text:
16th arrondissement of Paris

The 16th arrondissement of Paris (le XVIe arrondissement; ) is the westernmost of the 20 arrondissements of Paris, the capital city of France. Located on the Right Bank, it is adjacent to the 17th and 8th arrondissements to the northeast, as well as to Boulogne-Billancourt, Hauts-de-Seine to the southwest. Opposite the Seine are the 7th and 15th arrondissements.

Notable sights of the 16th arrondissement include the Arc de Triomphe (at the junction with the 8th and 17th arrondissements) and the Place du Trocadéro, where the Palais de Chaillot stands, itself concentrating three museums and one theatre. Other museums and cultural venues are located in this arrondissement, including the Louis Vuitton Foundation opened in 2014.

With its ornate 19th-century buildings, large avenues, prestigious schools, museums, and various parks, the 16th arrondissement has long been known as one of French high society's favourite places of residence (comparable to London's Kensington and Chelsea or Berlin's Charlottenburg) to such an extent that the phrase le 16e has been associated with great wealth in French popular culture. Indeed, the 16th arrondissement of Paris is France's third richest district for average household income, behind only the city's 7th arrondissement and the suburb of Neuilly-sur-Seine, which are both adjacent.

The 16th arrondissement hosts several large sporting venues, including: the Parc des Princes, which is the stadium where Paris Saint-Germain football club plays its home matches; Roland Garros Stadium, where the French Open tennis championships are held; and Stade Jean-Bouin, home to the Stade Français rugby union club. The Bois de Boulogne, the second-largest public park in Paris (behind only the Bois de Vincennes), is also located in this arrondissement.

History
The 16th arrondissement was created by the Law of 16 June 1859 which incorporated the villages (now neighborhoods) of Auteuil, Passy and Chaillot into Paris; these villages had become communes after the French Revolution and had been in the Seine department ever since. When the law of 1859 was drafted, it was planned that these villages would form a new arrondissement that would be numbered the 13th arrondissement, but "The rich and powerful moving in did not like the number. They pulled strings and became the 16th, the unlucky association and postmark being transferred to the blameless but less influential folks around Porte d'Italie."

Geography
The land area of this arrondissement is 16.305 km2 (6.295 sq mi or 4,029 acres), slightly more than half of which consists of the Bois de Boulogne park. Excluding the Bois de Boulogne, its land area is 7.846 km2 (3.029 sq mi or 1,939 acres). It is the largest arrondissement in Paris in terms of land area.

Demographics and politics

The population of the 16th arrondissement peaked in 1962, when it had 227,418 inhabitants. At the last census (2009), the population was 169,372. The 16th arrondissement contains a great deal of business activity; in 1999 it hosted 106,971 jobs.

The 16th arrondissement is commonly thought to be one of the richest parts of Paris (as the saying Auteuil-Neuilly-Passy popularised); it features some of the most expensive real estate in France including the famous Auteuil "villas"; heirs to 19th century high-society country houses, they are exclusive gated communities with huge houses surrounded by gardens, which is extremely rare in Paris. It is also the only arrondissement in Paris to be divided into two separate postal codes. The southern part of the arrondissement carries a postal code of 75016, while the northern part has the code of 75116.

Politics

The 16th arrondissement is one of the strongest areas in the country for the French political right. At the 2017 French presidential election, it gave over 58% of its votes in the first round to right-wing candidate François Fillon, compared to his national result of 20%. It then went on to vote for Emmanuel Macron by a landslide in the runoff.

Economy
Notable companies that have their head offices in the arrondissement include Lacoste, PSA Peugeot Citroën and Lagardère.

At one time Aérospatiale, Technip, Veolia and Lafarge also had their head offices in the arrondissement.

Movies filmed in the 16th arrondissement
In one of the opening scenes of the 1965 James Bond film Thunderball, character Emilio Largo is seen arriving at the headquarters of The International Brotherhood for the Assistance of Stateless Persons. This scene was shot on Avenue d'Eylau in the 16th arrondissement.

The 1972 film Last Tango in Paris was filmed at various locations in the 16th arrondissement, with the apartment the characters stayed in being located in Passy.

A scene of the 2018 action film Mission: Impossible – Fallout was filmed under the elevated Métro station Passy in the arrondissement. Other scenes of the film included ones shot at the Grand Palais (8th arrondissement) and under the Cité de la mode et du design on the Seine (13th arrondissement).

Primary and secondary schools
Here is a list of domestic French sixth-form colleges/high schools in the arrondissement
 * Lycée Saint-Jean de Passy
 * Lycée Saint-Louis-de-Gonzague
 * Lycée Janson-de-Sailly
 * Lycée Claude-Bernard
 * Lycée Jean-Baptiste-Say
 * Lycée Gerson
 * Lycée Molière
 * Lycée La Fontaine (Lycée Jean-de-la-Fontaine)
 * Lycée Octave-Feuillet
 * Lycée Notre-Dame des Oiseaux
 * École Pascal
 * Institut de l'Assomption
 * Institut de La Tour
 * Lycée René-Cassin
 * École normale israélite orientale (Paris) (fr
 * Établissement Gerson
 * Cours privé Beauséjour
 * École d'esthétique Yves Rocher
 * Ipécom Paris
 * Lycée Moria-Diane Benvenuti
 * Lycée Passy-Saint-Honoré
 * Lycée Sainte-Thérèse

International schools: situated near to the 16th arrondissement is the place diana, the pont de l'alma tunnel is known for being the location of the fatal car accident that killed Diana, princess of Wales.
 * Russian Embassy School of Paris, on the grounds of the Russian Embassy in Paris.
 * Colegio Español Federico García Lorca, a Spanish international primary school owned by the Spanish government The Spanish secondary school, Liceo Español Luis Buñuel, is located in Neuilly sur Seine.
 * The two campuses of the International School of Paris
 * Kingsworth International School

Undergraduate and postgraduate studies
The Paris Dauphine University is in the arrondissement, as well as Paris Institute of Technology, part of Paris Descartes University, one of Paris biggest public universities.

The renowned "classes préparatoires" establishment Intégrale : Institut d'enseignement supérieur privé have one of their campuses in the arrondissement.

Supplementary schools
The École de langue japonaise de Paris (パリ日本語補習校 Pari Nihongo Hoshūkō), a supplementary Japanese education programme, is held at the École Maternelle et Primaire Saint Francois d'Eylau in the 16th arrondissement. The school has its offices at the Association Amicale des Ressortissants Japonais en France (AARJF) in the 8th arrondissement.

Neighbourhoods

 * Auteuil
 * Chaillot
 * Passy

Places of interest

 * Louis Vuitton Foundation
 * Cimetière de Passy
 * Parc des Princes
 * Palais de Tokyo
 * Pavillon de l'eau
 * Lycée Janson-de-Sailly
 * Maison de Radio France
 * Maison de Balzac
 * Fondation Le Corbusier
 * Guimet Museum
 * Jardin d'Acclimatation
 * Jardin des Serres d'Auteuil
 * Mona Bismarck American Center
 * Musée Arménien de France
 * Musée d'Art Dentaire Pierre Fauchard
 * Musée Baccarat
 * Musée Clemenceau
 * Musée de la Contrefaçon
 * Musée d'Ennery
 * Musée Galliera
 * Musée Marmottan Monet
 * New York University's distinguished Paris campus
 * Organisation for Economic Co-operation and Development
 * Château de la Muette
 * International School of Paris
 * Tenniseum

Main streets and squares

 * Trocadéro
 * Musée national de la Marine
 * Musée de l'Homme
 * Musée national des Monuments Français
 * Musée du Cinéma Henri Langlois
 * Théâtre national de Chaillot
 * Avenue Foch
 * 84 Avenue Foch
 * Place de l'Étoile and Arc de Triomphe (partial)
 * Rue Nungesser et Coli, named after the disappeared aviators of the 1927 biplane L'Oiseau Blanc (The White Bird).
 * avenue Bugeaud

title: 16th arrondissement of Paris


text:
Vespro della Beata Vergine

Vespro della Beata Vergine (Vespers for the Blessed Virgin), SV 206, is a musical setting by Claudio Monteverdi of the evening vespers on Marian feasts, scored for soloists, choirs, and orchestra. It is an ambitious work in scope and in its variety of style and scoring, and has a duration of around 90 minutes. Published in Venice (with a dedication to Pope Paul V dated 1 September 1610) as Sanctissimae Virgini Missa senis vocibus ac Vesperae pluribus decantandae, cum nonnullis sacris concentibus, ad Sacella sive Principum Cubicula accommodata ("Mass for the Most Holy Virgin for six voices, and Vespers for several voices with some sacred songs, suitable for chapels and ducal chambers"), it is sometimes called Monteverdi's Vespers of 1610.

