2370 lines
60 KiB
C++
2370 lines
60 KiB
C++
#ifndef TNT_LINALG_H_
|
|
#define TNT_LINALG_H_
|
|
|
|
#include <algorithm>
|
|
// for min(), max() below
|
|
|
|
#include <cmath>
|
|
// for abs(), sqrt() below
|
|
|
|
|
|
namespace TNT {
|
|
namespace Linear_Algebra {
|
|
|
|
using namespace std;
|
|
|
|
/**
|
|
<P>
|
|
For a symmetric, positive definite matrix A, this function
|
|
computes the Cholesky factorization, i.e. it computes a lower
|
|
triangular matrix L such that A = L*L'.
|
|
If the matrix is not symmetric or positive definite, the function
|
|
computes only a partial decomposition. This can be tested with
|
|
the is_spd() flag.
|
|
|
|
<p>Typical usage looks like:
|
|
<pre>
|
|
Matrix<double> A(n,n);
|
|
Matrix<double> L;
|
|
|
|
...
|
|
|
|
Cholesky<double> chol(A);
|
|
|
|
if (chol.is_spd())
|
|
L = chol.getL();
|
|
|
|
else
|
|
cout << "factorization was not complete.\n";
|
|
|
|
</pre>
|
|
|
|
|
|
<p>
|
|
(Adapted from JAMA, a Java Matrix Library, developed by jointly
|
|
by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
|
|
|
|
*/
|
|
|
|
template <class Real>
|
|
class Cholesky
|
|
{
|
|
Matrix<Real> L_; // lower triangular factor
|
|
int isspd; // 1 if matrix to be factored was SPD
|
|
|
|
public:
|
|
|
|
Cholesky();
|
|
Cholesky(const Matrix<Real> &A);
|
|
Matrix<Real> getL() const;
|
|
Vector<Real> solve(const Vector<Real> &B);
|
|
Matrix<Real> solve(const Matrix<Real> &B);
|
|
int is_spd() const;
|
|
|
|
};
|
|
|
|
template <class Real>
|
|
Cholesky<Real>::Cholesky() : L_(Real(0.0)), isspd(0) {}
|
|
|
|
/**
|
|
@return 1, if original matrix to be factored was symmetric
|
|
positive-definite (SPD).
|
|
*/
|
|
template <class Real>
|
|
int Cholesky<Real>::is_spd() const
|
|
{
|
|
return isspd;
|
|
}
|
|
|
|
/**
|
|
@return the lower triangular factor, L, such that L*L'=A.
|
|
*/
|
|
template <class Real>
|
|
Matrix<Real> Cholesky<Real>::getL() const
|
|
{
|
|
return L_;
|
|
}
|
|
|
|
/**
|
|
Constructs a lower triangular matrix L, such that L*L'= A.
|
|
If A is not symmetric positive-definite (SPD), only a
|
|
partial factorization is performed. If is_spd()
|
|
evalutate true (1) then the factorizaiton was successful.
|
|
*/
|
|
template <class Real>
|
|
Cholesky<Real>::Cholesky(const Matrix<Real> &A)
|
|
{
|
|
|
|
|
|
int m = A.num_rows();
|
|
int n = A.num_cols();
|
|
|
|
isspd = (m == n);
|
|
|
|
if (m != n)
|
|
{
|
|
L_ = Matrix<Real>(Real(0.0));
|
|
return;
|
|
}
|
|
|
|
L_ = Matrix<Real>(n,n);
|
|
|
|
|
|
// Main loop.
|
|
for (int j = 0; j < n; j++)
|
|
{
|
|
Real d = Real(0.0);
|
|
for (int k = 0; k < j; k++)
|
|
{
|
|
Real s = Real(0.0);
|
|
for (int i = 0; i < k; i++)
|
|
{
|
|
s += L_[k][i]*L_[j][i];
|
|
}
|
|
L_[j][k] = s = (A[j][k] - s)/L_[k][k];
|
|
d = d + s*s;
|
|
isspd = isspd && (A[k][j] == A[j][k]);
|
|
}
|
|
d = A[j][j] - d;
|
|
isspd = isspd && (d > Real(0.0));
|
|
L_[j][j] = sqrt(d > Real(0.0) ? d : Real(0.0));
|
|
for (int k = j+1; k < n; k++)
|
|
{
|
|
L_[j][k] = Real(0.0);
|
|
}
|
|
}
|
|
}
|
|
|
|
/**
|
|
|
|
Solve a linear system A*x = b, using the previously computed
|
|
cholesky factorization of A: L*L'.
|
|
|
|
@param b matrix with as many rows as A and any number of columns.
|
|
@return x so that L*L'*x = b. If b is nonconformat, or if A
|
|
was not symmetric posidtive definite, a null (Real(0.0))
|
|
array is returned.
|
|
*/
|
|
template <class Real>
|
|
Vector<Real> Cholesky<Real>::solve(const Vector<Real> &b)
|
|
{
|
|
int n = L_.num_rows();
|
|
if (b.dim() != n)
|
|
return Vector<Real>();
|
|
|
|
|
|
Vector<Real> x = b;
|
|
|
|
|
|
// Solve L*y = b;
|
|
for (int k = 0; k < n; k++)
|
|
{
|
|
for (int i = 0; i < k; i++)
|
|
x[k] -= x[i]*L_[k][i];
|
|
x[k] /= L_[k][k];
|
|
|
|
}
|
|
|
|
// Solve L'*X = Y;
|
|
for (int k = n-1; k >= 0; k--)
|
|
{
|
|
for (int i = k+1; i < n; i++)
|
|
x[k] -= x[i]*L_[i][k];
|
|
x[k] /= L_[k][k];
|
|
}
|
|
|
|
return x;
|
|
}
|
|
|
|
|
|
/**
|
|
|
|
Solve a linear system A*X = B, using the previously computed
|
|
cholesky factorization of A: L*L'.
|
|
|
|
@param B A Matrix with as many rows as A and any number of columns.
|
|
@return X so that L*L'*X = B. If B is nonconformat, or if A
|
|
was not symmetric posidtive definite, a null (Real(0.0))
|
|
array is returned.
|
|
*/
|
|
template <class Real>
|
|
Matrix<Real> Cholesky<Real>::solve(const Matrix<Real> &B)
|
|
{
|
|
int n = L_.num_rows();
|
|
if (B.num_rows() != n)
|
|
return Matrix<Real>();
|
|
|
|
|
|
Matrix<Real> X = B;
|
|
int nx = B.num_cols();
|
|
|
|
// Solve L*y = b;
|
|
for (int j=0; j< nx; j++)
|
|
{
|
|
for (int k = 0; k < n; k++)
|
|
{
|
|
for (int i = 0; i < k; i++)
|
|
X[k][j] -= X[i][j]*L_[k][i];
|
|
X[k][j] /= L_[k][k];
|
|
}
|
|
}
|
|
|
|
// Solve L'*X = Y;
|
|
for (int j=0; j<nx; j++)
|
|
{
|
|
for (int k = n-1; k >= 0; k--)
|
|
{
|
|
for (int i = k+1; i < n; i++)
|
|
X[k][j] -= X[i][j]*L_[i][k];
|
|
X[k][j] /= L_[k][k];
|
|
}
|
|
}
|
|
|
|
|
|
|
|
return X;
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
Computes eigenvalues and eigenvectors of a real (non-complex)
|
|
matrix.
