Karina Johansen
email: karinakjohansen@hotmail.com

Rolf Jens Holm-Knudsen

Birgitte Charabi

Michael Seltz Kristensen

Lars Simon Rasmussen

Cannot ventilate-cannot intubate an infant: Surgical tracheotomy or transtracheal cannula?

Keywords: difficult airway, animal model, airway algorithm, emergency tracheotomy, children, transtracheal cannula

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Summary

Background: An unanticipated difficult airway is very uncommon in infants.

The recommendations for managing the can't ventilate-can't intubate ('CVCI') situation in infants and small children are based on difficult airway algorithms for adults. These algorithms usually recommend placement of a transtracheal cannula or performing a surgical tracheotomy as a last resort. In this study we compared the success rate and time used for inserting a transtracheal cannula versus performing a modified surgical tracheotomy in a piglet model.

Methods:

We used 10 three-weeks-old euthanized piglets, weighing eight kilograms each. Thirty physicians had a timed attempt of inserting a transtracheal cannula for jet ventilation. A maximum time of 120 sec was allowed. Ten physicians had a timed attempt of performing a modified surgical tracheotomy after a short introduction by an ENT surgeon. The allowed time for this procedure was 240 sec.

Results: Placement of the transtracheal cannula was successful for eight of 30 physicians. Median time for successful insertion was 68 sec. Surgical tracheotomy was successful for eight of 10 physicians. Median time for successful tracheotomy was 89 sec. There was a significantly higher success rate for surgical tracheotomy (P=0.007).

Conclusions:

We found placement of a transtracheal cannula to be significantly less successful than the modified surgical tracheotomy in a piglet

Introduction

In healthy infants an unanticipated difficult airway is very uncommon. The horrifying situation in which ventilation, oxygenation, and intubation are impossible, is fortunately extremely rare and most anaesthetists will rarely -if ever -encounter this scenario. In this situation when all other means have failed, an invasive technique to ensure oxygenation must be considered. In adult airway algorithms, it is usually recommended to do a needle or surgical cricothyroidotomy, but in infants or small children the recommendations are less clear [START_REF] Henderson | Difficult Airway Society guidelines for management of the unanticipated difficult intubation[END_REF][START_REF]Practice Guidelines for Management of the Difficult Airway. An Updated report by the American Society of Anesthesiologists Task Force on Management of the Difficult Airway[END_REF]. Only one national recommendation exists for children and the last step in the 'can't ventilate-can't intubate' situation ('CVCI') is "a rapid tracheal access or transtracheal jet ventilation" [START_REF] Frova | Recommendations for airway control and difficult airway management in paediatric patients[END_REF]. The technique to achieve this is not specified.

As the situation is extremely rare, no comparative studies exist and probably very few anaesthetists have any experience in invasive techniques that allow emergency airway access in small children thus a suitable training model of this situation is both necessary and difficult to make. Practicing on a manikin may help in decision-making and team work, but it is far from reality to palpate and place a needle on a silicone manikin. An animal model will be more realistic, but only one infant model using kittens has been presented in the literature [START_REF] Mclaughlin | Emergency pediatric tracheostomy: a usable technique and model for instruction[END_REF].

The aim of this study was to compare the success rate and time used for placing a transtracheal cannula versus performing a modified surgical tracheotomy in a piglet model.

Materials and methods

Piglets were chosen for the model as the authors have a broad experience in using pig necks as an adult model for practicing emergency airway access.

The airway of eight kilogram piglets have dimensions comparable to small children. When the piglets are placed on their backs, the trachea and larynx are identified as soft structures as in children.

Ten three-weeks-old piglets, weighing eight kilograms were euthanized by a veterinarian.

Thirty physicians who attended a difficult airway workshop, agreed to participate after being informed about the study.

The participants had prior to the workshop attended lectures on transtracheal jet ventilation and paediatric emergency tracheotomy. During the workshop an ENT surgeon introduced the participants to the tracheotomy technique and the equipment to be used.

For the transtracheal cannula placement the technique recommended by The European Resuscitation Council guidelines was used [START_REF]Airway and ventilation[END_REF]. A water filled 10 ml syringe with a standard 16 or 18 GA intravenous cannula (Venflon™ Pro) attached was used to puncture the trachea just inferior to the cricoid cartilage percutaneously at an angle of 45 0 caudally (Figure 1). After verification of correct position by air aspiration, the needle was removed and the cannula was connected via a three-way stopcock to a high-pressure oxygen source.

Inspiration or "jetting" was performed by occluding the free limb of the stopcock by a finger causing oxygen to be insufflated via the cannula for one second followed by four sec for "expiration" (Figure 2). The participants were timed from the start of palpation on the piglets' neck.

Jet ventilation was started when the participant was convinced that the cannula was placed in the trachea. Verification of cannula placement was performed clinically by watching movement of the thorax. If in doubt, the airflow was increased and/or the escape of air prevented via the upper airway.

An attempt was considered a failure if a functioning cannula was not in place within 120 sec from start, or if "jetting" was attempted with a non-functioning cannula. The time limit was set to make it more "realistic": If the trachea is not cannulated within two minutes of starting the procedure the hypoxia will be life-threatening. In addition, injury to several adjacent structures would likely have occurred, why it would be meaningless to pursue the same strategy.

For the modified surgical tracheotomy we selected tools readily available in theatre. A scalpel, a pair of scissors with a sharp tip and three towel forceps (Figure 3). The larynx and proximal trachea were identified by palpation, and a vertical incision was performed through the skin and subcutaneous tissue from the upper part of the larynx to the sternal notch. The strap muscles were grasped with two towel forceps and separated in the midline giving access to palpate and identify the trachea. Bleeding makes visual identification of the trachea more difficult and the participants were encouraged not to look for but to palpate the tracheal rings in order to indentify the trachea. The trachea was stabilised by grasping it from side to side with a towel forceps, and while the sharp tip of the scissors was inserted between two tracheal rings one to two cm distal to the larynx, the trachea was lifted anteriorly in order not to damage the posterior wall or resect the cartilage (Figure 4). With the scissors a vertical cut in the midline of the trachea in the distal direction was made, and the endotracheal tube was inserted. Placement was verified by successful ventilation of the lungs or by fibroscopy.

A tracheotomy attempt was considered a failure if no endotracheal tube was placed within 240 sec since this procedure would result in a definitive airway. If more than four minutes are used for the emergency tracheotomy a good outcome will be less likely. First all the participants, three per piglet, had a timed attempt of placing a transtracheal cannula for jet ventilation. Ten randomly selected participants subsequently had a timed attempt of doing a modified surgical tracheotomy and inserting a cuffed endotracheal tube size

Statistics

Continuous data are reported with median and range, and proportions are reported with 95% confidence intervals.

