// 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));
}
}
void SetImage::pave(const IntervalVector& x, double epsilon) {
Linside.clear();
Lboundary.clear();
PdcImageSubset p_fin(f,x,p_in);
assert(x.size()==n);
stack<IntervalVector> Ldomain;
IntervalVector xtilde(n);
IntervalVector ytilde(n);
LargestFirst lf(epsilon);
Ldomain.push(x);
while (! Ldomain.empty()) {
xtilde = Ldomain.top();
Ldomain.pop();
c_out.contract(xtilde);
if (xtilde.is_empty()) {
continue;
}
// use natural extension
ytilde=f.eval_vector(xtilde);
// improve with centered form
ytilde&=f.eval_vector(xtilde.mid())+f.jacobian(xtilde)*(xtilde-xtilde.mid());
if (p_in.test(xtilde)==YES && p_fin.test(cart_prod(xtilde,ytilde))==YES)
Linside.push_back(ytilde);
else if (xtilde.max_diam()<=epsilon)
Lboundary.push_back(ytilde);
else {
pair<IntervalVector,IntervalVector> boxes=lf.bisect(xtilde);
Ldomain.push(boxes.first);
Ldomain.push(boxes.second);
}
}
}
QString expando(const QString cmd)
{
QString rv;
QString word, c, endc;
QProcessEnvironment env=QProcessEnvironment::systemEnvironment();
rv.reserve(cmd.length()+1024);
int state=0;
int escape=0;
for (int i=0;i<cmd.length();i++)
{
c=cmd.mid(i,1);
if (escape!=0)
{
if (escape==1) rv+=c; else word+=c;
escape=0;
continue;
}
switch (state)
{
case 2: // looking for word sep
if (c=="\\")
{
escape=2;
continue;
}
if (c[0].isLetterOrNumber()||c[0]=='-'||c[0]=='_') // is this list complete?
{
word+=c;
continue;
}
rv+=env.value(word,"");
state=0;
// fall through
case 0: // no env
if (c=="\\")
{
escape=1;
continue;
}
if (c!="$")
{
rv+=c;
continue;
}
word="";
state=1;
continue;
case 1: // found $
if (c=="("||c=="{"||c=="[")
{
endc=")";
if (c=="{") endc="}";
if (c=="[") endc="]";
state=3;
continue;
}
if (c!="\\") word+=c;
state=2;
continue;
case 3: // looking for )
if (c=="\\")
{
escape=2;
continue;
}
if (c==endc)
{
if (word.mid(0,9)=="QLPROMPT:")
{
bool ok;
QString title="qlaunch prompt", prompt="", deftext="";
int n;
prompt=word.mid(9); // put $QLPROMPT:prompt:default:title (all but prompt optional)
n=prompt.indexOf(':');
if (n!=-1)
{
deftext=prompt.mid(n+1);
prompt=prompt.mid(0,n);
n=deftext.indexOf(':');
if (n!=-1)
{
title=deftext.mid(n+1);
deftext=deftext.mid(0,n);
}
}
QString itext=QInputDialog::getText(NULL,title,prompt,QLineEdit::Normal,deftext,&ok);
if (ok) rv+=itext;
}
else rv+=env.value(word,"");
state=0;
continue;
}
word+=c;
continue;
}
}
if (state==2||state==3) rv+=env.value(word,""); // pick up any left over word at the end
return xtilde(rv);
}
