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welcome to the third part of the multivariate polynomials in c++ using templates series. this part will explain how to compute partial derivatives using partial template specialization tricks.

the idea is very simple. given f \in R[X_0, \dots, X_{n-1}], written as f = \sum_{i=0}^k a_i X_0^i with a_i \in R[X_1, \dots, X_{n-1}], we have

\frac{\partial f}{\partial X_0} = \sum_{i=1}^k a_i X_0^{i-1}


\frac{\partial f}{\partial X_j} = \sum_{i=0}^k \biggl( \frac{\partial}{\partial X_j} a_i \biggr) X_0^i \quad \text{for } j > 0.

this allows us to implement partial derivatives iteratively. we assume that the number of the indeterminate by which one should differentiate is already known during compile time. the implementation will be done by a templated function

1template<int N, int n, class T>
2inline poly<n, T> derivative(const poly<n, T> & p);

the purpose of the template parameter N is to say that \frac{\partial p}{\partial X_N} should be computed. given a polynomial p, the function can be called simply by derivative<N>(p). to implement derivative<N>(p), we would like to again use partial template specialization to distinguish between the cases N == 0 and N > 0. unfortunately, partial template specialization is not available for template functions. hence, we need to introduce a helper template class:

1template<int N, int n, class T>
2class derivative_computer;

it should contain one static function, static inline void computeDerivative(const poly<n, T> & src, poly<n, T> & dest); which allows to compute the partial derivative \frac{\partial}{\partial X_N} of the polynomial in src and store it into dest. then derviative<N>(p) can simply be implemented by

1template<int N, int n, class T>
2inline poly<n, T, std::allocator<T> > derivative(const poly<n, T, std::allocator<T> > & p)
4    poly<n, T> res;
5    derivative_computer<N, n, T>::computeDerivative(p, res);
6    return res;

now let us focus on the class template derivative_computer<N, n, T>. the general implementation, for N > 0, uses the second formula from above and iteratively calls derivative_computer<N-1, n-1, T>::computeDerivative():

 1template<int N, int n, class T>
 2class derivative_computer
 5    static inline void computeDerivative(const poly<n, T> & src, poly<n, T> & dest)
 6    {
 7        dest.d_value.resize( + 1);
 8        for (int i = 0; i <=; ++i)
 9            derivative_computer<N - 1, n - 1, T>::computeDerivative(src[i], dest[i]);
10        dest.normalize();
11    }

now we are left with the case that N = 0. the implementation here is also simple:

 1template<int n, class T>
 2class derivative_computer<0, n, T>
 5    static inline void computeDerivative(const poly<n, T> & src, poly<n, T> & dest)
 6    {
 7        dest.d_value.resize( >= 0 ? : 0);
 8        for (int i = 1; i <=; ++i)
 9            dest[i - 1] = src[i] * (T)i;
10        dest.normalize();
11    }

here, we cast i to an object of type T before multiplication. from this point on, no further iteration is needed.

we are left with the case that some (malicious or just confused) user calls derivative<N>(p) for a polynomial p of type poly<n, T> with n <= N. in that case, the compiler tries to instanciate derivative_computer<k, 0, T>::computeDerivative() and tries to treat poly<0, T> as a polynomial. this will result in strange error messages, and it would be nice if there is something more readable. for that reason, i added another two specializations:

 1template<class T>
 2class derivative_computer<0, 0, T>
 5    class ERROR_N_must_be_strictly_less_than_n_in_derivative_template;
 7    static inline void computeDerivative(const poly<0, T> &, poly<0, T> &)
 8    {
 9        ERROR_N_must_be_strictly_less_than_n_in_derivative_template();
10    }
13template<int N, class T>
14class derivative_computer<N, 0, T>
17    class ERROR_N_must_be_strictly_less_than_n_in_derivative_template;
19    static inline void computeDerivative(const poly<0, T> &, poly<0, T> &)
20    {
21        ERROR_N_must_be_strictly_less_than_n_in_derivative_template();
22    }

these two definitions compile fine, but as soon as one of them is instanciated by the compiler, it will try to find out something about the class ERROR_N_must_be_strictly_less_than_n_in_derivative_template and not succeed, and spit out some error message hopefully containing the string ERROR_N_must_be_strictly_less_than_n_in_derivative_template. for example, if i replace the line computing the partial derivative of the intersection equation in poly-test.cpp by std::cout << derivative<2>(l) << "\n";, g++ will complain as follows:

