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the critical phenomena , typical of many body systems , have attracted remarkable scientific interest since a long time . the monte carlo  ( mc ) stochastic techniques  @xcite serve to estimate thermodynamic averages from samples of the configuration space . many advanced simulation approaches are rooted in elementary updates of metropolis algorithm  @xcite , which generates a markovian chain of spin configurations . most of the efforts try to improve mc method within the critical region . the efficiency of mc method have been increased by the development of sampling and data processing techniques . the reweighting technique applied anew to classical statistical systems  @xcite enables to perform a random sampling for one distribution function , and to calculate quantities of interest for similar distributions . to be more concrete , let us assume that a sample of @xmath0 configurations ( @xmath1 ) of quantity @xmath2 have been accumulated for a fixed temperature @xmath3 . the samples @xmath4 are used to construct a continuous temperature @xmath5 dependence of a single histogram approximation @xmath6 of the canonical thermodynamic average @xmath7 . the interpolation called `` _ reweighting on the fly _ ''  @xcite is based on the formula @xmath8 } { \displaystyle \sum_{q=1}^{n } \exp \left [ \frac{e_q}{t_t}\left(1 - \frac{t_t}{t } \right ) \right]}\ , , \label{reweight1}\ ] ] where @xmath9 is the energy  ( in boltzmann constant units ) of configuration @xmath10 . because of the finite length of the mc run , eq . ( [ reweight1 ] ) provides reliable results only for a relatively narrow range of @xmath5 values around @xmath3 . when @xmath5 differs too much from @xmath3 , it causes an increase of the statistical errors and may lead to unreliable results . the reliable range of @xmath5 values also decreases as the system size increases  @xcite . a qualitative study in ref . has shown that reweighting causes a systematic shift in the height and the location of the specific heat peak , which depends on the relative position of @xmath3 and @xmath5 . it turns out that the shift decreases as  @xmath11 only for @xmath0 much larger than the number of degrees of freedom . systematic errors , eventually affected by sampling autocorrelations , may also take place at finite @xmath0 . the first quantitative study of the statistical errors in reweighted data  @xcite has demonstrated that theoretical calculation of the error is both difficult and time consuming so it may not be justified for all studies . in this paper we present rather general iterative treatment based on the single histogram reweighting which implicitly takes into account both the systematic errors of reweighting and the statistical errors to generate stochastic fluctuations near a critical point . a well known systematic error , present in the histogram method , is an overestimation ( underestimation ) of @xmath12 when @xmath13 ( @xmath14 ) , respectively  @xcite . this fact and the general idea of feedback regulation  @xcite are used to construct an iterative algorithm . the aim of the suggested method is to estimate the critical temperature or other parameters of interest , as well as to perform an analysis and partial elimination of their errors . the method combines single reweighting , linear filtering and an iterative treatment . the incorporation of iterations is inspired by a general class of probabilistic models  @xcite . the iterations generate data that are further analyzed in terms of a suggested autoregressive moving average  ( arma ) model . the single - spin flip metropolis algorithm is applied here in order to obtain averages for moderate lattice sizes . the cluster methods are not used due to the efficiency reasons discussed in  ref .  . the efficiency of the algorithm should also be increased by hybrid schemes  @xcite . however , because of many side effects that can show up , a separate study of hybrid schemes is needed . the paper is organized as follows . in sec . [ general_scheme ] the iterative self - organized algorithm is suggested . in sec . [ sec : model ] the autoregressive phenomenological model of the stochastic iterative search process is proposed and the corresponding relations for second - order statistics are derived . the model is used to parametrize mc data obtained for ferromagnetic 2d  ising spin models on @xmath15 square lattices  ( see sec . [ sec : details ] ) . we start by defining a process where temperature is driven by some response function . the stochastic process can be formally described by the second - order recursive formula @xmath16 it generates sequence @xmath17 . the stochastic function @xmath18 , which provides a _ preliminary histogrammatic _ estimate @xmath19 of unknown critical temperature @xmath20 , is defined implicitly by @xmath21 the key point of the reweighting is to localize the extremal value of the response function @xmath22 evaluated for @xmath0 mc samples . commonly , the response function can be defined by the first and second moments of energy or magnetization , respectively . the role of filtering , defined by the plasticity parameter @xmath23 in eq . ( [ adap1 ] ) , is to weaken fluctuations  @xcite of the @xmath19 