Framing 'Islamic Art' for aesthetic interiors: revisiting the 1878 Paris exhibition Moya Carey and Mercedes Volait

The 1878 Exposition Universelle in Paris ran from May 1 to October 31, 1878, and was dedicated to the universal history of labour. This overarching theme included technical skills of fine craftsmanship as well as modern mechanical industry: the resulting displays were very diverse. The international representation was also disparate, reflecting the geopolitical situation of the era, and the specific perspective of France's Third Republic with its colonies, rivals, allies and interests. Within this framework, the exhibition featured a substantial scope of art and architecture from the Middle East [START_REF]de l'emploi des figures[END_REF] and North Africa: Khedival Egypt, Qajar Iran, Frenchcontrolled Algeria, and the kingdoms of Morocco and Tunisia. This was the most extensive such a range to be offered to an international audience at that time, within the contemporary sequence of international exhibitions spanning from London in 1851, to Vienna in 1873 and Philadelphia in 1876. This article will discuss a series of distinct installations distributed across a complex site which spanned the river Seine, for the 1878 Paris event (Frezouls 1878). [START_REF]Les arts musulmans: de l'ornementation arabes dans les oeuvres des maitres italiens[END_REF] All are related to the visual culture of the Arab, Turkish and Iranian worlds, from the advent of Islam up to modern times. In other words, they fall under the conventional rubric of 'Islamic Art', whatever the acknowledged shortcomings of this art-historiographical term. As will be discussed below, the various presentations followed different formats. Modern manufactured goods were in exhibition halls for commercial sale, temporary outdoor structures exhibited distinctive 'national' architectural modes, and interior galleries showed more highly-regarded historical objects on loan from private art collectors (or amateurs). Iran, Egypt, Tunisia and Morocco participated as nations, while Algeria was presented as a colonial possession of France. The British representation included a substantial display for India, which nonetheless (for reasons to be discussed below) included material from Iran, as well as Afghanistan and Central Asia. Ottoman Turkey did not participate, nor did recently-unified Germany: geopolitical realities could not guarantee a wholly international presence, only seven years after the devastating Franco-Prussian war, and months after the Russo-Turkish conflict. National governments overlapped with commercial firms throughout, and indeed the programme's successful delivery relied upon the deft coordination of diplomatic courtesy and private enterprise. Thus some displays were commercial initiatives under national auspices, such as 'L'Égypte des khalifes', a gallery nominally sponsored (or rather authorised) by the Khedive of Egypt, Isma`il Pasha (r.1863-79). The purpose-built 'Pavillon de la Perse' was constructed by master-builders dispatched from Qajar Iran, while a separate display of Iran's contemporary manufactures (within the main exhibition building) was coordinated under direct orders from Nasir al-Din Shah (r.1848-96). Ever-present too were commercial undertakings, with retailers offering modern imports, at nationally-allocated stalls within the vast Palais du Champ de Mars, and outdoors in the Trocadéro park. Finally, one further enterprise was the 'Galerie orientale' (or 'Salle orientale'), a loan exhibition curated by the Paris art-dealer Albert Goupil (1840-1884) inside the Palais du Trocadéro, where significant historical pieces in French private collections were on view. These were chiefly from Safavid Iran and Mamluk Egypt and Syria. Goupil and others were in the process of developing an increasingly public scholarly discourse about these objects, coupling academic research with aesthetic collecting, which made the displayed artefacts all the more significant on the art market.

While the display of Islamic art at national and international exhibitions has received increased attention and research, [START_REF] Çelik | Displaying the Orient: Architecture of Islam at Nineteenth-Century World's Fairs[END_REF] comparatively little is known about the artefacts displayed at this specific Paris event, aside from a few celebrated examples. [START_REF] Lavoix | La Galerie orientale au Trocadéro, à Monsieur Charles Schéfer[END_REF] To date, scholarly literature on the inclusion of Middle Eastern nations at international exhibitions has chiefly addressed Paris 1867 [START_REF] Nikou | National Architecture and International Politics: Pavilions of the Near Eastern Nations in the Paris International Exhibition of 1867[END_REF] and Chicago 1893; 6 furthermore, these installations have been analysed mostly through the prism of the 'architecture of Islam', and the national pavilions, which omits those artefacts presented within commercial contexts.

The purpose of this article is to analyse these installations at the 1878 Paris exhibition, by interpreting a wide range of primary sources, and by considering the contemporary context of wider collecting practices, in both Paris and Cairo. It aims to highlight in particular the architectural dimension which would characterise art collecting in the late nineteenth century: diverse fittings, furnishings and salvaged fragments were reformulated as private interior space, and typically the resulting visually-eclectic 'themed rooms' consolidated a social identity for collector networks. In Paris, the central influencers were individuals who had already coordinated such installations while living well in Cairo: Gaston de Saint-Maurice (1831-1905), Alphonse Delort de Gléon (1843-1899) and architect Ambroise Baudry (1838-1906) had held court appointments serving the Khedive of Egypt. All three had built remarkable private houses in Cairo, hôtels particuliers fitted with architectural fragments, plaster casts from historic architecture and modern reformulations of Mamluk woodwork. This fit an existing vogue for selective architectural historicism within cosmopolitan Egyptian society, such as the Alhambrastyle pavilion at Gezira Palace, built by Khedive Isma`il to mark the opening of the Suez Canal. [START_REF] Crinson | Empire Building, Orientalism and Victorian Architecture[END_REF] With the fall of the Khedive (under British financial pressure), these individuals would return to France, bringing extensive house collections back with them. [START_REF] Volait | Fous du Caire: Architectes, excentriques et amateurs d'art en Égypte (1867-1914[END_REF] Already developing in Paris, was a different group for whom Middle Eastern art collecting, as displayed in immersive settings in their homes, also evidenced foreign travel, and furthermore denoted intellectual integrity. A pivotal figure was Charles Schéfer (1820-1898), a scholar and linguist, who had collected manuscripts and art objects throughout his diplomatic career in Beirut, Alexandra, Izmir and Damascus, and now taught in Paris. Within a small friendship network crucially linking Schéfer at the École des langues orientales vivantes (est. 1795) and the art journal Gazette des Beaux-Arts (est. 1859), there had begun the gradual publication of illustrated academic research on Middle Eastern visual culture, drawing primarily on Paris-based collections. This was the elite circle whose private loans filled the 1878 Galerie Orientale with Safavid carpets, Mamluk mosque lamps, Indian armour, Spanish ivory and other valuable examples, displayed together with Cairo artefacts supplied by Delort de Gléon. Critically, these diverse artworks were architecturally contained and controlled, reduced to aspects of a private domestic salon, of exclusive access. All of these social and intellectual resonances would soon be carefully exploited on the market, most notably by the art firm Goupil et Cie.

The Paris exhibition in 1878 therefore included two particularly influential and timely displays, of Islamic art objects that were presently to be made available for purchase: from Egypt's national display, Gaston de Saint-Maurice's remarkable fittings and house-contents assembled in Cairo, and the separate Galerie Orientale of artworks from multiple private lenders in Paris. Shortly afterwards, both installations would influence the acquisition and display policies of two European design museums: 9 the South Kensington Museum (today the V&A) in London, and the Musée des Arts Décoratifs in Paris. Several South Kensington curators attended the 1878 Paris exhibition: susceptible to elitist nuances, they would retrospectively seek to acquire much of the Middle Eastern material then on display, including the house collection of Gaston de Saint-Maurice (purchased in 1884). This energy transformed the Middle Eastern holdings at South Kensington, but pertinently also the manner in which the collections were exhibited: from 1880, South Kensington began to turn towards reconstructions of intimate domestic interiors, such as 'Damascus Rooms'. [START_REF] Carey | Appropriating Damascus Rooms: Vincent Robinson, Caspar Purdon Clarke and Commercial Strategy in Victorian London[END_REF] With Goupil in Paris, exceptional examples of early Safavid carpets had been wall-mounted: national design museums now began to treat this textile category with a similar aesthetic status to fine art paintings, encouraging direct examination of their designs. Albert Goupil himself would be more directly involved with the Société du musée des arts décoratifs in Paris, becoming its director in 1880, and establishing an 'oriental' department in 1882: at this date, the Société amalgamated with the Union centrale des beaux-arts appliqués à l'industrie, and became the Musée des Arts décoratifs. He also gifted a Safavid silk carpet to the National Manufactory of the Gobelins, as a model pattern for weavers. Following a similar trajectory, South Kensington Museum authorities also acknowledged the likely reference value of Safavid carpet designs for British textile industries, issuing (from 1884) affordable colourprinted Art Portfolio template series, and beginning to purchase historical Iranian carpets. In this sense, it is proposed that in 1878 these various Paris displays demonstrated shifting issues of social and institutional practices, as well as recording certain Middle Eastern design histories.

The Rue des Nations Étrangères and the Palais du Champ de Mars

Many of the manufactured goods imported from North Africa and the Middle East were located off the 'Rue des nations étrangères' (Street of Foreign Nations). Laid out as an artificial street under the glass roof of the gigantic Palais du Champ de Mars, this avenue ran north to south alongside the central Gallery of Fine Arts. Four house-fronts representing Morocco, Iran, Tunisia and Thailand ('Siam') packed into the tight 10m-long façade that was allocated to them at the far end of the street; examples of their contemporary manufactures were exhibited in the stands behind.