The primary end point was the time for obtaining access to the airway, either by insertion of the transtracheal cannula or by performing surgical tracheotomy. This was compared using the Mann-Whitney test as we considered the attempts to be independent. The secondary end point was the success rate for the procedure. This was compared with Fisher's Exact test, based on the assumption that the two procedures were independent.

We illustrated the time and success using Kaplan-Meier statistics.

A p-value less than 0.05 was considered significant. The successful placement of a transtracheal cannula tended to be faster (P=0.08) (Table 1, Figure 5), but the success rate for surgical tracheotomy was significantly higher (P=0.007).

Results

Introduction

The fiberoptic examination was performed in five of the 10 attempts of surgical tracheotomy and confirmed correct placement of the endotracheal tube in all five cases. The tracheal diameter was eight mm (6-9mm)

Discussion

We found that anaesthetists had significantly less success in placing a transtracheal cannula within 120 sec than in performing a modified surgical tracheotomy within 240 sec.

In infants and small children the dimensions of the cricothyroid membrane are small and the cricoid cartilage is narrow, which makes it difficult or impossible to pass an endotracheal tube through the membrane without risk of fracturing the cartilages of the larynx (6). This combined with the difficulty in identifying the membrane in infants because of the short neck, the cephalad position of In our study surprisingly many of the participants had difficulty inserting a transtracheal cannula in the piglet model, despite the relative ease of palpating the larynx and upper part of the trachea. Only 27% succeeded within the time limit of 120 sec. In the 14 attempts assessed via the endoscope we also noticed that the anaesthetist had difficulty determining whether the cannula was in the tracheal lumen or not. In one case the cannula was correctly placed but kinked and could not be used. This could probably have been avoided by using a more kink resistant device as for instance the "Ravussin catheter". The success rate of transtracheal cannula was low and there is a risk of serious complications such as air embolism, extensive cervical emphysema, pneumothorax and We found that a short introduction to a modified surgical tracheotomy proved to be sufficient to make eight of 10 participants able to perform successfully on the model within the predefined time limit. The median time for successful tracheotomy was 89 sec. Although this is 21 sec more than for the successful insertion of a transtracheal cannula, the success rate was three times higher.

In addition, the tracheotomy is a secure airway, whereas the transtracheal cannula is only a temporary airway, which easily displaces and must be substituted urgently with another one.

The tools available were chosen to be as simple and few as possible to simulate a realistic scenario. A scalpel is always readily available in theatre and a pair of scissors and the towel forceps are standard equipment in our anaesthetic trolley. We chose to make the tracheal incision with the scissors instead of using the scalpel, to make it easier for a surgically untrained person to control the size of the tracheal incision.

This piglet model of course has limitations. It is not possible to create the same extreme level of stress that a severely hypoxic child will cause. But the stress will probably be equally disadvantageous in both procedures. As the piglets were dead, there was no bleeding during the procedure. Bleeding must be expected in a real situation and will add to the stress and make the procedures more difficult. In order to compensate for this we encouraged the participant not to depend on vision during identification of the trachea and In adults a transtracheal cannula is a well-established way of keeping an acceptable oxygenation during a 'CVCI' situation until a more secure airway is established [START_REF] Heard | The formulation and introduction of a 'can't intubate, can't ventilate' algorithm into clinical practice[END_REF][START_REF] Benumof | The importance of transtracheal jet ventilation in the management of the difficult airway[END_REF]. This is probably the explanation for recommending this procedure as a last resort in the emergency airway management in infants and small children as well (6, 7) [START_REF] Walker | The management of difficult intubation in children[END_REF][START_REF] Mace | Needle cricothyrotomy[END_REF].

The evidence for this recommendation does nevertheless not exist. Only a few papers, including one case report and one small series of elective percutaneous transtracheal jet ventilation, using the specially designed "Ravussin catheter", in children have been published [START_REF] Ravussin | Percutaneous transtracheal ventilation for laser endoscopic procedures in infants and small children with laryngeal obstruction: report of two cases[END_REF][START_REF] Depierraz | Percutaneous transtracheal jet ventilation for paediatric endoscopic laser treatment of laryngeal and subglottic lesions[END_REF]. The transtracheal cannulas were placed during controlled circumstances under direct endoscopic visual guidance in a well-oxygenated child. Despite this, serious complications occurred, including bilateral pneumothorax. This situation is very different form the chaotic situation that rules at the time point, when the anaesthetist finally as the last resort decides to attempt placing a transtracheal cannula in a severely hypoxic child.

In the literature we have not found any studies documenting success or risks associated with needle cricothyroidotomy or transtracheal needle access in a 'CVCI' situation in infants. Only one case with a successful placement of a In this case the cannula was placed via a pre-existing tracheal fistula [START_REF] Smith | Transtracheal ventilation in paediatric patients. Case reports[END_REF].

Currently the recommendations are based on expert's opinion and reviews of articles focused on adult airway management (8) [START_REF] Walker | The management of difficult intubation in children[END_REF]. Weiss et al [START_REF] Weiss | Proposal for the management of the unexpected difficult pediatric airway[END_REF] recently proposed an algorithm for the management of the unexpected difficult paediatric airway based on the Difficult Airway Society's (DAS) recommendations for adults [START_REF] Henderson | Difficult Airway Society guidelines for management of the unanticipated difficult intubation[END_REF]. As their rescue technique in the 'CVCI' situation they reserve cannula cricothyroidotomy for children older than eight, but they recommend surgical cricothyroidotomy in patients of all ages.

The 'CVCI' situation in a small child without anticipated difficult airway is extremely rare. As with other rare situations, ie malignant hyperthermia, it is important to be prepared and practice the situation on models or in simulators to improve the chance of survival of the patient. Not only the technical skills but also decision making should be trained.

Thorough preoperative assessment, clinical education, simulator training, and implementation of national society recommendations are all important in order to reduce the occurrence of a failed airway [START_REF] Weiss | Proposal for the management of the unexpected difficult pediatric airway[END_REF]. We think this should be We cannot draw any firm conclusions from this small study, but it may stimulate to some thoughts and more studies, involving animal models, on emergency airways in small children. We found the placement of a transtracheal needle to be more difficult and less successful than the modified surgical tracheotomy in a piglet model. We question if placement of a transtracheal cannula should be recommended in infants in a can't ventilatecan't intubate situation. Table 1. Time and success rate for obtaining access to the piglet airway, either by insertion of the transtracheal cannula or by performing surgical tracheotomy.

  question if placement of a transtracheal cannula should be recommended in infants in a can't ventilate-can't intubate situation. Keywords: difficult airway; animal model; airway algorithm; emergency tracheotomy; children; transtracheal cannula.