1In file included from poly-test.cpp:2:0:
2polynomial.hpp: In static member function ‘static void derivative_computer<N, 0, T>::computeDerivative(const poly<0, T>&, poly<0, T>&) [with int N = 1, T = double]’:
3polynomial.hpp:1200:13:   instantiated from ‘static void derivative_computer<N, n, T>::computeDerivative(const poly<n, T>&, poly<n, T>&) [with int N = 2, int n = 1, T = double]4polynomial.hpp:1247:5:   instantiated from ‘poly<n, T> derivative(const poly<n, T>&) [with int N = 2, int n = 1, T = double]5poly-test.cpp:19:33:   instantiated from here
6polynomial.hpp:1227:9: error: invalid use of incomplete type ‘struct derivative_computer<1, 0, double>::ERROR_N_must_be_strictly_less_than_n_in_derivative_template’
7polynomial.hpp:1223:11: error: declaration of ‘struct derivative_computer<1, 0, double>::ERROR_N_must_be_strictly_less_than_n_in_derivative_template’
8make: *** [poly-test.o] Error 1

this is not exactly readable, but the uppercase ERROR quickly lets you look at the important part of that message. (note that i removed the allocators from the error messages, so that it is in sync with the above writing. if you try it out yourself with the source code, you'll see a slightly more complicated error message.)

this completes part three of the series. in the next part, i will discuss how to evaluate polynomials efficiently using templates.

welcome to the second part of the multivariate polynomials in c++ using templates series. this part will explain how to store multivariate polynomials via templates, how to implement storage and operations.

multivariate polynomial rings in mathematics.

as you might recall from your algebra courses, R[x_0, \dots, x_{n-1}] denotes the (ring, in case R itself is a ring) of multivariate polynomials in the n indeterminates (also called variables) X_0, \dots, X_{n-1} with coefficients in R. note that i numerate the indeterminates beginning with zero, which is rather intuitive for a c++ programmer, though often not for a mathematician. there are different ways to treat such polynomials; for example, one can work with multiindices i = (i_0, \dots, i_{n-1}) \in \N^n; for such an i, let a_i be an element of R and define X^i := X_0^{i_0} \cdots X_{n-1}^{e_{n-1}}; then a polynomial f \in R[X_0, \dots, X_{n-1}] can be represented as f = \sum_{i \in \N^n} a_i X^i. clearly, one does not want to work with infinite objects, whence one restricts to a subset of \N^n, for example of type [0, (k_0, \dots, k_{n-1})] \cap \N^n = \{ i \in \N^n \mid 0 \le i_j \le k_j \text{ for } j = 0, \dots, n-1 \}. this is one way of representing multivariate polynomials, and fine for several applications, but not the most easy one to implement in c++ as a template depending on n.

another way to treat multivariate polynomials is to iteratively write R[X_0, \dots, X_{n-1}] = (R[X_1, \dots, X_{n-1}])[X_0] (one could also use R[X_0, \dots, X_{n-1}] = (R[X_0, \dots, X_{n-2}])[X_{n-1}], but i will stick to the other as it is more natural for the implementation i'm describing). hence, a polynomial f \in R[X_0, \dots, X_{n-1}] is interpreted as an <i>univariate</i> polynomial with coefficients in the ring R[X_1, \dots, X_{n-1}]. this allows a recursive definition which is well-suited for templates.

implementing the data structure using templates.

now let us leave the realm of mathematics for a while and talk about c++. the ring R mentioned above will be replaced by an abstract type T, which will be a template parameter. note that in the discussion below, we will ignore allocators, and just talk about them at the very end of this article. our main template type will be template<int n, class T> class poly;, which essentially contains a std::vector<poly<n - 1, T> > to store coefficients. (the storage is in fact more delicate, we will discuss this below.)

clearly, this is not enough yet. the compiler has no clue that we are talking about multivariate polynomials, and that n should always be non-negative, and it also does not knows that poly<0, T> is essentially an element of T. the way to solve this is partial template specialization: we create a specialized version of poly<n, T> for the case n == 0. the declaration is as follows:

 1template<int n, class T>
 2class poly; // forward declaration
 4template<class T>
 5class poly<0, T> // specialize for n = 0
 8    T d_value;
10    ...
13template<int n, class T>
14class poly // the general implementation
17    std::vector<poly<n - 1, T> > d_value;
19    ...

the specialization for n == 0 has a rather simple implementation. besides a constructor accepting an object of type T, with default value T(), it has a casting operator to T, provides an isZero() predicate, assignment and arithmetic operations, output to std::ostreams, and using std::swap() to swap two poly<0, T> objects, it has a simple “evaluate” function implemented as operator() which is essentially the same as the cast operator operator T().

the general implemenation is more interesting. there is a method normalize() which ensures that d_value.back() is the leading coefficient and d_value.size() - 1 the degree of the polynomial; note that we use -1 for the degree of the zero polynomial. these conditions are used for example in the implementation of isZero(), leading() and degree(). for most operations, the polynomial itself makes sure it is normalized, but when accessing a non-const polynomial with operator[], it might change its degree. after that, one better calls normalize(). it would also be possible to do this automatically using more template tricks, but i wanted to leave some responsibility with the user of the template. :-)

the arithmetic operations are pretty straightforward to implement; one has to resize d_value and later call normalize() to make sure nothing bad happens. for multiplication, one creates a temporary which contains the product (the maximal degree of the product is known, but normalize() should be run afterwards since there is no garuantee that the ring containing the objects of type T is zero-divisor free). as an example, adding two polynomials is accomplished by the following operator:

