public static AasenDecomposition LTLDecompose3(SymmetricMatrix M)
{
// Aasen's method, now with pivoting
int n = M.Dimension;
double[] a = new double[n];
double[] b = new double[n - 1];
SquareMatrix L = new SquareMatrix(n);
for (int i = 0; i < n; i++) L[i, i] = 1.0;
// working space for d'th column of H = T L^T
double[] h = new double[n];
// first row
a[0] = M[0, 0];
if (n > 1) b[0] = M[1, 0];
for (int i = 2; i < n; i++) L[i, 1] = M[i, 0] / b[0];
PrintLTLMatrices(h, a, b, L);
// second row
if (n > 1) {
a[1] = M[1, 1];
if (n > 2) b[1] = M[2, 1] - L[2, 1] * a[1];
for (int i = 3; i < n; i++) L[i, 2] = (M[i, 1] - L[i, 1] * a[1]) / b[1];
}
PrintLTLMatrices(h, a, b, L);
for (int d = 0; d < n; d++) {
Console.WriteLine("d = {0}", d);
// compute h (d'th row of T L^T)
if (d == 0) {
h[0] = M[0, 0];
} else if (d == 1) {
h[0] = b[0];
h[1] = M[1, 1];
} else {
h[0] = b[0] * L[d, 1];
h[1] = b[1] * L[d, 2] + a[1] * L[d, 1];
h[d] = M[d, d] - L[d, 1] * h[1];
for (int i = 2; i < d; i++) {
h[i] = b[i] * L[d, i + 1] + a[i] * L[d, i] + b[i - 1] * L[d, i - 1];
h[d] -= L[d, i] * h[i];
}
}
// compute alpha (d'th diagonal element of T)
if ((d == 0) || (d == 1)) {
a[d] = h[d];
} else {
a[d] = h[d] - b[d - 1] * L[d, d - 1];
}
Console.WriteLine("before pivot");
PrintMatrix(M);
PrintMatrix(L);
// find the pivot
if (d < (n - 1)) {
int p = d + 1;
double q = M[p, d];
for (int i = d + 2; i < n; i++) {
if (Math.Abs(M[i, d]) > Math.Abs(q)) {
p = i;
q = M[i, d];
}
}
Console.WriteLine("pivot = {0}", p);
// symmetricly permute the pivot element to M[d+1,d]
if (p != d + 1) {
// symmetricly permute the pivot element to M[d+1, d]
// we have to be a bit careful here, because some permutations will be done
// automatically by our SymmetricMatrix class due to symmetry
for (int i = 0; i < n; i++) {
if ((i == p) || (i == d + 1)) continue;
double t = M[d + 1, i];
M[d + 1, i] = M[p, i];
M[p, i] = t;
}
double tt = M[d + 1, d + 1];
M[d + 1, d + 1] = M[p, p];
M[p, p] = tt;
// also reorder the affected previously computed elements of L
for (int i = 1; i <= d; i++) {
double t = L[d + 1, i];
L[d + 1, i] = L[p, i];
L[p, i] = t;
}
Console.WriteLine("after pivot");
PrintMatrix(M);
PrintMatrix(L);
}
}
// compute beta (d'th subdiagonal element of T)
if (d < (n - 1)) {
b[d] = M[d + 1, d];
for (int i = 0; i <= d; i++) {
Console.WriteLine("n={0} d={1} i={2}", n, d, i);
b[d] -= L[d + 1, i] * h[i];
}
}
// compute (d+1)'th column of L
for (int i = d + 2; i < n; i++) {
L[i, d + 1] = M[i, d];
for (int j = 0; j <= d; j++) L[i, d + 1] -= L[i, j] * h[j];
L[i, d + 1] = L[i, d + 1] / b[d];
}
PrintLTLMatrices(h, a, b, L);
}
Console.WriteLine("Reconstruct");
SymmetricMatrix T = new SymmetricMatrix(n);
for (int i = 0; i < n; i++) {
T[i, i] = a[i];
}
for (int i = 0; i < (n - 1); i++) {
T[i + 1, i] = b[i];
}
SquareMatrix A = L * T * L.Transpose();
PrintMatrix(A);
SymmetricMatrix D = new SymmetricMatrix(n);
for (int i = 0; i < n; i++) {
D[i, i] = a[i];
}
for (int i = 0; i < (n - 1); i++) {
D[i + 1, i] = b[i];
}
for (int c = 1; c < (n - 1); c++) {
for (int r = c + 1; r < n; r++) {
D[r, c - 1] = L[r, c];
}
}
AasenDecomposition LTL = new AasenDecomposition(D);
return (LTL);
}