3. 5

 5 mm ID.The participants were told about the maximum time allowed prior to the study.Before starting the procedure, in half of the piglets the tip of an Olympus Lf-2 flexible fiberscope, connected to an Olympus CLE cold light source was placed at the level of the vocal cords by means of direct laryngoscopy. In this way the subglottic space and the trachea were observed during the entire procedure. The observations obtained via the fiberscope were not available to the person performing the transtracheal procedure nor to the person doing the timing and clinical confirmation of the cannula placement.

8

 8 After the workshop the trachea of the piglets were transected just below the cricoid cartilage and the tracheal diameter was measured.

  of the flexible scope allowed adequate visualization of the subglottic space and the trachea during the entire procedure in the five piglets, in which this was attempted. Placement of a transtracheal cannula was successful in eight of 30 (27%) attempts within 120 sec. The median time of the eight successful insertions was 68 sec (35-95 sec) until jet ventilation was established. Visualization using the fiberscope was carried out in 14 of these 30 attempts. The examination revealed a discrepancy between the clinical verification of the cannula placement and the actual placement in four of the 14 (29%) included attempted jet ventilation despite paratracheal placement, a kinked catheter, and also a correctly placed cannula, but no jet ventilation because of uncertainty of the cannula placement. A modified surgical tracheotomy was performed successfully in eight of 10 (80%) participants within 240 sec. The median time for successful insertion of the endotracheal tube with this procedure was 89 sec (71-200 sec). The ten participants performing a tracheotomy had a similar success rate of transtracheal cannula placement as the other participants (two out of 10).

  softness of the cartilages, and the prominent hyoid bone, which is easily mistaken for the thyroid cartilage, makes it unattractive to perform cricothyroidotomy in small children[START_REF] Cote | Pediatric transtracheal and cricothyrotomy airway devices for emergency use: which are appropriate for infants and children?[END_REF].The alternatives left in a 'CVCI' situation are to perform an emergency tracheotomy or insert a transtracheal cannula. The tracheotomy is a definitive airway, which can be used for ventilation, whereas the transtracheal cannula only is a temporary measure, which makes it possible to oxygenate the child with modified jet ventilation until surgical expertise arrives to perform a tracheotomy. Despite this, since anaesthetists are very familiar with cannula techniques and not surgically trained, the transtracheal cannula insertion will be the most appealing choice to the majority of anaesthetists.

  damage to the oesophagus if jet ventilation is attempted using a misplaced cannula.

  find the tracheal rings. The strengths of this model are that the palpation for the landmarks and the sensations during cannulation and surgery are very realistic and much better than when using manikins. The piglets size corresponds to an infant of 6 months to 3 years.

  a six year old, 20 kg Treacher Collins girl was found.

  combined with further development of an animal model for training and practicing surgical emergency airway access in infants and children. The setup described by Heard et al. (8) seems ideal, but the use of live animals is very demanding and costly and therefore probably not an option in most departments.

Figure 5 .

 5 Figure 5. Kaplan-Meier analysis of time and success for performing

  Fig 1 Transtracheal cannula placement. Fig 2 Three-way stopcock used for jet ventilation. Fig 3 Tools for emergency tracheotomy. Scalpel, nurses scissors, 3 towel forceps. Fig 4 Trachea is presented and stabilised and the strap muscles held aside with three towel forceps. A hole has been cut with in two tracheal rings with the scissors.


Greqam

Daria Onori
email: daria.onori@uclouvain.be.

Specialization and Growth

Keywords: Closed form solutions, Welfare, Competition, Degree of returns to specialization. JEL Classification: C61, L16, L4, O31, O33

In this paper we consider a simple model of horizontal differentiation and derive the closed form solutions for the level of the variables in the decentralized economy and in the social planner case. This enables us to analyze consumers' welfare as a function of the parameter representing market power. We surprisingly find that, when the total labor force is greater than a certain level, the welfare function is an inverted-N shape in the decentralized economy and monotonically decreasing in the centralized economy. This suggests that there is another effect which interacts with market power: the degree of returns to specialization.

Introduction

In this paper we analyze consumers' welfare as a function of a parameter representing the degree of competition. We consider the simplest endogenous growth R&D based model reported in Barro and Sala-i-Martin (2004), Ch.6, which is a simplified version of [START_REF] Romer | Endogenous technical change[END_REF] model.

In this setup R&D aims to expand the number of existing varieties of the intermediate goods and it uses labor. It is also assumed that intermediate goods are not durable.

By starting from the decentralized economy analysis we are able to derive the expression for consumers' welfare since we provide the closed form solutions for the trajectories of all the key variables. Usually economists have studied non-linear dynamical systems of endogenous growth models in a qualitative way, that is using the phase diagram and completed this I wish to thank R. Boucekkine, C. Camacho, A. Venditti, E. Marchetti, and the partecipants in seminars and conferences at Université catholique de Louvain and GREQAM Marseilles. Grateful acknowledgement is made for financial support by GREQAM. The usual disclaimer applies.

1 analysis with a quantitative one, linearizing around the steady state after having applied the so called reduction dimension strategy. This technique consists in the definition of new variables in terms of ratio between the original ones and it is generally not sufficient to be fully conclusive about the dynamic properties of the original variables in levels. We solve instead the dynamical system analytically. In particular we derive the differential equation which describes the dynamics of the amount of labor used in R&D. We obtain a Riccati differential equation which admits two steady states: one is the balanced growth path solution of Barro and Salai-Martin Ch. 6, and it is unstable, the other, which is equal to the total labor force, which is stable. After solving it we provide the explicit solution for the trajectory of consumption. We solve for the optimal trajectories by using the transversality condition and we find that for it to be satisfied the labor force allocated in R&D at time zero must be set equal to its balanced growth path solution. This proves that the model does not exhibit transitional dynamics. We can also determine the initial value for consumption by using market clearing condition for the final good at time zero. This allows us to compute the value of consumers' welfare, a result that is not usual to be found in literature, given that models usually do not allow for closed form solutions. We then solve the social planner's maximization problem and derive also in this case the level of consumption and the related welfare function.