|
|
<P>
|
|
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.
|
|
|
|
<P>
|
|
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
|
|
<pre>
|
|
|
|
u + iv . . . . .
|
|
. u - iv . . . .
|
|
. . a + ib . . .
|
|
. . . a - ib . .
|
|
. . . . x .
|
|
. . . . . y
|
|
</pre>
|
|
then D looks like
|
|
<pre>
|
|
|
|
u v . . . .
|
|
-v u . . . .
|
|
. . a b . .
|
|
. . -b a . .
|
|
. . . . x .
|
|
. . . . . y
|
|
</pre>
|
|
This keeps V a real matrix in both symmetric and non-symmetric
|
|
cases, and A*V = V*D.
|
|
|
|
|
|
|
|
<p>
|
|
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.
|
|
|
|
<p>
|
|
(Adapted from JAMA, a Java Matrix Library, developed by jointly
|
|
by the Mathworks and NIST (see http://math.nist.gov/javanumerics/jama),
|
|
which in turn, were based on original EISPACK routines.
|
|
**/
|
|
|
|
template <class Real>
|
|
class Eigenvalue
|
|
{
|
|
|
|
|
|
/** Row and column dimension (square matrix). */
|
|
int n;
|
|
|
|
int issymmetric; /* boolean*/
|
|
|
|
/** Arrays for internal storage of eigenvalues. */
|
|
|
|
Vector<Real> d; /* real part */
|
|
Vector<Real> e; /* img part */
|
|
|
|
/** Array for internal storage of eigenvectors. */
|
|
Matrix<Real> V;
|
|
|
|
/* Array for internal storage of nonsymmetric Hessenberg form.
|
|
@serial internal storage of nonsymmetric Hessenberg form.
|
|
*/
|
|
Matrix<Real> H;
|
|
|
|
|
|
/* Working storage for nonsymmetric algorithm.
|
|
@serial working storage for nonsymmetric algorithm.
|
|
*/
|
|
Vector<Real> 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 = Real(0.0);
|
|
Real h = Real(0.0);
|
|
for (int k = 0; k < i; k++) {
|
|
scale = scale + abs(d[k]);
|
|
}
|
|
if (scale == Real(0.0)) {
|
|
e[i] = d[i-1];
|
|
for (int j = 0; j < i; j++) {
|
|
d[j] = V[i-1][j];
|
|
V[i][j] = Real(0.0);
|
|
V[j][i] = Real(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] = Real(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 = Real(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] = Real(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] = Real(1.0);
|
|
Real h = d[i+1];
|
|
if (h != Real(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 = Real(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] = Real(0.0);
|
|
}
|
|
}
|
|
for (int j = 0; j < n; j++) {
|
|
d[j] = V[n-1][j];
|
|
V[n-1][j] = Real(0.0);
|
|
}
|
|
V[n-1][n-1] = Real(1.0);
|
|
e[0] = Real(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] = Real(0.0);
|
|
|
|
Real f = Real(0.0);
|
|
Real tst1 = Real(0.0);
|
|
Real eps = pow(2.0,-52.0);
|
|
for (int l = 0; l < n; l++) {
|
|
|
|
// Find small subdiagonal element
|
|
|
|
tst1 = max(tst1,abs(d[l]) + abs(e[l]));
|
|
int m = l;
|
|
|
|
// Original while-loop from Java code
|
|
while (m < n) {
|
|
if (abs(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, static_cast<Real>(Real(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 = Real(1.0);
|
|
Real c2 = c;
|
|
Real c3 = c;
|
|
Real el1 = e[l+1];
|
|
Real s = Real(0.0);
|
|
Real s2 = Real(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 (abs(e[l]) > eps*tst1);
|
|
}
|
|
d[l] = d[l] + f;
|
|
e[l] = Real(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 = Real(0.0);
|
|
for (int i = m; i <= high; i++) {
|
|
scale = scale + abs(H[i][m-1]);
|
|
}
|
|
if (scale != Real(0.0)) {
|
|
|
|
// Compute Householder transformation.
|
|
|
|
Real h = Real(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 = Real(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 = Real(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 ? Real(1.0) : Real(0.0));
|
|
}
|
|
}
|
|
|
|
for (int m = high-1; m >= low+1; m--) {
|
|
if (H[m][m-1] != Real(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 = Real(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 (abs(yr) > abs(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 = Real(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 = Real(0.0);
|
|
for (int i = 0; i < nn; i++) {
|
|
if ((i < low) || (i > high)) {
|
|
d[i] = H[i][i];
|
|
e[i] = Real(0.0);
|
|
}
|
|
for (int j = max(i-1,0); j < nn; j++) {
|
|
norm = norm + abs(H[i][j]);
|
|
}
|
|
}
|
|
|
|
// Outer loop over eigenvalue index
|
|
|
|
int iter = 0;
|
|
while (n >= low) {
|
|
|
|
// Look for single small sub-diagonal element
|
|
|
|
int l = n;
|
|
while (l > low) {
|
|
s = abs(H[l-1][l-1]) + abs(H[l][l]);
|
|
if (s == Real(0.0)) {
|
|
s = norm;
|
|
}
|
|
if (abs(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] = Real(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(abs(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 != Real(0.0)) {
|
|
d[n] = x - w / z;
|
|
}
|
|
e[n-1] = Real(0.0);
|
|
e[n] = Real(0.0);
|
|
x = H[n][n-1];
|
|
s = abs(x) + abs(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 = Real(0.0);
|
|
w = Real(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 = abs(H[n][n-1]) + 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 = 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 = abs(p) + abs(q) + abs(r);
|
|
p = p / s;
|
|
q = q / s;
|
|
r = r / s;
|
|
if (m == l) {
|
|
break;
|
|
}
|
|
if (abs(H[m][m-1]) * (abs(q) + abs(r)) <
|
|
eps * (abs(p) * (abs(H[m-1][m-1]) + abs(z) +
|
|
abs(H[m+1][m+1])))) {
|
|
break;
|
|
}
|
|
m--;
|
|
}
|
|
|
|
for (int i = m+2; i <= n; i++) {
|
|
H[i][i-2] = Real(0.0);
|
|