Less expectedly, objects from the Middle East could also be found within Britain's India section ('Exposition des Indes'). Placed in a prominent position within the 'grand vestibule' of the main exhibition hall, the Indian Court was an intricate wooden structure, painted red with copperplated domes at either end [Figure 1]. This was not architectural salvage from India, but the design of Caspar Purdon Clarke (1846-1911), a South Kensington-trained architect who was also Official Commercial Agent for this section. The President of Britain's organizing Commission for the Paris Exhibition was the Prince of Wales (Queen Victoria's son Albert Edward, later Edward VII). In 1875-6, he had toured India's royal courts, returning with an extensive collection of diplomatic gifts -which were now sent to the Paris exhibition.

Together with this royal loan, British commercial firms were strongly represented in the Indian section: one press reviewer equally admired the 'ravishing' textiles presented by the maharaja of Kashmir, and the 'superb' carpets offered by London-based Vincent J. Robinson (1829-1910) (and two other importers), all apparently viewed together inside the pavilion. [START_REF] Creux | Pavillon anglo-indien[END_REF] Also described as a 'temple', the pavilion seems to have been reserved for displaying textiles only. 12 [Figure 2]. 'Persian and Turcoman' carpets were particularly admired, 'for their great excellence of quality and design'. 13 For example, Robinson displayed three striking carpets now attributed to c.1600 Safavid Isfahan, but then often termed Afghan or Herati: 'The Afghanistan carpet exhibited by Messrs. Vincent Robinson & Co. is probably really of Mashhad manufacture. It is a rare example of the antique Persian style in carpets'. 14 This praise fits a broader critical enthusiasm for the historical Iranian carpets on display elsewhere in the Paris exhibition. Arguably Birdwood's warm words were linked with the conducive architectural surroundings in which they were encountered in Paris: the arcaded interior of the Indian-style pavilion created an intimate aesthetic context, within which visitors now felt encouraged to look more closely and appreciatively at the beautiful objects before them. Was an architectural presentation affecting viewers, in response to these impressive and relatively private spaces? [START_REF]of Middle Eastern objects persuasively 'framed' within the Damascus Room installed in his London premises[END_REF] The Trocadéro Park: Tunisia, Morocco, Iran, Algeria and Egypt The British Indian pavilion was unusual in that it was indoors, within the vestibule of the main exhibition hall. Outdoors, the architecture of Middle East and North Africa was prominent in the Trocadéro park across the river. A series of national pavilions was built around these gardens, tellingly dwarfed by the immense modern structures to either side. [START_REF]The palace's […] siting, size, and form as a whole created an image of France as a protective father/master with his arms encircling the colonial village (Zeynep Çelik and Leila Kinney[END_REF] This was a diverse and rather uneven group, with different facilities offered within temporary structures, built to demonstrate the typical architectural idiom of each country. Press commentary confirms that these displays were enjoyed as exoticized amenities for shopping and leisure, and also treated with curiosity. The Tunisian pavilion hosted a popular music café and three commercial exhibitors, one being the 'bazaar' of Abdallah ben Athia, selling examples of modern Tunisian craftsmanship, including textiles, leatherwork and mother-of-pearl-inlaid furniture. The Pavilion of Morocco was an example of a typical house, modelled on a grand example in Tangiers. Three shops were attached to it, where visitors could watch filigree jewellery being handmade, taste hashish or sip local liqueurs. 17 The Persian Pavilion was modelled upon the newly-built (1874-75) Eshratabad royal palace in Qajar Tehran, as Nasir al-Din Shah exclaimed on visiting. [START_REF] Creux | Le pavillon persan[END_REF] One upper room was minutely vaulted with mirror-mosaic, or aineh-kari, a fashionable decorative element of Qajar architectural interiors, which was discussed in the press with enthusiasm as 'dazzling' and 'fairylike'. [START_REF] Anon | Letters from Paris[END_REF] Under the official direction of Iran's imperial commissioner, Mirza Javad-Khan, the building had been constructed by two Tehrani workers, named in the press as 'Assa Ali' and 'Seda Ali', overseen by an architect or master-builder named in one press reference as Ustad Husayn `Ali. [START_REF] Creux | On South Kensington's request for the two 'Persian master masons' to visit London, see Abraham Thomas, 'The Orient and Ornament at the South Kensington Museum[END_REF] [Figure 3].

By far the largest section in the Trocadéro park was that of Algeria, deliberately emphasising France's status as a colonial power. 21 The Algerian representation was organized by its governorgeneral Antoine Chanzy and installed in an imposing structure with several historical crossreferences. Beyond the main building, independent commercial stands displayed more craftwork, such as highly-decorated chibouks, coffers (sanduq), carpets and jewellery. In total, the Algerian section numbered about 2,000 exhibitors. 22 The Egyptian pavilion stood on the opposite side of the Trocadéro park. Initially intended to match the ancient Egyptian exhibits it was to have housed, this building was given a Pharaonic style on the specifications of the renowned Egyptologist Auguste Mariette (1821-1881), or Mariette-Bey. 23 The impressive temple façade with square corner towers magisterially confirms the strong hold of Pharaonic imagery upon modern Egyptian visual identity. Within, the exhibits were to demonstrate economic progress achieved under the rule of Isma`il Pasha (r.1863-79), and had been sent either by the Khedival schools and factories or by foreign-owned businesses established in Egypt. One corner tower was dedicated to photography, while the second contained displays of modern Egyptian craft industries, including cabinet-making (Egypt's wood industry employing a total of 16,000 people). The principal exhibit of the latter section was the furniture [Figure 4] presented by Giuseppe (or Joseph in non-Italian sources) Parvis (1831-1909).

A cabinetmaker from Lombardy, Parvis had been based in Cairo since 1859: he had developed a successful factory specializing in 'Arabesque'-style furniture, and employed 50 workers (twothirds of them Egyptian) 24 . As one observer put it, revivalist production raised hopes that it might 'produce the happy result of serving the conservation and promotion of the beautiful designs of Arab art, which tended to disappear in Egypt'. [START_REF] Delchevalerie | L'Égypte agricole, industrielle et artistique', in Études sur l'exposition universelle Paris 1878[END_REF] Two additional halls were devoted to a panorama of the Suez Canal (opened in 1869), and to the 1877 abolition of slave-trading in Egypt. Aside from these promotions of modern progress, Egypt had a further presence in the galleries inside the Trocadéro palace.

A Palace for Loan Collections

Among the many innovations at the 1878 Paris exhibition, was an entirely new building, the Palais du Trocadéro, located on the far bank of the river Seine, and facing the Palais du Champ de Mars. This consisted of a central section with a large concert hall, and two wings curved in a horseshoe, containing long galleries. In deliberate counterpoint to the modern manufacturing display halls opposite, the Trocadéro was to be dedicated to the past, with an extensive survey exhibition of 'ancient art'. Entitled 'l'exposition historique de l'art ancien', this was also referred to as the 'galerie de l'art rétrospectif'. [START_REF] Frigoult De Liesville | L'exposition historique de l'art ancien[END_REF] The intention was to show the history of art production, century by century, in its widest variety, from the most humble to the most magnificent contexts. Nonetheless the intended survey rested upon a Eurocentric binary separation: the eastern wing of the Trocadéro palace was planned for European art, and the western wing to the arts of Asia, Africa, Oceania and the Americas. The show's full title was to be 'L'exposition historique de l'art ancien et de l'ethnographie des peuples étrangers à l'Europe'. Calls for national participation were however ineffective, and only Belgium and Spain responded in time. Within the host nation, regional museums and church treasuries contributed material, but the main Paris collections (such as the Louvre, Cluny and the Bibliothèque nationale) did not. Private collectors were more easily mobilized, but this swiftly complicated when major lenders insisted their collections be exhibited together. Eventually, the eastern wing was used for artworks (of any kind) owned by French (or France-based) collectors, and foreign submissions, including objects sent by Belgium, Sweden, Finland and Spain, were accommodated in the western wingoccasionally referred to as 'galerie ethnographique'. The resulting displays were thus partnational and part-individual, and furthermore partly 'artistic' and partly 'ethnographic'. One press critic responded to this compromise with amusement, querying the inconsistent taxonomies, with French collectors lending 'foreign' art (and vice versa), and asking why metalwork and ceramics were classified 'ethnographic' if produced in Spain, China and Belgium, and yet counted as 'art history' if made in France, Italy and Germany. [START_REF] Bonnaffé | Au Trocadéro. Causerie[END_REF] The Egyptian Rooms at the Palais du Trocadéro Egypt was among the countries that joined late. Financial conditions had drastically changed for the country; spending with the magnificence seen for the 1867 Paris exhibition was no longer an option. According to Khedival correspondence in the National Archives of Egypt, Egypt's participation was decided on December 21, 1876 and the budget set at a minimum. Auguste Mariette was hired as General Commissioner on January 4, 1877. [START_REF]National Archives of Egypt, Dar al-Watha'iq[END_REF] He was instructed by the Khedive to depict Egypt through its material culture in three main phases of its civilisation, namely the Pharaonic, the Arab-Ottoman and the Khedival ('la dynastie régnante'). [START_REF] Mariette-Bey | La Galerie de l'Égypte ancienne à l'exposition rétrospective du Trocadéro, description sommaire[END_REF] Installing at a late stage, this chronological sequence was divided in two: one section was hosted in the outdoor pavilion (already described above), and the other occupied a series of three galleries, right at the entrance of the western wing of the Palace. These Trocadéro rooms were to depict Ancient Egypt, Islamic Egypt and Modern Egypt, the contemporary Khedival era being itself divided between 'Equatorial Egypt' (modern Sudan) and central Egypt. The underlying concept was to illustrate the crafts that contributed to a global history of labour, the overarching theme of the 1878 exhibition.