The previous results enable us to undertake a comparative static analysis. In this model an increase in market power, represented by a reduction of the parameter α, reduces the decentralized economy's growth rate, since it causes a reallocation of the fixed labor force toward the final good sector, slowing down the R&D production, which in turns constitutes the engine of growth. This result places this model in the debate on the relationship between competition and growth in R&D-Based Growth Models. The first generation of Schumpeterian growth models of vertical differentiation (Grossman and Helpman 1991, Aghion and Howitt 1992, Barro and Sala-i-Martin 2004) predicts that innovation and hence growth should decline with competition, because more competition reduces the rents that reward successful innovators (this is the so called Schumpeterian effect). This discourages firms from investing in R&D, thus reducing the innovation rate and as a consequence the long run growth rate of the economy. However the empirical literature of the Nineties (Gerosky 1995, Blundell, Griffith and Van Reenen 1999) suggests a positive correlation between competition and growth. On the contrary models of horizontal differentiation are less conclusive on the predictions. The model considered in this paper (Romer's 1990 model) predicts a positive relationship between competition and growth, while Grossman and Helpman's (1991, Chapter 3) model predicts a negative relationship (see [START_REF] Bucci | Market Power and Aggregate Economic Growth in Models with Endogenous Technological Change[END_REF] for a review). Recently it has been shown some evidence in favor of a bell-shaped relationship between competition and growth [START_REF] Aghion | Competition and innovation: an inverted-U relationship[END_REF]. Few are the explanations provided by the literature. For example [START_REF] Aghion | Competition and innovation: an inverted-U relationship[END_REF] explain the bell-shaped form by means of the interaction between the Schumpeterian effect and the escape competition effect, while for [START_REF] Bucci | An Inverted-U Relationship between Product Market Competition and Growth in an Extended Romerian Model[END_REF] the interaction is between a resource allocation effect and the Schumpeterian effect1 . Finally, Onori's (2010) explanation relies on the presence of a spillover effect stemming from the intermediate good sector that may prevail over the Schumpeterian effect.

On the other hand, the social planner's growth rate is independent from market power index because she corrects this market failure. Hence we would expect a positive effect of an increase in α on consumers' welfare in the decentralized economy and no effect on social planner's welfare. However we surprisingly find that, when the total labor force is greater than a certain level, consumers' welfare in the decentralized economy is an inverted-N shaped function of α, while the welfare level of the centralized economy is a monotonically decreasing function of α. This suggests that there exists an additional effect which interacts with market power decrease. This effect is negative and is represented by the decrease of the degree of returns to specialization. The degree of returns to specialization measures the degree to which the society benefits from "specializing" production between a larger number of intermediates. Hence in the social planner's case an increase in α simply reduces the degree of returns to specialization, so explaining the monotonically decreasing welfare function. In the decentralized economy the negative effect of a reduction of the degree of returns to specialization may prevail over the positive effect of a reduction of market power, giving raise to an inverted-N shaped function.

The structure of the paper is the following. In section 2 we briefly present the model and we derive the closed form solutions for labor force employed in R&D and for consumption in the decentralized economy and in the social planner's case. Section 3 provides the welfare analysis, finally section 4 concludes.

The model

The decentralized economy

In this section we briefly describe the model of horizontal differentiation shown in Ch. 6 of [START_REF] Barro | Economic growth[END_REF], in the decentralized economy case.

There are three different productive sectors. The undifferentiated final good is produced by using labor and intermediated goods and it is sold in a perfectly competitive market2 . The technology is represented by a Cobb-Douglas production function:

Y (t) = (L -L R (t)) 1-α N (t) j=1 (x j (t)) α
where (L -L R (t)) is the work force used in the final good production3 , x j (t) is the quantity employed of the jth intermediate good and 0 < α < 1. The total labor force L is constant over time.

The first order conditions of final good producers' determine the demand function for each intermediate good and the wage rate:

x j (t) = α p j 1 1-α (L -L R (t)) w (t) = (1 -α) Y (t) / (L -L R (t)) .
R&D firms face a two-stage decision process. In the first stage, they decide whether to devote resources to invent a new brand. They will invest until the point in which the expected flow of future profits equals the R&D cost. In the second stage, the inventors, provided by a perpetual patent, enter the intermediate good sector and chose the optimal price at which to sell their new product. Hence the intermediate goods industry is imperfectly competitive and each producer uses the same technology: one unit of the final good is needed to produce one unit of intermediate good. Hence the profit of each monopolist is given by π j (t) = p j x j (t)x j (t) and the monopolistic price is p j = 1 α , ∀j. Using these informations we obtain the produced quantity of each intermediate good

x j (t) = x (t) = (L -L R (t)) α 2 1-α , ∀j.
The R&D sector is perfectly competitive and there is a positive spillover effect attached to the available stock of disembodied knowledge (approximated by the existing number of designs, N (t)) in discovering a new product variety: that is to produce a new variety of good it is necessary to employ η N (t) units of labor and η > 0 represents a cost parameter. Hence the dynamics of the number of brands is

Ṅ (t) = L R (t) η N (t) .
As the research sector is perfectly competitive, new firms will enter each R&D race until the profit is zero. As Barro and Sala-i-Martin analysis is only focused on the balanced growth path, L R and the interest rate are constant, so the research arbitrage condition writes as

ˆ+∞ t 1 -α α (L -L R ) α 2 1-α e -r(ν-t) dν = η (1 -α) α 2α 1-α .
The left hand side represents the benefit of each innovator while the right and side is the cost of doing R&D4 . It can be rewritten as

r = α (L -L R ) η
Finally the representative household maximizes utility over an infinite horizon. It consumes the final good and her savings finance R&D

max C(t) U = ´+∞ 0 C(t) 1-σ 1-σ e -ρt dt s. t. ȧ (t) = r (t) a (t) + w (t) L -C (t)
The optimality conditions are thus given by

       Ċ(t) C(t) = 1 σ (r (t) -ρ) ȧ (t) = r (t) a (t) + w (t) L -C (t) lim t→+∞ C (t) -σ a (t) e -ρt = 0
In a general equilibrium setup the following relation between the consumer's assets and the number of varieties exists:

a (t) = N (t) η (1 -α) α 2α 1-α .
Given the market clearing condition for consumption good and the fact that the aggregate output is proportional to the number of varieties (

Y (t) = N (t) (L -L R (t)) α 2α 1-α
), in the balanced growth path (BGP hereafter) all the variables grow at the same positive and constant growth rate:

g Y = g C = g X = g N = g = LR η
where LR is the steady state solution for the labor employed in R&D and it is given by

LR = η α + σ α η L -ρ .