 1    inline poly operator + (const poly & q) const
 2    {
 3        poly r(*this);
 4        if (r.d_value.size() < q.d_value.size())
 5            r.d_value.resize(q.d_value.size());
 6        for (unsigned i = 0; i < q.d_value.size(); ++i)
 7            r[i] += q[i];
 8        r.normalize();
 9        return r;
10    }

first, a copy of the first argument (*this) is made. then, its size is adjusted to make sure there's enough room. then, simply all coefficients of the second argument are added, and finally the result is normalize()d and returned. as another example, here's polynomial multiplication:

 1    inline poly operator * (const poly & p) const
 2    {
 3        int d = + degree();
 4        if (d < -1)
 5            d = -1;
 6        poly r(true, d + 1);
 7        if (!isZero() && !p.isZero())
 8            for (unsigned i = 0; i < r.d_value.size(); ++i)
 9                for (unsigned j = 0; j < d_value.size(); ++j)
10                    if (i < j + p.d_value.size())
11                        r[i] += d_value[j] * p[i - j];
12        r.normalize();
13        return r;
14    }

note that as we are inside the template, we can write poly instead of poly<n, T>. moreover, here, use is made of a special constructor poly(bool, unsigned size) which initializes the polynomial with size coefficients. the bool is just to ensure that the compiler won't confuse this constructor with any of the others. this constructor is made private.

printing the polynomial to a string is slightly more involved, as the correct names should be used for the indeterminates. for this, i use a method void print(std::ostream & s, int N = n) const which keeps track of the maximal number of indeterminates. the operator<< is then implemented as follows:

1    friend inline std::ostream & operator << (std::ostream & s, const poly & p)
2    {
3        p.print(s);
4        return s;
5    }

the implementation of print(...) works recursively, and N keeps track of n of the top-level instance. the indeterminate at the current level has index N - n.

finally, let me announce evaluation of polynomials. this will be done with operator(). the implementation inside poly<n, T> is very easy:

1    template<class S>
2    inline poly_evaluator<n, T, S> operator() (const S & x) const
3    {
4        return poly_evaluator<n, T, S>(*this, x);
5    }

the interesting part, namely the implementation of the poly_evaluator<n, T, S> template, will be discussed in part four of this series.

making the storage more intelligent.

in the above code, we use a simple std::vector<poly<n - 1, T> > to store the coefficients, which are (except for n=1) itself polynomials. for example, doing a simple resize of d_value in poly<11, T> will result in maybe many objects of type poly<10, T> to be created, copied and destroyed, and this in turn causes maybe many more objects of type poly<9, T> to be created, copied and destroyed, and so on. this is not exactly optimal. one solution is to replace std::vector<poly<n, T> > by std::vector<poly<n, T>*>, i.e. store pointers to polynomials instead of polynomials itself. this will make resizes very efficient, since only pointers have to moved around. unfortunately, it has a drawback: in the case n == 1, this is often not very optimal, as if T==int, we are storing pointers to ints, which are usually at least the same size as the ints themselves. this leads to inefficient code. one solution is to introduce another partial specialization for n=1, but this results in a lot of duplicated code. a more clever solution is to create another template, like template<class T, bool usePointers> class poly_ivector;, which stores pointers to T and makes sure objects are properly created and destroyed in case usePointers == true, and which is simply std::vector<T> in disguise in case usePointers == false. it provides a simple interface, offering the number of stored elements (unsigned size() const), to resize the array (void resize(unsigned)), accessing elements via operator[], and accessing the last element via back(). finally, it offers a void swap(poly_ivector<T, usePointers> &) method to swap the contents of two poly_ivector<T, usePointers> instances. if you are interested in the implementation of poly_ivector<T, bool>, please look at the source code.


in the real implementation, i've added basic support for allocators. these allow to provide own allocation functions (malloc/free) for memory used to store the coefficients. note that my template is not working perfectly with allocators having states, but for state-less allocators, it works fine. if you need to use it with allocators having states, better check if the code does exactly what you want.

i used the polynomial class with a self-written allocator which provides thread local memory management. this was necessary for the aforementioned toy project, for which i wanted to do massive parallel computation (128 threads). it turned out that most threads were waiting for others, as there was a lot of allocation/deallocation going on and the global memory manager, protected by a mutex, made everyone wait. after throwing in a not very efficient, but working memory manager with thread-local storage for the management data, the program ran perfectly parallel.