sequence . let us assume the invariant probability distribution function  ( pdf ) of the form @xmath24 where @xmath25 is the kronecker s symbol and @xmath26  is the arithmetic average  ( which differs from standard thermodynamic average @xmath27 ) . from now on @xmath20 is associated with pseudocritical temperature @xmath28 of the specific heat . for gaussian - like @xmath29 we suppose @xmath30 as mentioned above , the specific heat can be estimated within a histogram reweighting approach using the fluctuation - dissipation relation @xmath31 at the thermodynamic limit @xmath32 , the iterative process becomes deterministic and tends to an unique fixed point @xmath33 . for finite @xmath0 , a convergence criterion have to be formulated in a pure statistical sense . the variance @xmath34 has been chosen as a suitable measure that reflects the convergence . the iterative process , described by eqs . ( [ eqrecur1 ] ) , ( [ adap1 ] ) , ( [ objec1 ] ) supplemented by eq . ( [ objec2 ] ) , consists of nonlinear and stochastic rules . as an auxiliary tool of analysis , let us introduce a phenomenological model to parameterize the simulated statistics . within the model , the explicit form of the stochastic function  [ see eq . ( [ eqrecur1 ] ) ] is @xmath35 where an additive noise @xmath36 describes uncertainty . the convergence rate towards @xmath20 is determined by the parameter @xmath37 , which contains information of the systematic reweighting error . from eqs . ( [ eqrecur1 ] ) , ( [ adap1 ] ) and eq . ( [ model_alg ] ) follows that @xmath38 where the noise term is controlled by the coefficient @xmath23 . the model discussed here can be considered as an application of arma(1,1 ) model  @xcite . the solution of  eq . ( [ model_2 ] ) can be written simply as @xmath39 where @xmath40 . the above formula , written for any @xmath36 dependence , is general . at this point , let us consider the statistical properties of gaussian noise for @xmath36 specified by @xmath41 where @xmath42 . within the limit of a second order statistics in @xmath3 , from eq . ( [ solutt1 ] ) and eq . ( [ firstxi1 ] ) , follows @xmath43 therefore , the stationary regime is defined by the invariance @xmath44 , which stays in agreement with eq . ( [ tavel1 ] ) . if the solution given by eq . ( [ solutt1 ] ) is substituted into eq . ( [ disp1 ] ) , one obtains @xmath45 which is equivalent to the recursion @xmath46 . terms proportional to @xmath47 describe a _ transient regime _ of the initial temperature @xmath48 , which the system forgets exponentially fast . terms @xmath49 and @xmath50 are deterministic and stochastic contribution to @xmath51 , respectively . the variance  @xmath51 remains finite for @xmath52 if @xmath53 , that is if @xmath54 . the transient regime can be characterized by @xmath55 whereas , to obtain the stationary regime , one needs to calculate @xmath56 it results that @xmath57 diverges at @xmath58 or equivalently @xmath59 . the characteristic transient time , @xmath60 follows from its definition @xmath61 . the transient time has a singularity for @xmath62 and @xmath63 . in this limit the model describes immediate convergence  ( @xmath64 ) and variance @xmath65 . a singular parametric line , @xmath66 , splits region @xmath53 into alternating  ( @xmath67 ) and unidirectional convergence  ( @xmath68 ) of @xmath3 near @xmath20 .    regarding the convergence properties , it is useful to introduce @xmath69 as the value of the _ highest initial convergence rate _ , defined by the condition @xmath70 . this yields @xmath71 for this extremal value is @xmath72 . finally , we introduce some supplementary remarks about the limit of the vanishing error . clearly , for @xmath73 we assume @xmath74 . the consequence of eqs . ( [ eq : var_infty ] ) and ( [ time_ch ] ) is @xmath75 , that implies @xmath76 . unfortunately , because of  @xmath77 this can not be attained in practice . in order to explore the proposed model by numerical simulation , we chose the 2d  ising model with the metropolis algorithm as testing ground . most of the data are accumulated on square lattices of linear dimension @xmath78 and periodic boundary conditions . the methodology is expected to be quite general and of straightforward application for both discrete and continuous systems . every iteration step @xmath79 of algorithm , based on eq . ( [ eqrecur1 ] ) and eq . ( [ adap1 ] ) , consists of the following main blocks  ( i)-(iv ) :    _ equilibration _ that includes @xmath80 mcss ( monte carlo steps per spin ) for constant @xmath3 . _ accumulation _ of @xmath0 pairs @xmath81 into histograms . in order to reduce autocorrelations , successive samples are separated by one mcss . _ localization _ of the maximum @xmath82 by a gradient algorithm . recall , that @xmath22 is computed from eq . ( [ reweight1 ] ) and eq . ( [ objec2 ] ) for @xmath12 and @xmath83 . _ prediction _ of next @xmath84 using the filtering given by eq . ( [ adap1 ] ) . [ initconv1 ]  illustrates the properties of _ transient regimes _ , which are affected by the choice of @xmath48 and @xmath23 . the @xmath23-dependences of @xmath85 are presented in  fig . [ fig : var_1](a ) . one may see that the agreement of the simulated data with eq . ( [ eq : var_1 ] ) is quite satisfactory . typical values of the parameters determined by the fit are gathered in table  [ tab : par ] .    