The closed form solutions

In this section we derive the optimal trajectories for the labor force employed in R&D and consumption. We prove that the model does not exhibit transitional dynamics.

In order to derive the dynamics of L R (t) we differentiate the research arbitrage condition

ˆ+∞ t 1 -α α (L -L R (ν)) α 2 1-α e -r(t,ν)(ν-t) dν = η (1 -α) α 2α 1-α
with respect to time. We remind that in this case the interest rate is the average over the period: r (t, ν) = 1 ν-t ´ν t r (ω) dω and the amount of human capital employed in R&D can vary over time.

Let us call the R&D benefit B (t):

B (t) ≡ ˆ+∞ t 1 -α α (L -L R (ν)) α 2 1-α e -r(t,ν)(ν-t) dν. So dB (t) dt = - 1 -α α (L -L R (t)) α 2 1-α + r (t) B (t) . (2.1)
We know by the research arbitrage condition that

B (t) = η (1 -α) α 2α 1-α , hence (2.1) becomes r (t) = α (L -L R (t)) η . (2.2)
We can rewrite the Euler equation as

Ċ (t) C (t) = r (t) -ρ σ = α(L-L R (t)) η -ρ σ . (2.3)
In order to obtain the differential equation describing the dynamics L R (t), we use the final good market clearing condition:

Y (t) = C (t) + N (t) x (t) ,
where X (t) = Nx (t) is the aggregate quantity of produced intermediate goods.

Since the quantity produced by each intermediate firm is given by

x (t) = (L -L R (t)) α 2
1-α , we can rewrite the final output as Y (t) = N (t) (L -L R (t)) α 2α 1-α . Using these informations, we can solve the final good market clearing condition with respect to consumption

C (t) = α 2α 1-α 1 -α 2 N (L -L R (t)) .
By differentiating it with respect to time we obtain

Ċ (t) = α 2α 1-α 1 -α 2 Ṅ (t) (L -L R (t)) -LR (t) N (t) .
Hence Ċ (t)

C (t) = Ṅ (t) N (t) - LR (t) L -L R (t) .
Since the growth rate of designs is

Ṅ (t) N (t) = L R (t)
η , the previous relation becomes

Ċ (t) C (t) = L R (t) η - LR (t) L -L R (t) . (2.4)
By equating (2.3) and (2.4) we get a first order differential equation in L R :

LR (t) = 1 η -1 - α σ L 2 R (t) + L + 2αL σ - ρη σ L R (t) + ρηL σ - αL 2 σ . (2.5)
This equation admits two steady states, which are given by

LR 1 = η α + σ α η L -ρ
which is the solution we get with the BGP analysis, and

LR 2 = L
so that in this case the economy would end up starving in the long run.

2.2.1

The trajectories of L R (t) and C (t)

In this section we derive the trajectories of L R (t) and C (t). Equation (2.5) is a Riccati differential equation. In order to solve it we need to guess a particular solution. The particular solutions are the ones that solves the second degree polynomial on the right hand side, that is the steady state solutions.

We start by guessing LR = LR 2 = L, hence the general solution is given by

L R (t) = L + de 1 η (L+ ρη σ )t - 1 + α σ L + ρη σ -1 . (2.6)
The integration constant d is given by

d = [L R (0) -L] -1 + 1 + α σ L + ρη σ
where L R (0), the initial value for the work force used in R&D, is not known in advance since

L R (t) is not a predetermined variable 5 . Equation (2.6) becomes L R (t) = L + (L R (0) -L) -1 + 1 + α σ L + ρη σ e 1 η (L+ ρη σ )t - 1 + α σ L + ρη σ -1 .
(2.7)

If we take the limit for t → +∞ we obtain

lim t→+∞ L R (t) = L = LR 2 Figure 2.1: Possible trajectories of L R (t)
that is the steady state LR 2 = L is stable while the BGP solution,

LR 1 = η α+σ α η L -ρ , is unstable 6 .
Graphically the possible trajectories, for each possible initial value L R (0), of L R (t) are given by Our aim is to determine the optimal trajectory among these.

For sake of simplicity the analysis will be conducted for the logarithmic utility case (σ = 1).

By using (2.4), (2.5), (2.7) we can find the expression for the trajectory of consumption:

C (t) = f exp L-ρη 2η(1+α) t 1+α L+ρη - (L R (0) -L) -1 + 1+α L+ρη exp 1 η (L + ρη) t -α 1+α = f exp LR 1 η t 1+α L+ρη - (L R (0) -L) -1 + 1+α L+ρη exp 1 η (L + ρη) t -α 1+α .
(2.8)

The integration constant f is given by

f = C (0) (L -L R (0)) α 1+α
(2.9)

where consumption at time zero, C (0), has to be determined using the optimality conditions.

Optimal trajectories

In the following proposition we prove the optimality of the trajectory L R (t) = LR 1 = αL-ηρ 1+α , ∀t, that is the economy immediately jumps on the BGP.

Proposition 1. The trajectory L R (t) = LR 1 = αL-ηρ 1+α , ∀t, obtained for L R (0) = LR 1 = αL-ηρ 1+α is optimal.
Proof. In order to be optimal the solutions for L R (t) and C (t) given by, respectively, (2.7) and (2.8) have to satisfy the transversality condition. It is given by

lim t→+∞ e -ρt 1 C (t) a (t) = 0.
The total of consumer's assets equal the market value of the intermediate firms:

a (t) = V N (t) = η (1 -α) α 2α 1-α N (t)
so the transversality condition can be rewritten as

lim t→+∞ η (1 -α) α 2α 1-α e -ρt 1 C (t) N (t) = 0. (2.10)
Plugging in equations (2.7) and (2.8) we obtain:

lim t→+∞ N (t) η (1 -α) α 2α 1-α exp [-ρt] 1 f exp LR 1 η t -1 1+α
L+ρηd exp

1 η (L + ρη) t α 1+α exp L η exp d exp 1 η (L+ρη)t -1+α L+ρη -1 η = 0 where d = [L R (0) -L] -1 + 1+α L+ρη and f = C (0) (L -L R (0)) 1-α 1+α . Then lim t→+∞ 1 + α L + ρη -d exp 1 η (L + ρη) t α 1+α = ±∞
depending on the sign of the constant d7 .