in this (short) series, i want to explain how to implement support for multivariate polynomials in c++ using templates. for a toy project i worked on mostly last weekend, i needed to work with polynomials in different ways:

  • create a polynomial in three variables with double coefficients;
  • computing the partial derivatives of this polynomial;
  • evaluate the above polynomials in many points (with double coordinates);
  • plug in different univariate polynomials (of form \alpha_i T + \beta_i) into the first polynomial to obtain a univariate polynomial whose zeros correspond to the intersection of the surface defined by the multivariate polynomial together with a parameterized line;
  • test whether a univariate polynomial has zeros in (0, \infty) using sturm chains.

for a given, simple enough polynomial, everything can be done “by hand”, but sooner or later, this is annoying and too restricting. and it gets really annoying for the second part, where i wanted to obtain the univariate polynomial. therefore, i thought about how to create a multivariate polynomial template which provides means to implement all of the above in an easy manner. in the end, i had a multivariate polynomial template with a nice interface, for which – as a small example – the following test program compiles:

 1#include <iostream>
 2#include "polynomial.hpp"
 4int main()
 6    poly<2, int> f = Monomial<int>(1, 2) + 3 * Monomial<int>(4, 5);
 7    std::cout << f << "\n";
 8    int e = f(4)(2);
 9    std::cout << e << "\n";
10    std::cout << derivative<1>(f) << "\n";
11    poly<1, int> g = f(2), h = f(3);
12    std::cout << g << " and " << h << "\n";
13    std::cout << GCD<double>(g, h) << "\n"; 
14    // cast polynomials to poly<1, double> first; otherwise,
15    // result will be incorrect since int is not a field
16    std::cout << GCD(g, h) << "\n";
17    // to prove our point, check for yourself that the
18    // result is X_0^5 instead of X_0^2
19    poly<1, double> l = f(Monomial<double>(1) - 1)(Monomial<double>(1) - 2);
20    std::cout << l << "\n";
21    std::cout << derivative<0>(l) << "\n";
22    std::cout << GCD(l, derivative<0>(l)) << "\n";

compiling it via g++ -O3 -Wall -o poly-test poly-test.cpp, it yielded the following output when run:

1(1 X_1^2 X_0 + 3 X_1^5 X_0^4)
3(2 X_1 X_0 + 15 X_1^4 X_0^4)
4(2 X_0^2 + 48 X_0^5) and (3 X_0^2 + 243 X_0^5)
51 X_0^2
61 X_0^5
7(-100 + 632 X_0 + -1781 X_0^2 + 2905 X_0^3 + -3006 X_0^4 + 2043 X_0^5 + -912 X_0^6 + 258 X_0^7 + -42 X_0^8 + 3 X_0^9)
8(632 + -3562 X_0 + 8715 X_0^2 + -12024 X_0^3 + 10215 X_0^4 + -5472 X_0^5 + 1806 X_0^6 + -336 X_0^7 + 27 X_0^8)

let me explain this example. first, a polynomial f = X_0 X_1^2 + 3 X_0^4 X_1^5 in two indeterminates with int coefficients is created as the linear combination of two monomials (X_0 X_1^2 and X_0^4 X_1^5), which is the first line of the output. then the polynomial is evaluated in (4, 2), and the result is printed as line 2. after that, the partial derivative of the polynomial f with respect to X_1 is computed and written as line 3. after that, the polynomial is evaluated in X_0 in 2 and 3, yielding two new polynomials g(X_0) = f(2, X_0) and h(X_0) = f(3, X_0) which are printed in line 4. in line 5, the monic greatest common divisor of g and h is computed, for which the polynomials are treated as polynomials with double coefficients. that this is necessary is demonstrated by the sixth line, in which the output is printed when ints are used to compute the greatest common divisor: clearly, the result is wrong. then, in line 7 of the output, we evaluate f at (T - 1, T - 2), yielding a univariate polynomial \ell(X_0) = f(X_0 - 1, X_0 - 2) with double coefficients. line 8 of the output shows its derivative \ell', and line 9 the monic greatest common divisor of \ell and \ell'.

note that the program is easy to read for people used to the notion of multivariate polynomials and c++.

in the second part of the series, i want to describe how to represent the polynomials. in the third part, i want to explain how partial derivatives are computed. the fourth part will describe how to do efficient evaluation, and the fifth part will explain the long division and euclidean algorithm implementations.

finally, the source code of the library polynomial.hpp, the test program poly-test.cpp and a simple makefile (using g++) can be downloaded as a zip file here (10 kb).

in this post, i want to explain how to create a simple neat javascript/css “fullscreen” image viewer, or more concretely, the one i implemented myself here on spielwiese. feel free to re-use the code. it is not garuanteed to work everywhere, but it works fine with firefox, konqueror and chromium/chrome, to my knowledge, and probably also with safari (as it is based on konqueror), and hopefully with opera and more modern versions of internet explorer. i didn’t include any hacks to make it work under older versions of internet explorer or other browsers, but feel free to enhance my version and add a link to yours in the comments here.
i assume that so far, you include your images using

