| public swap ( int i1, int j1, int i2, int j2 ) : void | ||
| i1 | int | |
| j1 | int | |
| i2 | int | |
| j2 | int | |
| return | void | |
public SVD(Matrix M)
{
Matrix A;
/* The implementation requires that rows > columns.
* If this is not the case, we decompose M^T instead.
* Swapping the resulting U and V gives the desired
* result for M as
*
* M^T = U S V^T (decomposition of M^T)
*
* M = (U S V^T)^T (transpose)
*
* M = (V^T^T S^T U^T) ((AB)^T = B^T A^T)
*
* M = V S U^T (idempotence of transposition,
* symmetry of diagonal matrix S)
*/
if (M.rows() >= M.columns())
{
A = new Matrix(M);
transpose_ = false;
}
else
{
A = Matrix.transpose(M);
transpose_ = true;
}
m_ = A.rows();
n_ = A.columns();
// we're sure that m_ >= n_
s_ = new Vector(n_);
U_ = new Matrix(m_, n_);
V_ = new Matrix(n_, n_);
Vector e = new Vector(n_);
Vector work = new Vector(m_);
int i, j, k;
// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.
int nct = Math.Min(m_ - 1, n_);
int nrt = Math.Max(0, n_ - 2);
for (k = 0; k < Math.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] != 0.0)
{
if (A[k, k] < 0.0)
{
s_[k] = -s_[k];
}
for (i = k; i < m_; i++)
{
A[i, k] /= s_[k];
}
A[k, k] += 1.0;
}
s_[k] = -s_[k];
}
for (j = k + 1; j < n_; j++)
{
if ((k < nct) && (s_[k] != 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 (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] != 0.0)
{
if (e[k + 1] < 0.0)
{
e[k] = -e[k];
}
for (i = k + 1; i < n_; i++)
{
e[i] /= e[k];
}
e[k + 1] += 1.0;
}
e[k] = -e[k];
if ((k + 1 < m_) & (e[k] != 0.0))
{
// Apply the transformation.
for (i = k + 1; i < m_; i++)
{
work[i] = 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];
}
}
}
// 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 n.
if (nct < n_)
{
s_[nct] = A[nct, nct];
}
if (nrt + 1 < n_)
{
e[nrt] = A[nrt, n_ - 1];
}
e[n_ - 1] = 0.0;
// generate U
for (j = nct; j < n_; j++)
{
for (i = 0; i < m_; i++)
{
U_[i, j] = 0.0;
}
U_[j, j] = 1.0;
}
for (k = nct - 1; k >= 0; --k)
{
if (s_[k] != 0.0)
{
for (j = k + 1; j < n_; ++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] = 1.0 + U_[k, k];
for (i = 0; i < k - 1; i++)
{
U_[i, k] = 0.0;
}
}
else
{
for (i = 0; i < m_; i++)
{
U_[i, k] = 0.0;
}
U_[k, k] = 1.0;
}
}
// generate V
for (k = n_ - 1; k >= 0; --k)
{
if ((k < nrt) & (e[k] != 0.0))
{
for (j = k + 1; j < n_; ++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] = 0.0;
}
V_[k, k] = 1.0;
}
// Main iteration loop for the singular values.
int p = n_, pp = p - 1;
int iter = 0;
double eps = Math.Pow(2.0, -52.0);
while (p > 0)
{
int kase;
// 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 (Math.Abs(e[k]) <= eps * (Math.Abs(s_[k]) + Math.Abs(s_[k + 1])))
{
e[k] = 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 ? Math.Abs(e[ks]) : 0) +
(ks != k + 1 ? Math.Abs(e[ks - 1]) : 0);
if (Math.Abs(s_[ks]) <= eps * t)
{
s_[ks] = 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] = 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];
}
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] = 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];
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 = Math.Max(
Math.Max(
Math.Max(
Math.Max(Math.Abs(s_[p - 1]),
Math.Abs(s_[p - 2])),
Math.Abs(e[p - 2])),
Math.Abs(s_[k])),
Math.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 = 0.0;
if ((b != 0.0) | (c != 0.0))
{
shift = Math.Sqrt(b * b + c);
if (b < 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];
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 (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] <= 0.0)
{
s_[k] = (s_[k] < 0.0 ? -s_[k] : 0.0);
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;
}
s_.swap(k, k + 1);
if (k < n_ - 1)
{
for (i = 0; i < n_; i++)
{
V_.swap(i, k, i, k + 1);
}
}
if (k < m_ - 1)
{
for (i = 0; i < m_; i++)
{
U_.swap(i, k, i, k + 1);
}
}
k++;
}
iter = 0;
--p;
}
break;
}
}
}
public SVD(Matrix M)
{
Matrix A;
/* The implementation requires that rows > columns.