Since all the remaining terms tend to a constant or to zero, for the transversality condition to be satisfied a necessary condition is to have d = 0 ⇐⇒ L R (0) = LR 1 . Given this, we can determine the other constant f using the final good market clearing condition at t = 0:

Y (0) = C (0) + N (0) X (0) C (0) = α 2α 1-α 1 -α 2 N (0) (L -L R (0)) C (0) = α 2α 1-α 1 -α 2 N (0) L -LR 1
where C (0) is given by equation (2.9):

C (0) = f (L -L R (0)) α 1+α = f L -LR 1 α 1+α Hence f = α 2α 1-α f L -LR 1 -α 1+α 1 -α 2 N (0) L -LR 1 = α 2α 1-α 1 -α 2 N (0) L -LR 1 1 1+α = α 2α 1-α 1 -α 2 N (0) L+ρη 1+α 1 1+α .
In Proposition 1 we have just found a necessary condition for the transversality condition to be satisfied. Actually there can be other values for the constant C (0) which satisfy it. To see this, we first compute exp [-ρt]

1 f exp LR 1 η t -1 = exp -ρ-αρ-αL+ρη 1+α t . Then combining it with 1+α L+ρη -d exp 1 η (L + ρη) t α 1+α , we require that exp -ρ -αρ -αL + ρη 1 + α t 1+α α + 1 η (L + ρη)t → 0.
In light of Proposition 1, the trajectory of consumption can be rewritten as

C (t) = α 2α 1-α (1 -α) N (0) (L + ρη) exp αL -ρη η (1 + α) t . (2.11)
Hence the initial level of consumption is optimally chosen in such a way that the economy jumps on the BGP at t = 0.

The centralized economy

In this section we derive the closed form solution for consumption of a benevolent social planner who seeks to maximize the representative consumer's utility. The social planner is constrained by the low of motion of designs and by the economy's budged constraint:

max {C(t),x(t),L R (t)} ´+∞ 0 ln C (t) e -ρt dt s. t. Y (t) = C (t) + N (t) x (t) Ṅ (t) = L R (t) η N (t) N (0) given
From the economy's resource constraint we can derive an expression for consumption level

C (t) = (L -L R (t)) 1-α N (t) x (t) α -N (t) x (t) (2.12)
so that the problem can be rewritten as

max {x(t),L R (t)} ´+∞ 0 ln (L -L R (t)) 1-α N (t) x (t) α -N (t) x (t) e -ρt dt Ṅ (t) = L R (t) η N (t) N (0) given
The Pareto optimal solution for x (t), L R (t) and the growth rate are given by

x SO = (L -L R ) α 1 1-α L R SO = L -ρη g SO = L η -ρ.
Hence equation (2.12) can be rewritten as

C (t) SO = 1 -α α α 1 1-α ρηN (0) exp L η -ρ t .
(2.13)

3 Welfare, competition, specialization and growth

In this section we analyze the effects of a change in α on consumers' welfare of the decentralized economy and compare them with the results of the centralized economy.

The parameter α plays two different roles in this model: first 1 α -1 is the mark up enjoyed by innovators, so an increase in α reduces their market power; second it determines the degree of returns to specialization 8 . Definition. (Degree of returns to specialization [START_REF] Benassy | Is there always too little research in endogenous growth with expanding product variety?[END_REF]). The degree of returns to specialization measures the degree to which the society benefits from "specializing" production between a larger number of intermediates N (t).

We now compute it for our economy. Since the intermediate industries are symmetric, the production function of the final good can be written as

Y (t) = (L -L R (t)) 1-α N (t) j=1 (x j (t)) α = (L -L R (t)) 1-α N (t) x (t) α .
Given that the aggregate quantity of intermediate goods is X (t) = N (t) x (t) the previous expression rewrites as

Y (t) = (L -L R (t)) 1-α N (t) 1-α X (t) α
where (1α) is the degree of returns to specialization. Hence an increase in α has a negative effect on it 9 .

The growth rates

For sake of comparison we report the expression of the growth rates of the decentralized economy and of the centralized economy:

g DE = αL -ρη η (1 + α) and g SO = L η -ρ
We immediately note that ∂g DE ∂α > 0 while the social planner's growth rate is independent from α. This means that the growth rate of the decentralized economy only depends on market power, given that the social planner eliminates this distortion. An increase in α rises the growth rate of the decentralized economy because a reduction of the mark ups of all the intermediate inputs makes it more profitable for the final good producer to substitute labor force with intermediate goods. As a consequence, given that the total labor force is constant, the quantity of human capital available for research increases and since this sector is the engine of growth, the growth rate rises.

On the contrary, we will see in the next section that the degree of returns to specialization plays a role on determining welfare. constraint consumption is given by

C (t) = (L -LR (t)) 1-α N (t) j=1 (xj (t)) α -N (t) x (t)
, so it depends also linearly on x (t). 9 We remark that for a given N (t) the production function exhibits constant returns to scale in L (t) and X (t).

The Welfare functions

In this section we prove that the relationship between welfare and α is monotonically decreasing in the centralized economy, while it can be monotonically decreasing or an inverted-N shape in the decentralized economy and we provide an economic interpretation.

Proposition 2. The relationship between consumers' welfare and α in the centralized economy is monotonically decreasing.

Proof. The optimal level of consumption for the centralized economy is given by

C (t) SO = 1 -α α α 1 1-α ρηN (0) exp L η -ρ t
hence welfare is

U * SO = 1 ρ ln 1 -α α α 1 1-α ρηN (0) + 1 ρ L η -ρ . (3.1)
If we derive it with respect to α we obtain

∂U * SO ∂α = 1 ρ 1 1 -α 1 1 -α ln α
which is always negative since α ∈ (0, 1).

The results for the decentralized economy are reported in the following Proposition:

Proposition 3. If L ≥ L, where L = λρη, λ ≡ f min -1 1-α 2
1-α ln α + 1 10 , the relationship between α and the consumer's welfare of the decentralized economy is an inverted-N shape, otherwise it is a decreasing function of α.

Proof. (Cf. Appendix).

In Section 3.1 we saw that the growth rate of the decentralized economy is an increasing function of α while the one of the centralized economy is independent from α. This result is linked to social planner's effort to correct the distortion induced by monopoly. The results showed in Proposition 2 and Proposition 3 suggest that there are two interplaying effect on welfare stemming from an increase in α:

• a positive effect: an increase in α reduces innovators' market power ("competition effect");

• a negative effect: an increase in α reduces the degree of returns to specialization ("degree of returns to specialization effect").