public static AasenDecomposition LTLDecompose3(SymmetricMatrix M)
{
// Aasen's method, now with pivoting
int n = M.Dimension;
double[] a = new double[n];
double[] b = new double[n - 1];
SquareMatrix L = new SquareMatrix(n);
for (int i = 0; i < n; i++)
{
L[i, i] = 1.0;
}
// working space for d'th column of H = T L^T
double[] h = new double[n];
// first row
a[0] = M[0, 0];
if (n > 1)
{
b[0] = M[1, 0];
}
for (int i = 2; i < n; i++)
{
L[i, 1] = M[i, 0] / b[0];
}
PrintLTLMatrices(h, a, b, L);
// second row
if (n > 1)
{
a[1] = M[1, 1];
if (n > 2)
{
b[1] = M[2, 1] - L[2, 1] * a[1];
}
for (int i = 3; i < n; i++)
{
L[i, 2] = (M[i, 1] - L[i, 1] * a[1]) / b[1];
}
}
PrintLTLMatrices(h, a, b, L);
for (int d = 0; d < n; d++)
{
Console.WriteLine("d = {0}", d);
// compute h (d'th row of T L^T)
if (d == 0)
{
h[0] = M[0, 0];
}
else if (d == 1)
{
h[0] = b[0];
h[1] = M[1, 1];
}
else
{
h[0] = b[0] * L[d, 1];
h[1] = b[1] * L[d, 2] + a[1] * L[d, 1];
h[d] = M[d, d] - L[d, 1] * h[1];
for (int i = 2; i < d; i++)
{
h[i] = b[i] * L[d, i + 1] + a[i] * L[d, i] + b[i - 1] * L[d, i - 1];
h[d] -= L[d, i] * h[i];
}
}
// compute alpha (d'th diagonal element of T)
if ((d == 0) || (d == 1))
{
a[d] = h[d];
}
else
{
a[d] = h[d] - b[d - 1] * L[d, d - 1];
}
Console.WriteLine("before pivot");
PrintMatrix(M);
PrintMatrix(L);
// find the pivot
if (d < (n - 1))
{
int p = d + 1;
double q = M[p, d];
for (int i = d + 2; i < n; i++)
{
if (Math.Abs(M[i, d]) > Math.Abs(q))
{
p = i;
q = M[i, d];
}
}
Console.WriteLine("pivot = {0}", p);
// symmetricly permute the pivot element to M[d+1,d]
if (p != d + 1)
{
// symmetricly permute the pivot element to M[d+1, d]
// we have to be a bit careful here, because some permutations will be done
// automatically by our SymmetricMatrix class due to symmetry
for (int i = 0; i < n; i++)
{
if ((i == p) || (i == d + 1))
{
continue;
}
double t = M[d + 1, i];
M[d + 1, i] = M[p, i];
M[p, i] = t;
}
double tt = M[d + 1, d + 1];
M[d + 1, d + 1] = M[p, p];
M[p, p] = tt;
// also reorder the affected previously computed elements of L
for (int i = 1; i <= d; i++)
{
double t = L[d + 1, i];
L[d + 1, i] = L[p, i];
L[p, i] = t;
}
Console.WriteLine("after pivot");
PrintMatrix(M);
PrintMatrix(L);
}
}
// compute beta (d'th subdiagonal element of T)
if (d < (n - 1))
{
b[d] = M[d + 1, d];
for (int i = 0; i <= d; i++)
{
Console.WriteLine("n={0} d={1} i={2}", n, d, i);
b[d] -= L[d + 1, i] * h[i];
}
}
// compute (d+1)'th column of L
for (int i = d + 2; i < n; i++)
{
L[i, d + 1] = M[i, d];
for (int j = 0; j <= d; j++)
{
L[i, d + 1] -= L[i, j] * h[j];
}
L[i, d + 1] = L[i, d + 1] / b[d];
}
PrintLTLMatrices(h, a, b, L);
}
Console.WriteLine("Reconstruct");
SymmetricMatrix T = new SymmetricMatrix(n);
for (int i = 0; i < n; i++)
{
T[i, i] = a[i];
}
for (int i = 0; i < (n - 1); i++)
{
T[i + 1, i] = b[i];
}
SquareMatrix A = L * T * L.Transpose();
PrintMatrix(A);
SymmetricMatrix D = new SymmetricMatrix(n);
for (int i = 0; i < n; i++)
{
D[i, i] = a[i];
}
for (int i = 0; i < (n - 1); i++)
{
D[i + 1, i] = b[i];
}
for (int c = 1; c < (n - 1); c++)
{
for (int r = c + 1; r < n; r++)
{
D[r, c - 1] = L[r, c];
}
}
AasenDecomposition LTL = new AasenDecomposition(D);
return(LTL);
}