<a href=”path/to/image.jpeg”><img src=”path/to/image-thumbnail.jpeg” /></a>.

this displays a thumbnail (called path/to/image-thumbnail.jpeg) and allows the user to click it, to be forwarded to the real image (called path/to/image.jpeg). maybe you also use an alt tag, or specify the size of the thumbnail. that doesn’t really matter.
the idea is to add a javascript onclick event to the <a> tag, which will become active once the link (i.e. the image) is clicked and show the “popup”. in case the user disabled javascript, everything will fall back to the old behaviour: the user will be redirected to the image and no harm is done by the addition of the new code. the above html code has to be modified as follows:

<a href=”path/to/image.jpeg” onclick=”showPicture(‘path/to/image.jpeg’)“><img src=”path/to/image-thumbnail.jpeg” /></a>

the addition is printed in bold. the drawback of this method is that you have to replicate the image’s name, but then, you already had to do that for the thumbnail, and if you have automated the whole process, changing your script to insert this part shouldn’t be any problem.
so now let me show you how to implement the showPicture() function.

 1<script type="text/javascript">
 3function showPicture(url) {
 4 = "hidden";
 5  el = document.createElement("div");
 6 = "background: black; color: white; position: fixed; left: 0px; top: 0px; width: 100%; height: 100%; text-align: center; z-index: 1000;";
 7  el.onclick = Function("this.parentNode.removeChild(this); = 'visible';");
 8  var code = '<div style="width: 100%; height: 100%; display: table; position: static; overflow: hidden;"><div style="display: table-cell; vertical-align: middle; width:100%;">';
 9  code += '<img onclick="this.parentNode.parentNode.parentNode.removeChild( this.parentNode.parentNode); = \'visible\';" ';
10  code += 'style="max-width:' + window.innerWidth + 'px; max-height: ' + (window.innerHeight - 20) + 'px;" src="' + url + '" /><br />';
11  code += 'click the photo to return to the blog. click <a href="' + url + '">here</a> to go to the image file. copyright &copy; by whoever.';
12  code += '</div></div>';
13  el.innerHTML = code; 
14  document.documentElement.appendChild(el);
15  return false;

let me explain what it does. lines 1, 2, 17 and 18 tell your browser that there’s a javascript coming up in an xhtml compatible way; if you are using plain html (instead of xhtml), you might want to change lines 2 and 17. now let’s continue with the main function, showPicture(), expecting one parameter called url.
first, the function disables scrolling for the main document (line 4), as otherwise the user can scroll around while viewing the picture (which won’t move). then it creates a new <div> tag (line 5) and eventually adds it as a child to the document root (line 14). the tag is equiped with a css style (line 6) telling the browser to display it at a position relative to the browser’s viewpoint, namely in the upper right corner with the size equal to the viewpoint size – meaning it covers the whole viewpoint. it is colored in black, with text in it being centered and written in white. then, in line 7, a onclick event is set for the tag. it simply waits until the tag is clicked (or anything inside that tag, to be precise) and removes the tag, which closes the “fullscreen” view. it also enables scrolling of the document (which was disabled in line 4). the lines 8 to 12 build up the html code which will be put into the <div> tag in line 13. in line 8, we create two <div> tags which will ensure a vertical centering in sane browsers. lines 9 and 10 inserts the image specified in url; here, i set the max-width and max-heigth parameters to essentially the viewport size, which ensures that the image will be rescaled to fit into the viewpoint. i substract a bit on the vertical size so there’s some space for the “disclaimer”. the “disclaimer” is added in line 11, it tells the user how to get rid of the fullscreen view, provides a link to the image file, and shows a copyright notice.
you can adjust the code as you like, for example you can change the colors and the disclaimer. also adding keyboard support might be nice, so you can get rid of the viewer when pressing escape. i guess it isn’t too hard to do, i’ll probably do that somewhen later. you can also move all the formatting into your .css file, by replacing the = “…”; by el.className = “imageviewer”; or however you want to call the css class for that <div>.
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i recently presented a bash script which schedules computational tasks on multi-core machines. in the meanwhile, i fixed a bug in the display, made the program more flexible, and started to use local variables instead of global variables only. the new version is also more intelligent: it tries to adjust the running times of its controlled processes so that the running times are not far apart.
here is the newest version:

