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   <title>Algorithms</title>

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   <body>

         <p>
            This page documents library components that are all basically just implementations of 
            mathematical functions or algorithms that don't fit in any of the other pages
            of the dlib documentation.  So this includes things like checksums, cryptographic hashes, 
            sorting, etc.
         </p>

   </body>

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   <menu width="150">
    <top>
      <section>
         <name>Tools</name>
         <item>bigint</item> 
         <item>disjoint_subsets</item> 
         <item>disjoint_subsets_sized</item> 
         <item nolink="true">
            <name>Quantum Computing</name>
            <sub>
               <item>quantum_register</item> 
               <item>gate</item> 
            </sub>
         </item>
         <item>hsort_array</item> 
         <item>isort_array</item> 
         <item>numeric_constants</item> 
         <item>put_in_range</item> 
         <item>qsort_array</item> 
         <item>split_array</item> 
         <item>integrate_function_adapt_simp</item> 
         <item>square_root</item> 
         <item nolink="true">
            <name>Set Utilities</name>
            <sub>
               <item>set_intersection_size</item> 
               <item>set_intersection</item> 
               <item>set_union</item> 
               <item>set_difference</item> 
            </sub>
         </item>
      </section>



      <section>
         <name>Statistics</name>
         <item>rand</item> 
         <item>median</item> 
         <item>running_stats</item> 
         <item>running_stats_decayed</item> 
         <item>running_scalar_covariance_decayed</item> 
         <item>running_gradient</item> 
         <item>running_scalar_covariance</item> 
         <item>mean_sign_agreement</item> 
         <item>correlation</item> 
         <item>covariance</item> 
         <item>r_squared</item> 
         <item>mean_squared_error</item> 
         <item>running_covariance</item> 
         <item>running_cross_covariance</item> 
         <item>random_subset_selector</item> 
         <item>randomly_subsample</item> 
         <item>find_upper_quantile</item> 
         <item>count_steps_without_decrease_robust</item> 
         <item>count_steps_without_decrease</item> 
         <item>count_steps_without_increase</item> 

      </section>

      <section>
         <name>Hashing</name>
         <item>md5</item> 
         <item>crc32</item> 
         <item>hash</item> 
         <item>count_bits</item> 
         <item>hamming_distance</item> 
         <item>murmur_hash3</item> 
         <item>murmur_hash3_128bit</item> 
         <item>gaussian_random_hash</item> 
         <item>uniform_random_hash</item> 
         <item>projection_hash</item> 
         <item>create_random_projection_hash</item> 
         <item>create_max_margin_projection_hash</item> 
         <item>hash_samples</item> 
         <item>hash_similar_angles_64</item> 
         <item>hash_similar_angles_128</item> 
         <item>hash_similar_angles_256</item> 
         <item>hash_similar_angles_512</item> 
      </section>

      <section>
         <name>Filtering</name>
         <item>kalman_filter</item> 
         <item>rls_filter</item> 
      </section>

    </top>  
   </menu>

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   <components>
   
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      <component>
         <name>hash_similar_angles_64</name>
         <file>dlib/lsh.h</file>
         <spec_file link="true">dlib/lsh/hashes_abstract.h</spec_file>
         <description>
            This object is a tool for computing locality sensitive hashes that give
            vectors with small angles between each other similar hash values.  In
            particular, this object creates 64 random planes which pass though the
            origin and uses them to create a 64bit hash.
         </description>

      </component>

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      <component>
         <name>hash_similar_angles_128</name>
         <file>dlib/lsh.h</file>
         <spec_file link="true">dlib/lsh/hashes_abstract.h</spec_file>
         <description>
            This object is a tool for computing locality sensitive hashes that give
            vectors with small angles between each other similar hash values.  In
            particular, this object creates 128 random planes which pass though the
            origin and uses them to create a 128bit hash.
         </description>

      </component>

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      <component>
         <name>hash_similar_angles_256</name>
         <file>dlib/lsh.h</file>
         <spec_file link="true">dlib/lsh/hashes_abstract.h</spec_file>
         <description>
            This object is a tool for computing locality sensitive hashes that give
            vectors with small angles between each other similar hash values.  In
            particular, this object creates 256 random planes which pass though the
            origin and uses them to create a 256bit hash.
         </description>