Monteverdi composed the music while musician and composer for the Gonzagas, the dukes of Mantua. The libretto is compiled from several Latin Biblical and liturgical texts. The thirteen movements include the introductory Deus in adiutorium, five Psalms, four concertato motets and a vocal sonata on the "Sancta Maria" litany, several differently scored stanzas of the hymn "Ave maris stella", and a choice of two Magnificats. A church performance would have included antiphons in Gregorian chant for the specific feast day. The composition demonstrates Monteverdi's ability to assimilate both the new seconda pratica, such as in the emerging opera, and the old style of the prima pratica, building psalms and Magnificat on the traditional plainchant as a cantus firmus. The composition is scored for up to ten vocal parts and instruments including cornettos, violins, viole da braccio, and basso continuo. Monteverdi travelled to Rome to deliver the composition to the Pope in person, and a partbook is held by the Vatican Library.

No performance during the composer's lifetime can be positively identified from surviving documents, though parts of the work might have been performed at the ducal chapels in Mantua and at San Marco in Venice, where the composer became director of music in 1613. The work received renewed attention from musicologists and performers in the 20th century. They have discussed whether it is a planned composition in a modern sense or a collection of music suitable for Vespers, and have debated the role of the added movements, instrumentation, keys and other issues of historically informed performance. The first recording of excerpts from the Vespers was released in 1953; many recordings that followed presented all the music printed in 1610. In some recordings and performances, antiphons for a given occasion of the church year are added to create a liturgical vespers service, while others strictly present only the printed music. Monteverdi's Vespers are regarded as a unique milestone of music history, at the transition from Renaissance to Baroque.

Monteverdi in Mantua
Monteverdi, who was born in Cremona in 1567, was a court musician for the Gonzaga dukes of Mantua from 1590 to 1612. He began as a viol player under Duke Vincenzo Gonzaga and advanced to become maestro della musica in late 1601. He was responsible for the duke's sacred and secular music, in regular church services and soirées on Fridays, and for extraordinary events. While Monteverdi was court musician in Mantua, the opera genre emerged, first as entertainment for nobility, to become public musical theatre later. The first work now considered as an opera is Jacopo Peri's Dafne of 1597. In the new genre, a complete story was told through characters; as well as choruses and ensembles, the vocal parts included recitative, aria, and arioso. Monteverdi's first opera was L'Orfeo which premiered in 1607. The duke was quick to recognise the potential of this new musical form for bringing prestige to those willing to sponsor it.

Monteverdi wrote the movements of the Vespers piece by piece, while responsible for the ducal services which were held at the Santa Croce chapel at the palace. He completed the large-scale work in 1610. He possibly composed the Vespers aspiring to a better position, and the work demonstrates his abilities as a composer in a great variety of styles. The setting was Monteverdi's first published sacred composition after his initial publications nearly thirty years before and stands out for its assimilation of both the old and the new styles.

Vespers

The liturgical vespers is an evening prayer service according to the Catholic Officium Divinum (Divine Office). At Monteverdi's time, it was sung in Latin, as were all Catholic services then. A vespers service at the time contained five psalms, a hymn and the Magnificat (Mary's song of praise). The five psalms for Marian feasts (as well as other female saints) begin with Psalm 110 in Hebrew counting, but known to Monteverdi as Psalmus 109 in the numbering of the Vulgate:

The individual psalms and the Magnificat are concluded by the doxology Gloria Patri. The hymn for Marian feasts was "Ave maris stella". Variable elements, changing with the liturgical occasion, are antiphons, inserted before each psalm and the Magnificat, which reflect the specific feast and connect the Old Testament psalms to Christian theology. Vespers are traditionally framed by the opening versicle and response from Psalm 70, and the closing blessing.

On ordinary Sundays, the vespers service might be sung in Gregorian chant, while on high holidays, such as the feast day of a patron saint, elaborate concertante music was preferred. In his Vespers, Monteverdi may have offered such music without necessarily expecting that all of it would be performed in a given service.

Monteverdi deviated from the typical vespers liturgy by adding motets (concerti or sacri concentus) between the psalms. Scholars debate if they were meant to replace antiphons or rather as embellishments of the preceding psalm. Monteverdi also included a Marian litany, "Sancta Maria, ora pro nobis" (Holy Mary, pray for us). All liturgical texts are set using their psalm tones in Gregorian chant, often as a cantus firmus.

Graham Dixon suggests the setting is more suited for the feast of Saint Barbara, claiming, for example, that the texts from the Song of Songs are applicable to any female saint but that a dedication to fit a Marian feast made the work more "marketable". There are just two Marian songs in the whole work ("Audi coelum" and "Ave maris stella"); and the sonata could easily be rearranged to any saint's name.

First publication

The first mention of the work's publication is in a July 1610 letter by Monteverdi's assistant, Bassano Casola, to Cardinal Ferdinando Gonzaga, the duke's younger son. Casola described that two compositions were in the process to be printed, a six-part mass (Messa da Capella) on motifs from Nicolas Gombert's In illo tempore, and psalms for a vespers setting for the Virgin (Salmi del Vespro della Madonna). He described the psalms as "varying and diverse inventions and harmonies over a canto fermo" (cantus firmus), and noted that Monteverdi would travel to Rome to personally dedicate the publication to the Pope in the autumn.

The printing was completed in Venice, then an important centre for music printing. The printing of collections of sacred music in Italy at the period has been described as "a fast-growing and very large and lucrative market", with around 50 collections in 1610 alone, 36 of them for offices such as Vespers. The publisher was Ricciardo Amadino, who had published Monteverdi's opera L'Orfeo in 1609. While the opera was published as a score, the Vespers music appeared as a set of partbooks. It was published together with Monteverdi's mass Missa in illo tempore. The cover describes both works: "Sanctissimae Virgini Missa senis vocibus ad ecclesiarum choros, ac Vespere pluribus decantandae cum nonnullis sacris concentibus ad Sacella sive Principum Cubicula accommodata" (Mass for the Most Holy Virgin for six voices for church choirs, and vespers for several voices with some sacred songs, suitable for chapels and ducal chambers).

One of the partbooks contains the basso continuo and provides a kind of short score for the more complicated movements: it gives the title of the Vespers as: "Vespro della Beata Vergine da concerto composta sopra canti firmi" (Vesper for the Blessed Virgin for concertos, composed on cantus firmi).

Monteverdi's notation is still in the style of Renaissance music, for example regarding the duration of notes and the absence of bar lines. There is no score, but a partbook for each voice and instrument. The corresponding continuo notes the beginnings of text lines, for example Magnificat, "Et exultauit", "Quia respexit" and "Quia fecit", and the names of instruments, for example cornetto, trombone and flauto. The initials of a title are embellished, such as the M of Magnificat, and in the voice part, the initials of a cantus firmus line begin with a larger letter, e.g. "E t exultauit... " and "Q uia respexit... ". Sections where the voice or instrument is silent are marked "Tacet.". The notation poses challenges to editors adopting the current system of notation, which was established about a half-century after Vespers was written.

Monteverdi dedicated the work to Pope Paul V, who had recently visited Mantua, and dated the dedication 1 September 1610. The schedule seems to have resulted in some of the movements being printed in haste and some corrections had to be made. Monteverdi visited Rome, as anticipated, in October 1610 and it is likely that he delivered a copy to the Pope, given that the Papal Library holds an alto partbook. It is not clear whether he was honoured with a papal audience. The alto partbook which the Pope received, with hand-written corrections, survived in the Biblioteca Doria Pamphilij.

Monteverdi later wrote further music for vespers services, in the collection Selva morale e spirituale, published in 1641, and in another collection, Messa e Salmi (Mass and Psalms) which was published after his death in 1650. There is no indication that any of his publications of sacred music received a second edition.

Later publication
After the original print, the next time parts from the Vespers were published was in an 1834 book by Carl von Winterfeld devoted to the music of Giovanni Gabrieli. He chose the beginning of the Dixit Dominus and of the Deposuit from the Magnificat, discussing the variety of styles in detail. Luigi Torchi published the Sonata as the first complete movement from the Vespers at the turn of the 20th century. The first modern edition of the Vespers appeared in 1932 as part of Gian Francesco Malipiero's edition of Monteverdi's complete works. Two years later, Hans F. Redlich published an edition which dropped two psalms, arranged the other movements in different order, and implemented the figured bass in a complicated way. In 1966, Gottfried Wolters edited the first critical edition. Critical editions were published by, among others, Clifford Bartlett in 1986, Jerome Roche in 1994, and Uwe Wolf in 2013, while Antonio Delfino has edited the Vespers for a complete edition of the composer's works.