for  @xmath86 , we observe that @xmath87 , which is in agreement with our expectations . the insert of fig . [ fig : var_1](b ) illustrates the dependence of @xmath88 with the special cases @xmath89 and @xmath90 , which clearly coincide with eq . ( [ eq : eta_m ] ) . additional numerical study reveals the discrepancy for @xmath91 , that is caused by a non - trivial decrease of @xmath37 by decreasing @xmath48 ( @xmath92 ) . the @xmath0-dependence depicted in fig . [ fig : var_1](b ) is constructed for an initial value @xmath93 . there is a certain @xmath0 value for which @xmath94 and iterations cross from alternating to unidirectional convergence . in other words , statistical errors compensate systematic ones . in agreement with ref . , we have verified statistical errors of mc averages to be proportional to  @xmath95 , which implies @xmath96 . actually , the fit provides @xmath97 . if the number of iterations is much larger than @xmath98 , the initial conditions are forgotten and thus invariant pdf s of temperature variable are formed  ( see inset in fig . [ fig : var_infty ] ) . the calculation of @xmath99  ( @xmath100 ) is controlled by @xmath23 . we have verified @xmath101 for the _ stationary regime_. the error increases monotonically with @xmath23 for a given @xmath0 as in fig . [ fig : var_infty ] . we have compared the simulation and analytical @xmath23-dependences of @xmath57 for @xmath102 and @xmath37 , given in table  [ tab : par ] , and those determined by a fit of eq . ( [ eq : var_infty ] ) for @xmath103 . the comparison reveals a discrepancy of the @xmath102 parameter determination . the @xmath102 parameter , determined by the fit of eq . ( [ eq : var_infty ] ) to numerical data , increases its value about @xmath104 . a further extension of the model , e.g. taking into account a weak colored noise , should eliminate this distinction .    according to  eq . ( [ tavel1 ] ) , the mean values of invariant pdf s  ( see inset in fig .  [ fig : var_infty ] ) were used to estimate @xmath105 . the method is straightforward applied for @xmath106 . subsequently , the values @xmath28 are interpolated by the finite size formula @xmath107 . this yields to estimate the thermal coefficient @xmath108 and the critical temperature as @xmath109 . the last estimation stays in a satisfactory agreement with exact value @xmath110^{-1}=0.56729 $ ] . the problem of critical indices can be handled effectively in the frame of a two - lattice iterative algorithm , too  @xcite . in that case , the objective function given by eq . ( [ objec1 ] ) has to be written in terms of the binder cumulant , @xmath111 , we considered , as an example , the exponent @xmath112 of the correlation length @xmath113}{\ln\left(l_1/l_2\right ) } \ , . \label{eq : nu}\ ] ] within the proposed algorithm , we generated a sequence of estimates  ( for each iteration step separately ) of the critical exponent for the lattice pair @xmath114 . considering the peak of nonsymmetric pdf of @xmath115 we determined @xmath116 with a relative error of @xmath117 . an iterative process that is useful around the critical point and includes metropolis single spin - flip algorithm , single - histogram reweighting and linear filtering is studied numerically . the properties of this iterative algorithm are studied for the 2d  ising ferromagnetic model on a square lattice . the total errors in the estimate of pseudocritical temperature are obtained . we demonstrated how the accuracy can be affected by  @xmath23 . however , one should be careful with an extremely small  @xmath23 to avoid an enormous increase of transient time . our analysis demonstrates that it is insufficient to use several reweighting trials to estimate total errors . the most interesting iterations are of this kind that compensate the systematic errors introduced by the histogram reweighting approach . despite of the success of the linearized model , which allows interpretation of numerical data , further improvement of the proposed methodology is still expected . for instance , one should take into account non - linear corrections in eq . ( [ model_2 ] ) and/or the colored noise  ( that is , with autocorrelations of error terms ) in  eq . ( [ eq : whitenoise ] ) . we believe that nontrivial dependences of @xmath37 and @xmath118  ( see e.g. inset of fig . 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we present iterative monte carlo algorithm for which the temperature variable is attracted by a critical point . 
 the algorithm combines techniques of single histogram reweighting and linear filtering . 
 the 2d  ising model of ferromagnet is studied numerically as an illustration . in that case , the iterations uncovered stationary regime with invariant probability distribution function of temperature which is peaked nearly the pseudocritical temperature of specific heat . the sequence of generated temperatures is analyzed in terms of stochastic autoregressive model . 
 the error of histogram reweighting can be better understood within the suggested model . 
 the presented model yields a simple relation , connecting variance of pseudocritical temperature and parameter of linear filtering .