s;
+ g = c * e[i];
+ h = c * p;
+ r = hypot(p,e[i]);
+ e[i+1] = s * r;
+ s = e[i] / r;
+ c = p / r;
+ p = c * d[i] - s * g;
+ d[i+1] = h + s * (c * g + s * d[i]);
+
+ // Accumulate transformation.
+
+ for (int k = 0; k < n; k++) {
+ h = V[k][i+1];
+ V[k][i+1] = s * V[k][i] + c * h;
+ V[k][i] = c * V[k][i] - s * h;
+ }
+ }
+ p = -s * s2 * c3 * el1 * e[l] / dl1;
+ e[l] = s * p;
+ d[l] = c * p;
+
+ // Check for convergence.
+
+ } while (fabs(e[l]) > eps*tst1);
+ }
+ d[l] = d[l] + f;
+ e[l] = 0.0;
+ }
+
+ // Sort eigenvalues and corresponding vectors.
+
+ for (int i = 0; i < n-1; i++) {
+ int k = i;
+ Real p = d[i];
+ for (int j = i+1; j < n; j++) {
+ if (d[j] < p) {
+ k = j;
+ p = d[j];
+ }
+ }
+ if (k != i) {
+ d[k] = d[i];
+ d[i] = p;
+ for (int j = 0; j < n; j++) {
+ p = V[j][i];
+ V[j][i] = V[j][k];
+ V[j][k] = p;
+ }
+ }
+ }
+ }
+
+ // Nonsymmetric reduction to Hessenberg form.
+
+ void orthes () {
+
+ // This is derived from the Algol procedures orthes and ortran,
+ // by Martin and Wilkinson, Handbook for Auto. Comp.,
+ // Vol.ii-Linear Algebra, and the corresponding
+ // Fortran subroutines in EISPACK.
+
+ int low = 0;
+ int high = n-1;
+
+ for (int m = low+1; m <= high-1; m++) {
+
+ // Scale column.
+
+ Real scale = 0.0;
+ for (int i = m; i <= high; i++) {
+ scale = scale + fabs(H[i][m-1]);
+ }
+ if (scale != 0.0) {
+
+ // Compute Householder transformation.
+
+ Real h = 0.0;
+ for (int i = high; i >= m; i--) {
+ ort[i] = H[i][m-1]/scale;
+ h += ort[i] * ort[i];
+ }
+ Real g = sqrt(h);
+ if (ort[m] > 0) {
+ g = -g;
+ }
+ h = h - ort[m] * g;
+ ort[m] = ort[m] - g;
+
+ // Apply Householder similarity transformation
+ // H = (I-u*u'/h)*H*(I-u*u')/h)
+
+ for (int j = m; j < n; j++) {
+ Real f = 0.0;
+ for (int i = high; i >= m; i--) {
+ f += ort[i]*H[i][j];
+ }
+ f = f/h;
+ for (int i = m; i <= high; i++) {
+ H[i][j] -= f*ort[i];
+ }
+ }
+
+ for (int i = 0; i <= high; i++) {
+ Real f = 0.0;
+ for (int j = high; j >= m; j--) {
+ f += ort[j]*H[i][j];
+ }
+ f = f/h;
+ for (int j = m; j <= high; j++) {
+ H[i][j] -= f*ort[j];
+ }
+ }
+ ort[m] = scale*ort[m];
+ H[m][m-1] = scale*g;
+ }
+ }
+
+ // Accumulate transformations (Algol's ortran).
+
+ for (int i = 0; i < n; i++) {
+ for (int j = 0; j < n; j++) {
+ V[i][j] = (i == j ? 1.0 : 0.0);
+ }
+ }
+
+ for (int m = high-1; m >= low+1; m--) {
+ if (H[m][m-1] != 0.0) {
+ for (int i = m+1; i <= high; i++) {
+ ort[i] = H[i][m-1];
+ }
+ for (int j = m; j <= high; j++) {
+ Real g = 0.0;
+ for (int i = m; i <= high; i++) {
+ g += ort[i] * V[i][j];
+ }
+ // Double division avoids possible underflow
+ g = (g / ort[m]) / H[m][m-1];
+ for (int i = m; i <= high; i++) {
+ V[i][j] += g * ort[i];
+ }
+ }
+ }
+ }
+ }
+
+
+ // Complex scalar division.