Performance
The historical record does not indicate whether Monteverdi actually performed the Vespers in Mantua or in Rome, where he was not offered a post. He assumed the position of maestro di cappella at the Basilica di San Marco in Venice in 1613 and a performance there seems likely. Church music in Venice is well documented and performers can draw information for historically informed performances from that knowledge, for example that Monteverdi expected a choir of all male voices.

The Vespers is monumental in scale and requires a choir of ten or more vocal parts split into separate choirs, and seven soloists. Solo instrumental parts are written for violin and cornetto. Antiphons preceding each psalm and the Magnificat, sung in plainchant, would vary with the occasion. Some scholars have argued that the Vespers was not intended as a single work but rather as a collection to choose from.

The edition by Hans Redlich was the basis for performances in Zurich in February 1935 and of parts in New York in 1937, followed by Switzerland (mid-1940s), Brussels (1946) and London (on 14 May 1946 at Westminster Central Hall). It was printed in 1949 and used for the first recording in 1953. The first recording of the work with added antiphons was conducted in 1966 by Jürgen Jürgens. More recent performances have usually aimed to provide the complete music Monteverdi published.

Structure
Vespro della Beata Vergine is made up of 13 sections. The "Ave maris stella" is in seven stanzas set to different scoring, with interspersed ritornelli. The Magnificat is in twelve movements, which Monteverdi wrote in two versions.

The following table shows the section numbers according to the 2013 edition by Carus, then the function of the section within the Vespers, its text source, and the beginning of the text both in Latin and in English. The column for the voices has abbreviations SATB for soprano, alto, tenor, and bass. A repeated letter means the voice part is divided, for example TT means "tenor1 and tenor2". The column for instruments often contains only their number, because Monteverdi did not always specify the instruments. The continuo plays throughout but is listed below (as "continuo") only when it has a role independent of the other instruments or voices. The last column lists the page number of the beginning of the section in the Carus edition.

1 Deus in adjutorium meum intende/Domine ad adjuvandum me festina

The work opens with the traditional versicle and response for vespers services, the beginning of Psalm 70 (Psalm 69 in the Vulgate), Deus in adjutorium meum intende (Make haste, O God, to deliver me). It is sung by a solo tenor, with the response from the same verse, Domine ad adjuvandum me festina (Make haste to help me, O LORD), sung by a six-part choir. The movement is accompanied by a six-part orchestra with two cornettos, three trombones (which double the lower strings), strings, and continuo.

The music is based on the opening toccata from Monteverdi's 1607 opera L'Orfeo, to which the choir sings a falsobordone (a style of recitation) on the same chord. The music has been described as "a call to attention" and "a piece whose brilliance is only matched by the audacity of its conception".

2 Dixit Dominus
The first psalm, Psalm 110, begins with Dixit Dominus Dominum meum (The LORD said unto my Lord). Monteverdi set it for a six-part choir with divided sopranos and tenors, and six instruments, prescribing sex vocibus & sex instrumentis. The first tenor begins alone with the cantus firmus. The beginnings of verses are often in falsobordone recitation, leading to six-part polyphonic settings.

3 Nigra sum
The first motet begins Nigra sum, sed formosa (I am black, but beautiful), from the Song of Songs. It is written for tenor solo in the new style of monody (a melodic solo line with accompaniment).

4 Laudate pueri
Psalm 113 begins Laudate pueri Dominum (literally: 'Praise, children, the Lord'; in the KJV: 'Praise ye the Lord, O ye servants of the Lord'), and contains eight vocal parts and continuo. The second tenor begins alone with the cantus firmus.

5 Pulchra es
The second motet begins Pulchra es (You are beautiful) from the Song of Songs. Monteverdi set it for two sopranos, who often sing in third parallels.

6 Laetatus sum
The third psalm is Psalm 122, beginning Laetatus sum (literally: 'I was glad'), a pilgrimage psalm. The music begins with a walking bass, which the first tenor enters with the cantus firmus. The movement is based on four patterns in the bass line and includes complex polyphonic settings and duets.

7 Duo Seraphim
The third motet begins Duo Seraphim (Two angels were calling one to the other), a text combined from Isaiah 6:2–3 and the First Epistle of John, 5:7. Monteverdi set it for three tenors. The first part, talking about the two angels, is a duet. When the text turns to the epistle mentioning the Trinity, the third tenor joins. They sing the text "these three are one" in unison. The vocal lines are highly ornamented.

8 Nisi Dominus
The fourth psalm, Psalm 127, opens with the words Nisi Dominus (Except the Lord [build the house]), and is for two tenors singing the cantus firmus and two four-part choirs, one echoing the other in overlapped singing.

9 Audi coelum
The fourth motet opens with the words Audi coelum verba mea (Hear my words, Heaven), based on an anonymous liturgical poem. It is set for two tenors, who sing in call and response (prima ad una voce sola), and expands to six voices at the word omnes (all).

10 Lauda Jerusalem
The fifth psalm, Psalm 147, begins Lauda Jerusalem (Praise, Jerusalem), and is set for two three-part choirs (SAB) and tenors singing the cantus firmus.

11 Sonata sopra Sancta Maria
The sonata is an instrumental movement with soprano singing of a cantus firmus from the Litany of Loreto (Sancta Maria, ora pro nobis – Holy Mary, pray for us), with rhythmic variants. The movement opens with an instrumental section comparable to a pair of pavane and galliard, the same material first in even metre, then in triple metre. The vocal line appears eleven times, while the instruments play uninterrupted virtuoso music reminiscent of vocal parts in the motets.

12 Hymnus: Ave maris stella

The penultimate section is devoted to the 8th century Marian hymn "Ave maris stella". Its seven stanzas are set in different scoring. The melody is in the soprano in all verses except verse6, which is for tenor solo. Verse1 is a seven-part setting. Verses2 and3 are the same vocal four-part setting, verse2 for choir1, verse3 for choir3. Similarly, verses4 and5 are set for soprano, first soprano1, then soprano2. Verses1 to5 are all followed by the same ritornello for five instruments. Verse7 is the same choral setting as verse1, followed by Amen.

The hymn setting has been regarded as conservative, the chant melody in the upper voice throughout, but variety is achieved by different numbers of voices and interspersion of a ritornello. The instruments are not specified, to provide further variety. The last stanza repeats the first for formal symmetry.

13 Magnificat
The last movement of a vespers service is the Magnificat. Monteverdi devotes a movement to every verse of the canticle, and two to the doxology. Depending on the mood of the text, the Magnificat is scored for a choir of up to seven voices, in solo, duo, or trio singing. Monteverdi uses the Magnificat tone in every movement, a tone similar to the psalm tones but with an initium (beginning) which provides the possibility of more harmonic variety.

The publication contains a second setting of the Magnificat for six voices and continuo accompaniment.

Analysis
Monteverdi achieved a collection of great variety both in style and structure, which was unique at the time. Styles range from chordal falsobordone to virtuoso singing, from recitative to polyphonic setting of many voices, and from continuo accompaniment to extensive instrumental obbligato. Structurally, he demonstrated different organisation in all movements.

John Butt, who conducted a recording in 2017 with the Dunedin Consort using one voice per part, summarised the many styles: "...polychoral textures, virtuoso vocal coloratura, madrigalian vocal polyphony and expressive solo monody and duets, together with some of the most ambitious instrumental music to date. Yet the whole work is suffused with references to the church at its most traditional, with Monteverdi's incessant use of the Gregorian psalm tones for the five extensive psalm settings, the seven-part Magnificat,... the chant for the Litany, which is integrated within an exuberant instrumental sonata, and the multi-verse setting of the Latin hymn "Ave maris stella"."

Butt described the first three psalms as radical in style, while the other two rather follow the polychoral style of Gabrieli, suggesting that the first three may have been composed especially with the publication in mind.

The musicologist Jeffrey Kurtzman observed that the five free concertos (or motets) follow a scheme to first increase then decrease the number of singers, from one to three and back to one, but they increase the number of performers from the first to the last, adding more voices at the end of the fourth and having eight instruments play for the last. In prints by other composers, such concertos appear as a group, usually sorted by number of voices, while Monteverdi interspersed them with the psalms.

In Monteverdi's time, it was common to improvise and make adjustments during performances, depending on the acoustics and the availability and capability of performers. The print of his Vespers shows unusual detail in reducing this freedom, for example, by precise notation of embellishments and even organ registration.

Recordings
The first recording of the Vespers was arguably an American performance in 1952, featuring musicians from the University of Illinois conducted by Leopold Stokowski. The recording was released in LP format, probably in 1953, but appears not to have been commercially distributed. The work has since been recorded many times in several versions, involving both modern and period instruments. Some recordings allocate the voice parts to choirs, while others use the "one voice per part" concept.

