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Questionhow svd(singular decompostion value) works for matrix? Pin
Isawyouoo22-Nov-17 4:00
MemberIsawyouoo22-Nov-17 4:00 
Hi all!
this algorithm written in c# :

// Derived from LINPACK code.
// Initialize.
double[][] A = Arg;
// m = Arg.RowDimension;
// n = Arg.ColumnDimension;
int nu = System.Math.Min(m, n);
double[] s = new double[System.Math.Min(m + 1, n)];
double[][] U = new double[m][];
for (int i = 0; i < m; i++)
{
U[i] = new double[nu];
}
double[][] V = new double[n][];
for (int i2 = 0; i2 < n; i2++)
{
V[i2] = new double[n];
}
double[] e = new double[n];
double[] work = new double[m];
bool wantu = true;
bool wantv = true;

// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.

int nct = System.Math.Min(m - 1, n);
int nrt = System.Math.Max(0, System.Math.Min(n - 2, m));
for (int k = 0; k < System.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 (int i = k; i < m; i++)
{
s[k] = System.Math.Sqrt(s[k] * s[k] + A[i][k] * A[i][k]);
}
if (s[k] != 0.0)
{
if (A[k][k] < 0.0)
{
s[k] = -s[k];
}
for (int i = k; i < m; i++)
{
A[i][k] /= s[k];
}
A[k][k] += 1.0;
}
s[k] = -s[k];
}
for (int j = k + 1; j < n; j++)
{
if ((k < nct) & (s[k] != 0.0))
{

// Apply the transformation.

double t = 0;
for (int i = k; i < m; i++)
{
t += A[i][k] * A[i][j];
}
t = (-t) / A[k][k];
for (int 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 (int 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 (int i = k + 1; i < n; i++)
{
e[k] = System.Math.Sqrt(e[k] * e[k] + e[i] * e[i]);
}
if (e[k] != 0.0)
{
if (e[k + 1] < 0.0)
{
e[k] = -e[k];
}
for (int 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 (int i = k + 1; i < m; i++)
{
work[i] = 0.0;
}
for (int j = k + 1; j < n; j++)
{
for (int i = k + 1; i < m; i++)
{
work[i] += e[j] * A[i][j];
}
}
for (int j = k + 1; j < n; j++)
{
double t = (-e[j]) / e[k + 1];
for (int i = k + 1; i < m; i++)
{
A[i][j] += t * work[i];
}
}
}
if (wantv)
{

// Place the transformation in V for subsequent
// back multiplication.

for (int i = k + 1; i < n; i++)
{
V[i][k] = e[i];
}
}
}
}

// Set up the final bidiagonal matrix or order p.

int p = System.Math.Min(n, m + 1);
if (nct < n)
{
s[nct] = A[nct][nct];
}
if (m < p)
{
s[p - 1] = 0.0;
}
if (nrt + 1 < p)
{
e[nrt] = A[nrt][p - 1];
}
e[p - 1] = 0.0;

// If required, generate U.

if (wantu)
{
for (int j = nct; j < nu; j++)
{
for (int i = 0; i < m; i++)
{
U[i][j] = 0.0;
}
U[j][j] = 1.0;
}
for (int k = nct - 1; k >= 0; k--)
{
if (s[k] != 0.0)
{
for (int j = k + 1; j < nu; j++)
{
double t = 0;
for (int i = k; i < m; i++)
{
t += U[i][k] * U[i][j];
}
t = (-t) / U[k][k];
for (int i = k; i < m; i++)
{
U[i][j] += t * U[i][k];
}
}
for (int i = k; i < m; i++)
{
U[i][k] = -U[i][k];
}
U[k][k] = 1.0 + U[k][k];
for (int i = 0; i < k - 1; i++)
{
U[i][k] = 0.0;
}
}
else
{
for (int i = 0; i < m; i++)
{
U[i][k] = 0.0;
}
U[k][k] = 1.0;
}
}
}

// If required, generate V.

if (wantv)
{
for (int k = n - 1; k >= 0; k--)
{
if ((k < nrt) & (e[k] != 0.0))
{
for (int j = k + 1; j < nu; j++)
{
double t = 0;
for (int i = k + 1; i < n; i++)
{
t += V[i][k] * V[i][j];
}
t = (-t) / V[k + 1][k];
for (int i = k + 1; i < n; i++)
{
V[i][j] += t * V[i][k];
}
}
}
for (int i = 0; i < n; i++)
{
V[i][k] = 0.0;
}
V[k][k] = 1.0;
}
}

// Main iteration loop for the singular values.

int pp = p - 1;
int iter = 0;
double eps = System.Math.Pow(2.0, -52.0);
while (p > 0)
{
int k, 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
="" e[k-1]="" negligible,="" k<p,="" and
="" s(k),="" ...,="" s(p)="" are="" not="" (qr="" step).
="" e(p-1)="" (convergence).

="" for="" (k="p" -="" 2;="" k="">= -1; k--)
{
if (k == -1)
{
break;
}
if (System.Math.Abs(e[k]) <= eps * (System.Math.Abs(s[k]) + System.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 ? System.Math.Abs(e[ks]) : 0.0) + (ks != k + 1 ? System.Math.Abs(e[ks - 1]) : 0.0);
if (System.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 (int j = p - 2; j >= k; j--)
{
double t = System.Math.Sqrt(s[j] * s[j] + f * 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 (int 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 (int j = k; j < p; j++)
{
double t = System.Math.Sqrt(s[j] * s[j] + f * 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 (int 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 = System.Math.Max(System.Math.Max(System.Math.Max(System.Math.Max(System.Math.Abs(s[p - 1]), System.Math.Abs(s[p - 2])), System.Math.Abs(e[p - 2])), System.Math.Abs(s[k])), System.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 = System.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 (int j = k; j < p - 1; j++)
{
double t = System.Math.Sqrt(f * f + g * 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 (int 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 = System.Math.Sqrt(f * f + g * 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 (int 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);
if (wantv)
{
for (int 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 (int 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 (int 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;
}
}


}
algo compute an svd decompostion, first he decompose it into qr, and what's next didn't knew how it do to get eigen values and eigen vectors ?
AnswerRe: how svd(singular decompostion value) works for matrix? Pin
Mike-MadBadger22-Nov-17 4:36
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GeneralRe: how svd(singular decompostion value) works for matrix? Pin
Isawyouoo22-Nov-17 5:00
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GeneralRe: how svd(singular decompostion value) works for matrix? Pin
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GeneralRe: how svd(singular decompostion value) works for matrix? Pin
Isawyouoo23-Nov-17 11:19
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