On entering the western wing from the main entrance on Place du Trocadéro, visitors stepped first into the Equatorial Egyptian room (the first of two sections on Modern Egypt). [START_REF]The exact sequence of rooms is described in Alfred Frigoult de Liesville, Coup d'oeil général sur l'exposition historique de l'art ancien (palais du Trocadéro[END_REF] Arms (panoplies of spears), musical instruments (drums), vessels, furniture and dress of the Sudanese people (recently colonised by Egypt), were displayed. Next came (central) Modern Egypt, with displays of textiles, jewellery, ceramics, arms and armour, and diverse utensils of recent manufacture. [START_REF] Liesville | [END_REF] Adjoining this was Ancient Egypt: Mariette had commissioned large reproductions of murals depicting hunting scenes or other aspects of daily life, to illustrate society under the Pharaohs. Selected pieces, mostly plastercast replicas, were to show the development of antique sculpture. Ancient examples of artist's palettes and paint-boxes illustrated a long craft history, adhering to the 1878 theme of workmanship. Under a central canopy in Pharaonic style, stood a bust of the current Khedive . [START_REF] Rhoné | L'Égypte antique au Trocadéro[END_REF] The last room was for 'Arab and Ottoman Egypt'. This display was noted as 'L'Égypte des khalifes' or 'L'Égypte du Moyen Age' in the programme literature. [START_REF] Rhoné | La galerie des Antiquités égyptiennes à l'Exposition[END_REF] Here, the Egyptian government had loaned three glass lamps and 'eight items from the Khedival Library'. [START_REF]National Archives of Egypt, Dar al-Watha'iq, Box 36 g Exhibitions[END_REF] Alongside Mamluk Qur'ans, six folios from the 1488 Bustan of the Persian poet Sa`di, illustrated in Timurid Herat, were also on display, including the renowned painting depicting Yusuf and Zulaykha, the Qur'anic account of the Biblical prophet Joseph escaping Potiphar's wife, during his time as a slave in Egypt. [START_REF]Exposition universelle de 1878[END_REF] The British director of the Egyptian Ministry of Public Instruction, the Arabist and numismatist Edward Thomas Rogers (1831-1884) (entitled Rogers Bey), loaned his collection of inscribed glass discs (then termed 'Cufic coins') that he had established were used as weights. [START_REF]His research was based on studying his own collection, and that of Rev. Greville Chester, see Edward Thomas Rogers, 'Unpublished Glass Weights and Measures[END_REF] Ambroise Baudry, who had been chief architect to the Khedive, contributed over 300 Ottoman tiles. The rest of display came predominantly from the personal collection of aristocrat Gaston de Saint-Maurice, who also curated this section of Egypt's display sequence. This gentleman had served at the Khedival court from 1868 to 1877 as grand equerry and was an avid Islamic art collector. No catalogue was published at the time, but visual and textual descriptions in the press [Figure 5], and moreover the extensive survival of the actual objects today, all help reconstruct its scope and rationale. Most of the Saint-Maurice exhibits would shortly enter public collections: in 1884, the South Kensington Museum finalized the acquisition of about 400 objects from Saint-Maurice. [START_REF]B2/3, Cahier manuscrit. 'Collection de faïences décoratives recueillies au Caire par les soins de M. Ambroise Baudry[END_REF] Other artefacts, such as the tiles loaned by Baudry, were purchased by the Louvre in 1898. 38 The packing inventory signed by Baudry in Cairo on March 12, 1878, most probably prior to sending his tiles off to Paris, numbered 331 pieces. 39 Not all of these necessarily went on open display, but certainly a substantial amount did, if we are to believe the visual evidence of the display. The brilliantly colourful tiles [Figure 6] were presented against large wall-mounted carpets that reportedly 'belittled their brightness'. [START_REF][END_REF] The clash of colours must have been striking: the light and bright blue tones of Baudry's tiles over the dark red and deep brown tones of Saint-Maurice's carpets and kilims [Figure 7]. Their critical success in combination confirms the contemporary taste for vibrant polychromy. Identified today as Syrian, Baudry's tiles came from recently-demolished Cairo houses and were still referred to then as 'Persian tiles' or 'Rhodian tiles'. [START_REF]The tiles had therefore already been salvaged from Syrian architecture, and brought to Egypt, see Rhoné[END_REF] Featuring Persian manuscripts, Anatolian carpets and Syrian tiles, L'Égypte des Khalifes was not solely a display about historical Egyptian dynasties therefore, but reflected the collections, national and individual, that had formed within the country. The variety also confirms that such goods had been in motion across the wider region, through commercial activity and other network channels reaching into Egypt.

L'Égypte des khalifes included many examples of woodwork fittings, starting with a large mashrabiyya balcony ('from a harem') (not visible in Figure 5), and examples of composite panelling. This reflects a trend among Islamic art collectors in Cairo, French and Egyptian alike: salvaged medieval fragments of ivory and wood were commonly reconstructed in their original geometric pattern layouts, and then incorporated into modern woodwork frames. [START_REF] Volait | Fous du Caire. 43 The South Kensington Museum bought heavily from 'the Meymar Collection' at Paris in 1869, including fragments of the Lajin minbar which were similarly re-assembled in a modern framework[END_REF] Eleven years previously, the architect-engineer Husayn Pasha Fahmi al-mi'mar, known as 'Dr Meymar' in the European sources, had exhibited similar salvaged examples at Egypt's pavilion constructed for the 1867 Paris exhibition. 43 Baudry's house in Cairo, built in 1875-76, was replete with composite wood furnishings of this type; one of them, bearing copies of an inscription naming the fourteenth-century Sultan Barquq, now stands in the Mamluk galleries of the Museum of Islamic art in Cairo. [START_REF] Crosnier | L'Égypte d'un architecte[END_REF] As a commissioner for Egypt to the 1876 Centennial Exhibition in Philadelphia, Baudry may have been behind the display of a reconstructed minbar door out of detached carved polygons, re-mounted onto modern woodwork, with inscriptions bearing again the name of Sultan Barquq. Valued then at $14,000, the door was considered a sensation, and was secured for Boston's newly-established Museum of Fine Arts. 45 For his newly-designed Cairo mansion, Saint-Maurice followed this same trend, and engaged directly with the salvage, re-use and incorporation of architectural fittings. [START_REF] Volait | Le remploi de grands décors mamelouks et ottomans dans une architecture moderne[END_REF] He owned numerous doors consisting of a modern woodwork panel inset with carved Mamluk-period ivory and ebony plaques installed in complex geometric patterns [Figure 8]; several sets are visible in the print illustration of L'Égypte des Khalifes.

Less expected perhaps was the quantity of architectural plaster casts displayed. These were moulded, in pinkish plaster ('terre cuite rosée'), 47 from relief-carved stone decoration from Mamluk Cairo monuments, such as the fourteenth-century mosque of Sultan Hassan and fifteenth-century architecture for Sultan Qaytbay [Figure 9]. One cast records only a quartersection of a dome, and is copied from a fifteenth-century house in Harat al-Rum (historically a Coptic quarter of Cairo); another replicates one quarter of a flat ceiling in the same house, possibly a previous dwelling of the Coptic Patriarchate [START_REF] Auber De Lapierre | Le Musée copte du Caire, une utopie architecturale[END_REF] Saint-Maurice had used both casts for the construction of the centrally-lit grand hall of his house in Cairo. Finally, specimens of inlaid metalwork were also on show. As a display, L'Égypte des Khalifes was therefore strongly representative of contemporary Revivalist trends in Cairo (echoing the work of furniture-makers such as Parvis), as well as being an exhibition of historical architectural design, fittings and furnishings.