      </component>

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      <component>
         <name>hash_similar_angles_512</name>
         <file>dlib/lsh.h</file>
         <spec_file link="true">dlib/lsh/hashes_abstract.h</spec_file>
         <description>
            This object is a tool for computing locality sensitive hashes that give
            vectors with small angles between each other similar hash values.  In
            particular, this object creates 512 random planes which pass though the
            origin and uses them to create a 512bit hash.
         </description>

      </component>

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      <component>
         <name>hash_samples</name>
         <file>dlib/graph_utils_threaded.h</file>
         <spec_file link="true">dlib/graph_utils/find_k_nearest_neighbors_lsh_abstract.h</spec_file>
         <description>
            This is a simple function for hashing a bunch of vectors using a
            locality sensitive hashing object such as <a href="#hash_similar_angles_128">hash_similar_angles_128</a>.  
            It is also capable of running in parallel on a multi-core CPU.
         </description>

      </component>

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      <component checked="true">
         <name>bigint</name>
         <file>dlib/bigint.h</file>
         <spec_file>dlib/bigint/bigint_kernel_abstract.h</spec_file>
         <description>
            This object represents an arbitrary precision unsigned integer.  It's pretty simple.  
            It's interface is just like a normal int, you don't have to tell it how much memory
            to use or anything unusual.  It just goes :)       
         </description>
         
         <implementations>
            <implementation>
               <name>bigint_kernel_1</name>
               <file>dlib/bigint/bigint_kernel_1.h</file>
               <description> 
                  This implementation is done using an array of unsigned shorts.  It is also reference counted.  
                  For further details see the above link.  Also note that kernel_2 should be 
      faster in almost every case so you should really just use that version of the bigint object.
               </description> 
    
               <typedefs>
                  <typedef>
                     <name>kernel_1a</name>
                     <description>is a typedef for bigint_kernel_1</description>
                  </typedef>
               </typedefs>                
               
            </implementation> 
                     
            <implementation>
               <name>bigint_kernel_2</name>
               <file>dlib/bigint/bigint_kernel_2.h</file>
               <description> 
                  This implementation is basically the same as kernel_1 except it uses the 
                  Fast Fourier Transform to perform multiplications much faster.  
               </description> 
    
               <typedefs>
                  <typedef>
                     <name>kernel_2a</name>
                     <description>is a typedef for bigint_kernel_2</description>
                  </typedef>
               </typedefs>                
               
            </implementation> 
                     
         </implementations>
                           
                        
      </component>
      
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      <component>
         <name>crc32</name>
         <file>dlib/crc32.h</file>
         <spec_file>dlib/crc32/crc32_kernel_abstract.h</spec_file>
         <description>
            This object represents the CRC-32 algorithm for calculating checksums.   
         </description>
         
      </component>
            
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      <component>
         <name>gaussian_random_hash</name>
         <file>dlib/hash.h</file>
         <spec_file link="true">dlib/general_hash/random_hashing_abstract.h</spec_file>
         <description>
              This function uses hashing to generate Gaussian distributed random values
              with mean 0 and variance 1.  
         </description>
         
      </component>
            
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      <component>
         <name>uniform_random_hash</name>
         <file>dlib/hash.h</file>
         <spec_file link="true">dlib/general_hash/random_hashing_abstract.h</spec_file>
         <description>
              This function uses hashing to generate uniform random values in the range [0,1).
         </description>
         
      </component>
            
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      <component>
         <name>murmur_hash3</name>
         <file>dlib/hash.h</file>
         <spec_file>dlib/general_hash/murmur_hash3_abstract.h</spec_file>
         <description>
            This function takes a block of memory and returns a 32bit hash.  The 
            hashing algorithm used is Austin Appleby's excellent 
            <a href="http://code.google.com/p/smhasher/">MurmurHash3</a>. 
         </description>
         