if (i > m+2) {
|
|
H[i][i-3] = Real(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] : Real(0.0));
|
|
x = abs(p) + abs(q) + abs(r);
|
|
if (x != Real(0.0)) {
|
|
p = p / x;
|
|
q = q / x;
|
|
r = r / x;
|
|
}
|
|
}
|
|
if (x == Real(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)
|
|
|
|
// Backsubstitute to find vectors of upper triangular form
|
|
|
|
if (norm == Real(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] = Real(1.0);
|
|
for (int i = n-1; i >= 0; i--) {
|
|
w = H[i][i] - p;
|
|
r = Real(0.0);
|
|
for (int j = l; j <= n; j++) {
|
|
r = r + H[i][j] * H[j][n];
|
|
}
|
|
if (e[i] < Real(0.0)) {
|
|
z = w;
|
|
s = r;
|
|
} else {
|
|
l = i;
|
|
if (e[i] == Real(0.0)) {
|
|
if (w != Real(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 (abs(x) > abs(z)) {
|
|
H[i+1][n] = (-r - w * t) / x;
|
|
} else {
|
|
H[i+1][n] = (-s - y * t) / z;
|
|
}
|
|
}
|
|
|
|
// Overflow control
|
|
|
|
t = 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 (abs(H[n][n-1]) > 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(Real(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] = Real(0.0);
|
|
H[n][n] = Real(1.0);
|
|
for (int i = n-2; i >= 0; i--) {
|
|
Real ra,sa,vr,vi;
|
|
ra = Real(0.0);
|
|
sa = Real(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] < Real(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 == Real(0.0)) && (vi == Real(0.0))) {
|
|
vr = eps * norm * (abs(w) + abs(q) +
|
|
abs(x) + abs(y) + 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 (abs(x) > (abs(z) + 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 = max(abs(H[i][n-1]),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 = Real(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 Matrix<Real> &A) {
|
|
n = A.num_cols();
|
|
V = Matrix<Real>(n,n);
|
|
d = Vector<Real>(n);
|
|
e = Vector<Real>(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 = Matrix<Real>(n,n);
|
|
ort = Vector<Real>(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 (Matrix<Real> &V_) {
|
|
V_ = V;
|
|
return;
|
|
}
|
|
|
|
/** Return the real parts of the eigenvalues
|
|
@return real(diag(D))
|
|
*/
|
|
|
|
void getRealEigenvalues (Vector<Real> &d_) {
|
|
d_ = d;
|
|
return ;
|
|
}
|
|
|
|
/** Return the imaginary parts of the eigenvalues
|
|
in parameter e_.
|
|
|
|
@param e_: new matrix with imaginary parts of the eigenvalues.
|
|
*/
|
|
void getImagEigenvalues (Vector<Real> &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
|
|
<pre>
|
|
|
|
u + iv . . . . .
|
|
. u - iv . . . .
|
|
. . a + ib . . .
|
|
. . . a - ib . .
|
|
. . . . x .
|
|
. . . . . y
|
|
</pre>
|
|
then D looks like
|
|
<pre>
|
|
|
|
u v . . . .
|
|
-v u . . . .
|
|
. . a b . .
|
|
. . -b a . .
|
|
. . . . x .
|
|
. . . . . y
|
|
</pre>
|
|
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 (Matrix<Real> &D) {
|
|
D = Matrix<Real>(n,n);
|
|
for (int i = 0; i < n; i++) {
|
|
for (int j = 0; j < n; j++) {
|
|
D[i][j] = Real(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];
|
|
}
|
|
}
|
|
}
|
|
};
|
|
|
|
|
|
/** LU Decomposition.
|
|
<P>
|
|
For an m-by-n matrix A with m >= n, the LU decomposition is an m-by-n
|
|
unit lower triangular matrix L, an n-by-n upper triangular matrix U,
|
|
and a permutation vector piv of length m so that A(piv,:) = L*U.
|
|
If m < n, then L is m-by-m and U is m-by-n.
|
|
<P>
|
|
The LU decompostion with pivoting always exists, even if the matrix is
|
|
singular, so the constructor will never fail. The primary use of the
|
|
LU decomposition is in the solution of square systems of simultaneous
|
|
linear equations. This will fail if isNonsingular() returns false.
|
|
*/
|
|
template <class Real>
|
|
class LU
|
|
{
|
|
|
|
private:
|
|
|
|
/* Array for internal storage of decomposition. */
|
|
Matrix<Real> LU_;
|
|
int m, n, pivsign;
|
|
Vector<int> piv;
|
|
|
|
|
|
Matrix<Real> permute_copy( const Matrix<Real> &A,
|
|
const Vector<int> &piv, int j0, int j1)
|
|
{
|
|
int piv_length = piv.dim();
|
|
|
|
Matrix<Real> X(piv_length, j1-j0+1);
|
|
|
|
|
|
for (int i = 0; i < piv_length; i++)
|
|
for (int j = j0; j <= j1; j++)
|
|
X[i][j-j0] = A[piv[i]][j];
|
|
|
|
return X;
|
|
}
|
|
|
|
Vector<Real> permute_copy( const Vector<Real> &A,
|
|
const Vector<int> &piv)
|
|
{
|
|
int piv_length = piv.dim();
|
|
if (piv_length != A.dim())
|
|
return Vector<Real>();
|
|
|
|
Vector<Real> x(piv_length);
|
|
|
|
|
|
for (int i = 0; i < piv_length; i++)
|
|
x[i] = A[piv[i]];
|
|
|
|
return x;
|
|
}
|
|
|
|
|
|
public :
|
|
|
|
/** LU Decomposition
|
|
@param A Rectangular matrix
|
|
@return LU Decomposition object to access L, U and piv.
|
|
*/
|
|
|
|
LU (const Matrix<Real> &A) : LU_(A), m(A.num_rows()), n(A.num_cols()),
|
|
piv(A.num_rows())
|
|
|
|
{
|
|
|
|
// Use a "left-looking", dot-product, Crout/Doolittle algorithm.
|
|
|
|
|
|
for (int i = 0; i < m; i++) {
|
|
piv[i] = i;
|
|
}
|
|
pivsign = 1;
|
|
Real *LUrowi = 0;;
|
|
Vector<Real> LUcolj(m);
|
|
|
|
// Outer loop.
|
|
|
|
for (int j = 0; j < n; j++) {
|
|
|
|
// Make a copy of the j-th column to localize references.
|
|
|
|
for (int i = 0; i < m; i++) {
|
|
LUcolj[i] = LU_[i][j];
|
|
}
|
|
|
|
// Apply previous transformations.
|
|
|
|
for (int i = 0; i < m; i++) {
|
|
LUrowi = LU_[i];
|
|
|
|
// Most of the time is spent in the following dot product.