The Galerie Orientale at the Palais de Trocadéro

For practical reasons of budget and readiness, relatively few national governments had (like Egypt) contributed to the historical art survey intended for the Trocadéro's extensive galleries, as well as the contemporary manufactures they may have displayed in the main hall across the river. Instead (as noted above), both gallery wings were completed by recourse to private collections, belonging to wealthy art connoisseurs living in France. This introduced two complications, which would necessarily adjust the Trocadéro's intended narrative sequence. Bonnaffé's light-hearted press sketch about private lenders ('les princes de la curiosité') concluded that it would surely be easier to classify (pre-Islamic Iranian) Sasanian coins than to coordinate so many private collections in one exhibition. 49 The first complication was that some private lenders (such as the Russian diplomat Alexander Basilewski) preferred to display their artworks in one cluster, rather than dispersed within the sweeping chronological order imposed by a historical survey. The second issue was the essentially mixed and biographical character of private collecting. Private collecting is often directed to the conditions of domestic display, which at this period favoured variety. Diversity conveyed an owner's cosmopolitanism, as well as his or her discerning and selective taste. [START_REF]For example the objects displayed by Émilien de Nieuwerkerke (1811-1892), the director of France's Imperial Museums under Napoléon III, in his apartments at the Louvre, on public viewing[END_REF] Thus, a private art collection in 1870s Paris was hardly likely to be limited to French or even European art. Sure enough, Liesville's Trocadéro sketch noted occasional Islamic artworks belonging to major lenders of European art, such as Basilewsky's Alhambra vase ('estimé un prix fou') while that of Lavoix noted the same collector's exceptional Kashan lustre vase ('un monument d'un intérêt capital et d'une beauté exceptionelle'). [START_REF] Liesville | L'exposition historique', 11; Lavoix 'La Galerie Orientale', 791. 52 Bonnaffé[END_REF] In mock despair, Bonnaffé complained that Middle Eastern carpets, inlaid metalwork and enamelled glass were to be found in several locations around the site. 52 Nonetheless, the Trocadéro arrangements acknowledged and isolated this specific collecting trend, then current in France: such was the purpose of the Galerie Orientale, presented in a separate gallery on the first floor of the palace. Within this single display, the objects were culturally diverse, predominantly from Spain, Egypt, Syria, Turkey, Iran and India, in different media and ranging from the tenth to the sixteenth centuries. Their essential isolation was significant: raised on a hill across the river from the main halls, identified primarily by privileged European ownership and then further separated in an upper room, the artworks in the Galerie Orientale were carefully demarcated as pre-modern, precious and different. Such homogenized Other-ness suggests a single display role within elitist contemporary collecting circles, as private domestic installation. The uniting social aspect was an identifiable cult of Orientalist 'taste', expressed through personal aesthetic judgement, confirmed by ownership of exquisite objects. In addition, this wealthy peer group of art lovers (amateurs) was connecting with French academic scholarship of Middle Eastern languages, and thereby promoting a different understanding of Middle Eastern visual culture. The key players were Charles Schéfer, Henri Lavoix (1820-92) and Albert Goupil, who were all friends: a diplomat-linguist, a numismatist-journalist and a photographer-dealer, respectively. [START_REF]Henri Lavoix was a curator at the department of medals and antiques, at the Bibliothèque Nationale de France, and wrote the catalogues for the Islamic coin collections[END_REF] Over the years, they had advanced academic interpretations of privately-owned material, reading inscriptions and citing historical sources, and embellished the profile of private collectors. Schéfer's own art collection was a primary resource. Lavoix and others drew from their own specialist knowledge to publish illustrated accounts of specific objects in the art press. 54 So when in 1878 for example Lavoix published images of two enamelled glass mosque lamps loaned by André [Figure 10], he could refer to ten more such lamps in other collections, provide full translations of all their inscriptions and explain both the blazon system and genealogy of the Mamluk sultans. As a coin specialist, such direct reporting was key to his textbased methodology. More generally, Lavoix's prose regularly cited texts of Islamic history, from Umayyad, `Abbasid, Fatimid and Mamluk sources. In particular he conveyed a deep enthusiasm for `Abbasid Baghdad -a gilded era of cosmopolitan privilege, aesthetically-minded patronage, and exquisite craft production -`Abbasid history he observed, was worthy of a great novel. [START_REF] Lavoix | La Galerie orientale', 773. This is praise indeed coming in late nineteenth-century France, in the epic times of Émile Zola, Victor Hugo and Alexandre Dumas. A familiar and fallible trait in historiography is to overlay one's own time with one's period of study. 56 Lavoix[END_REF] Aside from Mamluk titulature, dynastic history was rarely specific to the artworks under discussion: instead Lavoix (like Goupil and others) was promoting an overarching narrative about Middle Eastern culture which drew from his own professional circuit. For him, this prism was the published corpus of edited translations of Middle Eastern historical literature, i.e. the research generated at the École des Langues Orientales Vivantes in Paris.

In different ways, the impact of Goupil's 1878 display had long been choreographed. Through press comment and the Trocadéro position, Lavoix and Goupil conveyed the impression that the Galerie Orientale now laid bare an intimate social network of well-travelled and distinguished friends. Lavoix wrote an article for La Gazette des Beaux-Arts, which is composed as an open letter to Charles Schéfer, affectionately remembering the former diplomat's travels in Syria, and his wonderful and influential art collection. 56 Lavoix also noted how Schéfer had anticipated the current collecting trend: 'les arts musulmans' had once seemed inaccessible to Europeans, because the language barrier meant that only trained scholars could access their meaning ('c'était affaire des savants'). The previous Paris exhibition in 1867, Lavoix recalled, had since marked a new shift in French taste, and he thanked ten collectors by name for allowing scholars access to their 'treasures', in the intervening years. 57 This 1878 narrative was ultimately also a calculated act of commercial exposure, guaranteed to stimulate demand and inflate market value, by emphasising the social status of collectors (or rather clients). Goupil would continue to develop the Paris market for star objects, presented in similar 'orientalist' architectural interiors, before his early death in 1884.

In 1878, over thirty individuals loaned to Goupil's display. [START_REF] Aumont | the only female lender)[END_REF] Ranging from manuscripts to carpets, these objects must have been wall-mounted or freestanding. One large display-case featured only the collections of Delort de Gléon and Goupil himself. 59 However there is no visual record of the gallery layout, nor any complete catalogue list. Instead separate images of six objects were published in Lavoix's account of the display, in La Gazette des Beaux-Arts: two enamelled glass mosque lamps belonging to Edouard André, a gilded copper mosque lamp from Mamluk Syria belonging to Charles Schéfer, a gilded and enamelled sword hilt from Spain belonging to the Marquis de Villaseca, 60 a Spanish ivory casket (dated 355H/965AD) then belonging to Goupil himself, and a repoussé gold bracelet belonging to the art-dealer Stanislas Baron. Such images underline a special status for 'star objects' that Lavoix emphasised were already familiar within the circle of collectors: with publication, these would now become increasingly well-known in reproduction, either as photographs, prints or colour plates based on designers' drawings. This visual celebrity focused attention upon the objects, signalling their intrinsic aesthetic as well as economic value: the Spanish ivory casket on loan from Goupil, would be reproduced again seven years later for the published report of his estate sale. [START_REF]The object was bought for the Union Central des Arts Décoratifs, where it remains today[END_REF] Contemporary designers were all the more likely to produce replicas of these celebrated objects, such as the footed glass bowl (reportedly bought by Schéfer from his barber in Damascus), sketched by Adalbert de Beaumont in 1866, and copied by Phillippe-Joseph Brocard's Paris glassworks in 1869. [START_REF] Bought | Moore and later bequeathed to the Metropolitan Museum[END_REF]

Presenting Safavid Carpets in Goupil's Salon

The 1878 Galerie Orientale was therefore part of an established trend in Paris, in which academic linguistic research and aesthetic art-collecting had maintained a symbiotic relationship. This produced a clear preference for art objects bearing inscriptions, which introduces a specific group in Goupil's salon. One of the revelations of the Paris exhibition, particularly for industrial museums, was the design quality of the Safavid Iranian carpets on show. Describing Goupil's Trocadéro display, Lavoix had praised the 'marvellous' carpets with inscriptions in the borders, belonging to Goupil, the Orientalist painter Jean-Léon Gérome (Goupil's brother-in-law) and the two Rothschilds. 63 His comment (together with other secondary accounts) plausibly links the 1878-displayed carpets (for which no direct record survives) with three late-sixteenth-century carpets which in 1885 were photographed wall-mounted in the Goupil family firm's Paris showroom, while multiple less-rated carpets covered the floor. Inscribed with Persian poetry, these metal-brocaded carpets have now been identified as Safavid court objects, used as diplomatic gifts in the 1560s. As photographed in 1885, they are seen on Goupil's salon walls flanked by detached door-panels (apparently Mamluk minbar doors) with geometric ivory inlaywork, of the type exhibited by Saint-Maurice in 1878, and Baudry in Philadelphia 1876 (as noted above) [Figure 11]. Wall-mounted in sequence, both types of objects are converted to a new architectural role: to enclose an interior space. The 1885 room was the thematic 'Salon Oriental' in the Goupil company's main premises on rue Chaptal in Paris, which was matched by their 'Salon Occidental' of European Renaissance art and furnishings. At the time, these valuable Safavid carpets were rated with fine oil painting, which explains their vertical display at Goupil's showroom, one of the earliest known instance of such an installation. [START_REF] Mantz | La collection Albert Goupil[END_REF] This aestheticizing fashion gained currency in the following decades in private interiors as well as in museum galleries, progressing in step with the rising commercial prices for 'classical' Middle Eastern carpets and even carpet fragments. [START_REF] Kive | 24619, and Paris: Musée des arts décoratifs, 4406. A fourth carpet, an uninscribed all-silk Kashan example, also hung in the Goupil salon (now Paris: Musée des Gobelins 1534) (all four identified in Susan Day, 'The Artist's Eye. Carpet and Textile Collections of the Orientalists[END_REF] Following Goupil's death in 1884, the three carpets were sold onwards to collections in France and America. 66 In the more immediate aftermath of the 1878 Paris exhibition though, it appears that more examples of this specific Safavid carpet category were acquired by other collectors: in London, George Salting had purchased one by 1883, when he loaned it on a long term basis to the South Kensington Museum. [START_REF] V&a | 402-1910 (acquired as part of the Salting Bequest). With relatively early South Kensington publicity, later art historiography has now labelled this entire (and diverse) group as 'Salting Carpets[END_REF] It may be significant that this critical reception for the so-called 'Salting Carpets' seems to commence only from 1878: the group is now understood as having been sold out of the Topkapi Palace treasury in Istanbul, during the city-wide duress caused by the Russo-Turkish war of 1877-8. [START_REF] Mills | The Salting Group: A History and A Clarification[END_REF] If so, this would wholly confirm the long-acknowledged effect of geopolitical instability upon the international art market: here, a transformative category of art object was reportedly exported because military action was anticipated, and feared. Two years after Paris, a junior curator at the South Kensington Museum remembered the critical impact of Goupil's display:

The remarkable loan collection of antique oriental carpets in the Trocadero Gallery of the Paris Exhibition of 1878 first presented these objects to public notice, and it was then discovered that a high class branch of fine art production had hitherto been utterly ignored by Museum administrations, although well known to amateurs, and commanding prices on a level with the most valued productions of the middle ages. [START_REF] Purdon | Report[END_REF] This epiphany would alter the acquisition policy for 'Persian carpets' at the South Kensington Museum, with the authorities finally agreeing to buy these expensive collectors' items. [START_REF]Only after this did South Kensington purchase Safavid-era carpets, such as the Ardabil Carpet (dated 1539-40) acquired in 1893 (Jennifer Wearden, 'Acquisition of Persian and Turkish Carpets by the South Kensington Museum[END_REF] Initially South Kensington borrowed important examples, from George Salting (as noted), and others. 71 The museum also published Art Portfolio series of loaned (as well as accessioned) carpets, providing affordable colour images for designers' reference. [START_REF] South Kensington | Portfolio of Persian Art[END_REF] In France, most of Goupil's carpets had also passed into national collections linked with design industries: he bequeathed one to the Gobelins, and the collector Jules Maciet would donate two from Goupil's 1888 estate sale directly to Musée des Arts Décoratifs. [START_REF] Day | Jules Maciet et le goût du tapis islamique[END_REF]

Conclusions: the aesthetic appeal of immersive displays

A number of conclusions can be drawn from this attempt to reconstruct the displays of Middle Eastern material culture at the 1878 international exhibition in Paris. The first addresses the evocative diversity of architectural settings within which artefacts were presented to the public: commercial stalls, national pavilions, themed exhibitions and loan displays. Formal categorization was still uncertain, and often haphazard: this is best illustrated by the way in which Middle Eastern items stood among Spanish and Belgian painting and sculpture in the 'ethnographic' wing of the Palais du Trocadéro.