      </component>
            
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      <component>
         <name>murmur_hash3_128bit</name>
         <file>dlib/hash.h</file>
         <spec_file link="true">dlib/general_hash/murmur_hash3_abstract.h</spec_file>
         <description>
            This function takes a block of memory and returns a 128bit hash.  The 
            hashing algorithm used is Austin Appleby's excellent 
            <a href="http://code.google.com/p/smhasher/">MurmurHash3</a>. 
         </description>
         
      </component>
            
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      <component>
         <name>kalman_filter</name>
         <file>dlib/filtering.h</file>
         <spec_file>dlib/filtering/kalman_filter_abstract.h</spec_file>
         <description>
                This object implements the Kalman filter, which is a tool for 
                recursively estimating the state of a process given measurements
                related to that process.  To use this tool you will have to 
                be familiar with the workings of the Kalman filter.  An excellent
                introduction can be found in the paper:
                <blockquote>
                    An Introduction to the Kalman Filter
                    by Greg Welch and Gary Bishop
                </blockquote>
         </description>
         
      </component>
            
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      <component>
         <name>rls_filter</name>
         <file>dlib/filtering.h</file>
         <spec_file>dlib/filtering/rls_filter_abstract.h</spec_file>
         <description>
            This object is a tool for doing time series prediction using 
            linear <a href="ml.html#rls">recursive least squares</a>.  In particular,
            this object takes a sequence of points from the user and, at each
            step, attempts to predict the value of the next point.
         </description>
         
      </component>
            
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      <component>
         <name>projection_hash</name>
         <file>dlib/lsh.h</file>
         <spec_file>dlib/lsh/projection_hash_abstract.h</spec_file>
         <description>
                This is a tool for hashing elements of a vector space into the integers.  
                It is intended to represent locality sensitive hashing functions such as 
                the popular <a href="#create_random_projection_hash">random projection hashing</a> method.
         </description>
         
      </component>
            
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      <component>
         <name>create_random_projection_hash</name>
         <file>dlib/lsh.h</file>
         <spec_file link="true">dlib/lsh/create_random_projection_hash_abstract.h</spec_file>
         <description>
            Creates a random projection based locality sensitive 
            <a href="#projection_hash">hashing function</a>.  The projection matrix
            is generated by sampling its elements from a Gaussian random number generator.
         </description>
         
      </component>
            
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      <component>
         <name>create_max_margin_projection_hash</name>
         <file>dlib/lsh.h</file>
         <spec_file link="true">dlib/lsh/create_random_projection_hash_abstract.h</spec_file>
         <description>
            Creates a random projection based locality sensitive 
            <a href="#projection_hash">hashing function</a>.  
               This is accomplished using a variation on the random hyperplane generation
               technique from the paper:
               <blockquote>
                    Random Maximum Margin Hashing by Alexis Joly and Olivier Buisson
               </blockquote>
                  In particular, we use a linear support vector machine to generate planes.
                  We train it on randomly selected and randomly labeled points from 
                  the data to be hashed.
         </description>
         
      </component>
            
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      <component>
         <name>hash</name>
         <file>dlib/hash.h</file>
         <spec_file>dlib/general_hash/hash_abstract.h</spec_file>
         <description>
            This is a set of convenience functions for invoking <a href="#murmur_hash3">murmur_hash3</a>
            on std::strings, std::vectors, std::maps, or <a href="linear_algebra.html#matrix">dlib::matrix</a> objects.
            <p>
               As an aside, the hash() for matrix objects is defined <a href="dlib/matrix/matrix_utilities_abstract.h.html#hash">here</a>.
               It has the same interface as all the others.
            </p>
         </description>
         
      </component>
            
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      <component>
         <name>count_bits</name>
         <file>dlib/hash.h</file>
         <spec_file link="true">dlib/general_hash/count_bits_abstract.h</spec_file>
         <description>
            This function counts the number of bits in an unsigned integer which are
            set to 1.
         </description>
         
      </component>
            
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      <component>
         <name>hamming_distance</name>
         <file>dlib/hash.h</file>
         <spec_file link="true">dlib/general_hash/count_bits_abstract.h</spec_file>
         <description>
            This function returns the hamming distance between two unsigned integers.
            That is, it returns the number of bits which differer in the two integers.
         </description>
         