If this is not the case, we decompose M^T instead.
Swapping the resulting U and V gives the desired
result for M as
M^T = U S V^T (decomposition of M^T)
M = (U S V^T)^T (transpose)
M = (V^T^T S^T U^T) ((AB)^T = B^T A^T)
M = V S U^T (idempotence of transposition,
symmetry of diagonal matrix S)
*/
if (M.rows() >= M.columns()) {
A = new Matrix(M);
transpose_ = false;
} else {
A = Matrix.transpose(M);
transpose_ = true;
}
m_ = A.rows();
n_ = A.columns();
// we're sure that m_ >= n_
s_ = new Vector(n_);
U_ = new Matrix(m_, n_);
V_ = new Matrix(n_, n_);
Vector e = new Vector(n_);
Vector work = new Vector(m_);
int i, j, k;
// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.
int nct = Math.Min(m_ - 1, n_);
int nrt = Math.Max(0, n_ - 2);
for (k = 0; k < Math.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] != 0.0) {
if (A[k, k] < 0.0) {
s_[k] = -s_[k];
}
for (i = k; i < m_; i++) {
A[i, k] /= s_[k];
}
A[k, k] += 1.0;
}
s_[k] = -s_[k];
}
for (j = k + 1; j < n_; j++) {
if ((k < nct) && (s_[k] != 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 (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] != 0.0) {
if (e[k + 1] < 0.0) {
e[k] = -e[k];
}
for (i = k + 1; i < n_; i++) {
e[i] /= e[k];
}
e[k + 1] += 1.0;
}
e[k] = -e[k];
if ((k + 1 < m_) & (e[k] != 0.0)) {
// Apply the transformation.
for (i = k + 1; i < m_; i++) {
work[i] = 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];
}
}
}
// 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 n.
if (nct < n_) {
s_[nct] = A[nct, nct];
}
if (nrt + 1 < n_) {
e[nrt] = A[nrt, n_ - 1];
}
e[n_ - 1] = 0.0;
// generate U
for (j = nct; j < n_; j++) {
for (i = 0; i < m_; i++) {
U_[i, j] = 0.0;
}
U_[j, j] = 1.0;
}
for (k = nct - 1; k >= 0; --k) {
if (s_[k] != 0.0) {
for (j = k + 1; j < n_; ++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] = 1.0 + U_[k, k];
for (i = 0; i < k - 1; i++) {
U_[i, k] = 0.0;
}
} else {
for (i = 0; i < m_; i++) {
U_[i, k] = 0.0;
}
U_[k, k] = 1.0;
}
}
// generate V
for (k = n_ - 1; k >= 0; --k) {
if ((k < nrt) & (e[k] != 0.0)) {
for (j = k + 1; j < n_; ++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] = 0.0;
}
V_[k, k] = 1.0;
}
// Main iteration loop for the singular values.
int p = n_, pp = p - 1;
int iter = 0;
double eps = Math.Pow(2.0, -52.0);
while (p > 0) {
int kase;
// 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 (Math.Abs(e[k]) <= eps * (Math.Abs(s_[k]) + Math.Abs(s_[k + 1]))) {
e[k] = 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 ? Math.Abs(e[ks]) : 0) +
(ks != k + 1 ? Math.Abs(e[ks - 1]) : 0);
if (Math.Abs(s_[ks]) <= eps * t) {
s_[ks] = 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] = 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];
}
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] = 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];
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 = Math.Max(
Math.Max(
Math.Max(
Math.Max(Math.Abs(s_[p - 1]),
Math.Abs(s_[p - 2])),
Math.Abs(e[p - 2])),
Math.Abs(s_[k])),
Math.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 = 0.0;
if ((b != 0.0) | (c != 0.0)) {
shift = Math.Sqrt(b * b + c);
if (b < 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];
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 (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] <= 0.0) {
s_[k] = (s_[k] < 0.0 ? -s_[k] : 0.0);
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;
}
s_.swap(k, k + 1);
if (k < n_ - 1) {
for (i = 0; i < n_; i++) {
V_.swap(i, k, i, k + 1);
}
}
if (k < m_ - 1) {
for (i = 0; i < m_; i++) {
U_.swap(i, k, i, k + 1);
}
}
k++;
}
iter = 0;
--p;
}
break;
}
}
}