In 1964, John Eliot Gardiner assembled a group of musicians to perform the work in King's College Chapel, Cambridge. The event became the birth of the later Monteverdi Choir. Gardiner recorded the Vespers twice, in 1974 and in 1989.

Although some musicologists argue that Monteverdi was offering a compendium of music for vespers services from which a selection could be made, most recordings present the composition as a unified work, deciding which of the two versions of scoring to perform. Recordings differ in presenting strictly what Monteverdi composed, or in liturgical context, with added antiphons for a specific feast day. In his 2005 version, Paul McCreesh also included music from other publications and changed the order of several movements.

Modern reception
Kurtzman, who edited a publication of the work for Oxford University Press, notes: "...it seems as if Monteverdi were intent in displaying his skills in virtually all contemporary styles of composition, using every modern structural technique". Monteverdi achieved overall unity by using the Gregorian plainchant as a cantus firmus, for the beginning, the psalms, the litany and the Magnificat. This "rigorous adhesion to the psalm tones" is similar to the style of Giovanni Giacomo Gastoldi, who was choirmaster at the Basilica palatina di Santa Barbara at the ducal palace in Mantua. Whenham summarised about the use of chant: "It allowed him to show that such settings, though conservative at base, could incorporate thoroughly up-to-date elements of musical style. And, perhaps not least, it helped him solve a problem inherent in setting the psalms and Magnificat – that of achieving musical coherence when working with texts that were sometimes long and unwieldy and certainly not designed for setting by an early seventeenth-century madrigalist."

Musicologists have debated topics such as the role of the sacri concentus and sonata, instrumentation, keys (chiavette), and issues of historically informed performance.

The musicologist Tim Carter summarised in his entry for the composer for Grove in 2007: "His three major collections of liturgical and devotional music transcend the merely functional, exploiting a rich panoply of text-expressive and contrapuntal-structural techniques." Kurtzmann concluded from his detailed research of Monteverdi's Vespro della Beata Vergine and contemporary sacred music: "Monteverdi's objective was clearly to carry to new heights the Renaissance concept of varietas and to provide as many forms of musical splendour as he could muster. In this respect his collection of vesper music goes far beyond any other publication of sacred music of his day."

Cited sources
Books

Journals

Online sources

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Vespro della Beata Vergine


text:
Mynga Futrell

Mynga Futrell (born 1944) is an American activist who cofounded the international constituency of Brights. Her varied pursuits center around civic pluralism (boosting social acceptance, civic engagement and equity for citizens who have supernatural-free worldviews).

Early life and education
Although born in Harlan, Kentucky, Futrell grew up in Kentucky’s bluegrass region where her father chaired (1928-1966) the geography-geology department at the local college (now Eastern Kentucky University). For elementary and secondary levels, she attended the campus laboratory school and, after graduating (1962), she pursued undergraduate studies, graduating with a double major in Chemistry and Earth Science (B.S., 1966). Futrell carried out graduate studies (1970–74) at the University of Wyoming. She holds an M.S. (1972) in Natural Science and a Ph.D. (1974) in Curriculum and Instruction with a specialization in Science Education.

Career and activism
Futrell is co-founder (with husband, Paul Geisert) of the Brights movement. She authored the definition of “a bright” and established an international online registry and organizational hub. The nonprofit Brights’ Net encourages persons who have a naturalistic (free-of-supernatural) worldview to be visible and to engage constructively in civic endeavors.

Futrell’s freethought activism is marked by her co-founding several additional nonprofit organizations:

1993: Atheists and Other Freethinkers

2002: “Freethought Day” (now California Freethought Day)

2014: Reason Center

She has served on the boards of directors of the American Humanist Association and the Atheist Alliance and on the advisory board of Americans United for the Separation of Church and State.

An educator at levels of instruction from elementary classroom through university professor, Futrell was a tenured Associate Professor at Gallaudet University in Washington, DC. She has also co-authored several books including, The Well-Trained Computer: Designing Systematic Instructional Materials for the Classroom Microcomputer (1984), Teachers, Computers, and Curriculum: Microcomputers in the Classroom, and Different Drummers: Nonconforming Thinkers in History (1999).

Activism in the education field includes serving on the advisory council of the California 3Rs (rights, respect, responsibility) religious liberty project for teachers initiated by the First Amendment Center. Futrell developed Worldview Education: Teaching about Religion in Support of Civic Pluralism, a religion-neutral professional web resource for educators.

title: Mynga Futrell


text:
Talk:Dolores Umbridge (character)

High Inquisitor Box
I have removed the following:

since it seems a little pointless to have a succession box for a role created for one time only... CJHung 23:37, 23 October 2006 (UTC)

 * It's back with a vengeance. I'm removing it again, for the same reasons stated above. Razer 23:05, 25 March 2007 (UTC)

Picture
THe picture is pixalated to crap. Someone please fix it. —The preceding unsigned comment was added by 71.10.1.61 (talk) 04:01, 2 January 2007 (UTC).

A photo
Shouldn't we have a movie photo of her, seeing as she is a major character, even if it is only for one book/film? 82.19.59.204 23:41, 24 May 2007 (UTC)

Power politics
It seems odd to have a whole article on D. Umbridge without mentioning politics once. After all, she is part of a governmental body, and an ambitious person with a strong hunger for power, playing authority politics with the most prestigious school of magic regardless of the effects on the students. I'd like to see this added to a decription of her character. --205.201.141.146 19:36, 7 June 2007 (UTC)

I agree. The website Amazon.com actually refers to her as "a bureaucratic tyrant" in it's review of the book so I think we should include a paragraph on the political aspect of her character.

What does this mean?
"It is later revealed that Umbridge herself had ordered Dementors to attack Harry in an attempt to silence him after several Ministry employees had mentioned but never done so. " I can't make sense of this sentence. Someone who is more familair with the books might want to rewrite it.--24.15.11.254 03:45, 14 June 2007 (UTC)

 * I think that's trying to get across that Umbridge had heard other Ministry employees talking about silencing Harry but she was the one who took action by ordering the Dementors to Little Whinging. Soapy Sunshine 20:58, 17 July 2007 (UTC)

Article title
I have moved (again) this page to Dolores Umbridge per policy (WP:NAME) and to reflect other Harry Potter character and literary character article names. While the character's full name may be Dolores Jane Umbridge, she is best known as Dolores Umbridge. Exploding Boy 23:24, 15 July 2007 (UTC)

Fair use rationale for Image:ImeldaStauntonPLT 334x450.jpg
Image:ImeldaStauntonPLT 334x450.jpg is being used on this article. I notice the image page specifies that the image is being used under fair use but there is no explanation or rationale as to why its use in this Wikipedia article constitutes fair use. In addition to the boilerplate fair use template, you must also write out on the image description page a specific explanation or rationale for why using this image in each article is consistent with fair use.

Please go to the image description page and edit it to include a fair use rationale. Using one of the templates at Fair use rationale guideline is an easy way to insure that your image is in compliance with Wikipedia policy, but remember that you must complete the template. Do not simply insert a blank template on an image page.

If there is other fair use media, consider checking that you have specified the fair use rationale on the other images used on this page. Note that any fair use images uploaded after 4 May, 2006, and lacking such an explanation will be deleted one week after they have been uploaded, as described on criteria for speedy deletion. If you have any questions please ask them at the Media copyright questions page. Thank you.

BetacommandBot 01:26, 24 July 2007 (UTC)

Elvira Umbridge
Where in the draft is her name written as "Elvira"?
 * Yeah, I don't see it either. - Arcayne   (cast a spell)  03:07, 13 August 2007 (UTC)
 * It's in title for NO 20, Time JAN: Extended powers of E LVIRA U MBRIDGE . - Fighting Fefnir 22:21, 13 August 2007 (UTC)
 * Come on, man, it's a scribble. Even if it were writ large in bold red crayon, what's the noteworthiness of it? Has JKR commented on some grand Elvira in her past that she wanted to stick it to? I've been published, and I can tell you that characters and minor points evolve all the time. It is simply cruft, and I am a CruftEater™. - Arcayne   (cast a spell)  23:07, 13 August 2007 (UTC)

title: Talk:Dolores Umbridge (character)


text:
Method of Fluxions

Method of Fluxions (De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed in 1671 and posthumously published in 1736.

Background
Fluxion is Newton's term for a derivative. He originally developed the method at Woolsthorpe Manor during the closing of Cambridge due to the Great Plague of London from 1665 to 1667. Newton did not choose to make his findings known (similarly, his findings which eventually became the Philosophiae Naturalis Principia Mathematica were developed at this time and hidden from the world in Newton's notes for many years). Gottfried Leibniz developed his form of calculus independently around 1673, 7 years after Newton had developed the basis for differential calculus, as seen in surviving documents like “the method of fluxions and fluents..." from 1666. Leibniz, however, published his discovery of differential calculus in 1684, nine years before Newton formally published his fluxion notation form of calculus in part during 1693.