+
+ Real cdivr, cdivi;
+ void cdiv(Real xr, Real xi, Real yr, Real yi) {
+ Real r,d;
+ if (fabs(yr) > fabs(yi)) {
+ r = yi/yr;
+ d = yr + r*yi;
+ cdivr = (xr + r*xi)/d;
+ cdivi = (xi - r*xr)/d;
+ } else {
+ r = yr/yi;
+ d = yi + r*yr;
+ cdivr = (r*xr + xi)/d;
+ cdivi = (r*xi - xr)/d;
+ }
+ }
+
+
+ // Nonsymmetric reduction from Hessenberg to real Schur form.
+
+ void hqr2 () {
+
+ // This is derived from the Algol procedure hqr2,
+ // by Martin and Wilkinson, Handbook for Auto. Comp.,
+ // Vol.ii-Linear Algebra, and the corresponding
+ // Fortran subroutine in EISPACK.
+
+ // Initialize
+
+ int nn = this->n;
+ int n = nn-1;
+ int low = 0;
+ int high = nn-1;
+ Real eps = pow(2.0,-52.0);
+ Real exshift = 0.0;
+ Real p=0,q=0,r=0,s=0,z=0,t,w,x,y;
+
+ // Store roots isolated by balanc and compute matrix norm
+
+ Real norm = 0.0;
+ for (int i = 0; i < nn; i++) {
+ if ((i < low) || (i > high)) {
+ d[i] = H[i][i];
+ e[i] = 0.0;
+ }
+ for (int j = max(i-1,0); j < nn; j++) {
+ norm = norm + fabs(H[i][j]);
+ }
+ }
+
+ // Outer loop over eigenvalue index
+
+ int iter = 0;
+ int totIter = 0;
+ while (n >= low) {
+
+ // NT limit no. of iterations
+ totIter++;
+ if(totIter>100) {
+ //if(totIter>15) std::cout<<"!!!!iter ABORT !!!!!!! "< low) {
+ s = fabs(H[l-1][l-1]) + fabs(H[l][l]);
+ if (s == 0.0) {
+ s = norm;
+ }
+ if (fabs(H[l][l-1]) < eps * s) {
+ break;
+ }
+ l--;
+ }
+
+ // Check for convergence
+ // One root found
+
+ if (l == n) {
+ H[n][n] = H[n][n] + exshift;
+ d[n] = H[n][n];
+ e[n] = 0.0;
+ n--;
+ iter = 0;
+
+ // Two roots found
+
+ } else if (l == n-1) {
+ w = H[n][n-1] * H[n-1][n];
+ p = (H[n-1][n-1] - H[n][n]) / 2.0;
+ q = p * p + w;
+ z = sqrt(fabs(q));
+ H[n][n] = H[n][n] + exshift;
+ H[n-1][n-1] = H[n-1][n-1] + exshift;
+ x = H[n][n];
+
+ // Real pair
+
+ if (q >= 0) {
+ if (p >= 0) {
+ z = p + z;
+ } else {
+ z = p - z;
+ }
+ d[n-1] = x + z;
+ d[n] = d[n-1];
+ if (z != 0.0) {
+ d[n] = x - w / z;
+ }
+ e[n-1] = 0.0;
+ e[n] = 0.0;
+ x = H[n][n-1];
+ s = fabs(x) + fabs(z);
+ p = x / s;
+ q = z / s;
+ r = sqrt(p * p+q * q);
+ p = p / r;
+ q = q / r;
+
+ // Row modification
+
+ for (int j = n-1; j < nn; j++) {
+ z = H[n-1][j];
+ H[n-1][j] = q * z + p * H[n][j];
+ H[n][j] = q * H[n][j] - p * z;
+ }
+
+ // Column modification
+
+ for (int i = 0; i <= n; i++) {
+ z = H[i][n-1];
+ H[i][n-1] = q * z + p * H[i][n];
+ H[i][n] = q * H[i][n] - p * z;
+ }
+
+ // Accumulate transformations
+
+ for (int i = low; i <= high; i++) {
+ z = V[i][n-1];
+ V[i][n-1] = q * z + p * V[i][n];
+ V[i][n] = q * V[i][n] - p * z;
+ }
+
+ // Complex pair
+
+ } else {
+ d[n-1] = x + p;