The relationship between welfare and α in the social planner case is negative and this means that only the negative effect of a reduction of the degree of returns to specialization plays a role, since the social planner corrects the market imperfection. While the relationship between welfare and α may be an inverted-N shape in the case of the decentralized economy. We can disentangle these two effects by looking at the derivative of the welfare function of the decentralized economy

∂U * DE ∂α = 1 ρ 1 1 -α 2 1 -α ln α + 1 + 1 ρη L + ρη (1 + α) 2 . (3.2)
It can be decomposed in two parts:

g (α) ≡ 1 1 -α 2 1 -α ln α + 1 and h (α) ≡ 1 ρη L + ρη (1 + α) 2 .
These two functions are represented in the figure below Since g (α) is negative, it clearly represents the degree of returns to specialization effect, while being h (α) positive, it can be identified as the competition effect. The presence of these two effects in the welfare function of the decentralized economy becomes much more evident if we compare equation (3.2) with the derivative of the welfare function in the social planner's case, as given by equation (3.1). The latter function shows the same behavior as g (α), while we have already underlined that the competition effect is not present in the centralized economy, hence we do not observe the part which depends positively on α.

The position of h (α) in the graph depends on the values of the parameters L, η and ρ: as it is showed by the sufficient condition provided by Proposition 3, the greater is L and the lower are η and ρ, the greater is the possibility that h (α) and the symmetric of g (α) cross twice 11 , hence the first derivative has two zeros and the welfare function is an inverted-N shape.

Graphically: L is the economy's total human capital. A high L means that there will be more labor force to be employed in R&D 12 . But this is exactly the scale effect, a well known shortcoming of this model. η is the R&D parameter cost. The smaller is η, the lower is the cost of R&D and the higher will be the resources engaged in expanding brands. Finally, ρ is the consumers' discount rate. The lower is ρ, the higher is consumers' willingness to save, and the higher is their investment in R&D 13 . These effects are assimilable to the effect of an increase of α on 11 This pushes h (α) up in the graphic.

12 Recall that it is:

∂ LR 1 ∂L > 0.
13 We remind that in this economy consumers finance R&D.

the growth rate of the decentralized economy through a reduction of the mark up and explain the positive sloped branch of the welfare function.

We can also say more about how the two effects determine the shape of the welfare function in the decentralized economy. This explanation is related to the second derivative of the welfare function. The competition effect has a positive effect on welfare but it is decreasing in α: as competition increases, even if human capital resources to be employed in R&D are a lot, innovators loose any incentive to invest in R&D because they are deprived of their market power. On the other hand the degree of returns to specialization is not monotonic in α. Remember Figure 3.1. When α is small, the degree of returns to specialization is high, however the weight of the intermediate goods on the final good production function is low: the importance of the technological sector in the economy is small and the produced quantity of each intermediate good x = (L -L R (t)) α 2 1-α is very small. For this reason the degree of returns to specialization effect is strongly negative. As α increases, even if the degree of returns to specialization is still negative, the function g (α) attains a maximum: the degree of returns to specialization is still high (the maximum is attained for α MAX 0.383012), but now the importance of the technological sector is higher than before and this explains why the degree of returns to specialization effect becomes less negative. When instead α is high enough, the degree of returns to specialization becomes smaller and smaller, while the impact of the intermediate sector is very high. This explains the decreasing branch of the function g (α). To sum up, in the first decreasing branch of the welfare function the degree of returns to specialization effect prevails over the competition effect because it is true that the degree of returns to specialization is high, but the importance of the intermediate sector is small. In the increasing branch of the welfare function it is possible to see the positive effect of an increase in competition counterbalancing the negative effect of the degree of returns to specialization effect. The last branch is negatively sloped because the the economy is close to a situation of perfect competition, the importance of the technological sector has become very high so the degree of returns to specialization dominates the competition effect, hence we see the same behavior of the welfare function as in the centralized economy case.

Numerical examples

In this section we provide some numerical examples in order to show the possible shapes that the welfare in the decentralized economy can exhibit. We normalize the total human capital L to one and we set the discount rate ρ equal to 0.03 14 . Moreover we normalize the initial number of designs to one and we let the R&D cost parameter free to vary because it is not possible to calibrate it by using the balanced growth path restrictions since all of them involve α and we cannot fix it. By the sufficient condition of Proposition 3 that assures an inverted-N 14 These are quite standard parametrizations. Given the expression of the first derivative (3.2) it is difficult to find the turning points analytically. We can provide a solution of equation (3.2) in a neighborhood of α = 1, which corresponds to the value of α for which the welfare function attains a maximum. By rewriting the first derivative as

ln α = - L + ρη ρη (1 + α) 2 (1 -α) -1 1 -α 2
and by setting α = 1 -, we obtain the following second order equation in 16 :

ρη 2 -(L + 5ρη) + 4ρη = 0 whose solutions are 1,2 = (L + 5ρη) ± (L + 5ρη) 2 -16ρ 2 η 2 2ρη = (L + 5ρη) ± L 2 + 9ρη 2 + 10Lρη 2ρη
.

15 In this model the mark up, as defined as Price-Marginal Cost Marginal Cost

, is equal to 1 α -1. The Lerner index, which is a good approximation of the price-cost margin, Price-Marginal Cost Pricet , is related to the mark up in the following way: Lerner Index= Mark Up 1+Mark Up . 16 We recall that ln(1 -) -.

Since

1 = (L + 5ρη) + (L + 5ρη) 2 -16ρ 2 η 2 2ρη > 1 2 = (L + 5ρη) - (L + 5ρη) 2 -16ρ 2 η 2 2ρη < 1.
1 implies that α < 0, while with 2 we get a solution for α in the interval (0, 1) which is given by:

α * = -3ρη -L + (L + 5ρη) 2 -16ρ 2 η 2 2ρη .
This expression provides a good approximation of the value of α which maximizes welfare in the decentralized economy, if it is in a neighborhood of 1. For example, in the η = 1 case, the true value of α which maximizes welfare is 0.8836 and α * = 0.8954.

Conclusions

In this paper we considered the model of horizontal differentiation reported in Barro and Salai-Martin (2004), Ch. 6, which is a simplified version of Romer's (1990) model and derived the closed form solutions for the level of the variables in the decentralized economy and in the social planner case. This enabled us to analyze consumers' welfare as a function of the parameter representing the degree of competition. As is well known in this model an increase in competition enhances the decentralized economy's growth rate because it causes a reallocation of human capital toward the R&D sector, while the centralized economy's growth rate is independent from the competition parameter because the social planner eliminates the distortion due to the presence of monopolistic competition. As a result of welfare analysis, instead, we surprisingly find that, when the total labor force is greater than a certain level, the welfare function is an inverted-N shape in the decentralized economy and monotonically decreasing in the centralized economy. This suggests that there is another effect which interacts with market power: the degree of returns to specialization effect.