Impact
The calculus notation in use today is mostly that of Leibniz, although Newton's dot notation for differentiation $$\dot{x}$$ is frequently used to denote derivatives with respect to time.

Rivalry with Leibniz
Newton's Method of Fluxions was formally published posthumously, but following Leibniz's publication of the calculus a bitter rivalry erupted between the two mathematicians over who had developed the calculus first, provoking Newton to reveal his work on fluxions.

Newton's development of analysis
For a period of time encompassing Newton's working life, the discipline of analysis was a subject of controversy in the mathematical community. Although analytic techniques provided solutions to long-standing problems, including problems of quadrature and the finding of tangents, the proofs of these solutions were not known to be reducible to the synthetic rules of Euclidean geometry. Instead, analysts were often forced to invoke infinitesimal, or "infinitely small", quantities to justify their algebraic manipulations. Some of Newton's mathematical contemporaries, such as Isaac Barrow, were highly skeptical of such techniques, which had no clear geometric interpretation. Although in his early work Newton also used infinitesimals in his derivations without justifying them, he later developed something akin to the modern definition of limits in order to justify his work.

title: Method of Fluxions


text:
Talk:Newton's method

Recent changes should be reverted
I am concerned that after a flurry of changes the current version Special:Permalink/1219909644 by @Fangong00 is significantly less clear and less accessible to non-expert readers than the previous version Special:Permalink/1218106312, as well as not really matching expected Wikipedia style.

In my opinion this whole collection of changes should be reverted, and any dramatic change along these lines should be first worked on as a draft (e.g. at User:Fangong00/Newton's method or Talk:Newton's method/draft rewrite) and then discussed before adoption here.

In my opinion, within the first few sections of this article we must focus on the most common and basic case of finding the zeros of a single-variable real-valued (or maybe also complex-valued) function, and everything about arbitrary multivariate functions, Banach spaces, etc. must be deferred to sections far down the article about more general contexts. The early sections must also avoid unnecessary jargon and must provide clear descriptive background rather than a wall of inscrutable technical details about this or that variant. –jacobolus (t) 18:12, 20 April 2024 (UTC)

 * I disagree. The first screen need to tell visitors what Newton's iteration is and what convergence conditions are.
 * Wikepedia is not a teaching platform, it is for quick reference. Fangong00 (talk) 13:57, 21 April 2024 (UTC)

Update: I tried reading the first few sections of Fangong00's version aloud to a friend, and we both found it almost impenetrable, so I decided to just revert the changes, pending discussion. Fangong00: I don't want to discourage you from trying to make improvements though. I'm sure there are things that could be expanded, reworked, etc. about the article as it was when you arrived. Feel free to make a rewrite draft, or to bring concrete criticisms of the existing article here for discussion. –jacobolus (t) 18:26, 20 April 2024 (UTC)

 * I agree with jacobolus. Bubba73 You talkin' to me? 02:39, 21 April 2024 (UTC)
 * I completely agree with jacobolus. Gumshoe2 (talk) 02:54, 21 April 2024 (UTC)
 * Also, after the page is back to a stable version, the Nash-Moser section (even though empty) should link to Nash–Moser theorem. Gumshoe2 (talk) 02:58, 21 April 2024 (UTC)
 * I agree with jacobolus. PatrickR2 (talk) 06:10, 21 April 2024 (UTC)
 * Please be specific. Which part is "impenetrable"? I can certainly try to improve it. Fangong00 (talk) 13:05, 21 April 2024 (UTC)
 * In your proposed version of the page, the section called Simpson's extension is pretty much identical in content to the existing section k variables, k functions together with the existing section in a Banach space. It's not good to have content repeated twice in different locations on the same page. And I believe the existing sections are more clearly written with non-mathematician readers in mind. What do you think is wrong with these sections? Gumshoe2 (talk) 13:39, 21 April 2024 (UTC)
 * Didactically, we want to understand the classical 1d situation before even mentioning other cases. —Kusma (talk) 13:39, 21 April 2024 (UTC)
 * As I understand, Wikepedia is a reference, not place to teach Newton's method. Users want to know quickly from the first screen what Newton's method is and what convergence conditions are.
 * I do plan to delete the duplicate part. Fangong00 (talk) 13:44, 21 April 2024 (UTC)
 * Wikipedia is an encyclopedia, namely neither a reference nor an instruction manual. The intention should not be to front-load the information that someone might be trying to look up. Gumshoe2 (talk) 13:54, 21 April 2024 (UTC)
 * I suppose people look up an encyclopedia for a quick reference. A page about Newton's method shouldn't give users an impression it is only a special case, as Newton-Raphson is. Hiding what Newton's iteration really is later shouldn't be the way. Fangong00 (talk) 14:03, 21 April 2024 (UTC)

I think the changes should be restored the article, at least in some form. The status quo article is pretty bad, and this added content is in a much better shape for improvement. For example, local quadratic convergence is stated clearly, and various important generalizations to the elementary version are stated up front in a consistent style. Thus the new article is more useful to people who want to use the method, as opposed to those just learning it for the first time. I think a separate "Introduction" section should be written, which covers the one variable theory that one learns about in Calculus, followed by the new stuff pretty much as written, modulo stylistic changes. Tito Omburo (talk) 12:16, 21 April 2024 (UTC)

 * Thank you. Fangong00 (talk) 13:03, 21 April 2024 (UTC)
 * I don't object at all to the content being on the page somewhere. (As I mentioned on wikiproject:math, it almost all already was on the page, in a less detailed version.) That's with the exception of the rank-r Newton method; is it clear that it deserves any space on the page? Gumshoe2 (talk) 13:22, 21 April 2024 (UTC)
 * I do agree that it is debatable on the inclusion of rank-r Newton's iteration since it is quite new. However, it is the first and ONLY iterative method that converges to positive dimensional solutions. It is published in Math Comp. Maybe it should be a one sentence item. Fangong00 (talk) 13:51, 21 April 2024 (UTC)
 * I am extremely skeptical of that. Such iterative methods go back to at least 1950. Gumshoe2 (talk) 14:35, 21 April 2024 (UTC)
 * I checked up that page. Are you sure there is an iterative method proposed in "Some mapping theorems" Fangong00 (talk) 14:50, 21 April 2024 (UTC)
 * The iteration is described in the proof of Theorem 1, also with a modification in the proof of Theorem 4. Gumshoe2 (talk) 14:56, 21 April 2024 (UTC)
 * I'll look that up. Thank you. It would be a shame for a method to be buried in a proof. Fangong00 (talk) 15:12, 21 April 2024 (UTC)
 * By a quick read, it deos not seem to be an iteration converging to positive dimentional solutions. The condition K(x) mapping to the whole space disqualify the Jacobian that is rank-deficient at such solutions. So the Jacobian can't map to the whole space. If it works, the linear mapping K is unknown so it is not a practical algorithm. Fangong00 (talk) 16:46, 21 April 2024 (UTC)
 * You may be skeptical. But I know a lot of numerical analysts who try compute nonisolated solutions. No one knows a single algorithm for that purpose so they all go extra steps to isolate the solutions first before rank-r Newton's iteration. Fangong00 (talk) 17:28, 21 April 2024 (UTC)
 * As for Graves' 1950 paper, it is about underdetermined systems, where the space of solutions has positive dimension. In that case it's often true that the linearization is surjective (i.e. K(x) maps to the whole space). It seems that rank deficiency would correspond instead to overdetermined systems, where the linearization would instead be injective and the solution space would have isolated points.
 * Regardless, what is important is that Wikipedia needs to contain only verifiable information; is there any reliable source which says (implicitly or explicitly) that this 2023 paper is significant? Gumshoe2 (talk) 17:35, 21 April 2024 (UTC)
 * That makes sense. Actually Newton's iteration (with inverse replaced by Moore-Penrose inverse) converges to positive dimensional solutions and it was proved by M. Chu in 1970s for underdetermined systems with surjective Jacobians. What I meant was that no known method for general rank-deficient Jacobians.
 * You have a great point. A 2023 paper may be too new to mention. Fangong00 (talk) 17:47, 21 April 2024 (UTC)
 * I disagree with the outlook of @Tito Omburo and @Fangong00
 * "Thus the new article is more useful to people who want to use the method, as opposed to those just learning it for the first time. "
 * Anyone wanting to apply Newton's method has the motivation to read deeper into the article and to look up alternative sources. Everyone else should not be forced to into a deep dive to get the basic concept. I've used some of these techniques in the past and yet I would stop reading at the second section of the new draft.
 * A later section such as "Formal concepts" (or any number of similar titles) with deeper content can have the same value for the in-depth reader. Johnjbarton (talk) 14:49, 21 April 2024 (UTC)
 * "Anyone wanting to apply Newton's method has the motivation to read deeper into the article and to look up alternative sources." Surely this is a problem with the order in which topics appear, not whether topics should be covered in the article at all.  (And before anyone says that the article already discussed the new additions to some extent: yes, but not as thoroughly nor as well-organized.)  Tito Omburo (talk) 17:14, 21 April 2024 (UTC)
 * Strongly disagree. A scientific practitioner may just want to try a known method and don't want/need to "read deeper into the article and to look up alternative sources". I know a lot of people, including myself, do that all the time. Fangong00 (talk) 17:23, 21 April 2024 (UTC)
 * @Tito Omburo Adding new topics / expanding existing topics was not the problem. The problem was that the first 3 sections of the article were pushed out of reach of the majority of the intended audience. It's entirely fine to make a section halfway through the article about the multidimensional case, with as much detail as desired. In cases where both experts and novices can be expected to be interested in the same topic, it is essential that the top of the article remains accessible for novices, and that the rest remains clearly motivated, put into context, organized with some narrative flow, etc. (In cases where one section of such an article, focused on an advanced topic, has enough to say about it that it starts ballooning in size, it can be helpful to leave a summary and split it into a new article focused on that subtopic. That is probably unnecessary here though unless someone wants to keep expanding this much further.) –jacobolus (t) 18:56, 21 April 2024 (UTC)
 * This is pretty much what I've been saying all along, that an introduction should be at the beginning, followed by more advanced material. Let me just ask: do you support adding this content if it is after the third section?   Tito Omburo (talk) 00:13, 22 April 2024 (UTC)
 * As far as I can tell nobody has any problem with expanding later sections about more advanced topics. –jacobolus (t) 00:44, 22 April 2024 (UTC)
 * The discussion is about wholesale reverting these changes, not moving them to a later section in a constructive fashion. So yes: people taking this position do have a problem with improving later sections of the article.  Also this revert exclusively affected primarily sections much later in the article.  Tito Omburo (talk) 10:52, 22 April 2024 (UTC)
 * You are mischaracterizing what happened and misunderstanding people's intentions. –jacobolus (t) 11:20, 22 April 2024 (UTC)