|
|
|
|
int kmax = min(i,j);
|
|
double s = Real(0.0);
|
|
for (int k = 0; k < kmax; k++) {
|
|
s += LUrowi[k]*LUcolj[k];
|
|
}
|
|
|
|
LUrowi[j] = LUcolj[i] -= s;
|
|
}
|
|
|
|
// Find pivot and exchange if necessary.
|
|
|
|
int p = j;
|
|
for (int i = j+1; i < m; i++) {
|
|
if (abs(LUcolj[i]) > abs(LUcolj[p])) {
|
|
p = i;
|
|
}
|
|
}
|
|
if (p != j) {
|
|
int k=0;
|
|
for (k = 0; k < n; k++) {
|
|
double t = LU_[p][k];
|
|
LU_[p][k] = LU_[j][k];
|
|
LU_[j][k] = t;
|
|
}
|
|
k = piv[p];
|
|
piv[p] = piv[j];
|
|
piv[j] = k;
|
|
pivsign = -pivsign;
|
|
}
|
|
|
|
// Compute multipliers.
|
|
|
|
if ((j < m) && (LU_[j][j] != Real(0.0))) {
|
|
for (int i = j+1; i < m; i++) {
|
|
LU_[i][j] /= LU_[j][j];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
|
|
/** Is the matrix nonsingular?
|
|
@return 1 (true) if upper triangular factor U (and hence A)
|
|
is nonsingular, 0 otherwise.
|
|
*/
|
|
|
|
int isNonsingular () {
|
|
for (int j = 0; j < n; j++) {
|
|
if (LU_[j][j] == 0)
|
|
return 0;
|
|
}
|
|
return 1;
|
|
}
|
|
|
|
/** Return lower triangular factor
|
|
@return L
|
|
*/
|
|
|
|
Matrix<Real> getL () {
|
|
Matrix<Real> L_(m,n);
|
|
for (int i = 0; i < m; i++) {
|
|
for (int j = 0; j < n; j++) {
|
|
if (i > j) {
|
|
L_[i][j] = LU_[i][j];
|
|
} else if (i == j) {
|
|
L_[i][j] = Real(1.0);
|
|
} else {
|
|
L_[i][j] = Real(0.0);
|
|
}
|
|
}
|
|
}
|
|
return L_;
|
|
}
|
|
|
|
/** Return upper triangular factor
|
|
@return U portion of LU factorization.
|
|
*/
|
|
|
|
Matrix<Real> getU () {
|
|
Matrix<Real> U_(n,n);
|
|
for (int i = 0; i < n; i++) {
|
|
for (int j = 0; j < n; j++) {
|
|
if (i <= j) {
|
|
U_[i][j] = LU_[i][j];
|
|
} else {
|
|
U_[i][j] = Real(0.0);
|
|
}
|
|
}
|
|
}
|
|
return U_;
|
|
}
|
|
|
|
/** Return pivot permutation vector
|
|
@return piv
|
|
*/
|
|
|
|
Vector<int> getPivot () {
|
|
return piv;
|
|
}
|
|
|
|
|
|
/** Compute determinant using LU factors.
|
|
@return determinant of A, or 0 if A is not square.
|
|
*/
|
|
|
|
Real det () {
|
|
if (m != n) {
|
|
return Real(0);
|
|
}
|
|
Real d = Real(pivsign);
|
|
for (int j = 0; j < n; j++) {
|
|
d *= LU_[j][j];
|
|
}
|
|
return d;
|
|
}
|
|
|
|
/** Solve A*X = B
|
|
@param B A Matrix with as many rows as A and any number of columns.
|
|
@return X so that L*U*X = B(piv,:), if B is nonconformant, returns
|
|
Real(0.0) (null) array.
|
|
*/
|
|
|
|
Matrix<Real> solve (const Matrix<Real> &B)
|
|
{
|
|
|
|
/* Dimensions: A is mxn, X is nxk, B is mxk */
|
|
|
|
if (B.num_rows() != m) {
|
|
return Matrix<Real>(Real(0.0));
|
|
}
|
|
if (!isNonsingular()) {
|
|
return Matrix<Real>(Real(0.0));
|
|
}
|
|
|
|
// Copy right hand side with pivoting
|
|
int nx = B.num_cols();
|
|
|
|
|
|
Matrix<Real> X = permute_copy(B, piv, 0, nx-1);
|
|
|
|
// Solve L*Y = B(piv,:)
|
|
for (int k = 0; k < n; k++) {
|
|
for (int i = k+1; i < n; i++) {
|
|
for (int j = 0; j < nx; j++) {
|
|
X[i][j] -= X[k][j]*LU_[i][k];
|
|
}
|
|
}
|
|
}
|
|
// Solve U*X = Y;
|
|
for (int k = n-1; k >= 0; k--) {
|
|
for (int j = 0; j < nx; j++) {
|
|
X[k][j] /= LU_[k][k];
|
|
}
|
|
for (int i = 0; i < k; i++) {
|
|
for (int j = 0; j < nx; j++) {
|
|
X[i][j] -= X[k][j]*LU_[i][k];
|
|
}
|
|
}
|
|
}
|
|
return X;
|
|
}
|
|
|
|
|
|
/** Solve A*x = b, where x and b are vectors of length equal
|
|
to the number of rows in A.
|
|
|
|
@param b a vector (Vector> of length equal to the first dimension
|
|
of A.
|
|
@return x a vector (Vector> so that L*U*x = b(piv), if B is nonconformant,
|
|
returns Real(0.0) (null) array.
|
|
*/
|
|
|
|
Vector<Real> solve (const Vector<Real> &b)
|
|
{
|
|
|
|
/* Dimensions: A is mxn, X is nxk, B is mxk */
|
|
|
|
if (b.dim() != m) {
|
|
return Vector<Real>();
|
|
}
|
|
if (!isNonsingular()) {
|
|
return Vector<Real>();
|
|
}
|
|
|
|
|
|
Vector<Real> x = permute_copy(b, piv);
|
|
|
|
// Solve L*Y = B(piv)
|
|
for (int k = 0; k < n; k++) {
|
|
for (int i = k+1; i < n; i++) {
|
|
x[i] -= x[k]*LU_[i][k];
|
|
}
|
|
}
|
|
|
|
// Solve U*X = Y;
|
|
for (int k = n-1; k >= 0; k--) {
|
|
x[k] /= LU_[k][k];
|
|
for (int i = 0; i < k; i++)
|
|
x[i] -= x[k]*LU_[i][k];
|
|
}
|
|
|
|
|
|
return x;
|
|
}
|
|
|
|
}; /* class LU */
|
|
|
|
|
|
/**
|
|
<p>
|
|
Classical QR Decompisition:
|
|
for an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n
|
|
orthogonal matrix Q and an n-by-n upper triangular matrix R so that
|
|
A = Q*R.