      </component>
            
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      <component>
         <name>rand</name>
         <file>dlib/rand.h</file>
         <spec_file>dlib/rand/rand_kernel_abstract.h</spec_file>
         <description>
            This object represents a pseudorandom number generator.
         </description>
         
      </component>
            
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      <component>
         <name>disjoint_subsets</name>
         <file>dlib/disjoint_subsets.h</file>
         <spec_file link="true">dlib/disjoint_subsets/disjoint_subsets_abstract.h</spec_file>
         <description>
                This object represents a set of integers which is partitioned into
                a number of disjoint subsets.  It supports the two fundamental operations
                of finding which subset a particular integer belongs to as well as 
                merging subsets. 
         </description>

      </component>
      
      
   <!-- ************************************************************************* -->

      <component>
         <name>disjoint_subsets_sized</name>
         <file>dlib/disjoint_subsets.h</file>
         <spec_file link="true">dlib/disjoint_subsets/disjoint_subsets_sized_abstract.h</spec_file>
         <description>
                This object is just like <a href="#disjoint_subsets">disjoint_subsets</a> except that it
                also keeps track of the size of each set.
         </description>

      </component>
      
      
   <!-- ************************************************************************* -->

      <component>
         <name>running_stats</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object represents something that can compute the running mean,
                variance, skewness, and kurtosis statistics of a stream of real numbers.  
         </description>

         <examples>
            <example>running_stats_ex.cpp.html</example>
            <example>kcentroid_ex.cpp.html</example>
         </examples>
                                 
      </component>
      
      
   <!-- ************************************************************************* -->

      <component>
         <name>running_stats_decayed</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object represents something that can compute the running mean and
                variance of a stream of real numbers.  It is similar to <a href="#running_stats">running_stats</a>
                except that it forgets about data it has seen after a certain period of
                time.  It does this by exponentially decaying old statistics. 
         </description>
      </component>
      
      
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      <component>
         <name>running_scalar_covariance_decayed</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object represents something that can compute the running covariance of
                a stream of real number pairs.  It is essentially the same as
                <a href="#running_scalar_covariance">running_scalar_covariance</a> except that it forgets about data it has seen
                after a certain period of time.  It does this by exponentially decaying old
                statistics. 
         </description>
      </component>
      
      
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      <component>
         <name>running_gradient</name>
         <file>dlib/statistics/running_gradient.h</file>
         <spec_file link="true">dlib/statistics/running_gradient_abstract.h</spec_file>
         <description>
                This object is a tool for estimating if a noisy sequence of numbers is
                trending up or down and by how much.  It does this by finding the least
                squares fit of a line to the data and then allows you to perform a
                statistical test on the slope of that line.
         </description>
      </component>
      
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      <component>
         <name>find_upper_quantile</name>
         <file>dlib/statistics/running_gradient.h</file>
         <spec_file link="true">dlib/statistics/running_gradient_abstract.h</spec_file>
         <description>
              Finds and returns the scalar value such that a user specified percentage of
              the values in a container are greater than said value.  For example, 0.5
              would find the median value in a container while 0.1 would find the value
              that lower bounded the 10% largest values in a container.
         </description>
      </component>
      
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      <component>
         <name>count_steps_without_increase</name>
         <file>dlib/statistics/running_gradient.h</file>
         <spec_file link="true">dlib/statistics/running_gradient_abstract.h</spec_file>
         <description>
            Given a potentially noisy time series, this function returns a count of how
            long the time series has gone without noticeably increasing in value.  It does
            this by adding the elements of the time series into a <a
            href="#running_gradient">running_gradient</a> object and counting how many
            elements, starting with the most recent, you need to examine before you
            are confident that the series has been increasing in value.    
         </description>
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>count_steps_without_decrease</name>
         <file>dlib/statistics/running_gradient.h</file>
         <spec_file link="true">dlib/statistics/running_gradient_abstract.h</spec_file>
         <description>
            Given a potentially noisy time series, this function returns a count of how
            long the time series has gone without noticeably decreasing in value.  It does
            this by adding the elements of the time series into a <a
            href="#running_gradient">running_gradient</a> object and counting how many
            elements, starting with the most recent, you need to examine before you
            are confident that the series has been decreasing in value.    
         </description>
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>count_steps_without_decrease_robust</name>
         <file>dlib/statistics/running_gradient.h</file>
         <spec_file link="true">dlib/statistics/running_gradient_abstract.h</spec_file>
         <description>
            This function behaves just like <a
            href="#count_steps_without_decrease">count_steps_without_decrease</a> except
            that it ignores times series values that are anomalously large.  This makes it
            robust to sudden noisy but transient spikes in the time series values.
         </description>
      </component>
      