Secondly, the range of exhibits characterized a new appreciation of Islamic art. While this had started in the 1860s, in 1885 Henri Lavoix would reflect that the 1878 exhibition had marked a shift in the public perception of historical Middle Eastern artworks, from obscure academic fare to objects of celebrated social distinction, now finally enshrined in private ownership and domestic display in 'themed rooms'. Goupil's 1878 Trocadéro display had been pivotal in changing this social attitude, and by 1885, Lavoix would praise a situation in which 'treasures were [now] in the service of erudition', because refined collectors had now assembled 'these precious documents' for scholars to study: the collectors 'are our first and most useful collaborators'. [START_REF] Lavoix | La Collection Albert Goupil. II. L'art oriental[END_REF] The confined architectural interiors of French domestic display had, by this reasoning, transformed French research on Middle Eastern art history: if so, these displays had certainly defined and also limited the scope of such scholarship.

Thirdly, most installations were architectural in reference, whether as actual salvaged fragments, reconstructed rooms from (French) private house collections, or as modern replicas staged for the Rue des Nations facades and the Trocadéro park pavilions alike: architectural framing played a crucial part in this reappraisal. Deriving from a decade lived in Cairo, Saint-Maurice's housecollection was both diverse and distinctly personal: its presentation at Trocadéro was admired by both amateurs (art-lovers) and the general public. When South Kensington curators saw the 1878 installations, they were persuaded not just by the art objects themselves, but by the intimate social character of private installations. This London museum purchased (much of) the Saint-Maurice collection (in 1884), and (from 1883 onwards) pursued carpets of the calibre of both Robinson and Goupil's displays -for the first time. Located in these specific 1878 displays, Iranian carpets signified imagined contexts of Indian royal architecture (echoed in the British India pavilion) and Paris salon society (in Goupil's exclusive gallery) respectively.

As an interior design trend, the installation of Middle Eastern architectural salvage was not entirely new in Europe. It pertained to a culture, and almost an industry, of medieval architectural reuse that can be traced back to the 1830s in France with the reclaiming and refurnishing of the abbey of Cluny by Alexandre du Sommerard (today the Musée de Cluny, in Paris). [START_REF]Les intérieurs orientalistes du comte de Saint-Maurice et d'Albert Goupil: des « Cluny arabe » au Caire et à Paris à la fin du XIX e siècle[END_REF] Neither was the phenomenon restricted to 'Islamophilia': the taste for Japonisme offers strong parallels, as it generated similar themed interiors in Japanese style. Perhaps significantly, both trends were typically pursued by wealthy bachelor aesthetes. [START_REF] Reed | Bachelor Quarters: Spaces of Japonisme in Nineteenth-century Paris[END_REF] It has been hypothesized that bachelorhood, and moreover the outright rejection of domesticity, were the key determinants for this specific, highly gendered, collecting culture. [START_REF]Not at home: The Suppression of Domesticity in Modern Art and Architecture[END_REF] Most of the collectors surveyed in this article fit this homosocial pattern.

In Khedival Cairo, historic salvage was already part of the modern state activity of demolitions through the old city, just as (paradoxically) modern Revivalist imitations were used to demonstrate the nation's future potential -at foreign exhibitions such as Paris 1878, and in local industry including Khedival court architecture. This trend may have accelerated removals from demolition sites, and the experimental reuse of salvaged fragments and plaster casts, re-fitted into modern furniture and private architectural settings. The Revivalist craft industry emerged in Cairo, at the same time that European designers and craftsmen were imitating Middle Eastern objects encountered at international exhibitions, as well as private collections, across Europe. Egypt's government-sponsored displays whether historic or modern, demonstrate this parallel process, with salvaged fragments and revival products typically exhibited side by side.

The 1878 Paris exhibition had therefore proposed 'Islamic art' as something architecturally defined and contained. In private circles, access to a privileged social world of 'themed rooms' was controlled, but increasingly available to academic scholarship, Revivalist designers and (now) the wider exhibition-going public. Possession of such distinguished objects now implied the privileged status of elite travel, intellectual knowledge and personal taste. As momentum grew, museum collections began to absorb the presentation modes of tasteful private interiors, along with their typical art contents, often acquiring diverse collections wholesale -along with salvaged architectural fragments, revivalist composites and plastercast replicas. Architectural context was the intimate and persuasive signifier, confirming the art status of a material culture broadly unfamiliar to European public opinion, and encapsulating (and overlooking) its complex diversity within a single 'themed room'.

Images

Figure 1:

Figure 1 :

 1 Figure 1: 'The Indian Court, Paris Exhibition, 1878', frontispiece illustration to George C.M. Birdwood, Handbook to the British Indian Section, London: Offices of the Royal Commission, 1878.

Figure 2 :

 2 Figure 2:

Figure 2 :

 2 Figure 2: Émile Bergerat, 'Maison Vincent Robinson & Co', Les Chefs-D'Oeuvre d'Art à l'Exposition Universelle de 1878, Paris: Ludovic Baschet, 1878, p.172.

Figure 3 :

 3 Figure 3:

Figure 3 :

 3 Figure 3: 'Perse: Palais Persan', in Glucq, L'Album de l'Exposition 1878, Paris: 1878, pl.61.

Figure 4 :

 4 Figure 4:

Figure 4 :

 4 Figure 4: Cabinet with revivalist inlaid decoration made by Giuseppe Parvis, Illustrated catalogue of the Paris International exhibition, London: Virtue, 1878, p. 8.

Figure 5 :

 5 Figure 5:

Figure 5 :

 5 Figure 5: Saint-Maurice's display, entitled L'Égypte des khalifes', here published as 'L'Exposition retrospective égyptienne dans le parc du Trocadéro', L'Illustration, 72/1872, 2 November 1878, p. 277.

Figure 6 :

 6 Figure 6:

Figure 6 :

 6 Figure 6: Two tiles, probably Syria, sixteenth century, from the Cairo collection of Ambroise Baudry, Paris: Louvre, OA 4047/66, 4047/89.

Figure 7 :

 7 Figure 7: Courtesy of the Trustees of the Victoria and Albert Museum

Figure 7 :

 7 Figure 7: Carpet, Qashqa'i, signed Yusuf `Ali, Fars province, Iran, c.1850-1875, London: V&A, 1013-1884.

Figure 8 :

 8 Figure 8:

Figure 8 :

 8 Figure 8: Door with inlaid geometric design, Cairo, fourteenth-century inlaid panels mounted within nineteenth-century frame, formerly in the house collection of Gaston de Saint-Maurice, Cairo, Edinburgh: National Museum of Scotland, A.1884.2.1.

Figure 9 :

 9 Figure 9: Courtesy of the Trustees of the Victoria and Albert Museum

Figure 9 :

 9 Figure 9: Plaster cast, moulded in London from the original Cairo cast of Gaston de Saint-Maurice, moulded from decorative stonework of the mosque of Sultan Hassan, London: V&A, REPRO.1884-827.

Figure 10 :

 10 Figure 10:

Figure 10 :

 10 Figure 10: Mamluk enamelled glass mosque lamp, from the collection of Edouard André, published in Henri Lavoix, 'La Galerie Orientale de Trocadero', Gazette des Beaux-Arts, 18/5, 1878, p. 777.

Figure 11 :

 11 Figure 11: © INHA, Dist. RMN-Grand Palais/Martine Beck-Coppola.

Figure 11 :

 11 Figure 11: Édmond Bénard, Goupil's salon oriental, photograph, c.1885, Paris, Bibliothèque de l'INHA, Intérieurs privés d'artistes. Ateliers d'artistes, 1880-1910, 4 Phot 021, n° 35.

