?? eigenvaluedecomposition.java
字號:
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 = Math.abs(x) + Math.abs(z);
p = x / s;
q = z / s;
r = Math.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 = Math.abs(H[n][n-1]) + Math.abs(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 = Math.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 = Math.abs(p) + Math.abs(q) + Math.abs(r);
p = p / s;
q = q / s;
r = r / s;
if (m == l) {
break;
}
if (Math.abs(H[m][m-1]) * (Math.abs(q) + Math.abs(r)) <
eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +
Math.abs(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++) {
boolean 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 = Math.abs(p) + Math.abs(q) + Math.abs(r);
if (x != 0.0) {
p = p / x;
q = q / x;
r = r / x;
}
}
if (x == 0.0) {
break;
}
s = Math.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 <= Math.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)
// Backsubstitute to find vectors of upper triangular form
if (norm == 0.0) {
return;
}
for (n = nn-1; n >= 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 (Math.abs(x) > Math.abs(z)) {
H[i+1][n] = (-r - w * t) / x;
} else {
H[i+1][n] = (-s - y * t) / z;
}
}
// Overflow control
t = Math.abs(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 (Math.abs(H[n][n-1]) > Math.abs(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--) {
double 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 * (Math.abs(w) + Math.abs(q) +
Math.abs(x) + Math.abs(y) + Math.abs(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 (Math.abs(x) > (Math.abs(z) + Math.abs(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 = Math.max(Math.abs(H[i][n-1]),Math.abs(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 <= Math.min(j,high); k++) {
z = z + V[i][k] * H[k][j];
}
V[i][j] = z;
}
}
}
/* ------------------------
Constructor
* ------------------------ */
/** Check for symmetry, then construct the eigenvalue decomposition
@param A Square matrix
@return Structure to access D and V.
*/
public EigenvalueDecomposition (Matrix Arg) {
double[][] A = Arg.getArray();
n = Arg.getColumnDimension();
V = new double[n][n];
d = new double[n];
e = new double[n];
issymmetric = true;
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 = new double[n][n];
ort = new double[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();
}
}
/* ------------------------
Public Methods
* ------------------------ */
/** Return the eigenvector matrix
@return V
*/
public Matrix getV () {
return new Matrix(V,n,n);
}
/** Return the real parts of the eigenvalues
@return real(diag(D))
*/
public double[] getRealEigenvalues () {
return d;
}
/** Return the imaginary parts of the eigenvalues
@return imag(diag(D))
*/
public double[] getImagEigenvalues () {
return e;
}
/** Return the block diagonal eigenvalue matrix
@return D
*/
public Matrix getD () {
Matrix X = new Matrix(n,n);
double[][] D = X.getArray();
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
D[i][j] = 0.0;
}
D[i][i] = d[i];
if (e[i] > 0) {
D[i][i+1] = e[i];
} else if (e[i] < 0) {
D[i][i-1] = e[i];
}
}
return X;
}
}
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