+ d[n] = x + p;
+ e[n-1] = z;
+ e[n] = -z;
+ }
+ n = n - 2;
+ iter = 0;
+
+ // No convergence yet
+
+ } else {
+
+ // Form shift
+
+ x = H[n][n];
+ y = 0.0;
+ w = 0.0;
+ if (l < n) {
+ y = H[n-1][n-1];
+ w = H[n][n-1] * H[n-1][n];
+ }
+
+ // Wilkinson's original ad hoc shift
+
+ if (iter == 10) {
+ exshift += x;
+ for (int i = low; i <= n; i++) {
+ H[i][i] -= x;
+ }
+ s = fabs(H[n][n-1]) + fabs(H[n-1][n-2]);
+ x = y = 0.75 * s;
+ w = -0.4375 * s * s;
+ }
+
+ // MATLAB's new ad hoc shift
+
+ if (iter == 30) {
+ s = (y - x) / 2.0;
+ s = s * s + w;
+ if (s > 0) {
+ s = sqrt(s);
+ if (y < x) {
+ s = -s;
+ }
+ s = x - w / ((y - x) / 2.0 + s);
+ for (int i = low; i <= n; i++) {
+ H[i][i] -= s;
+ }
+ exshift += s;
+ x = y = w = 0.964;
+ }
+ }
+
+ iter = iter + 1; // (Could check iteration count here.)
+
+ // Look for two consecutive small sub-diagonal elements
+
+ int m = n-2;
+ while (m >= l) {
+ z = H[m][m];
+ r = x - z;
+ s = y - z;
+ p = (r * s - w) / H[m+1][m] + H[m][m+1];
+ q = H[m+1][m+1] - z - r - s;
+ r = H[m+2][m+1];
+ s = fabs(p) + fabs(q) + fabs(r);
+ p = p / s;
+ q = q / s;
+ r = r / s;
+ if (m == l) {
+ break;
+ }
+ if (fabs(H[m][m-1]) * (fabs(q) + fabs(r)) <
+ eps * (fabs(p) * (fabs(H[m-1][m-1]) + fabs(z) +
+ fabs(H[m+1][m+1])))) {
+ break;
+ }
+ m--;
+ }
+
+ for (int i = m+2; i <= n; i++) {
+ H[i][i-2] = 0.0;
+ if (i > m+2) {
+ H[i][i-3] = 0.0;
+ }
+ }
+
+ // Double QR step involving rows l:n and columns m:n
+
+ for (int k = m; k <= n-1; k++) {
+ int notlast = (k != n-1);
+ if (k != m) {
+ p = H[k][k-1];
+ q = H[k+1][k-1];
+ r = (notlast ? H[k+2][k-1] : 0.0);
+ x = fabs(p) + fabs(q) + fabs(r);
+ if (x != 0.0) {
+ p = p / x;
+ q = q / x;
+ r = r / x;
+ }
+ }
+ if (x == 0.0) {
+ break;
+ }
+ s = sqrt(p * p + q * q + r * r);
+ if (p < 0) {
+ s = -s;
+ }
+ if (s != 0) {
+ if (k != m) {
+ H[k][k-1] = -s * x;
+ } else if (l != m) {
+ H[k][k-1] = -H[k][k-1];
+ }
+ p = p + s;
+ x = p / s;
+ y = q / s;
+ z = r / s;
+ q = q / p;
+ r = r / p;
+
+ // Row modification
+
+ for (int j = k; j < nn; j++) {
+ p = H[k][j] + q * H[k+1][j];
+ if (notlast) {
+ p = p + r * H[k+2][j];
+ H[k+2][j] = H[k+2][j] - p * z;
+ }
+ H[k][j] = H[k][j] - p * x;
+ H[k+1][j] = H[k+1][j] - p * y;
+ }
+
+ // Column modification
+
+ for (int i = 0; i <= min(n,k+3); i++) {
+ p = x * H[i][k] + y * H[i][k+1];
+ if (notlast) {
+ p = p + z * H[i][k+2];
+ H[i][k+2] = H[i][k+2] - p * r;
+ }
+ H[i][k] = H[i][k] - p;
+ H[i][k+1] = H[i][k+1] - p * q;
+ }
+
+ // Accumulate transformations
+
+ for (int i = low; i <= high; i++) {
+ p = x * V[i][k] + y * V[i][k+1];
+ if (notlast) {
+ p = p + z * V[i][k+2];