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      <component>
         <name>running_covariance</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object is a simple tool for computing the mean and
                covariance of a sequence of vectors.  
         </description>

      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>running_cross_covariance</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object is a simple tool for computing the mean and
                cross-covariance matrices of a sequence of pairs of vectors.  
         </description>

      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>running_scalar_covariance</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
                This object is a simple tool for computing the covariance of a 
                sequence of scalar values.  
         </description>
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>mean_sign_agreement</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the probability that 
               matching elements of two std::vectors have the same sign.
         </description>
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>correlation</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the correlation between 
               matching elements of two std::vectors.
         </description>
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>covariance</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the covariance between 
               matching elements of two std::vectors.
         </description>
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>r_squared</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the R squared coefficient between 
               matching elements of two std::vectors.
         </description>
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>mean_squared_error</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/statistics_abstract.h</spec_file>
         <description>
               This is a function for computing the mean squared error between 
               matching elements of two std::vectors.
         </description>
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>random_subset_selector</name>
         <file>dlib/statistics.h</file>
         <spec_file>dlib/statistics/random_subset_selector_abstract.h</spec_file>
         <description>
               This object is a tool to help you select a random subset of a large body of data.  
               In particular, it is useful when the body of data is too large to fit into memory.
         </description>

                                 
      </component>
      
      
   <!-- ************************************************************************* -->

      <component>
         <name>randomly_subsample</name>
         <file>dlib/statistics.h</file>
         <spec_file link="true">dlib/statistics/random_subset_selector_abstract.h</spec_file>
         <description>
            This is a set of convenience functions for 
            creating <a href="#random_subset_selector">random subsets</a> of data.
         </description>
                                 
      </component>
      
   <!-- ************************************************************************* -->

      <component>
         <name>hsort_array</name>
         <file>dlib/sort.h</file>
         <spec_file link="true">dlib/sort.h</spec_file>
         <description>
            hsort_array is an implementation of the heapsort algorithm.  It will sort anything that has an 
            array like operator[] interface.  
         </description>
                                 
      </component>
      
   <!-- ************************************************************************* -->
      
      <component>
         <name>put_in_range</name>
         <file>dlib/algs.h</file>
         <spec_file link="true">dlib/algs.h</spec_file>
         <description>
            This is a simple function that takes a range and a value and returns the given
            value if it is within the range.  If it isn't in the range then it returns the  
            end of range value that is closest.
         </description>
                                 
      </component>
               
   <!-- ************************************************************************* -->
      
      <component>
         <name>isort_array</name>
         <file>dlib/sort.h</file>
         <spec_file link="true">dlib/sort.h</spec_file>
         <description>
            isort_array is an implementation of the insertion sort algorithm.  It will sort anything that has an 
            array like operator[] interface.   
         </description>
                                 
      </component>
               
      
   <!-- ************************************************************************* -->
      
      <component>
         <name>numeric_constants</name>
         <file>dlib/numeric_constants.h</file>
         <spec_file>dlib/numeric_constants.h</spec_file>
         <description>
            This is just a header file containing definitions of common numeric constants such as pi and e.
         </description>
      </component>
      
   <!-- ************************************************************************* -->
      
      <component>
         <name>qsort_array</name>
         <file>dlib/sort.h</file>
         <spec_file link="true">dlib/sort.h</spec_file>
         <description>
            qsort_array is an implementation of the QuickSort algorithm.  It will sort anything that has an array like 
            operator[] interface.  If the quick sort becomes unstable then it switches to a heap sort.  This 
            way sorting is guaranteed to take at most N*log(N) time.
         </description>
                                 