Appendix. Proof of Proposition 3

Given the solution for the level of consumption: Deriving it with respect to α gives

C (t) DE = α
∂U * DE ∂α = 1 ρ 1 1 -α 2 1 -α ln α + 1 + 1 ρη L + ρη (1 + α) 2 . ( .1)
The sign of the curly bracket is ambiguous since ln α < 0 for α ∈ (0, 1). In order to study it we define

f (α) ≡ 1 1 -α 2 1 -α ln α + 1 + 1 ρη L + ρη (1 + α) 2 .
This function is continuous for α ∈ (0, 1) and lim α→0 f (α) = lim α→1 f (α) = -∞. However

given the fact that it depends on the parameters of the model, L, η and ρ, we are not able to say anything about the existence of some intersection with the αaxis17 and, if they exist, how many they are. To this end, we define:

g (α) ≡ 1 1 -α 2 1 -α ln α + 1 ( .2)
and

h (α) ≡ 1 ρη L + ρη (1 + α) 2 . ( .3)
The function g (α) is continuous in (0, 1), always negative, and concave. Moreover lim α→0 g (α) = lim α→1 g (α) = -∞ . The function h (α) is continuous in (0, 1), always positive, decreasing and convex. Moreover lim α→0 h (α) = L+ρη ρη and lim α→1 h (α) = L+ρη 4ρη (see Figure 3.1 in the text).

In order to understand if there exist some values of α such that the sum of the two functions is zero we take the symmetric of g (α) with respect to the αaxis, -g (α). If -g (α) and h (α) cross somewhere, this means that there exists a value of α such that f (α) = 0. Three situations can arise, depending on the configuration of the parameters of the model L, η and ρ (see Figure 3.2 in the text):

Case 1: h (α) and -g (α) do not intersect, hence the relationship between welfare and α is monotonically decreasing; Case 2: h (α) and -g (α) are tangent, hence the relationship between welfare and α is monotonically decreasing and it just shows a change of concavity; Case 3: h (α) and -g (α) cross twice, hence the relationship between welfare and α is an inverted-N shape.

Given that h(0) = L+ρη ρη and h (1) = L+ρη 4ρη , Case 3 is more likely to occur the greater is L and the smaller are η and ρ 18 . To find a sufficient condition for Case 3 to occur we compute h (α) for the value of α such that -g (α) attains a minimum, α MIN ≡ min -

Figure 3 . 1 :

 31 Figure 3.1: Degree of returns to specialization effect (g (α)) and Competition effect (h (α))

Figure 3 . 2 :

 32 Figure 3.2: Interaction between Degree of returns to specialization effect and Competition effect

Figure 3 . 3 :

 33 Figure 3.3: The sign of the first derivative (L = 1, ρ = 0.03, N (0) = 1)

Figure 3 . 4 :

 34 Figure 3.4: η = 10: The welfare function is monotonically decreasing

Figure 3 . 5 : η = 5 :

 355 Figure 3.5: η = 5: The welfare function is an inverted-N shape

Figure 3 . 6 :

 36 Figure 3.6: η = 1: The welfare function is an inverted-N shape

2α 1 -

 1 α (1α) N (0) (L + ρη) exp αLρη η (1 + α) twe can compute the representative consumer's welfare:

  ln α + 1 19 . If h (α MIN ) ≥ -g(α MIN ) 20 , the two functions intersect twice. This happens for L ≥ L where L = λρη, λ ≡ f min ln α + 1 21 .

[START_REF] Bucci | An Inverted-U Relationship between Product Market Competition and Growth in an Extended Romerian Model[END_REF] generalizes the model considered in the present paper to the case in which labor is used also in the production of the intermediate goods.

We take it as the numeraire.

It can be interpreted as human capital.

Since in this model labor is homogeneous and perfectly mobile, the wage rate is the same in the final good sector and in the R&D sector.

Notice that this solution is possible if and only if LR (0) = L.

These results are confirmed by the linearization around each steady state.

The constant d is negative for LR 1 < LR (0) < LR 2 , it is positive for 0 < LR (0) < LR 1 , it is equal to zero for LR (0) = LR 1 , it is infinite for LR (0) = LR 2 .

In this framework α cannot represent the consumer's love for variety since by the aggregate resource

λ 5.54394.

This would mean that the function U * DE changes its slope.

We will talk about the role of these parameters in Section 3.2.2. 19 This value is approximately equal to αMIN 0.383012. 20 where -g(αMIN ) 3.42127. 21 λ 5.54394.


A Proposal of Residual Reduction method with preconditioning based on IDR Theorem

Seiji Fujino, Takashi Sekimoto, Yusuke Onoue

To cite this version: Takashi Sekimoto2 and Yusuke Onoue 2 IDR(s) method [START_REF] Sonneveld | IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems[END_REF] and extended variants of IDR(s) method [START_REF] Kusakabe | A proposal of Jacobi method based on extended Induced Dimension Reduction Theorem aimed at high convergence rate[END_REF] based on the IDR Theorem have been proposed one after another. The authors also proposed IDR(s)-based iterative method with iteration matrix of SOR (Successive Over-Relaxation) method in order to accelerate its convergence [START_REF] Fujino | Application of the IDR Theorem to Stationary Iterative Methods and their Performance Evaluation[END_REF]. In my talk, we propose an algorithm of IDR(s)-based Residual Reduction method using precondition with Eisenstat trick [START_REF] Eisenstat | Efficient implementation of a class of preconditioned conjugate gradient methods[END_REF]. We refer to IDR(s)-R2 method. We present summary of performance of IDR(s)-R2 method using some preconditions with Eisenstat trick in Table 1. We realize that preconditioned IDR(s)-R2 method outperforms with the conventional iterative methods.

Table 1 :

 1 Summary of performance of preconditioned IDR(s)-R2 method.

	method	precond.	no. of	no. of	no. of	total
			convergence nonconvergence fastest case	(%)
		none	11/18	7/18	0/18	0
		Gauss-Seidel-L	18	0	8
	IDR(s)-R2 Eisenstat-L	17	1	1	14(61%)
		Eisenstat-UL	17	1	5
	BiCGStab ILU(0)	15	3	5	5(22%)
	BiCGSafe ILU(0)	15	3	4	4(17%)

Research Institute for Information Technology, Kyushu University

Graduate School of Information Science and Electrical Engineering, Kyushu University 1