+ V[i][k+2] = V[i][k+2] - p * r;
+ }
+ V[i][k] = V[i][k] - p;
+ V[i][k+1] = V[i][k+1] - p * q;
+ }
+ } // (s != 0)
+ } // k loop
+ } // check convergence
+ } // while (n >= low)
+ //if(totIter>15) std::cout<<"!!!!iter "<= 0; n--) {
+ p = d[n];
+ q = e[n];
+
+ // Real vector
+
+ if (q == 0) {
+ int l = n;
+ H[n][n] = 1.0;
+ for (int i = n-1; i >= 0; i--) {
+ w = H[i][i] - p;
+ r = 0.0;
+ for (int j = l; j <= n; j++) {
+ r = r + H[i][j] * H[j][n];
+ }
+ if (e[i] < 0.0) {
+ z = w;
+ s = r;
+ } else {
+ l = i;
+ if (e[i] == 0.0) {
+ if (w != 0.0) {
+ H[i][n] = -r / w;
+ } else {
+ H[i][n] = -r / (eps * norm);
+ }
+
+ // Solve real equations
+
+ } else {
+ x = H[i][i+1];
+ y = H[i+1][i];
+ q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
+ t = (x * s - z * r) / q;
+ H[i][n] = t;
+ if (fabs(x) > fabs(z)) {
+ H[i+1][n] = (-r - w * t) / x;
+ } else {
+ H[i+1][n] = (-s - y * t) / z;
+ }
+ }
+
+ // Overflow control
+
+ t = fabs(H[i][n]);
+ if ((eps * t) * t > 1) {
+ for (int j = i; j <= n; j++) {
+ H[j][n] = H[j][n] / t;
+ }
+ }
+ }
+ }
+
+ // Complex vector
+
+ } else if (q < 0) {
+ int l = n-1;
+
+ // Last vector component imaginary so matrix is triangular
+
+ if (fabs(H[n][n-1]) > fabs(H[n-1][n])) {
+ H[n-1][n-1] = q / H[n][n-1];
+ H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
+ } else {
+ cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
+ H[n-1][n-1] = cdivr;
+ H[n-1][n] = cdivi;
+ }
+ H[n][n-1] = 0.0;
+ H[n][n] = 1.0;
+ for (int i = n-2; i >= 0; i--) {
+ Real ra,sa,vr,vi;
+ ra = 0.0;
+ sa = 0.0;
+ for (int j = l; j <= n; j++) {
+ ra = ra + H[i][j] * H[j][n-1];
+ sa = sa + H[i][j] * H[j][n];
+ }
+ w = H[i][i] - p;
+
+ if (e[i] < 0.0) {
+ z = w;
+ r = ra;
+ s = sa;
+ } else {
+ l = i;
+ if (e[i] == 0) {
+ cdiv(-ra,-sa,w,q);
+ H[i][n-1] = cdivr;
+ H[i][n] = cdivi;
+ } else {
+
+ // Solve complex equations
+
+ x = H[i][i+1];
+ y = H[i+1][i];
+ vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
+ vi = (d[i] - p) * 2.0 * q;
+ if ((vr == 0.0) && (vi == 0.0)) {
+ vr = eps * norm * (fabs(w) + fabs(q) +
+ fabs(x) + fabs(y) + fabs(z));
+ }
+ cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
+ H[i][n-1] = cdivr;
+ H[i][n] = cdivi;
+ if (fabs(x) > (fabs(z) + fabs(q))) {
+ H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
+ H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
+ } else {
+ cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);
+ H[i+1][n-1] = cdivr;
+ H[i+1][n] = cdivi;
+ }
+ }
+
+ // Overflow control
+
+ t = max(fabs(H[i][n-1]),fabs(H[i][n]));
+ if ((eps * t) * t > 1) {
+ for (int j = i; j <= n; j++) {