-- Dear User:jacobolus, Bubba73, PatrickR2 and Gumshore2:

Why do you not want a vistor know what Newton's method is on the first screen? This is the central issue and we need to focus on that, please.

The Newton-Raphson is only a special case of Newton's method. It is a disservice to hide what Newton's method is in numerical analysis. Fangong00
 * As I mentioned on wikiproject:math, I don't think there would be any issue with modifying the lead section to highlight that there are different formulations of Newton's method, including the standard multidimensional version which you are calling "Simpson's extension." The lead section should also say that there are a number of different theorems which guarantee convergence of Newton's method to solutions. It isn't hard for readers to use the table of contents to click to where in the article they want to go in order to get details.
 * Also, I hope you are aware of the three-revert rule. I believe you are already in violation of it; if you continue to revert the page it is very possible that an admin will temporarily block you. Gumshoe2 (talk) 14:42, 21 April 2024 (UTC)
 * It might be okay to add a sentence or two to the end of the lead section, as well as expanding later sections. However, what you did was remove all of the readable introductory explanation from the top of the article, which made it very difficult for most readers to understand. I urge you to find a non-mathematician friend, print out the first 3–4 sections of your version of the article and the first 3–4 sections of the previous version, and ask them which one makes more sense. –jacobolus (t) 15:31, 21 April 2024 (UTC)
 * I took your suggestion. Fangong00 (talk) 16:03, 21 April 2024 (UTC)
 * @Fangong00 Can you please work on your proposal in a draft outside of the article, e.g. in your user namespace, as a subpage of this talk page, or proposing particular sentences or paragraphs directly here in the talk page? Then you won't end up just getting reverted repeatedly.
 * Your updated changes to the lead section still make it worse, in my opinion. Perhaps you can recruit someone who both knows deeply about this topic and has strong expository writing skills to help keep changes legible. –jacobolus (t) 16:17, 21 April 2024 (UTC)
 * I found your last paragraph interesting, to say the least. So I'd like to ask: Do you know why the claim "preserves the quadratic convergence rate" in the section "Slow convergence for roots of multiplicity greater than 1" is actually false despite an impeccable proof? Few people know it is false and even fewer know why.
 * You would know it is false if you ever test it. Fangong00 (talk) 00:38, 22 April 2024 (UTC)
 * I can't figure out what point you are trying to make. The convergence rate to higher-multiplicity roots, a topic I am certainly no expert in, is irrelevant to our conversation here. (But if this particular claim is wrong, then it should by all means be fixed.) If your point is just that you (and probably several other participants in this discussion) know more about this topic than I do, I am happy to concede that. I am not by any means an expert in root-finding algorithms, nonlinear optimization, or whatever. –jacobolus (t) 00:49, 22 April 2024 (UTC)
 * You know what point I am making, and I made it. There is no shame for not knowing that.
 * The claim is practically false because, in numerical computation, tiny number divided by tiny number results in a random number generator, like m*f(x)/f'(x) near a multiple root where f(x)=f'(x)=0. The theorem has a hidden condition: f(x) and f'(x) can be exactly computed. The condition shouldn't be hidden because it can not be met in numerical analysis. The proof is impeccable with that hidden condition assumed, of cause. Fangong00 (talk) 01:21, 22 April 2024 (UTC)
 * No I really do not understand what point you are trying to make. It seems like some kind of passive aggressive status thing. Please just be clear and direct and say what you mean. –jacobolus (t) 01:27, 22 April 2024 (UTC)
 * Do we agree no one is supposed to act like having some kind of status here? Fangong00 (talk) 02:10, 22 April 2024 (UTC)
 * I cannot make heads or tails of this subthread. As far as I can tell it has nothing to do with the rest of the conversation. –jacobolus (t) 02:18, 22 April 2024 (UTC)
 * It is interesting you avoided answering. Besides the point I already made, the status quo page you advocate is bad scientifically and in presentation, IMHO, maybe as bad as or worse than you think my corrections are. I'll list more errors when I get a chance. I do agree it is time to stop this subthread. Fangong00 (talk) 03:10, 22 April 2024 (UTC)
 * I didn't reply because it seems off topic and I don't entirely understand what you mean by "no one is supposed to act like having some kind of status". I will say that (a) each of us is just an editor (and just a person for that matter), and Wikipedia pages' content is decided by consensus (cf. Consensus) rather than credentials or reputation inside or outside of Wikipedia, (b) playing word games, trying to catch people with "gotchas", and speculating about other editors' motives or anything beyond what they directly did or said is usually a distracting waste of time, and (c) everyone should try to follow Etiquette.
 * status quo page you advocate – I did not ever "advocate" the previous page. All I said is that the previous version has essential features that should not be removed without consensus, and that any consensus for such substantial changes can only be achieved through discussion.
 * Please do make that list of errors, that sounds great. That was what I asked of you at the start: to (a) work on your draft in user space or as a subpage of this talk page instead of revert warring, and to (b) make a concrete list of criticisms / proposals here on this talk page. That other editors can concretely reply to those point by point. But please put it into a new talk page section instead of this deeply nested comment thread, and try not to address it personally toward any particular editor. –jacobolus (t) 05:04, 22 April 2024 (UTC)

(ec)
 * Fangong00 - introduce it simply.  Banach spaces and all of that doesn't belong in the introduction.  It doesn't even belong in the body of the article, until all of the main use of Newton's method has been covered.  Bubba73 You talkin' to me? 16:04, 21 April 2024 (UTC)

 * I disagree. Newton's method in Banach spaces is very important and is standard material. So I think it should be mentioned in the lead section, although I would put it closer to the end thereof, instead of in the very first sentence (where it is now).
 * I think Fangong00's new edit is possible to work with and improve, although many of my previous objections stand. ("Simpson's extension" is not standard terminology, the sections on it and Gauss-Newton iteration are a little harder to read than the way they were previously expressed, and the rank-r Newton iteration is not notable enough to mention.) Gumshoe2 (talk) 16:19, 21 April 2024 (UTC)
 * OK, then I think that the last sentence of the introduction could say that it can be generalized to Bsnach spaces. Then the coverage of that should be near the end of the body.  Bubba73 You talkin' to me? 22:48, 21 April 2024 (UTC)