We use the term Middle East to encompass the region of west Asia spanning Turkey, the Arab world (including Syria, Iraq and Egypt), Iran and Afghanistan. See discussion in Hassan Hanafi, 'The Middle East, in Whose World? (Primary Reflections)', in The Middle East in a Globalized World. Papers from the Fourth Nordic Conference on Middle Eastern Studies, Oslo 1998, eds. B.O. Utvik and K.S. Vikor (Bergen: Nordic Society for Middle Eastern Studies,

2000), 1-9. 2 Frezouls (1878), Plan-guide définitif de l'Exposition internationale de 1878..., Paris: Bibliothèque nationale de France, département Cartes et plans, GE C-7272 (A)',


Xinzhe Wu
email: xinzhe.wu@ed.univ-lille1.fr

Serge G Petiton
email: serge.petiton@univ-lille1.fr

Yutong Lu
email: yutong.lu@nscc-gz.cn

A Parallel Generator of Non-Hermitian Matrices computed from Given Spectra

Keywords: Parallel, Non-Hermitian Matrix, Matrix Generation, Spectrum

Iterative linear algebra methods are the important parts of the overall computing time of applications in various fields since decades. Recent research related to social networking, big data, machine learning and artificial intelligence has increased the necessity for non-hermitian solvers associated with much larger sparse matrices and graphs. The analysis of the iterative method behaviors for such problems is complex, and it is necessary to evaluate their convergence to solve extremely large non-Hermitian eigenvalue and linear problems on parallel and/or distributed machines. This convergence depends on the properties of spectra. Then, it is necessary to generate large matrices with known spectra to benchmark the methods. These matrices should be non-Hermitian and non-trivial, with very high dimension. This paper highlights a scalable matrix generator that uses the user-defined spectrum to construct largescale sparse matrices and to ensure their eigenvalues as the given ones with high accuracy. This generator is implemented on CPUs and multi-GPU platforms. Good strong and weak scaling performance is obtained on several supercomputers. We also propose a method to verify its ability to guarantee the given spectra.

Introduction

The eigenvalue problem can be defined as finding some pairs (λ, u) with λ ∈ C and u ∈ C m of matrix A ∈ C m×m , that satisfy the relation Au = λu. Many applications from various fields can be expressed as eigenvalue problems or linear system problems. In numerical simulations, Schrödinger equations, molecular simulations, geology, etc. are usually analyzed by solving eigenvalue problems and linear systems. In machine learning and pattern recognition, both supervised and unsupervised learning algorithms, such as principal component analysis (PCA), Fisher discriminant analysis (FDA), and clustering, often require solving eigenvalue problems. An insufficient accuracy and a failure of the solvers usually result in, respectively, a poor approximation to original problems and a failure of entire algorithms. A good selection of eigenvalue and linear system solvers becomes especially essential. Researchers urgently require test matrices to benchmark the numerical performance and parallel efficiency of these methods.

Nowadays, the size of eigenvalue/linear system problems and the supercomputer systems continues to scale up. The whole ecosystem of High Performance Computing (HPC), especially the linear algebra applications, should be adjusted to larger computing platforms. Under this background, there are four special requirements for the test matrices to evaluate the numerical algorithms: 1) the spectra must be known and can be customized; 2) sparse, non-Hermitian and non-trivial; 3) a very high dimension, including the non-zero element numbers and/or the matrix dimension to evaluate the algorithms on large-scale systems; 4) the controllable sparsity patterns. The matrix generator should be implemented in parallel to profit from the distributed memory clusters.

Since the eigenvalue/linear system solvers and some of their preconditioners are sensitive to a specific part of the spectra, the test matrices with customized spectra can help to analyze and provide numerically robust solvers. In practice, the spectrum is one of the important factors which influence the convergence of different solvers. Although the impact of spectral distribution of linear system on the Krylov solvers is complicated and cannot be ignored even for the normal matrix [START_REF] Liesen | Krylov subspace methods: principles and analysis[END_REF], the existence of test matrix generator with customized eigenvalues can still guide the study of numerical method. Moreover, the purpose of most preconditioners such as ILU, Jacobi, and SOR is to convert the distribution of related spectrum to another by right or left-multiplying a preconditioning matrix. The spectral distribution of the preconditioned matrix might speed up the convergence. As an example, X. Wu [START_REF] Wu | A distributed and parallel asynchronous unite and conquer method to solve large scale non-hermitian linear systems[END_REF] et al. implemented a Unite and Conquer hybrid method for solving linear systems with the combination of a Krylov linear system solver, an eigenvalue solver, and a Least Square polynomial method proposed by Y. Saad [START_REF] Saad | Least squares polynomials in the complex plane and their use for solving nonsymmetric linear systems[END_REF] in 1987. In the preconditioning part of this method, the dominant eigenvalues are used to accelerate the convergence. It is extremely necessary to evaluate the influence of the distribution of dominant eigenvalues on the acceleration. In addition, some scientific communities may be interested in matrices with clustered, conjugated eigenvalues or other special spectral distributions. It is important to develop a very large set of non-Hermitian test matrices whose eigenvalues can be customized.

The properties of being sparse, non-Hermitian and non-trivial together can add many mathematical features for the test matrices. Additionally, they should have very high dimensions for experiments on large scale platforms, which means that the proposed generation method should be easy to parallelize. Furthermore, since the enormous matrices are generated in parallel, their different slices are already distributed over separate computing units. These data can be used di-rectly to evaluate the solvers, without having to load the large matrix from the file system. It can save time and increase the efficiency of the applications.

In this paper, we present a Scalable Matrix Generator from Given Spectra (SMG2S) to benchmark the linear/eigenvalue solvers on large-scale platforms. Firstly, it has been implemented in parallel based on PETSc (Portable, Extensible Toolkit for Scientific Computation) for homogenous platforms and PETSc+CUDA (Compute Unified Device Architecture) for multi-GPU heterogeneous machines. Then an open source package4 is available, with specific communication optimization based on MPI and C++ [START_REF] Wu | SMG2S Manual v1[END_REF]. Its scalability and ability to maintain the given spectra have been evaluated on different supercomputers.

This paper is organized as follows: Section 2 talks about the related work on the test matrix collections. Section 3 gives an overview of the proposed algorithm for matrix generation. The parallel implementation of SMG2S is discussed in Section 4. In Section 5, we evaluate its strong and weak scalability on two supercomputers. The eigenvalue accuracy of SMG2S is tested with different spectral distributions in Section 6. Finally, we conclude in Section 7.

Related Work

It's rare but there are already several efforts to supply test matrix collections. SPARSEKIT [START_REF] Saad | Sparsekit: a basic tool kit for sparse matrix computation (version2)[END_REF] implemented by Y. Saad contains various simple matrix generation subroutines. The Galeri package of Trilinos provides to generate simple well-known finite element matrices in parallel. Z. Bai [START_REF] Bai | A test matrix collection for nonhermitian eigenvalue problems[END_REF] presented a collection of test matrices for developing numerical algorithms for solving nonsymmetric eigenvalue problems. There are also two widely spread matrix providers, the Tim Davis [START_REF] Davis | The university of florida sparse matrix collection[END_REF] and Matrix Market collections [START_REF] Boisvert | Matrix market: a web resource for test matrix collections[END_REF]. They both contain many matrices from scientific fields with various mathematical characteristics. But the spectra of matrices in these collections are fixed, and cannot be customized. M.T. Chu [START_REF] Chu | Structured inverse eigenvalue problems[END_REF] provides an overview of the inverse eigenvalues problems concerning the reconstruction of a structured matrix from prescribed spectrum without parallel implementation. A test matrix generation suite with given spectra was introduced by J. Demmel [START_REF] Demmel | A test matrix generation suite[END_REF] in 1989 to benchmark the routines of dense matrices in LA-PACK 5 . Their method uses an orthogonal matrix to transfer a diagonal matrix with given spectrum into dense with the same spectrum, and then transform the dense matrix into an unsymmetric one by Householder transform. This method is not suitable and efficient to test the solvers for the sparse matrix, because it requires O(n 3 ) time and O(n 2 ) storage even for generating a small bandwidth matrix. Furthermore, it was implemented for the shared memory systems rather than larger distributed memory systems, thus it is difficult to generate largescale test matrices targetting for extreme-scale clusters. That is the motivation for us to propose SMG2S which can generate large-scale non-Hermitian matrices with given spectra in parallel. This method requires much less time and storage, and can be easily parallelized on modern distributed memory systems.

Matrix Generation Algorithm

In this section, we introduce the proposed matrix generation algorithm. First of all, in Section 3.1, we present a summary of its mathematical framework based on the preliminary theorem proposed by H. Gachlier et al. [START_REF] Galicher | Generate very large sparse matrices starting from a given spectrum[END_REF].

Matrix Generation Method

Theorem 1. Let's consider a collection of matrices

M (t) ∈ C n×n , n ∈ N * . If M (t) verifies:    dM (t) dt = AM (t) -M (t)A, M (t = 0) = M 0 .
Then M (t) and M 0 are similar. M (t) has the same eigenvalues as M 0 .

Based on this theorem proposed by H. Gachlier [START_REF] Galicher | Generate very large sparse matrices starting from a given spectrum[END_REF], a matrix M 0 with given spectra can be transferred to another one M (t) that satisfies T heorem 1 and keeps the spectra of M 0 . Due to page limitation, we do not give the definitive proof of this theorem. We propose a matrix generation method by selecting many parameters such as the matrices A and M 0 .

Denote a linear operator of matrix M determined by matrix A as

A A = AM -M A, ∀A ∈ C n×n , M ∈ C n×n , n ∈ N * .
Here AM and M A are the matrix-matrix multiplication operation of matrices A and M . By solving the differential equation in T heorem 1, we can firstly get the formule of M (t) with the exponential operator and then extend it by the Taylor series formula:

       M (t) = e A A (M0)t , M (t) = ∞ k=0 t k k! ( A A ) k (M 0 ). (1)
Through the loop

M i+1 = M i + 1 i! ( A A ) i (M 0 ), i ∈ (0, +∞), a initial matrix M 0 ∈ C n×n can
be transfered into a new non-trivial and non-Hermitian matrix M +∞ ∈ C n×n , which has the same spectra but different eigenvectors with M 0 .