+ H[j][n-1] = H[j][n-1] / t;
+ H[j][n] = H[j][n] / t;
+ }
+ }
+ }
+ }
+ }
+ }
+
+ // Vectors of isolated roots
+
+ for (int i = 0; i < nn; i++) {
+ if (i < low || i > high) {
+ for (int j = i; j < nn; j++) {
+ V[i][j] = H[i][j];
+ }
+ }
+ }
+
+ // Back transformation to get eigenvectors of original matrix
+
+ for (int j = nn-1; j >= low; j--) {
+ for (int i = low; i <= high; i++) {
+ z = 0.0;
+ for (int k = low; k <= min(j,high); k++) {
+ z = z + V[i][k] * H[k][j];
+ }
+ V[i][j] = z;
+ }
+ }
+ }
+
+public:
+
+
+ /** Check for symmetry, then construct the eigenvalue decomposition
+ @param A Square real (non-complex) matrix
+ */
+
+ Eigenvalue(const TNT::Array2D &A) {
+ n = A.dim2();
+ V = Array2D(n,n);
+ d = Array1D(n);
+ e = Array1D(n);
+
+ issymmetric = 1;
+ for (int j = 0; (j < n) && issymmetric; j++) {
+ for (int i = 0; (i < n) && issymmetric; i++) {
+ issymmetric = (A[i][j] == A[j][i]);
+ }
+ }
+
+ if (issymmetric) {
+ for (int i = 0; i < n; i++) {
+ for (int j = 0; j < n; j++) {
+ V[i][j] = A[i][j];
+ }
+ }
+
+ // Tridiagonalize.
+ tred2();
+
+ // Diagonalize.
+ tql2();
+
+ } else {
+ H = TNT::Array2D(n,n);
+ ort = TNT::Array1D(n);
+
+ for (int j = 0; j < n; j++) {
+ for (int i = 0; i < n; i++) {
+ H[i][j] = A[i][j];
+ }
+ }
+
+ // Reduce to Hessenberg form.
+ orthes();
+
+ // Reduce Hessenberg to real Schur form.
+ hqr2();
+ }
+ }
+
+
+ /** Return the eigenvector matrix
+ @return V
+ */
+
+ void getV (TNT::Array2D &V_) {
+ V_ = V;
+ return;
+ }
+
+ /** Return the real parts of the eigenvalues
+ @return real(diag(D))
+ */
+
+ void getRealEigenvalues (TNT::Array1D &d_) {
+ d_ = d;
+ return ;
+ }
+
+ /** Return the imaginary parts of the eigenvalues
+ in parameter e_.
+
+ @pararm e_: new matrix with imaginary parts of the eigenvalues.
+ */
+ void getImagEigenvalues (TNT::Array1D &e_) {
+ e_ = e;
+ return;
+ }
+
+
+/**
+ Computes the block diagonal eigenvalue matrix.
+ If the original matrix A is not symmetric, then the eigenvalue
+ matrix D is block diagonal with the real eigenvalues in 1-by-1
+ blocks and any complex eigenvalues,
+ a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex
+ eigenvalues look like
+
+
+ u + iv . . . . .
+ . u - iv . . . .
+ . . a + ib . . .
+ . . . a - ib . .
+ . . . . x .
+ . . . . . y
+
+ then D looks like
+
+
+ u v . . . .
+ -v u . . . .
+ . . a b . .
+ . . -b a . .
+ . . . . x .
+ . . . . . y
+
+ This keeps V a real matrix in both symmetric and non-symmetric
+ cases, and A*V = V*D.
+
+ @param D: upon return, the matrix is filled with the block diagonal
+ eigenvalue matrix.
+
+*/
+ void getD (TNT::Array2D &D) {
+ D = Array2D