 * @Gumshoe2 What's this Banach thing? I just want to know about Newton's method. Johnjbarton (talk) 16:43, 21 April 2024 (UTC)
 * The whole debate is about what visitors want to know. You may not want to know Newton's method on Banach spaces but there are certainly some visitors do, and may need it. Fangong00 (talk) 16:49, 21 April 2024 (UTC)
 * @Fangong00 Fine, so put that detail done further in the article, not in the intro and first sections. Johnjbarton (talk) 17:44, 21 April 2024 (UTC)
 * Just as with differentiation in general, it is best to first understand the one dimensional version, although the Banach space version is no different if viewed from the right angle. Best nit to dwell on it much in an introductory article. —Kusma (talk) 16:49, 21 April 2024 (UTC)
 * This is not a teaching platform. Someone trying to solve an equation in Banach space should get the information on the opening screen. I guess the original page was written by a teacher, not by a researcher or scientific computing practitioner. Fangong00 (talk) 17:01, 21 April 2024 (UTC)
 * Wikipedia is both a general and a specialist encyclopaedia. Articles should be targeted at the widest range of interested people. For something as relatively basic as Newton's method, that is undergraduates. —Kusma (talk) 17:05, 21 April 2024 (UTC)
 * Mentioning its capabilities is exactly commodating "widest range". Your suggestion serves only undergraduates, who can skip the Banach part as they wish.
 * I don't understand why people want to hide the capacities of a great algorithm. Fangong00 (talk) 17:13, 21 April 2024 (UTC)
 * Why do people want to deny other visitors' access to crucial information simply because they don't need? Fangong00 (talk) 17:16, 21 April 2024 (UTC)
 * It's much clearer for a newcomer if the relevant part of the lead section says "multivariate function" instead of "mapping between vector spaces or Banach spaces". The latter is being made gratuitously obscure for pretty much no benefit. Wikipedia has a huge problem of chasing readers away from technical articles because they find them inscrutable. To the extent possible we want to push back on that, even at the expense of occasional mild imprecision or inconvenience for advanced readers. –jacobolus (t) 19:10, 21 April 2024 (UTC)
 * To my personal taste, a lead section along the following lines would be ideal, with the hope that it's readable even if some of the words are unfamiliar. (I'd also prefer that even the single-variable formulas are pushed out of the lead section and into the first section.)
 * Newton's method, also known as Newton's iteration, is an iterative method which uses the tangent lines of a single-variable function to iteratively define a sequence of numbers. Although there are exceptional cases, often this sequence is infinitely iterable and converges to a zero of the function. This can be studied empirically, although there are a number of mathematical theorems which provide conditions that guarantee the convergence of Newton's method to a zero.
 * More generally, Newton's method has also been studied, both theoretically and empirically, in a number of multivariable contexts. In this context, the multivariable generalization of the tangent line known as the tangent plane is used to define the iteration. The multivariable Newton's method leverages the solvability of systems of linear equations to find solutions of nonlinear systems of equations. Computer implementations of this are useful in many practical problems of numerical analysis to solve complicated equations. This forms a powerful tool in the field of optimization, where Newton's method is useful for finding critical points. A number of variants of the standard Newton method have also been developed and are widely used in these applied contexts.
 * In pure mathematics, the multivariable Newton's method can naturally be put into the even broader context of Banach spaces. While the multivariable Newton method allows the points of the iteration to be tuples of numbers, this generalized Newton's method allows the points of the iteration to be, for example, functions. This flexibility is theoretically useful in parts of functional analysis and partial differential equations.
 * Gumshoe2 (talk) 19:58, 21 April 2024 (UTC)
 * I don't like the starting sentences. I would instead say something more like "Newton's method is an algorithm for numerically finding a solution of a nonlinear equation using a succession of linear approximations." I would recommend explicitly describing how the single-variable case works in the lead section, using both a picture and mathematical notation. We should explicitly describe the main idea: to find a root of a function, we make an initial guess, use knowledge of the function's value and derivative to construct a linear approximation near that guess, calculate the root of the linear approximation, then use that as our next guess. –jacobolus (t) 20:36, 21 April 2024 (UTC)
 * One thing that I think would help a lot is an animated example image at the top of the article. We don't absolutely have to lead with formulas, but we should try to include a concrete explanation of what the main idea is behind Newton's method. –jacobolus (t) 18:57, 22 April 2024 (UTC)
 * I agree, the starting sentences that I wrote there are pretty weak. Personally I'd like to avoid formulas in the lead section if possible, since they would have to be repeated pretty much immediately in the first section anyway. But animations would be very good, are they easy to make? Gumshoe2 (talk) 00:09, 23 April 2024 (UTC)
 * "Someone trying to solve an equation in Banach space should get the information on the opening screen."
 * (I assume that by "opening screen" you mean the beginning of the page.) There is absolutely no need for a page to be structured in this way. Why would it be so difficult for someone to use the table of contents to navigate perhaps very deep into the page to get what they need? Gumshoe2 (talk) 17:39, 21 April 2024 (UTC)
 * Still don't get it. Why do you trying so hard to prevent one sentence in the opening graph mentioning the capabilities? Most visitors do not look at table of contents. Fangong00 (talk) 17:55, 21 April 2024 (UTC)
 * I fully support the lead section giving a largely nontechnical summary of contents, as per Manual of Style/Lead section, and as such I support that the Banach space context should be mentioned there in a sentence. I am against deeper information or technical detail appearing at the beginning of the article; there is no problem with it being pushed towards the end. Gumshoe2 (talk) 18:00, 21 April 2024 (UTC)
 * Yes. Sometimes the best way to balance the various requirements that a lead section ought to satisfy — both accessibility and summarizing the article that follows — is a brief synopsis of highly technical material towards the end. XOR'easter (talk) 19:02, 21 April 2024 (UTC)
 * A few-sentence paragraph at the end of the lead section of clear prose explaining the importance of the multidimensional version of Newton's method would be fine ("Newton's method has been extended to multivariate functions, where it has several variants, and is widely used in application X, Y, and Z" or whatever).
 * I would leave the term "Banach spaces" out of the lead section entirely, especially when that isn't really discussed in any detail later on. Gratuitous jargon should be avoided, especially in the lead section. Readers interested in generalizations are not going to be confused by skimming down the page. –jacobolus (t) 19:05, 21 April 2024 (UTC)
 * I would add to this proposed lede that the ideal convergence rate is extremely rapid, but that devising conditions to ensure ideal convergence is a difficult problem. Chaos also deserves to be mentioned.  Tito Omburo (talk) 11:06, 22 April 2024 (UTC)
 * By "ensure ideal convergence," you mean that it's difficult to choose the initial point well? Gumshoe2 (talk) 00:10, 23 April 2024 (UTC)

Questionable examples
I'm a little skeptical of two of the examples in Newton's Method section. Could anyone clarify or provide a relevant reference? If not, I think these examples should be removed. Gumshoe2 (talk) 00:06, 23 April 2024 (UTC)
 * 1) The example in the "discontinuous derivative" section. I agree with all of it except the key phrase "and Newton's method will diverge almost everywhere in any neighborhood of it," which seems by no means clearly established. The unboundedness of $$f(x)/f'(x)$$ near the root seems to be, in principle, perfectly compatible with many/most Newton iterations converging to the root. A simulation with randomly chosen initial points seems to back this up.
 * 2) The second half of the "zero derivative" section. What is the meaning of "nearly double" in this context; does it mean that the root is close to a point where the derivative vanishes? Six also seems like a very moderate number of iterations to require. In all, I just don't know what this example is trying to illustrate.

 * Also, although I don't have any issue with the "no second derivative" section, I haven't been able to find any standard textbook reference where this example or a similar one is given. Does anyone here know one? Gumshoe2 (talk) 00:12, 23 April 2024 (UTC)
 * Since nobody has spoken up, I've removed these examples. I've also combined (and somewhat rewrote) the three different example sections into a single one, which I think is helpful for reading. Gumshoe2 (talk) 14:19, 28 April 2024 (UTC)

Proof of quadratic convergence for Newton's iterative method
Does anyone think it's necessary to give a full proof of this as in the present version? It's possible to summarize it reasonably and convincingly in about two lines, and there are plenty of standard books to reference where the full proof appears. Gumshoe2 (talk) 14:28, 28 April 2024 (UTC)

 * I agree. Also, the preceding subsection is also confusing because of too much technical details. Imo, one must reduce the proofs and explanations to functions of class C2, with the simple mention that there are generalizations to slighty larger classes of functions. D.Lazard (talk) 16:12, 28 April 2024 (UTC)

"Poor initial estimate"
The following paragraph was added in 2016:
 * A large error in the initial estimate can contribute to non-convergence of the algorithm. To overcome this problem one can often linearise the function that is being optimized using calculus, logs, differentials, or even using evolutionary algorithms, such as the Stochastic Funnel Algorithm. Good initial estimates lie close to the final globally optimal parameter estimate.  In Nonlinear Regression the SSE equation is only "close to" parabolic in the region of the final parameter estimates.  Initial estimates found here will allow the Newton-Raphson method to quickly converge. It is only here that the Hessian of the SSE is positive and the first derivative of the SSE is close to zero.