It is not reasonable to generate a matrix by infinity times of iterations, thus a good selection of matrix A which can make (A A ) i tends to 0 in limited steps is very necessary. In this paper, we define A as a nilpotent matrix, which means that there exists an integer k such that: A i = 0 for all i ≥ k. Such k is called the nilpotency of A. In fact, the selection of nilpotent matrix will influence the sparsity pattern of the upper band of the generated matrix.

The exact shape of A is given in Fig. 1a. Inside a n × n matrix A, its entries are default 0, except on the upper diagonal of the distance p from the diagonal. In this diagonal, its entries start with d consecutive 1 and then a 0, this term repeats until the end. Matrix A can be nilpotent by well choosing the parameters p, d and n. The determination of A to be nilpotent or not is complicated. Firstly, d should be divisible by p, secondly, it should exist the integers e and f ∈ N * that makes (d + 1)e be divisible by i + pf , for all i ∈ 1, 2, • • • , d p . But the cases that p = 1 or p = 2 are very simple, which can completely fulfil our demands. If p = 1, with d ∈ N * , or p = 2 with d ∈ N * to be even, the nilpotency of A and the upper band's bandwidth of generated matrix are respectively d + 1 and 2pd. Obviously, there is another constraint that the matrix size n should be greater or equal to the upper band's width 2pd. For p = 2, if d is odd, the matrix A will not be nilpotent, thus we do not take it into account.

Algorithm

As shown in Algorithm 1, the procedure of SMG2S is simple. Firstly, it reads an array Spec in ∈ C n , as the given eigenvalues. Then it inserts entries in h lower diagonals of the initial matrix M 0 randomly or with selected strategies, and sets its diagonal to be Spec in , and scales it with (2d)!. Meanwhile, it generates a nilpotent matrix A with the parameters d and p. The final matrix M t can be generated as M t = 1 (2d)! M 2d , where M 2d is the result after 2d times of loop

M i+1 = M i + ( 2d k=i+1 k)( A A ) i (M 0 )
. The slight modification of the loop formula is to reduce the potential rounding errors coming from numerous division operations on modern computer systems.

Algorithm 1 Matrix Generation Method

1: function matGen(input:Specin ∈ C n , p, d, h, output: Mt ∈ C n×n ) 2:
Insert the entries in h lower diagonals of Mo ∈ C n×n 3:

Insert Specin on the diagonal of M0 and M0 = (2d)!M0 4:

Generate nilpotent matrix A ∈ C n×n with selected parameters d and p 5:

for i = 0, • • • , 2d -1 do 6: Mi+1 = Mi + ( 2d k=i+1 k)( AA) i (M0) 7: Mt = 1 (2d)! M 2d
For M t , if M 0 is a lower triangular matrix having h non zero diagonals, it will be a band diagonal matrix, whose number of new diagonals in the upper triangular zone will be at most 2pd -1. Thus the maximal number of the bandwidth of matrix M t is: width = h + 2pd -1, as in Fig. 1b. In general, researchers use these matrices to test the iterative methods for sparse linear systems. The h lower diagonals of the initial matrix can be set to be sparse, which ensures the sparsity of the final generated matrix, as shown in Fig. 2. Moreover, the permutation matrix can also be applied to further change the sparsity of the generated matrix.

Parallel Implementation

In this section, we will introduce parallel implementations of SMG2S on homogeneous and heterogeneous clusters. Firstly, we implemented SMG2S based on PETSc on the CPUs, and based on MPI, CUDA, and PETSc on multi-GPU. We chose PETSc because it provides several ways to verify the generated matrices and the basic operations optimized for different computer architectures. After the PETSc-based SMG2S validation, an open source parallel software with specific optimized communication is also implemented based on MPI and C++.

Basic Implementation on CPUs

For the initial implementation, we chose PETSc instead of ScaLAPACK because we want to evaluate the solvers for sparse linear systems. The kernels of SMG2S are the sparse matrix-matrix multiplication (SpGEMM) AM and M A, and the matrix-matrix addition (AYPX operation) as AM -M A. All the sparse matrices during the generation procedure are stored by the block diagonal Compressed Sparse Row (CSR) format which is supported in default by PETSc. This matrix format keeps separately the block diagonal and off-diagonal parts of a matrix on each process into two sequence CSR format matrices. We use the matrix operations SpGEMM and AYPX provided by PETSc to facilitate the implementation.

Implementation on mutil-CPU

PETSc does not support SpGEMM and AXPY operations on multiple GPUs, so we implement them based on PETSc data structures, MPI, CUDA and, cuS-

)×_ `= )×_ Host (CPU) Host (CPU) Device (GPU) MPI MPI CUDA MPI & CUDA ) abc d , ) eff d , _ abc d , _ eff d ) abc g , ) eff g , _ abc g , _ eff g ) abc h , ) eff h , _ abc h , _ eff h ) abc d , ) eff d , _ iej d , _ ekl d ) abc g , ) eff g , _ iej g , _ ekl g ) abc h , ) eff h , _ iej h , _ ekl h `d = ) abc d _ iej d + ) eff d _ ekl d `g = ) abc g _ iej g + ) eff g _ ekl g `h = ) abc h _ iej h + ) eff h _ ekl h
Fig. 3: The structure of a CPU-GPU implementation of SpGEMM, where each GPU is attached to a CPU. The GPU is in charge of the computation, while the CPU handles the MPI communication among processes.

PARSE. The implementation is given in Fig. 3, by an example of A × B. Firstly, same as in PETSc, A and B are stored by the block CSR format, noted as A i dia , A i of f , B i dia and B i of f the sequence matrices on process i. Then B i dia and B i of f

are combined together as a novel sequence matrix as B i loc on each process i. With MPI functionalities, each CPU gather all the remote data of matrix B from the other processes, and construct them to a new sequence matrix B i oth . These matrices from each process are copied to one attached GPU, and calculate

C i = A i dia B i loc + A i of f B i oth .
The matrix operations on each GPU device is supported by the cuSPARSE. The final result C can be obtained by gathering all slices C i from all the devices.

Specific Optimized Communication Implementation on CPUs

In fact, the parallel SpGEMM kernel's communication can be specifically optimized based on the particular property of nilpotent matrix A. Since A is determined by three parameters p, d and n as we mentioned in Section 3.1, it is not necessary to explicitly implement this parallel matrix. We note this nilpotent matrix as A(p, d, n). Denote J(i, j) the entry in row i and column j of matrix J; J(i, :) all the entries of row i; and J(:, j) all the entries of column j. As shown in Fig. 4, the right-multiplcation A(p, d, n) will cause all the entries of the first n-p columns of M to shift right by an offset p. Denote M A the result gotten by the for p + 1 ≤ i < t do 4:

for 0 ≤ j < n do 5:

if M (i, j) = 0 then 6:

AM k (i -p, j) = M k (i, j) 7:

for 0 ≤ i < p do 8:

if k = 0 then 9:

isend ith row M k (i) to k -1 10: if k = m -1 then 11: irecv ith row M k (i) from k + 1 12: AM k (t -p + i) = M k (i)
Algorithm 3 Parallel MPI MA Implementation for 0 ≤ i < t do 4:

for p + 1 ≤ j < n do 5:

if M k (i, j) = 0 then 6:

M A k (i, j + p) = M k (i, j)
For the parallel implementation on distributed memory systems, the three parameters p, d and n can be shared by all MPI processes, then operations AM and M A are different from a general parallel SpGEMM. Firstly, the matrix M is one-dimensional distributed by row across m MPI process. As shown in Fig. 4b, for M A, there is no communication inter different MPI processes since the data are moved inside each row. Ensure that n m ≥ p, for AM , the intercommunication of MPI takes place when the MPI process k (k ∈ 1, • • • , m -1) should send the first p rows of their sub-matrix to the closest previous MPI process numbering k -1. The communication complexity for each process is O(np). When generating the band matrix with low bandwidth b, it tends to be a O(bp) with p = 1 or 2. The MPI-based optimization implementations of AM and M A are respectively given by Algorithm 2 and 3. The communication inter MPI process is implemented by the asynchronous sending and receiving functions. In this algorithm, M k , M A k and AM k imply the sub-matrices on process k with t rows.

The rows and columns of these sub-matrices in Algorithm 2 and 3 are all indexed by the local indices.

The communication-optimized SMG2S is implemented based on MPI and C++. The submatrix on each process is stored in ELLPACK format, using the key-value map containers provided by C++. The key-value map implementation facilitates the indexing and moving of the rows and columns. We did not implement a GPU version of SMG2S with this kind of communication optimization since its core is the data movement among different computing units, which is not well suitable for the multi-GPU architecture.

Performance Evaluation

Hardware Environment

In experiments, we install SMG2S on the supercomputers Tianhe-2 and Romeo. Tianhe-2 is installed at National Super Computer Center in Guangzhou, China, with 16000 compute nodes. Each node composes 2 Intel Ivy Bridge 12 cores @ 2.2 GHz. Romeo is located at University of Reims Champagne-Ardenne, France, which is a heterogeneous system with 130 BullX R421 nodes. Each node composes 2 Intel Ivy Bridge 8 cores @ 2.6 GHz and 2 NVIDIA Tesla K20x GPUs. In this section, we will use double-precision real and complex values to evaluate the strong and weak scalability of SMG2S's different implementations on CPU and multiple GPUs. All the test matrices in this paper are generated with the h set to be 10 and d to be 7. The details of the weak scaling experiments are given in Table 1. The matrix size of the strong scaling experiments on T ianhe-2 with CPUs, ROM EO with CPUs and ROM EO with GPUs are respectively 1.6 × 10 7 , 3.2 × 10 6 and 8.0 × 10 5 . The results are given in Fig. 5. The weak scaling for the PETSc implementation of SMG2S on Tianhe-2 trends to be bad when MPI processes number is larger than 768, where the communication overhead becomes dominant for computation. But for the communication optimized SMG2S, both the strong and weak scaling perform well when the MPI process number is larger than 768. The experiments show that SMG2S implemented with GPUs can still have good strong and weak scalability. In conclusion, SMG2S has always good strong scaling performance when d and h are much smaller than the dimension of the matrix n, because it turns to be a O(n) problem. The weak scalability is good enough for most cases. The reason is that the nilpotent matrix A in SpGEMM is simple with not many non-zero elements, therefore there is not enormous communication among different computing units. The weak scalability has its drawback in case that the computing unit number come to be huge for the SMG2S implementation based on PETSc, where the communication overhead become dominant. The special implementation of communicationoptimized SMG2S makes his strong and weak scalability better. It is also shown that the double precision complex type matrix generation takes almost two times time over the double precision real type for the basic SMG2S implementation, but the time consumption of complex and real type matrix generation of optimized SMG2S seems similar. The reason is that there is no numerical values multiplication anymore in the optimized implementation of SMG2S.