|
|
<P>
|
|
The QR decompostion always exists, even if the matrix does not have
|
|
full rank, so the constructor will never fail. The primary use of the
|
|
QR decomposition is in the least squares solution of nonsquare systems
|
|
of simultaneous linear equations. This will fail if isFullRank()
|
|
returns 0 (false).
|
|
|
|
<p>
|
|
The Q and R factors can be retrived via the getQ() and getR()
|
|
methods. Furthermore, a solve() method is provided to find the
|
|
least squares solution of Ax=b using the QR factors.
|
|
|
|
<p>
|
|
(Adapted from JAMA, a Java Matrix Library, developed by jointly
|
|
by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
|
|
*/
|
|
|
|
template <class Real>
|
|
class QR {
|
|
|
|
|
|
/* Array for internal storage of decomposition.
|
|
@serial internal array storage.
|
|
*/
|
|
|
|
Matrix<Real> QR_;
|
|
|
|
/* Row and column dimensions.
|
|
@serial column dimension.
|
|
@serial row dimension.
|
|
*/
|
|
int m, n;
|
|
|
|
/* Array for internal storage of diagonal of R.
|
|
@serial diagonal of R.
|
|
*/
|
|
Vector<Real> Rdiag;
|
|
|
|
|
|
public:
|
|
|
|
/**
|
|
Create a QR factorization object for A.
|
|
|
|
@param A rectangular (m>=n) matrix.
|
|
*/
|
|
QR(const Matrix<Real> &A) /* constructor */
|
|
{
|
|
QR_ = A;
|
|
m = A.num_rows();
|
|
n = A.num_cols();
|
|
Rdiag = Vector<Real>(n);
|
|
int i=0, j=0, k=0;
|
|
|
|
// Main loop.
|
|
for (k = 0; k < n; k++) {
|
|
// Compute 2-norm of k-th column without under/overflow.
|
|
Real nrm = 0;
|
|
for (i = k; i < m; i++) {
|
|
nrm = hypot(nrm,QR_[i][k]);
|
|
}
|
|
|
|
if (nrm != Real(0.0)) {
|
|
// Form k-th Householder vector.
|
|
if (QR_[k][k] < 0) {
|
|
nrm = -nrm;
|
|
}
|
|
for (i = k; i < m; i++) {
|
|
QR_[i][k] /= nrm;
|
|
}
|
|
QR_[k][k] += Real(1.0);
|
|
|
|
// Apply transformation to remaining columns.
|
|
for (j = k+1; j < n; j++) {
|
|
Real s = Real(0.0);
|
|
for (i = k; i < m; i++) {
|
|
s += QR_[i][k]*QR_[i][j];
|
|
}
|
|
s = -s/QR_[k][k];
|
|
for (i = k; i < m; i++) {
|
|
QR_[i][j] += s*QR_[i][k];
|
|
}
|
|
}
|
|
}
|
|
Rdiag[k] = -nrm;
|
|
}
|
|
}
|
|
|
|
|
|
/**
|
|
Flag to denote the matrix is of full rank.
|
|
|
|
@return 1 if matrix is full rank, 0 otherwise.
|
|
*/
|
|
int isFullRank() const
|
|
{
|
|
for (int j = 0; j < n; j++)
|
|
{
|
|
if (Rdiag[j] == 0)
|
|
return 0;
|
|
}
|
|
return 1;
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
Retreive the Householder vectors from QR factorization
|
|
@returns lower trapezoidal matrix whose columns define the reflections
|
|
*/
|
|
|
|
Matrix<Real> getHouseholder (void) const
|
|
{
|
|
Matrix<Real> H(m,n);
|
|
|
|
/* note: H is completely filled in by algorithm, so
|
|
initializaiton of H is not necessary.
|
|
*/
|
|
for (int i = 0; i < m; i++)
|
|
{
|
|
for (int j = 0; j < n; j++)
|
|
{
|
|
if (i >= j) {
|
|
H[i][j] = QR_[i][j];
|
|
} else {
|
|
H[i][j] = Real(0.0);
|
|
}
|
|
}
|
|
}
|
|
return H;
|
|
}
|
|
|
|
|
|
|
|
/** Return the upper triangular factor, R, of the QR factorization
|
|
@return R
|
|
*/
|
|
|
|
Matrix<Real> getR() const
|
|
{
|
|
Matrix<Real> R(n,n);
|
|
for (int i = 0; i < n; i++) {
|
|
for (int j = 0; j < n; j++) {
|
|
if (i < j) {
|
|
R[i][j] = QR_[i][j];
|
|
} else if (i == j) {
|
|
R[i][j] = Rdiag[i];
|
|
} else {
|
|
R[i][j] = Real(0.0);
|
|
}
|
|
}
|
|
}
|
|
return R;
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/**
|
|
@return Q the (ecnomy-sized) orthogonal factor (Q*R=A).
|
|
*/
|
|
|
|
Matrix<Real> getQ() const
|
|
{
|
|
int i=0, j=0, k=0;
|
|
|
|
Matrix<Real> Q(m,n);
|
|
for (k = n-1; k >= 0; k--) {
|
|
for (i = 0; i < m; i++) {
|
|
Q[i][k] = Real(0.0);
|
|
}
|
|
Q[k][k] = Real(1.0);
|
|
for (j = k; j < n; j++) {
|
|
if (QR_[k][k] != 0) {
|
|
Real s = Real(0.0);
|
|
for (i = k; i < m; i++) {
|
|
s += QR_[i][k]*Q[i][j];
|
|
}
|
|
s = -s/QR_[k][k];
|
|
for (i = k; i < m; i++) {
|
|
Q[i][j] += s*QR_[i][k];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
return Q;
|
|
}
|
|
|
|
|
|
/** Least squares solution of A*x = b
|
|
@param b right hand side (m-length vector).
|
|
@return x n-length vector that minimizes the two norm of Q*R*X-B.
|
|
If B is non-conformant, or if QR.isFullRank() is false,
|
|
the routine returns a null (0-length) vector.
