/* ------------------------
Constructor
* ------------------------ */
///<summary>QR Decomposition, computed by Householder reflections.</summary>
///<param name="A">Rectangular matrix</param>
///<returns>Structure to access R and the Householder vectors and compute Q.</returns>
public QRDecomposition(Matrix A)
{
// Initialize.
QR = A.ArrayCopy();
m = A.RowDimension;
n = A.ColumnDimension;
Rdiag = new double[n];
// Main loop.
for (int k = 0; k < n; k++) {
// Compute 2-norm of k-th column without under/overflow.
double nrm = 0;
for (int i = k; i < m; i++) {
nrm = Maths.Hypot(nrm,QR[i][k]);
}
if (nrm != 0.0) {
// Form k-th Householder vector.
if (QR[k][k] < 0) {
nrm = -nrm;
}
for (int i = k; i < m; i++) {
QR[i][k] /= nrm;
}
QR[k][k] += 1.0;
// Apply transformation to remaining columns.
for (int j = k+1; j < n; j++) {
double s = 0.0;
for (int i = k; i < m; i++) {
s += QR[i][k]*QR[i][j];
}
s = -s/QR[k][k];
for (int i = k; i < m; i++) {
QR[i][j] += s*QR[i][k];
}
}
}
Rdiag[k] = -nrm;
}
}
/* ------------------------
Constructor
* ------------------------ */
/** LU Decomposition
@param A Rectangular matrix
@return Structure to access L, U and piv.
*/
public LUDecomposition(Matrix A)
{
// Use a "left-looking", dot-product, Crout/Doolittle algorithm.
LU = A.ArrayCopy();
m = A.RowDimension;
n = A.ColumnDimension;
piv = new int[m];
for (int i = 0; i < m; i++) {
piv[i] = i;
}
pivsign = 1;
double[] LUrowi;
double[] LUcolj = new double[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 = Math.Min(i,j);
double s = 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 (Math.Abs(LUcolj[i]) > Math.Abs(LUcolj[p])) {
p = i;
}
}
if (p != j) {
for (int k = 0; k < n; k++) {
double t = LU[p][k]; LU[p][k] = LU[j][k]; LU[j][k] = t;
}
int k2 = piv[p]; piv[p] = piv[j]; piv[j] = k2;
pivsign = -pivsign;
}
// Compute multipliers.
if (j < m && LU[j][j] != 0.0) {
for (int i = j+1; i < m; i++) {
LU[i][j] /= LU[j][j];
}
}
}
}
/* ------------------------
Constructor
* ------------------------ */
///<summary>Construct the singular value decomposition</summary>
///<param name="Arg">Rectangular matrix</param>
///<returns>Structure to access U, S and V.</returns>
public SingularValueDecomposition(Matrix Arg)
{
// Derived from LINPACK code.
// Initialize.
double[][] A = Arg.ArrayCopy();
m = Arg.RowDimension;
n = Arg.ColumnDimension;
/* Apparently the failing cases are only a proper subset of (m<n),
so let's not throw error. Correct fix to come later?
if (m<n) {
throw new System.ArgumentException("Jama SVD only works for m >= n"); }
*/
int nu = Math.Min(m,n);
s = new double [Math.Min(m+1,n)];
U = new double [m][];
for (int i = 0; i < m; i++)//Added by KJ
{
U[i] = new double[m];
}
V = new double [n][];
for (int i = 0; i < n; i++)//Added by KJ
{
V[i] = 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 = Math.Min(m-1,n);
int nrt = Math.Max(0,Math.Min(n-2,m));
for (int 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 (int i = k; i < m; i++) {
s[k] = Maths.Hypot(s[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] = Maths.Hypot(e[k],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 = 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 = Math.Pow(2.0,-52.0);
double tiny = Math.Pow(2.0,-966.0);
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
// 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]) <=
tiny + 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]) : 0.0) +
(ks != k+1 ? Math.Abs(e[ks-1]) : 0.0);
if (Math.Abs(s[ks]) <= tiny + 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 = Maths.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 (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 = Maths.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 (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 = 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 (int j = k; j < p-1; j++) {
double t = Maths.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 (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 = Maths.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 (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;
}
}
}
///<Sumary>Least squares solution of A*X = B</Sumary>
///<param name="B">A Matrix with as many rows as A and any number of columns.</param>
///<returns> X that minimizes the two norm of Q*R*X-B.</returns>
public Matrix Solve(Matrix B)
{
if (B.RowDimension != m) {
throw new ArgumentException("Matrix row dimensions must agree.");
}
if (!IsFullRank()) {
throw new Exception("Matrix is rank deficient.");
}
// Copy right hand side
int nx = B.ColumnDimension;
double[][] X = B.ArrayCopy();
// Compute Y = transpose(Q)*B
for (int k = 0; k < n; k++) {
for (int j = 0; j < nx; j++) {
double s = 0.0;
for (int i = k; i < m; i++) {
s += QR[i][k]*X[i][j];
}
s = -s/QR[k][k];
for (int i = k; i < m; i++) {
X[i][j] += s*QR[i][k];
}
}
}
// Solve R*X = Y;
for (int k = n-1; k >= 0; k--) {
for (int j = 0; j < nx; j++) {
X[k][j] /= Rdiag[k];
}
for (int i = 0; i < k; i++) {
for (int j = 0; j < nx; j++) {
X[i][j] -= X[k][j]*QR[i][k];
}
}
}
return (new Matrix(X,n,nx).GetMatrix(0,n-1,0,nx-1));
}
///<summary>Solve A*X = B</summary>
///<param name="B">A Matrix with as many rows as A and any number of columns.</param>
///<returns>X so that L*L'*X = B</returns>
///<exception cref="ArgumentException">System.ArgumentException Matrix row dimensions must agree.</exception>
///<exception cref="Exception">System.Exception Matrix is not symmetric positive definite.</exception>
public Matrix Solve(Matrix B)
{
if (B.RowDimension != n) {
throw new ArgumentException("Matrix row dimensions must agree.");
}
if (!isspd) {
throw new Exception("Matrix is not symmetric positive definite.");
}
// Copy right hand side.
double[][] X = B.ArrayCopy();
int nx = B.ColumnDimension;
// Solve L*Y = B;
for (int k = 0; k < n; k++) {
for (int j = 0; j < nx; j++) {
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 k = n-1; k >= 0; k--) {
for (int j = 0; j < nx; j++) {
for (int i = k+1; i < n ; i++) {
X[k][j] -= X[i][j]*L[i][k];
}
X[k][j] /= L[k][k];
}
}
return new Matrix(X,n,nx);
}