  3initProfile() {
  4    PROFILEFN=bigprimerunner-$PROFILE.profile
  5    CORES=`grep "^CORES " $PROFILEFN`
  9    eval STARTUP=$STARTUP
 12# Startup
 14if [ "$1" != "" ]
 16    PROFILE=$1
 18    PROFILE=`hostname`
 20if [ "$2" != "" ]
 25if [ "$CORES" == "" ]
 27    echo "Cannot load profile $PROFILEFN!"
 28    exit
 30echo Cores: $CORES
 31echo Load modifier: $LOADMODIFIER
 33# The command to execute
 36computeFreecores() {
 37    FREECORES=0
 38    local DAY=`date +%w`
 39    local LINE=`grep "^$DAY " $PROFILEFN`
 40    local LINE=${LINE/$DAY }
 41    local HOUR=`date +%k`
 42    for ((i=0;i<$HOUR;++i));
 43    do
 44        local LINE=${LINE#* }
 45    done
 46    local LINE=${LINE/ *}
 47    eval FREECORES=$LINE
 48    # Also determine how many jobs should be started
 51    eval STARTUP=$STARTUP
 54killProcess() { # One argument: PID of process to kill
 55    local PID=$1
 56    local FILE=`lsof -p $PID -F n 2>/dev/null | grep primedatabase | grep -v "\.nfs"`
 57    kill $PID 2> /dev/null
 58    local A=${FILE#n*}
 59    local A=${A/ (nfs*}
 60    if [ "$A" != "" ]
 61    then
 62        rm $A
 63        echo Killed $PID with open file $A
 64    else
 65        echo Killed $PID with no open file
 66    fi
 69stopsignal() {
 70    local PIDS=`jobs -p`
 71    echo
 72    echo
 73    echo Terminating...
 74    echo Killing: $PIDS
 75    for PID in $PIDS;
 76    do
 77        killProcess $PID
 78    done
 79    echo done.
 80    exit
 83trap 'stopsignal' 2
 87echo "Starting $STARTUP instances (in $BINDIR)"
 89filterRunning() { # Removes all PIDs from the arguments which are currently stopped
 90    ps -o pid= -o s= $* | grep R | sed -e "s/R//"
 93filterStopped() { # Removes all PIDs from the arguments
 94    ps -o pid= -o s= $* | grep T | sed -e "s/T//"
 97determineToAdd() {
 98    computeFreecores
 99    local LOAD=`uptime`
100    local LOAD=${LOAD#*average: }
101    local LOAD=${LOAD/,*}
102    local LOAD=${LOAD/.*}
104    local JOBS=`jobs -p`
105    local JOBS=`filterRunning $JOBS`
106    echo "Load: $[LOAD+LOADMODIFIER], Intended number of free cores: $FREECORES, Running: `echo $JOBS | wc -w`, Started: `jobs -p | wc -l` (should be $STARTUP)"
109continueOne() {
110    local JOBS=`jobs -p`
111    local JOBS=`filterStopped $JOBS`
112    if [ "$JOBS" != "" ]
113    then
114        local PID=`ps -o pid= --sort +time $JOBS | head -1`
115        echo Continuing $PID...
116        kill -SIGCONT $PID
117    fi
120stopOne() {
121    local JOBS=`jobs -p`
122    local JOBS=`filterRunning $JOBS`
123    if [ "$JOBS" != "" ]
124    then
125        local PID=`ps -o pid= --sort -time $JOBS | head -1`
126        echo Stopping $PID...
127        kill -SIGSTOP $PID
128    fi
131killOne() {
132    local JOBS=`jobs -p`
133    if [ "$JOBS" != "" ]
134    then
135        local PID=`ps -o pid= --sort -time $JOBS | head -1`
136        killProcess $PID
137    fi
140launchOne() {
141    echo "Launching \"$COMMAND\"..."
142    $COMMAND &
143    sleep 1.5
146computeTotaltimeInSecs() {
147    # Input: $1
148    # Output: $TOTALSECS
149    local I=$1
150    local SECS=${I##*:}
151    local REST=${I%:*}
152    local MINS=${REST##*:}
153    local REST=${REST%:*}
154    local HOURS=${REST##*-}
155    local DAYS=`expr "$REST" : '\([0-9]*-\)'`
156    local DAYS=${DAYS%-}
157    if [ "$DAYS" == "" ]
158    then
159        local DAYS=0
160    fi
161    if [ "$HOURS" == "" ]
162    then
163        local HOURS=0
164    fi
165    if [ "$MINS" == "" ]
166    then
167        local MINS=0
168    fi
169    echo "((($DAYS * 24) + $HOURS) * 60 + $MINS) * 60 + $SECS" | bc
172adjustProcesses() {
173    local JOBS=`jobs -p`
174    local JOBS=`filterRunning $JOBS`
175    if [ "$JOBS" != "" ]
176    then
177        local STOPPID=`ps -o pid= --sort -time $JOBS | head -1`
178        local JOBS=`jobs -p`
179        local JOBS=`filterStopped $JOBS`
180        if [ "$JOBS" != "" ]
181        then
182            local CONTPID=`ps -o pid= --sort +time $JOBS | head -1`
183            # Compute times
184            local I=`ps -o time= $STOPPID`
185            local STOPSEC=`computeTotaltimeInSecs $I`
186            local I=`ps -o time= $CONTPID`
187            local CONTSEC=`computeTotaltimeInSecs $I`
188            # Compare times
189            local CT=`echo $CONTSEC+60*5 | bc`
190            if [ $STOPSEC -gt $CT ]
191            then
192                echo Stopping $STOPPID and continuing $CONTPID
193                kill -SIGSTOP $STOPPID
194                kill -SIGCONT $CONTPID
195            fi
196        fi
197    fi
200# Start programs in the background
202for ((i=1;i<=STARTUP;++i));
204    launchOne
205    if [ $i -gt $ADD ]
206    then
207        sleep 1
208        kill -SIGSTOP %$i
209    fi
212# Start mainloop
213while [ 1 ]
215    sleep 60
217    # Determine how many processes should be added/removed
218    determineToAdd
220    # Stop/continue processes
221    if [ $ADD -gt 0 ]
222    then
223        # Add processes
224        echo ADD:$ADD
225        for ((i=0;i<ADD;++i))
226        do
227            continueOne
228        done
229    fi
230    if [ $ADD -lt 0 ]
231    then
232        REM=$[-ADD]
233        # Remove processes
234        echo REMOVE:$REM
235        for ((i=0;i<REM;++i))
236        do
237            stopOne
238        done;
239    fi
241    # Launch new processes or kill running ones
242    CURRLAUNCHED=`jobs -p | wc -l`
243    if [ $STARTUP != $CURRLAUNCHED ]
244    then
245        if [ $STARTUP -lt $CURRLAUNCHED ]
246        then
247            echo kill: $STARTUP $CURRLAUNCHED
248            for ((i=STARTUP;i<CURRLAUNCHED;++i));
249            do
250                killOne
251            done;
252        else
253            echo add: $CURRLAUNCHED $STARTUP
254            for ((i=CURRLAUNCHED;i<STARTUP;++i));
255            do
256                launchOne
257            done;
258        fi
259    fi
260    sleep 2
262    # Adjust
263    adjustProcesses
posted in: computer