|
|
*/
|
|
|
|
Vector<Real> solve(const Vector<Real> &b) const
|
|
{
|
|
if (b.dim() != m) /* arrays must be conformant */
|
|
return Vector<Real>();
|
|
|
|
if ( !isFullRank() ) /* matrix is rank deficient */
|
|
{
|
|
return Vector<Real>();
|
|
}
|
|
|
|
Vector<Real> x = b;
|
|
|
|
// Compute Y = transpose(Q)*b
|
|
for (int k = 0; k < n; k++)
|
|
{
|
|
Real s = Real(0.0);
|
|
for (int i = k; i < m; i++)
|
|
{
|
|
s += QR_[i][k]*x[i];
|
|
}
|
|
s = -s/QR_[k][k];
|
|
for (int i = k; i < m; i++)
|
|
{
|
|
x[i] += s*QR_[i][k];
|
|
}
|
|
}
|
|
// Solve R*X = Y;
|
|
for (int k = n-1; k >= 0; k--)
|
|
{
|
|
x[k] /= Rdiag[k];
|
|
for (int i = 0; i < k; i++) {
|
|
x[i] -= x[k]*QR_[i][k];
|
|
}
|
|
}
|
|
|
|
|
|
/* return n x nx portion of X */
|
|
Vector<Real> x_(n);
|
|
for (int i=0; i<n; i++)
|
|
x_[i] = x[i];
|
|
|
|
return x_;
|
|
}
|
|
|
|
/** Least squares solution of A*X = B
|
|
@param B m x k Array (must conform).
|
|
@return X n x k Array that minimizes the two norm of Q*R*X-B. If
|
|
B is non-conformant, or if QR.isFullRank() is false,
|
|
the routine returns a null (Real(0.0)) array.
|
|
*/
|
|
|
|
Matrix<Real> solve(const Matrix<Real> &B) const
|
|
{
|
|
if (B.num_rows() != m) /* arrays must be conformant */
|
|
return Matrix<Real>(Real(0.0));
|
|
|
|
if ( !isFullRank() ) /* matrix is rank deficient */
|
|
{
|
|
return Matrix<Real>(Real(0.0));
|
|
}
|
|
|
|
int nx = B.num_cols();
|
|
Matrix<Real> X = B;
|
|
int i=0, j=0, k=0;
|
|
|
|
// Compute Y = transpose(Q)*B
|
|
for (k = 0; k < n; k++) {
|
|
for (j = 0; j < nx; j++) {
|
|
Real s = Real(0.0);
|
|
for (i = k; i < m; i++) {
|
|
s += QR_[i][k]*X[i][j];
|
|
}
|
|
s = -s/QR_[k][k];
|
|
for (i = k; i < m; i++) {
|
|
X[i][j] += s*QR_[i][k];
|
|
}
|
|
}
|
|
}
|
|
// Solve R*X = Y;
|
|
for (k = n-1; k >= 0; k--) {
|
|
for (j = 0; j < nx; j++) {
|
|
X[k][j] /= Rdiag[k];
|
|
}
|
|
for (i = 0; i < k; i++) {
|
|
for (j = 0; j < nx; j++) {
|
|
X[i][j] -= X[k][j]*QR_[i][k];
|
|
}
|
|
}
|
|
}
|
|
|
|
|
|
/* return n x nx portion of X */
|
|
Matrix<Real> X_(n,nx);
|
|
for (i=0; i<n; i++)
|
|
for (j=0; j<nx; j++)
|
|
X_[i][j] = X[i][j];
|
|
|
|
return X_;
|
|
}
|
|
|
|
|
|
};
|
|
|
|
|
|
/** Singular Value Decomposition.
|
|
<P>
|
|
For an m-by-n matrix A with m >= n, the singular value decomposition is
|
|
an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
|
|
an n-by-n orthogonal matrix V so that A = U*S*V'.
|
|
<P>
|
|
The singular values, sigma[k] = S[k][k], are ordered so that
|
|
sigma[0] >= sigma[1] >= ... >= sigma[n-1].
|
|
<P>
|
|
The singular value decompostion always exists, so the constructor will
|
|
never fail. The matrix condition number and the effective numerical
|
|
rank can be computed from this decomposition.
|
|
|
|
<p>
|
|
(Adapted from JAMA, a Java Matrix Library, developed by jointly
|
|
by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
|
|
*/
|
|
template <class Real>
|
|
class SVD
|
|
{
|
|
|
|
|
|
Matrix<Real> U, V;
|
|
Vector<Real> s;
|
|
int m, n;
|
|
|
|
public:
|
|
|
|
|
|
SVD (const Matrix<Real> &Arg) {
|
|
|
|
|
|
m = Arg.num_rows();
|
|
n = Arg.num_cols();
|
|
int nu = min(m,n);
|
|
s = Vector<Real>(min(m+1,n));
|
|
U = Matrix<Real>(m, nu, Real(0));
|
|
V = Matrix<Real>(n,n);
|
|
Vector<Real> e(n);
|
|
Vector<Real> work(m);
|
|
Matrix<Real> A(Arg);
|
|
int wantu = 1; /* boolean */
|
|
int wantv = 1; /* boolean */
|
|
int i=0, j=0, k=0;
|
|
|
|
// Reduce A to bidiagonal form, storing the diagonal elements
|
|
// in s and the super-diagonal elements in e.
|
|
|
|
int nct = min(m-1,n);
|
|
int nrt = max(0,min(n-2,m));
|
|
for (k = 0; k < max(nct,nrt); k++) {
|
|
if (k < nct) {
|
|
|
|
// Compute the transformation for the k-th column and
|
|
// place the k-th diagonal in s[k].
|
|
// Compute 2-norm of k-th column without under/overflow.
|
|
s[k] = 0;
|
|
for (i = k; i < m; i++) {
|
|
s[k] = hypot(s[k],A[i][k]);
|
|
}
|
|
if (s[k] != Real(0.0)) {
|
|
if (A[k][k] < Real(0.0)) {
|
|
s[k] = -s[k];
|
|
}
|
|
for (i = k; i < m; i++) {
|
|
A[i][k] /= s[k];
|
|
}
|
|
A[k][k] += Real(1.0);
|
|
}
|
|
s[k] = -s[k];
|
|
}
|
|
for (j = k+1; j < n; j++) {
|
|
if ((k < nct) && (s[k] != Real(0.0))) {
|
|
|
|
// Apply the transformation.
|
|
|
|
double t = 0;
|
|
for (i = k; i < m; i++) {
|
|
t += A[i][k]*A[i][j];
|
|
}
|
|
t = -t/A[k][k];
|
|
for (i = k; i < m; i++) {
|
|
A[i][j] += t*A[i][k];
|
|
}
|
|
}
|
|
|
|
// Place the k-th row of A into e for the
|
|
// subsequent calculation of the row transformation.
|
|
|
|
e[j] = A[k][j];
|
|
}
|
|
if (wantu & (k < nct)) {
|
|
|
|
// Place the transformation in U for subsequent back
|
|
// multiplication.
|
|
|
|
for (i = k; i < m; i++) {
|
|
U[i][k] = A[i][k];
|
|
}
|
|
}
|
|
if (k < nrt) {
|
|
|
|
// Compute the k-th row transformation and place the
|
|
// k-th super-diagonal in e[k].
|
|
// Compute 2-norm without under/overflow.