      </component>
            
      
   <!-- ************************************************************************* -->
      
      <component>
         <name>split_array</name>
         <file>dlib/array.h</file>
         <spec_file link="true">dlib/array/array_tools_abstract.h</spec_file>
         <description>
            This function is used to efficiently split <a href="containers.html#array">array</a>
            like objects into two parts.  It uses the global swap() function instead
            of copying to move elements around, so it works on arrays of non-copyable
            types.
         </description>
      </component>
      
   <!-- ************************************************************************* -->
      
      <component>
         <name>integrate_function_adapt_simp</name>
         <file>dlib/numerical_integration.h</file>
         <spec_file link="true">dlib/numerical_integration/integrate_function_adapt_simpson_abstract.h</spec_file>
         <description>
          Computes an approximation of the integral of a real valued function using the
          adaptive Simpson method outlined in 
          <blockquote>
             Gander, W. and W. Gautshi, "Adaptive
             Quadrature -- Revisited" BIT, Vol. 40, (2000), pp.84-101
          </blockquote>
         </description>
         <examples>
            <example>integrate_function_adapt_simp_ex.cpp.html</example>
         </examples>
                                 
      </component>
            
      
   <!-- ************************************************************************* -->
      
      <component>
         <name>md5</name>
         <file>dlib/md5.h</file>
         <spec_file>dlib/md5/md5_kernel_abstract.h</spec_file>
         <description>
            This is an implementation of The MD5 Message-Digest Algorithm as described in rfc1321.   
         </description>
                                 
      </component>
            
      
      
   <!-- ************************************************************************* -->
      
      <component>
         <name>median</name>
         <file>dlib/algs.h</file>
         <spec_file link="true">dlib/algs.h</spec_file>
         <description>
            This function takes three parameters and finds the median of the three.  The median is swapped into
            the first parameter and the first parameter ends up in one of the other two, unless the first parameter was
            the median to begin with of course. 
         </description>
                                 
      </component>
            
   <!-- ************************************************************************* -->
      
      <component>
         <name>square_root</name>
         <file>dlib/algs.h</file>
         <spec_file link="true">dlib/algs.h</spec_file>
         <description>        
            square_root is a function which takes an unsigned long and returns the square root of it or
            if the root is not an integer then it is rounded up to the next integer.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>set_intersection</name>
         <file>dlib/set_utils.h</file>
         <spec_file link="true">dlib/set_utils/set_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#set">set</a> objects and 
            gives you their intersection.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>set_union</name>
         <file>dlib/set_utils.h</file>
         <spec_file link="true">dlib/set_utils/set_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#set">set</a> objects and 
            gives you their union.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>set_difference</name>
         <file>dlib/set_utils.h</file>
         <spec_file link="true">dlib/set_utils/set_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#set">set</a> objects and 
            gives you their difference.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>set_intersection_size</name>
         <file>dlib/set_utils.h</file>
         <spec_file link="true">dlib/set_utils/set_utils_abstract.h</spec_file>
         <description>        
            This function takes two <a href="containers.html#set">set</a> objects and tells you
            how many items they have in common.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>quantum_register</name>
         <file>dlib/quantum_computing.h</file>
         <spec_file link="true">dlib/quantum_computing/quantum_computing_abstract.h</spec_file>
         <description>        
                This object represents a set of quantum bits.  It can be used
                with the quantum <a href="#gate">gate</a> object to simulate
                quantum algorithms.
         </description>

         <examples>
            <example>quantum_computing_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>gate</name>
         <file>dlib/quantum_computing.h</file>
         <spec_file link="true">dlib/quantum_computing/quantum_computing_abstract.h</spec_file>
         <description>        
            This object represents a quantum gate that operates on a 
            <a href="#quantum_register">quantum_register</a>.   
         </description>
         <examples>
            <example>quantum_computing_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->

   </components>

   <!-- ************************************************************************* -->


</doc>

