Existence of a Solution for Matrix Equations
Thanks to this site
Let 
be an 
matrix, 
is 
and 
is 
![Ax = b\\ rank(A) = r\\ rank(\left[ A b \right] ) = r_a](http://mynotes.ws/wp-content/ql-cache/quicklatex.com-051f64bd79c2d88a127372f2756058a4_l3.png "Rendered by QuickLaTeX.com")
If 
then there are an infinite number of solutions.
If 
then there is one unique solution.
If 
then there are no solutions.
Therefore, when is a linear combination of the columns in , then 
(since is not independent of the other columns in , it will not add to the rank) and there is at least one solution. But if is not a linear combination of the columns in then there are no solutions.
Installing Drupal
yum install mysql-server, httpd, and drupal. Follow the directions here for setting up the database. In case that link changes, see the following:
mv drupal-x.x /var/www/
cp sites/default/default.settings.php sites/default/settings.php
chmod a+w sites/default/settings.php
chmod a+w sites/default
mysqladmin -u username -p create databasename
mysql -u username -p
GRANT SELECT, INSERT, UPDATE, DELETE, CREATE, DROP, INDEX, ALTER, CREATE TEMPORARY TABLES, LOCK TABLES
ON databasename.*
TO 'username'@'localhost' IDENTIFIED BY 'password';
Change lines in /etc/httpd/conf/httpd.conf:
DocumentRoot "/var/www/drupal-x.x"
...
<Directory "/var/www/drupal-x.x">
Adding a manifest to gcc-compiled .dll that links to msvcr90
First, create a manifest file, called $(DLLNAME).dll.manifest
<assembly xmlns="urn:schemas-microsoft-com:asm.v1" manifestVersion="1.0">
<trustInfo xmlns="urn:schemas-microsoft-com:asm.v3">
<security>
<requestedPrivileges>
<requestedExecutionLevel level="asInvoker" uiAccess="false"></requestedExecutionLevel>
</requestedPrivileges>
</security>
</trustInfo>
<dependency>
<dependentAssembly>
<assemblyIdentity type="win32" name="Microsoft.VC90.CRT" version="9.0.21022.8" processorArchitecture="*" publicKeyToken="1fc8b3b9a1e18e3b"></assemblyIdentity>
</dependentAssembly>
</dependency>
</assembly>
Then make a small $(DLLNAME).rc file:
#include "winuser.h"
2 RT_MANIFEST "$(DLLNAME).dll.manifest"
Then compile this using windres in Makefile:
WINDRES = windres.exe
$(DLLNAME).o: $(DLLNAME).rc
$(WINDRES) --input $(DLLNAME).rc --output $(DLLNAME).o --output-format=coff
and link as usual:
$(BIN): $(OBJ) $(DLLNAME).o
$(CPP) $(SHARED) $(OBJ) $(DLLNAME).o $(LINKFLAGS) $(LIBS) -o $(BIN)
Thanks to this site and this one.
Extending Linux Logical Volume
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First, find the volume in /dev/. On the server, this was in /dev/cciss/:
# ls
c0d0 c0d0p1 c0d0p2 c0d1 c0d1p1
c0d0 was already in the Logical Volume; we want to add c0d1.
# pvcreate /dev/cciss/c0d1p1
Physical volume "/dev/cciss/c0d1p1" successfully created
Then
vgextend VolGroup00 /dev/cciss/c0d1p1
Volume group "VolGroup00" successfully extended
Need to have installed system-config-lvm. This is a graphical tool where you can go into the Logical View of VolGroup00, “Edit Properties”, and reset the size (“Use remaining”). Click “OK” and wait for the volume to get bigger.
Amount of Space on Drive Used by Certain Files
For the total disk space of files matching “searchstring”
ls -l | awk '/searchstring/ {space = space + $5} END {print space}'
Or for the total of all files:
ls -l | awk '{space = space + $5} END {print space}'
R Optimization
Check eigenvalues of hessian of optimized sum of squares to check for singular gradient matrix. A singular gradient matrix has infinite solutions, so the best you can do is the optimized set of values plus a linearly scaled vector. The vector is equal to the eigenvector associated with the eigenvalue that is numerically indistinguishable from zero.
So if you have
optimfunc <- function(x, data, sevcalcfunc, optimgoal, ...) {
calcsev <- sevcalcfunc(x, data, ...)
sumsqerr <- sum((optimgoal - calcsev)^2)
return(sumsqerr)
}
and some function, then you can optimize
system.time(optimsimple <- optim(c(.3, -12614.716, .1), optimfunc, hessian = TRUE,data = reg.data, sevcalcfunc = calclosssimple, optimgoal = reg.data$maxloss))
and
eigen(optimsimple$hessian)
$values
[1] 1.498617e+16 1.868481e+14 8.412726e+04
Since the 3rd eigenvalue is very small compared to the first two, adding any constant z times the 3rd eigenvector to the optimized solution doesn't really change the value of the optimized sum of squares, and can therefore be considered a solution as well.
eigen(optimsimple$hessian)$vectors[,3]
[1] 1.956755e-06 1.000000e+00 4.573782e-06
> optimfunc(optimsimple$par, reg.data, calclosssimple, reg.data$maxloss)
[1] 3.793891e+14
> optimfunc(optimsimple$par + 1000 * eigen(optimsimple$hessian)$vectors[,3], reg.data, calclosssimple, reg.data$maxloss)
[1] 3.794212e+14
> optimfunc(optimsimple$par - 1000 * eigen(optimsimple$hessian)$vectors[,3], reg.data, calclosssimple, reg.data$maxloss)
[1] 3.794211e+14
Search and replace in multiple files
Thanks to this site
for fl in *.txt; do
mv $fl $fl.old
sed ’s/FINDSTRING/REPLACESTRING/g’ $fl.old > $fl
#rm -f $fl.old
done
Clear Memory Cache
Clear memory cache:
sync; echo 3 > /proc/sys/vm/drop_caches
Thanks to this site
Compile R with ATLAS
First, install yum-utils so you can run yum-builddeps:
yum install yum-utils
Then build all R dependencies with:
yum-builddep R
Add EPEL to your code repositories list, yum install atlas and wget the latest R code
rpm -Uvh http://download.fedora.redhat.com/pub/epel/5/i386/epel-release-5-4.noarch.rpm
yum install atlas-devel.x86_64
wget http://cran.r-project.org/src/base/R-2/R-2.11.1.tar.gz
Gunzip and tar into /usr/local/lib64, then compile R from inside R-version/
./configure --with-blas="-L/usr/lib64/atlas -lptf77blas -lpthread -latlas" --enable-R-shlib --enable-BLAS-shlib
make
make check
make install
And test:
set.seed(10)
x <- matrix(runif(1000000), 1000,1000)
y <- matrix(runif(1000000), 1000,1000)
system.time(z <- x %*% y)
#base
# user system elapsed
# 4.036 0.008 4.039
#atlas
# user system elapsed
# 0.428 0.020 0.457
set.seed(10)
x <- matrix(runif(10000000), 1000,10000)
y <- matrix(runif(10000000), 10000,1000)
system.time(z <- x %*% y)
#base
# user system elapsed
# 39.302 0.016 39.309
#atlas
# user system elapsed
# 2.840 0.028 2.873
Valuations under Funding Costs, Counterparty Risk and Collateralization.
http://www.christian-fries.de/finmath/discounting/Discounting.pdf