|
|
e[k] = 0;
|
|
for (i = k+1; i < n; i++) {
|
|
e[k] = hypot(e[k],e[i]);
|
|
}
|
|
if (e[k] != Real(0.0)) {
|
|
if (e[k+1] < Real(0.0)) {
|
|
e[k] = -e[k];
|
|
}
|
|
for (i = k+1; i < n; i++) {
|
|
e[i] /= e[k];
|
|
}
|
|
e[k+1] += Real(1.0);
|
|
}
|
|
e[k] = -e[k];
|
|
if ((k+1 < m) & (e[k] != Real(0.0))) {
|
|
|
|
// Apply the transformation.
|
|
|
|
for (i = k+1; i < m; i++) {
|
|
work[i] = Real(0.0);
|
|
}
|
|
for (j = k+1; j < n; j++) {
|
|
for (i = k+1; i < m; i++) {
|
|
work[i] += e[j]*A[i][j];
|
|
}
|
|
}
|
|
for (j = k+1; j < n; j++) {
|
|
double t = -e[j]/e[k+1];
|
|
for (i = k+1; i < m; i++) {
|
|
A[i][j] += t*work[i];
|
|
}
|
|
}
|
|
}
|
|
if (wantv) {
|
|
|
|
// Place the transformation in V for subsequent
|
|
// back multiplication.
|
|
|
|
for (i = k+1; i < n; i++) {
|
|
V[i][k] = e[i];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Set up the final bidiagonal matrix or order p.
|
|
|
|
int p = min(n,m+1);
|
|
if (nct < n) {
|
|
s[nct] = A[nct][nct];
|
|
}
|
|
if (m < p) {
|
|
s[p-1] = Real(0.0);
|
|
}
|
|
if (nrt+1 < p) {
|
|
e[nrt] = A[nrt][p-1];
|
|
}
|
|
e[p-1] = Real(0.0);
|
|
|
|
// If required, generate U.
|
|
|
|
if (wantu) {
|
|
for (j = nct; j < nu; j++) {
|
|
for (i = 0; i < m; i++) {
|
|
U[i][j] = Real(0.0);
|
|
}
|
|
U[j][j] = Real(1.0);
|
|
}
|
|
for (k = nct-1; k >= 0; k--) {
|
|
if (s[k] != Real(0.0)) {
|
|
for (j = k+1; j < nu; j++) {
|
|
double t = 0;
|
|
for (i = k; i < m; i++) {
|
|
t += U[i][k]*U[i][j];
|
|
}
|
|
t = -t/U[k][k];
|
|
for (i = k; i < m; i++) {
|
|
U[i][j] += t*U[i][k];
|
|
}
|
|
}
|
|
for (i = k; i < m; i++ ) {
|
|
U[i][k] = -U[i][k];
|
|
}
|
|
U[k][k] = Real(1.0) + U[k][k];
|
|
for (i = 0; i < k-1; i++) {
|
|
U[i][k] = Real(0.0);
|
|
}
|
|
} else {
|
|
for (i = 0; i < m; i++) {
|
|
U[i][k] = Real(0.0);
|
|
}
|
|
U[k][k] = Real(1.0);
|
|
}
|
|
}
|
|
}
|
|
|
|
// If required, generate V.
|
|
|
|
if (wantv) {
|
|
for (k = n-1; k >= 0; k--) {
|
|
if ((k < nrt) & (e[k] != Real(0.0))) {
|
|
for (j = k+1; j < nu; j++) {
|
|
double t = 0;
|
|
for (i = k+1; i < n; i++) {
|
|
t += V[i][k]*V[i][j];
|
|
}
|
|
t = -t/V[k+1][k];
|
|
for (i = k+1; i < n; i++) {
|
|
V[i][j] += t*V[i][k];
|
|
}
|
|
}
|
|
}
|
|
for (i = 0; i < n; i++) {
|
|
V[i][k] = Real(0.0);
|
|
}
|
|
V[k][k] = Real(1.0);
|
|
}
|
|
}
|
|
|
|
// Main iteration loop for the singular values.
|
|
|
|
int pp = p-1;
|
|
int iter = 0;
|
|
double eps = pow(2.0,-52.0);
|
|
while (p > 0) {
|
|
int k=0;
|
|
int kase=0;
|
|
|
|
// Here is where a test for too many iterations would go.
|
|
|
|
// This section of the program inspects for
|
|
// negligible elements in the s and e arrays. On
|
|
// completion the variables kase and k are set as follows.
|
|
|
|
// kase = 1 if s(p) and e[k-1] are negligible and k<p
|
|
// kase = 2 if s(k) is negligible and k<p
|
|
// kase = 3 if e[k-1] is negligible, k<p, and
|
|
// s(k), ..., s(p) are not negligible (qr step).
|
|
// kase = 4 if e(p-1) is negligible (convergence).
|
|
|
|
for (k = p-2; k >= -1; k--) {
|
|
if (k == -1) {
|
|
break;
|
|
}
|
|
if (abs(e[k]) <= eps*(abs(s[k]) + abs(s[k+1]))) {
|
|
e[k] = Real(0.0);
|
|
break;
|
|
}
|
|
}
|
|
if (k == p-2) {
|
|
kase = 4;
|
|
} else {
|
|
int ks;
|
|
for (ks = p-1; ks >= k; ks--) {
|
|
if (ks == k) {
|
|
break;
|
|
}
|
|
double t = (ks != p ? abs(e[ks]) : 0.) +
|
|
(ks != k+1 ? abs(e[ks-1]) : 0.);
|
|
if (abs(s[ks]) <= eps*t) {
|
|
s[ks] = Real(0.0);
|
|
break;
|
|
}
|
|
}
|
|
if (ks == k) {
|
|
kase = 3;
|
|
} else if (ks == p-1) {
|
|
kase = 1;
|
|
} else {
|
|
kase = 2;
|
|
k = ks;
|
|
}
|
|
}
|
|
k++;
|
|
|
|
// Perform the task indicated by kase.
|
|
|
|
switch (kase) {
|
|
|
|
// Deflate negligible s(p).
|
|
|
|
case 1: {
|
|
double f = e[p-2];
|
|
e[p-2] = Real(0.0);
|
|
for (j = p-2; j >= k; j--) {
|
|
double t = hypot(s[j],f);
|
|
double cs = s[j]/t;
|
|
double sn = f/t;
|
|
s[j] = t;
|
|
if (j != k) {
|
|
f = -sn*e[j-1];
|
|
e[j-1] = cs*e[j-1];
|
|
}
|
|
if (wantv) {
|
|
for (i = 0; i < n; i++) {
|
|
t = cs*V[i][j] + sn*V[i][p-1];
|
|
V[i][p-1] = -sn*V[i][j] + cs*V[i][p-1];
|
|
V[i][j] = t;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
break;
|
|
|
|
// Split at negligible s(k).