Strong and Weak Scalability Results and Analysis

Speedup Results and Analysis

The speedup of both SMG2S on multi-GPU and communication-optimized SMG2S on the CPUs compared with the PETSc-based implementation on CPU are also tested on Romeo. According to the previous evaluation that complex and real value types have always good scalability, we select the double precision complex values for the speedup evaluation. The details of experiments are also given in Table 1c. The results are shown in Fig. 6. We can find that the GPU version of SMG2S has almost 1.9× speedup over the PETSc CPU version. The communication-optimized SMG2S on CPUs has about 8× speedup over the basic PETSc CPU version.

Accuracy Verification

In the last section, we presented the good parallel performance of SMG2S, then it is necessary to verify if the generated matrices are able to keep the given spectra with enough accuracy. Generally, the iterative eigenvalue solvers such as the Arnoldi or other Krylov methods [START_REF] Petiton | Parallel subspace method for non-hermitian eigenproblems on the connection machine (cm2)[END_REF] are applied to approximate the dominant eigenvalues. But the accuracy verification is an opposite case. Now there exists a value, we want to check if it is an eigenvalue of a given matrix. These iterative methods cannot directly and efficiently deal with this kind of verification. In this section, we present a method for the accuracy verification using the Shifted Inverse Power method, which was easily implemented in parallel. Speedup/4CPUs

Verification Method

The Power method is an algorithm to approximate the greatest eigenvalue. Meanwhile, the Inverse Power method is a similar iterative algorithm to find the smallest eigenvalue. The middle eigenvalues can be obtained by the Shifted Inverse Power method [START_REF] Hernandez | Single vector iteration methods in slepc[END_REF]. The Shifted Inverse Power method is able to compute the eigenpair whose eigenvalue is the nearest to a given value in a few iterations.

In details, for checking if the given value λ is the eigenvalue of a matrix, we select a shifted value σ which is close enough to λ. An eigenpair (λ , v ) with the relation Av = λ v can be approximated in very few steps by Shifted Inverse Power method, with λ is the closest eigenvalue to σ. Since σ is very close to λ, it should be that λ and λ are the same eigenvalue of a system, and v should be the eigenvector related to λ. In reality, even if the computed eigenvalue is very close to the true one, the related eigenvector may be quite inaccurate. For the right eigenpairs, the formula Av ≈ λv should be satisfied. Based on this relation, we define the relative error as Formula (2) to quantify the accuracy.

error = ||Av -λv || 2 ||Av || 2 (2)
If λ = λ, this error should be 0, if not, this generated matrix will not have an exact eigenvalue as λ. In real experiments, the exact solution cannot always be guaranteed with the arithmetic rounding errors of floating operations during the generation. A threshold could be set for accepting it or not.

Experimental Results

In the experiments, we test the accuracy of SMG2S with four selected cases among the various tests of different spectral distributions. Fig. 7a and Fig. 7b are cases of clustered eigenvalues with different scales. Fig. 7c is a special case with the dominant part of eigenvalues clustered in a small region. Fig. 7d is a case that composes the conjugate and closest pair eigenvalues. These figures compare the difference between the given spectra (noted as initial eigenvalues in the figures) and the approximated ones (noted as computed eigenvalues) by the Shifted Inverse Power Method. Clearly, the matrices generated by SMG2S can keep almost all the given eigenvalues in the four cases even they are very clustered and close. The acceptance threshold is set to be 1.0 × 10 -3 .

This acceptance for cases of Fig. 7a, Fig. 7b, Fig. 7c and Fig. 7d are respectively 93%, 100%, 94% and 100%. The maximum error for them are respectively 3 × 10 -2 , 7 × 10 -5 , 3 × 10 -2 and 3 × 10 -7 . After the tests, we conclude that SMG2S is able to keep accurately the given spectra even for the very clustered and closest eigenvalues. In some cases, a very little number of too clustered eigenvalues may result in the inaccuracy of given ones, but in general, the generated matrix can fufil the need to evaluate the linear system and eigenvalue solvers. Fig. 7: Verification using Different Types of Spectra.

Arithmetic Precision Analysis

Any floating operations will introduce rounding errors, which is not negligible for the generation of large matrices. Regarding the non-Hermitian matrix, its eigenvalues may be extremely sensitive to perturbation. This sensibility is bounded by bound(λ) ≤ ||E|| 2 Cond(λ), with Cond(λ) the condition number of related eigenvalue λ which can be excessively high for the non-Hermitian matrix and ||E|| 2 the Euclidean norm of errors [START_REF] Saad | Numerical Methods for Large Eigenvalue Problems: Revised Edition[END_REF]. One solution is to use the integer values for the matrix generation, since only integers and the operations +, -, and × on the microprocessor can make absolutely exact computations. As shown in Algorithm 1, most of the operations in SMG2S are +, -and ×, except the step 7 with a division operation. Without step 7, we could introduce a special SMG2S fully using integers to avoid the risks of rounding errors. The spectra of the generated matrix will be (2d)! times scaled up over the given one.

Conclusion and Pespectives

In this paper, we presented a scalable matrix generator and its parallel implementation on homogeneous and heterogeneous clusters. It allows generating large-scale non-Hermitian matrices with customized eigenvalues to evaluate the impact of spectra on the linear/eigenvalue solvers on large-scale platforms. The experiments proved its good scalability and the ability to keep the given spectra with acceptable accuracy. For large matrices, the I/O operation on supercomputers is always a bottleneck even with the high bandwidth. The matrices generated in parallel by SMG2S, with data already allocated on different processes, can be used directly to evaluate the numerical methods without concerning the I/O operation. The interfaces of SMG2S to C, Python and scientific libraries PETSc and Trilinos are provided. Interface to Fortran will be implemented in future.

Fig. 1 :

 1 Fig. 1: (a) gives the nilpotent Matrix, with p off-diagonal offset, d number of continuous 1, and n matrix dimension; (b) shows the matrix generation example.

Fig. 2 :

 2 Fig. 2: Matrix Generation Sparsity Pattern Example.The operation complexity of SMG2S is max(O(hdn), O(d 2 n)). The worst case would require O(n 3 ) operations and O(n 2 ) memory storage with large d and h. However, if we want to generate a band matrix with a small bandwidth, or if the lower band of M 0 is sparse, it becomes a O(n) problem with good potential scalability and consuming O(n) memory storage.

Fig. 4 :

 4 Fig. 4: (a) AM operation; (b) MA operation.

1 :

 1 function MA(input: matrix M , matrix row number n, p, d, proc number m; output: matrix M A) 2: Distribute t row blocks M k of M to process k 3:

  CPU strong scaling on Tianhe-2.

  CPU weak scaling on Tianhe-2.

  double (optimized) real double (optimized) (c) CPU strong scaling on ROMEO.

  double (optimized) real double (optimized) (d) CPU weak scaling on ROMEO.

  GPU strong scaling on ROMEO.

  GPU weak scaling on ROMEO.

Fig. 5 :

 5 Fig. 5: Strong and weak scaling results of SMG2S on different platforms. A base 2 logarithmic scale is used for X-axis, and a base 10 logarithmic scale for Y-axis.

Fig. 6 :

 6 Fig. 6: Weak scaling speedup comparison of GPUs on ROMEO.

2 -

 2 Clustered Eigenvalues I: acceptance = 93%, max error = 2 × 10 -Dominant Clustered Eigenvalues: acceptance = 94%, max error = 3 × 10 -2 Conjugate and Closest Eigenvalues: acceptance = 100%, max error = 3×10 -7

Table 1 :

 1 Details for weak scaling and speedup evaluation.(a) Matrix size for the CPU weak scaling tests on T ianhe-2.

	CPU number	48	96	192	384		768	1536
	matrix size	1 × 10 6	2 × 10 6	4 × 10 6	8 × 10 6	1.6 × 10 7	3.2 × 10 7
	CPU number	16	32	64	128	256
	matrix size	4 × 10 5
	CPU or GPU number	16	32	64		128	256
	matrix size		2 × 10 5	4 × 10 5	8 × 10 5	1.6 × 10 6	3.2 × 10 6

(b) Matrix size for the CPU weak scaling on ROM EO. 8 × 10 5 1.6 × 10 6 3.2 × 10 6 6.4 × 10 6 (c) Matrix size for the GPU weak scaling and speedup evaluation on ROM EO.

Released package download: https://smg2s.github.io/download.html

Linear Algebra PACKage

This work was partially supported by the HPC Center of Champagne-Ardenne Romeo. It is funded by the project MYX of French National Research Agency (ANR) (Grant No. ANR-15-SPPE-003) under the SPPEXA framework.