It is both very unclear and, based on the username of the editor, seems to be added as an advertisement for their own research. For now I've removed it, possibly someone can salvage quality material from it. Gumshoe2 (talk) 15:09, 28 April 2024 (UTC)

 * Remove this paragraph: if the initial estimate induce non-convergence or convergence to a wrong local optimum, the best method if to choose another initial estimate. It is unclear whether this paragraph is about the choice of a better initial estimate. On the opposite, is seems to suggest to replace the function with an approximation of it, which is a very bad idea, as it may change the optimum dramatically. Moreover, the choice a good initial estimate depends on the nature of the problem, and I doubt that there exist general methods for that, that are better than a random choice. D.Lazard (talk) 15:48, 28 April 2024 (UTC)

Computable numbers
I'm removing the following paragraph:
 * When Newton's method can be applied to a transcendental equation, and converges to a solution of the equation, this implies that the solution is a computable number that is exactly represented by the pair formed by an initial approximation and an algorithm for increasing the accuracy of any approximation.

As stated, it seems definitely wrong. If $c$ is an uncomputable number, then the root of $x - c$ is $c$, and Newton's method initialized anywhere converges to the root (and even does so in a single step). Gumshoe2 (talk) 15:43, 28 April 2024 (UTC)

 * The paragraph becomes true if "transcentdental equation" is replaced with "equation defined by a computable function". This is the simplest way to prove that algebraic numbers are computable. D.Lazard (talk) 17:38, 28 April 2024 (UTC)
 * I don't know much on this topic, but the wikipage computable function suggests that the domain and range are only natural numbers, or tuples thereof. Is this the wrong context?
 * But it does make sense to me that Newton's method, along with other root-finding algorithms, can prove that algebraic numbers are computable. Is there a standard reference for this? I haven't been able to find it. Gumshoe2 (talk) 19:15, 28 April 2024 (UTC)
 * Sorry, by trying to give sense to the removed paragraph, I did (wrong) original research. For computability with real numbers, one must look on the work by Stephen Smale and his followers. Also, for applying Newton's method to a transcendental equation, one requires to work with an increasing approximation of the equation and the involved derivatives. This seems to not be an easy task to define this propertly. This has probably been done in constructive mathematics, but I do not know this subject well.
 * By the way, in section . it must be said that it is the quadratic convergence that insures that the computation of a multiplicative inverse has the same bit complexity as multiplication, even if fast multiplication is used (beacause of the quadratic convergence and the property of geometric series, all iterations but the last one take together no more time than the last iteration). This is also Newton's method that is used to show that matrix inversion has the same complexity as matrix multiplication, independently of the used multiplication algortihm. (As usual, I an unable to give references, but this should be found in most textbooks on complexity theory). D.Lazard (talk) 20:38, 30 April 2024 (UTC)

Systems of equations article draft
A draft article now exist for solving systems of equations, "Newton's method for systems of nonlinear equations". If approved, it should replace the existing sub-heading on the subject. While waiting for review, I would be helpful if interested parties were to weigh in with additional edits to the draft on subjects such as mathematical proofs, convergence theorems, etc. It will make a more credible draft and help with the review process. Netshine2 (talk) 01:50, 10 May 2024 (UTC)

 * Where is the draft available? I'm a little skeptical that it needs its own article, but certainly the present coverage should be expanded. Gumshoe2 (talk) 02:32, 10 May 2024 (UTC)
 * If someone wants to write one, I think this subject definitely can support its own article. There is a ton to say about it, enough to overflow 1–2 sections here. A new article would IMO be a much better solution than trying to generalize the first several sections of this article to cover all cases up front, but depending on the amount drafted, it could also remain a section. (I'm not sure whether a wordy explicit title would work better, or something more concise like Multivariate Newton's method.) –jacobolus (t) 02:51, 10 May 2024 (UTC)
 * I think there could be other solutions -- like you, I don't think the first several sections of this article should be generalized in place. I think it should be possible to more or less keep the present section layout on this page. But it all depends on the particular content in question.
 * As for titles, one issue is that, with the exception of freshman calculus texts, the "multivariate" Newton's method is the standard context that Newton's method is presented and used in. I think giving it a special page title like "Multivariate Newton's method" would make it appear unnecessarily niche. (Also see only 47 hits on Google Scholar for "multivariate Newton's method", 30 for "multivariable Newton's method") Gumshoe2 (talk) 03:06, 10 May 2024 (UTC)
 * If the consensus is to leave the Multivariate content in the current article, I am good with that. However, my thinking is that the Multivariate content will grow, and the existing article focuses heavily on single variable.  The new draft article still needs an applications section, which I can contribute to, mathematical sections to prove out the details which math experts can contribute to, etc.  Since, as you point out, half the draft article is already in the existing section, I can place the remainder in the existing section, too, at least until the new article issue is finalized.   I am somewhat concerned that the remaining content, being primarily based on life experiences, may not meet verification standards, but we can try it.  Any further thoughts on this subject?
 * As for the name, "Multivariate", the draft currently uses terminology from old text books. The theory still works, but naming conventions do change over time.  I will update the draft to conform to current naming conventions. Netshine2 (talk) 15:58, 10 May 2024 (UTC)
 * To be clear: I have no deep insight or particular care about what title/terminology to use. I was just throwing out an idea of a more concise title. But a longer one can also be fine. –jacobolus (t) 16:47, 10 May 2024 (UTC)
 * Thank you for the suggestion. I liked it.  I was unable to change the title in the draft, so I made a talk entry for it. Netshine2 (talk) 00:42, 11 May 2024 (UTC)
 * Seems to be at Draft:Newton's method for systems of nonlinear equations. –jacobolus (t) 02:58, 10 May 2024 (UTC)
 * I can comment later on the content, but I think that can all be well absorbed into this page. (The first half is already here.) Gumshoe2 (talk) 03:10, 10 May 2024 (UTC)

Example is not referenced.
The section under Multidimensional methods > Example has no sources. A recent edit by an IP user changed one number without any edit summary. Without any way to verify the content, I reverted the edit.

My change was reverted by @Netshine2 claiming that the IP user was correct. That's great, and I'm glad the content may now be correct. However we still have no way to verify it.

I think the content has marginal notability and absent a reference it may be a small kind of original research. Is there a case to keep this content? Johnjbarton (talk) 19:50, 29 May 2024 (UTC)

 * The references for the example would be the link to "Jacobian" matrix in the above section, and the multidimensional definition of Newton's method in the section above that. It is assumed that the derivatives in the example are covered by the reference to "Jacobian", since the Jacobian matrix is a matrix of partial derivatives.
 * The presence of the example serves to clarify the definitions in the section above. Users not familiar with Newton's method as it applies to actually doing something with it can easily see how the defining equations above work to obtain a valid converged solution.  The equations are validated by keeping them simple and breaking them down into small enough pieces such that anyone skilled in Mathematics can validate the converging processes each step at a time. I do not believe the term "original research" really applies here because the example is simple to create and reproduce by anyone skilled in mathematics.  The only challenge would be from user's not-so-skilled in mathematics, which is what the example is intended for.  I have no objection to replacing the example with an existing published simple example, provided it is simple enough for users to follow, and complex enough for users to grasp the usefulness of Newton's method for multidimensional applications, but I have to point out that published example can contain typos, too. Netshine2 (talk) 20:21, 29 May 2024 (UTC)
 * Thanks! (please us the reply button on Talk pages). I made a minor edit to the reference.
 * ✅ Johnjbarton (talk) 00:10, 30 May 2024 (UTC)
 * Looks Good! I'll have to remember the footnotes for future reference.  Looks like we did good job of making the Wikipedia page better. 24.121.207.148 (talk) 00:27, 30 May 2024 (UTC)

 * I did some additional research and was able to find a published reference that used a similar example to ours, but it used 3 equations instead of our 2. For simplicity, I would prefer that our example keep the number of equations down to 2.  Note that even if we were to use the reference example as-is, it still needs to be validated for typos and accuracy, which I very capable of doing.  Please look the reference example, and let us know if you believe we should use it.  The reference is easier to follow if you create a free account for yourself with the reference site. Netshine2 (talk)

title: Talk:Newton's method


text:
Hugh (disambiguation)

Hugh may refer to:

 * Hugh, a given name (including a list of people and fictional characters with the given name and mononym)
 * Hugh, an acronym for Hill Under Graham Height, which is a category of hills in the British Isles
 * Hugh Lake, in Quebec, Canada
 * Hugh (robot), an artificial intelligence robot librarian
 * Hugh, Northern Territory, a locality in Australia

title: Hugh (disambiguation)