void Print(const UpperTriangularMatrix& X)
{
++PCN;
cout << "\nMatrix type: " << X.Type().Value() << " (";
cout << X.Nrows() << ", ";
cout << X.Ncols() << ")\n\n";
if (X.IsZero()) { cout << "All elements are zero\n" << flush; return; }
int nr=X.Nrows(); int nc=X.Ncols();
for (int i=1; i<=nr; i++)
{
int j;
for (j=1; j<i; j++) cout << "\t";
for (j=i; j<=nc; j++) cout << X(i,j) << "\t";
cout << "\n";
}
cout << flush; ++PCZ;
}
// produces the Cholesky decomposition of EAE where A = chol.t() * chol
// and E produces a LEFT circular shift of the rows and columns from
// 1,...,k-1,k,k+1,...l,l+1,...,p to
// 1,...,k-1,k+1,...l,k,l+1,...,p to
void left_circular_update_Cholesky(UpperTriangularMatrix &chol, int k, int l)
{
int nRC = chol.Nrows();
int i, j;
// I. compute shift of column k to the lth position
Matrix cholCopy = chol;
// a. grab column k
ColumnVector columnK = cholCopy.Column(k);
// b. shift columns k+1,...l to the LEFT
for(j = k+1; j <= l; ++j)
cholCopy.Column(j-1) = cholCopy.Column(j);
// c. copy the elements of columnK into the lth column of cholCopy
cholCopy.Column(l) = 0.0;
for(i = 1; i <= k; ++i)
cholCopy(i,l) = columnK(i);
// II. apply and compute Given's rotations
int nGivens = l-k;
ColumnVector cGivens(nGivens); cGivens = 0.0;
ColumnVector sGivens(nGivens); sGivens = 0.0;
for(j = k; j <= nRC; ++j)
{
ColumnVector columnJ = cholCopy.Column(j);
// apply the previous Givens rotations to columnJ
int imax = j - k; if (imax > nGivens) imax = nGivens;
for(int i = 1; i <= imax; ++i)
{
int gIndex = i;
int topRowIndex = k + i - 1;
GivensRotationR(cGivens(gIndex), sGivens(gIndex),
columnJ(topRowIndex), columnJ(topRowIndex+1));
}
// compute a new Given's rotation when j < l
if(j < l)
{
int gIndex = j-k+1;
columnJ(j) = pythag(columnJ(j), columnJ(j+1), cGivens(gIndex),
sGivens(gIndex));
columnJ(j+1) = 0.0;
}
cholCopy.Column(j) = columnJ;
}
chol << cholCopy;
}
// produces the Cholesky decomposition of EAE where A = chol.t() * chol
// and E produces a RIGHT circular shift of the rows and columns from
// 1,...,k-1,k,k+1,...l,l+1,...,p to
// 1,...,k-1,l,k,k+1,...l-1,l+1,...p
void right_circular_update_Cholesky(UpperTriangularMatrix &chol, int k, int l)
{
int nRC = chol.Nrows();
int i, j;
// I. compute shift of column l to the kth position
Matrix cholCopy = chol;
// a. grab column l
ColumnVector columnL = cholCopy.Column(l);
// b. shift columns k,...l-1 to the RIGHT
for(j = l-1; j >= k; --j)
cholCopy.Column(j+1) = cholCopy.Column(j);
// c. copy the top k-1 elements of columnL into the kth column of cholCopy
cholCopy.Column(k) = 0.0;
for(i = 1; i < k; ++i) cholCopy(i,k) = columnL(i);
// II. determine the l-k Given's rotations
int nGivens = l-k;
ColumnVector cGivens(nGivens); cGivens = 0.0;
ColumnVector sGivens(nGivens); sGivens = 0.0;
for(i = l; i > k; i--)
{
int givensIndex = l-i+1;
columnL(i-1) = pythag(columnL(i-1), columnL(i),
cGivens(givensIndex), sGivens(givensIndex));
columnL(i) = 0.0;
}
// the kth entry of columnL is the new diagonal element in column k of cholCopy
cholCopy(k,k) = columnL(k);
// III. apply these Given's rotations to subsequent columns
// for columns k+1,...,l-1 we only need to apply the last nGivens-(j-k) rotations
for(j = k+1; j <= nRC; ++j)
{
ColumnVector columnJ = cholCopy.Column(j);
int imin = nGivens - (j-k) + 1; if (imin < 1) imin = 1;
for(int gIndex = imin; gIndex <= nGivens; ++gIndex)
{
// apply gIndex Given's rotation
int topRowIndex = k + nGivens - gIndex;
GivensRotationR(cGivens(gIndex), sGivens(gIndex),
columnJ(topRowIndex), columnJ(topRowIndex+1));
}
cholCopy.Column(j) = columnJ;
}
chol << cholCopy;
}
// produces the Cholesky decomposition of A - x.t() * x where A = chol.t() * chol
void downdate_Cholesky(UpperTriangularMatrix &chol, RowVector x)
{
int nRC = chol.Nrows();
// solve R^T a = x
LowerTriangularMatrix L = chol.t();
ColumnVector a(nRC); a = 0.0;
int i, j;
for (i = 1; i <= nRC; ++i)
{
// accumulate subtr sum
Real subtrsum = 0.0;
for(int k = 1; k < i; ++k) subtrsum += a(k) * L(i,k);
a(i) = (x(i) - subtrsum) / L(i,i);
}
// test that l2 norm of a is < 1
Real squareNormA = a.SumSquare();
if (squareNormA >= 1.0)
Throw(ProgramException("downdate_Cholesky() fails", chol));
Real alpha = sqrt(1.0 - squareNormA);
// compute and apply Givens rotations to the vector a
ColumnVector cGivens(nRC); cGivens = 0.0;
ColumnVector sGivens(nRC); sGivens = 0.0;
for(i = nRC; i >= 1; i--)
alpha = pythag(alpha, a(i), cGivens(i), sGivens(i));
// apply Givens rotations to the jth column of chol
ColumnVector xtilde(nRC); xtilde = 0.0;
for(j = nRC; j >= 1; j--)
{
// only the first j rotations have an affect on chol,0
for(int k = j; k >= 1; k--)
GivensRotation(cGivens(k), -sGivens(k), chol(k,j), xtilde(j));
}
}
// produces the Cholesky decomposition of A + x.t() * x where A = chol.t() * chol
void update_Cholesky(UpperTriangularMatrix &chol, RowVector x)
{
int nc = chol.Nrows();
ColumnVector cGivens(nc); cGivens = 0.0;
ColumnVector sGivens(nc); sGivens = 0.0;
for(int j = 1; j <= nc; ++j) // process the jth column of chol
{
// apply the previous Givens rotations k = 1,...,j-1 to column j
for(int k = 1; k < j; ++k)
GivensRotation(cGivens(k), sGivens(k), chol(k,j), x(j));
// determine the jth Given's rotation
pythag(chol(j,j), x(j), cGivens(j), sGivens(j));
// apply the jth Given's rotation
{
Real tmp0 = cGivens(j) * chol(j,j) + sGivens(j) * x(j);
chol(j,j) = tmp0; x(j) = 0.0;
}
}
}