ever had the problem that you have access to a big machine (with many cores), and you want to run many (tens of thousands) small computations, but you want to make sure that not too many cores are used?
i’ve had this problem, and since i now have a pretty nice (i think so) solution, i thought that maybe more people are interested in it. so here’s my setup. i have a program, let’s call it primefinder, which, for a certain input n (where n is a natural number ≤ 21000), computes a prime of n bits with special properties. the program loops over all possible n, and checks for each n if a file exists. if it does not, it creates it (with zero content), computes the prime (which can take between minutes and days), writes the prime into the file and continues with the next file. this simple task distribution technique allows me to run the program in parallel on different machines (since the files are in a nfs folder) with many instances on each machine. now at our institute, we have a big computation machine (64 cores) and four user machines (on which the users work, each 32 cores). since the user machines are often not intensively used (and that only during certain times of the day), i want to use these as well. but there should be enough cores free, so the users won’t notice that there are computations going on in the background. on the computation server, also other people want to run something, so there should also be some free cores. optimally, my program would somehow decide how many cores are used by others, and use the rest. or most of them, to leave some free, especially on the user machines.
after a suggestion by our it guys, i started writing a bash script which controls the instances of my program on the same machine. the first version used the time of the day to determine the number of processes. everything was computed in terms of the number of cores of the machine, the load (with a load modifier applied, since some machines have uninterruptable processes running which do not effectively do something, and which won’t go away until the next reboot) and the hour of the day. but it is not easy to find a good scheme which yields good results on all machines. something which works well on the user machines is wasting processor time on the computation server.
so today i rewrote the program to use profiles. a profile contains information on the number of cores (this is necessary since the computation server has hyperthreading enabled, and thus returns twice the number of cores), the number of processes to be started, and the number of cores to be left free during each hour and day of a week. so on weekends or nights, i choose lower numbers for the free cores for the user machines, while for the computational server the number is always 1.
a profile can look like this (this is from a user machine, the file is called primefinderrunner-user.profile for later reference):

30 $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8]
41 $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/4] $[CORES/4] $[CORES/4] $[CORES/8]
52 $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/4] $[CORES/4] $[CORES/4] $[CORES/8]
63 $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/4] $[CORES/4] $[CORES/4] $[CORES/8]
74 $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/4] $[CORES/4] $[CORES/4] $[CORES/8]
85 $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/2] $[CORES/4] $[CORES/4] $[CORES/4] $[CORES/8]
96 $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/8] $[CORES/16] $[CORES/16] $[CORES/16] $[CORES/16]

the line with prefix CORES gives the number of cores. the line prefixed by STARTUP gives the number of processes to run (at most); here, we use 7/8 of the number of cores. the lines prefixed by a number between 0 (sunday) and 6 (saturday) have 24 entries following: every entry (seperated by exactly one space, as the prefix itself is separated by exactly one space from the entries!) says how many cores should be free at each time of the day. usually during night (up to 7 am) at least 1/16 of the total number of cores should be free, while during workday (8 am to 7 pm) half of the cores should be free. of course, the numbers are different for weekends (saturday and sunday) than for the other working days.
now the script itself looks like this (for reference, the filename is