|
|
|
|
case 2: {
|
|
double f = e[k-1];
|
|
e[k-1] = Real(0.0);
|
|
for (j = k; j < p; j++) {
|
|
double t = hypot(s[j],f);
|
|
double cs = s[j]/t;
|
|
double sn = f/t;
|
|
s[j] = t;
|
|
f = -sn*e[j];
|
|
e[j] = cs*e[j];
|
|
if (wantu) {
|
|
for (i = 0; i < m; i++) {
|
|
t = cs*U[i][j] + sn*U[i][k-1];
|
|
U[i][k-1] = -sn*U[i][j] + cs*U[i][k-1];
|
|
U[i][j] = t;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
break;
|
|
|
|
// Perform one qr step.
|
|
|
|
case 3: {
|
|
|
|
// Calculate the shift.
|
|
|
|
double scale = max(max(max(max(
|
|
abs(s[p-1]),abs(s[p-2])),abs(e[p-2])),
|
|
abs(s[k])),abs(e[k]));
|
|
double sp = s[p-1]/scale;
|
|
double spm1 = s[p-2]/scale;
|
|
double epm1 = e[p-2]/scale;
|
|
double sk = s[k]/scale;
|
|
double ek = e[k]/scale;
|
|
double b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
|
|
double c = (sp*epm1)*(sp*epm1);
|
|
double shift = Real(0.0);
|
|
if ((b != Real(0.0)) || (c != Real(0.0))) {
|
|
shift = sqrt(b*b + c);
|
|
if (b < Real(0.0)) {
|
|
shift = -shift;
|
|
}
|
|
shift = c/(b + shift);
|
|
}
|
|
double f = (sk + sp)*(sk - sp) + shift;
|
|
double g = sk*ek;
|
|
|
|
// Chase zeros.
|
|
|
|
for (j = k; j < p-1; j++) {
|
|
double t = hypot(f,g);
|
|
double cs = f/t;
|
|
double sn = g/t;
|
|
if (j != k) {
|
|
e[j-1] = t;
|
|
}
|
|
f = cs*s[j] + sn*e[j];
|
|
e[j] = cs*e[j] - sn*s[j];
|
|
g = sn*s[j+1];
|
|
s[j+1] = cs*s[j+1];
|
|
if (wantv) {
|
|
for (i = 0; i < n; i++) {
|
|
t = cs*V[i][j] + sn*V[i][j+1];
|
|
V[i][j+1] = -sn*V[i][j] + cs*V[i][j+1];
|
|
V[i][j] = t;
|
|
}
|
|
}
|
|
t = hypot(f,g);
|
|
cs = f/t;
|
|
sn = g/t;
|
|
s[j] = t;
|
|
f = cs*e[j] + sn*s[j+1];
|
|
s[j+1] = -sn*e[j] + cs*s[j+1];
|
|
g = sn*e[j+1];
|
|
e[j+1] = cs*e[j+1];
|
|
if (wantu && (j < m-1)) {
|
|
for (i = 0; i < m; i++) {
|
|
t = cs*U[i][j] + sn*U[i][j+1];
|
|
U[i][j+1] = -sn*U[i][j] + cs*U[i][j+1];
|
|
U[i][j] = t;
|
|
}
|
|
}
|
|
}
|
|
e[p-2] = f;
|
|
iter = iter + 1;
|
|
}
|
|
break;
|
|
|
|
// Convergence.
|
|
|
|
case 4: {
|
|
|
|
// Make the singular values positive.
|
|
|
|
if (s[k] <= Real(0.0)) {
|
|
s[k] = (s[k] < Real(0.0) ? -s[k] : Real(0.0));
|
|
if (wantv) {
|
|
for (i = 0; i <= pp; i++) {
|
|
V[i][k] = -V[i][k];
|
|
}
|
|
}
|
|
}
|
|
|
|
// Order the singular values.
|
|
|
|
while (k < pp) {
|
|
if (s[k] >= s[k+1]) {
|
|
break;
|
|
}
|
|
double t = s[k];
|
|
s[k] = s[k+1];
|
|
s[k+1] = t;
|
|
if (wantv && (k < n-1)) {
|
|
for (i = 0; i < n; i++) {
|
|
t = V[i][k+1]; V[i][k+1] = V[i][k]; V[i][k] = t;
|
|
}
|
|
}
|
|
if (wantu && (k < m-1)) {
|
|
for (i = 0; i < m; i++) {
|
|
t = U[i][k+1]; U[i][k+1] = U[i][k]; U[i][k] = t;
|
|
}
|
|
}
|
|
k++;
|
|
}
|
|
iter = 0;
|
|
p--;
|
|
}
|
|
break;
|
|
}
|
|
}
|
|
}
|
|
|
|
|
|
void getU (Matrix<Real> &A)
|
|
{
|
|
int minm = min(m+1,n);
|
|
|
|
A = Matrix<Real>(m, minm);
|
|
|
|
for (int i=0; i<m; i++)
|
|
for (int j=0; j<minm; j++)
|
|
A[i][j] = U[i][j];
|
|
|
|
}
|
|
|
|
/* Return the right singular vectors */
|
|
|
|
void getV (Matrix<Real> &A)
|
|
{
|
|
A = V;
|
|
}
|
|
|
|
/** Return the one-dimensional array of singular values */
|
|
|
|
void getSingularValues (Vector<Real> &x)
|
|
{
|
|
x = s;
|
|
}
|
|
|
|
/** Return the diagonal matrix of singular values
|
|
@return S
|
|
*/
|
|
|
|
void getS (Matrix<Real> &A) {
|
|
A = Matrix<Real>(n,n);
|
|
for (int i = 0; i < n; i++) {
|
|
for (int j = 0; j < n; j++) {
|
|
A[i][j] = Real(0.0);
|
|
}
|
|
A[i][i] = s[i];
|
|
}
|
|
}
|
|
|
|
/** Two norm (max(S)) */
|
|
|
|
double norm2 () {
|
|
return s[0];
|
|
}
|
|
|
|
/** Two norm of condition number (max(S)/min(S)) */
|
|
|
|
double cond () {
|
|
return s[0]/s[min(m,n)-1];
|
|
}
|
|
|
|
/** Effective numerical matrix rank
|
|
@return Number of nonnegligible singular values.
|
|
*/
|
|
|
|
int rank ()
|
|
{
|
|
double eps = pow(2.0,-52.0);
|
|
double tol = max(m,n)*s[0]*eps;
|
|
int r = 0;
|
|
for (int i = 0; i < s.dim(); i++) {
|
|
if (s[i] > tol) {
|
|
r++;
|
|
}
|
|
}
|
|
return r;
|
|
}
|
|
};
|
|
|
|
} /* namespace Linear_Algebra */
|
|
} /* namespace TNT */
|
|
|
|
#endif
|
|
// TNT_LINALG_H_
|