  3initProfile() {
  4    PROFILEFN=primefinderrunner-$PROFILE.profile
  5    CORES=`grep "^CORES " $PROFILEFN`
  9    eval STARTUP=$STARTUP
 13if [ "$1" != "" ]
 15    PROFILE=$1
 17    PROFILE=`hostname`
 19if [ "$2" != "" ]
 24if [ "$CORES" == "" ]
 26    echo "Cannot load profile $PROFILEFN!"
 27    exit
 29echo Cores: $CORES
 30echo Load modifier: $LOADMODIFIER
 32computeFreecores() { 
 33    # two arguments: day (0..6) and hour (0..23)
 34    FREECORES=0
 35    DAY=`date +%w`
 36    LINE=`grep "^$DAY " $PROFILEFN`
 37    LINE=${LINE/$DAY }
 38    HOUR=`date +%k`
 39    for ((i=0;i<$HOUR;++i));
 40    do
 41        LINE=${LINE#* }
 42    done
 43    LINE=${LINE/ *}
 44    eval FREECORES=$LINE
 49stopsignal() {
 50    for PID in `jobs -p`;
 51    do
 52        FILE=`lsof -p $PID -F n 2>/dev/null | grep primedatabase | grep -v "\\.nfs"`
 53        A=${FILE#n*}
 54        A=${A/ (nfs*}
 55        echo killing $PID with open file $A
 56        rm $A
 57        kill $PID
 58    done
 59    exit
 62trap 'stopsignal' 2
 64echo "Starting $STARTUP instances"
 66determineToAdd() {
 67    computeFreecores
 68    LOAD=`uptime`
 69    LOAD=${LOAD#*average: }
 70    LOAD=${LOAD/,*}
 71    LOAD=${LOAD/.*}
 73    echo Load: $[LOAD-LOADMODIFIER], Intended number of free cores: $FREECORES
 76# Start programs in the background
 78for ((i=1;i<=STARTUP;++i));
 80    primefinder &amp;
 81    sleep 2
 83sleep 20
 84if [ $ADD -lt 0 ]
 86    ADD=0
 88for ((i=ADD+1;i<=STARTUP;++i));
 90    kill -SIGSTOP %$i
 94RUNNINGSTART=1 # The first one running
 95RUNNINGSTOP=$CURRRUNNING # The last one running
 97startOne() {
 98    # Assume that $CURRRUNNING < $STARTUP
104stopOne() {
105    # Assume that $CURRRUNNING > 0
111# Start mainloop
112while [ 1 ]
114    sleep 60
116    # Determine how many threads should be added/removed
117    determineToAdd
118    if [ $ADD -gt 0 ]
119    then
120        if [ $[ADD+CURRRUNNING] -gt $STARTUP ]
121        then
123        fi
124        # Add processes
125        echo ADD:$ADD
126        for ((i=0;i<ADD;++i))
127        do
128            startOne
129        done
130    fi
131    if [ $ADD -lt 0 ]
132    then
133        REM=$[-ADD]
134        # Clip
135        if [ $REM -gt $CURRRUNNING ]
136        then
137            REM=$CURRRUNNING
138        fi
139        # Remove processes
140        echo REMOVE:$REM
141        for ((i=0;i<REM;++i))
142        do
143            stopOne
144        done
145    fi
146    sleep 60

the script first starts all instances, then stops the ones which are too many, and then starts the main loop. in the main loop, it waits 60 seconds (for the average load to adjust to the new process count), and then decides how many cores should be left free, and what that means for the number of processes (add/remove some). note that the profile file is read every minute, so it can be changed any time without any need to re-run the whole thing.
in case the script is stopped (with control+c), all primefinder processes are killed and their open file is deleted. to determine the open file, i use lsof with some greps. you have to adjust and test that line before using this script!
note that this script is quite a hack, and far from perfect. and it is somehow system dependent, or at least “setup dependent” since it has certain assumptions on the executables, on how the output of lsof looks like, … so better make sure it works before you use it, especially on bigger systems. also note that in the beginning, all instances are ran (they are started with a two second delay between two instances), and then everything is run for 20 seconds before the first adjustment (i.e. stopping processes which are too many) are made. if you share the system with other people, this might already annoy others when they try to measure timings of their programs (especially if hyperthreading is enabled).

posted in: computer