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David McKay Publications

David McKay Publications

David McKay Publications published some of the Ace Comics (1937 series), Blondie Comics, Dick Tracy, Mandrake the Magician (1938) and several others.

External links


- [http://www.milehighcomics.com/comicindex/Publisher-David-Mckay-Publications-DMKP.html all titles] Category: publishers Category: comics

Ace (comics)

Ace is a female member of the Royal Flush Gang, which is group of villains modeled on the cards in a royal flush in poker. Her power is mind control. The Royal flush gang has fought Batman and Batman Beyond's Terry McGinnis and featured regularly in the cartoon "Justice League". There have been atleast two incarnations of the gang.Ace had the same power in both.Her powers ist she is a superstrong android. She can also create illusions, and drive people insane just by looking at them. Later, her powers developed into the ability to actually alter reality Category:Justice League villains

Blondie (comic strip)

Blondie is a popular comic strip created by Murat Bernard "Chic" Young and syndicated by King Features Syndicate. It has been published in newspapers since September 8, 1930. Chic Young drew Blondie until his death in 1973, when the control of the strip passed to his son Dean Young. Dean Young has collaborated with a number of artists on the strip, including Jim Raymond, Mike Gersher, Stan Drake, Denis Lebrun and most recently John Marshall. In 1995, the strip was one of 20 included in the Comic Strip Classics series of commemorative postage stamps. In 2005, Blondie celebrates its 75th anniversary with a three-month-long story line, featuring various surprises including cameo appearances of characters from other comic strips such as B.C., Beetle Bailey, The Family Circus, For Better or For Worse, Garfield, Hägar the Horrible, and The Wizard of Id, among others. It remains popular, appearing in more than 2300 newspapers in 55 countries and translated into 35 languages. In addition, Blondie and/or Dagwood were featured in Dennis the Menace, Hi and Lois, B.C., Family Circus, Baby Blues, Hagar the Horrible, and Curtis (comic strip).

Characters

Originally, Blondie focused on the adventures of Blondie Boopadoop, a carefree flapper girl who spent her days in dance halls. On February 17, 1933, after much fanfare and buildup, Miss Boopadoop was married to her boyfriend Dagwood Bumstead, the son of a wealthy industrialist. Unfortunately for the Bumsteads, Dagwood was disowned by his upper-crust family for marrying beneath his class, and ever since, he has been slaving away at the office of the J. C. Dithers Construction Company, named after his tyrannical boss Julius Caesar Dithers. A running gag in the strip is the impossibly tall sandwich Dagwood fixes for a snack, which came to be known as a Dagwood sandwich. Blondie and Dagwood live in suburbia, next door to Herb and Tootsie Woodley. The Bumstead family has grown, with the addition of a son named Alexander (originally "Baby Dumpling"), a daughter named Cookie, and a dog named Daisy. Alexander and Cookie have grown up into teenagers who uncannily resemble their parents. Other regular characters include Mr. Beasley (the mailman) and Elmo Tuttle (a pesky neighborhood kid).

Changing times

While the look of Blondie has been carefully preserved, a number of details have been altered to keep up with changing times. Blondie herself is no longer a housewife. She and Tootsie Woodley started a catering business in 1991. Dagwood still knocks heads with his boss, Mr. Dithers, but now he does it in his capacity as webmaster for J. C. Dithers Construction Company, and he now begins each morning racing to meet his carpool rather than catching a city bus to work.

Other media

Blondie has occasionally graduated from the comics page to other media. There was a series of comic books starring characters from Blondie from 1937 to 1976. In the 1930s, Blondie had her own weekly radio show. In 1938, the film Blondie was made, with Penny Singleton and Arthur Lake as Blondie and Dagwood, reprising their roles from the radio. Following the success of this film, a whole series of over twenty Blondie movies were made, and then a television series in 1957 and later in 1968, each which lasted one season. There were two Blondie and Dagwood TV specials in 1987, with Loni Anderson and Frank Welker in the title roles.

External links


- [http://www.blondie.com/ Blondie and Dagwood - The Official Website]
- [http://www.kingfeatures.com/features/comics/blondie/about.htm King Features Blondie website]
- [http://www.toonopedia.com/blondie.htm Article on Blondie] from Toonopedia Category:Comic strips

Dick Tracy

Dick Tracy is a popular character in American pop culture. The character of Dick Tracy is a hard hitting, fast shooting, and supremely intelligent police detective who has matched wits with a variety of often grotesquely ugly villains. Dick Tracy was created by cartoonist Chester Gould in 1931 for a newspaper comic strip also entitled Dick Tracy. The strip, which made its debut appearance on October 4th, 1931, was distributed by the Chicago Tribune Syndicate. Gould wrote and drew the strip until 1977.

The comic strip

Early years

Chester Gould introduced a raw violence to comic strips, reflecting the violence of 1930s Chicago. Gould also did his best to keep up with the latest in crime fighting techniques and, while Tracy often ends a case in a shootout, he uses forensic science, advanced gadgetry, and plain hard thinking to track the bad guy down. It has been suggested that this comic strip was the first example of the police procedural mystery story. Others have noted that actual mystery plots were relatively rare in the stories since the comic strip format is a difficult one for that kind of plot. The real focus, they argue, is the chase with the criminal seen committing the crime and Dick Tracy figuring out the case and relentlessly pursuing the criminal who becomes increasingly desperate as the detective closes in. The strip's villains are arguably the strongest appeal of the story. Tracy's world is decidedly black and white where the bad guys are sometimes so evil, their very flesh is deformed to announce their sins to the world. The evil sometimes is raw and coarse like the criminally insane Selbert Depool ("looped" spelled backwards, typical Gould). At other times it is suave like the arrogant Shoulders, who can't help thinking that all women like him. It can even border on genius like the Nazi spy Pruneface who is not only a machine design engineer but also dabbles with a chemical nerve gas. nerve gas However, by far the most popular villain was Flattop Jones, a freelance hitman who had a large head that was as flat as an aircraft carrier's flight deck. In a classic storyline, Flattop was hired by black marketeers to murder Tracy and he came within a hairsbreath of accomplishing that before deciding to blackmail his employers for more money before he did the deed. This proved to be a fatal mistake since it gave Tracy time to signal for help and he eventually defeated his assassin in a spectacular fight scene even as the police were storming the hideout. When Flattop was eventually killed, fans went into public mourning. Reflecting some of the era that also produced film noir, Gould tapped into the existential despair of the criminals as small crimes lead to bigger ones and plans slip out of control and events happen sometimes for no reason at all because life can be unpredictable and cruel. Treachery is everywhere as henchmen are killed ruthlessly by their bosses and bosses are betrayed by jilted girlfriends and good people in the wrong place at the wrong time are gunned down. film noir

Evolution of the strip

Gould changed Tracy with the times, sometimes with mixed results as with the introduction of science fiction elements such as the two-way wrist radio which proved to be the first of a variety of personal wrist communicators and other futuristic gadgets provided by the eccentric industrialist, Diet Smith. This eventually led to what Gould thought was the logical conclusion in the 1960s of the Space Coupe, a spacecraft with a magnetic propulsion system. This started a much-derided science fiction period that had Tracy and friends having adventures on the moon and meeting the late Moon Maid and her race in 1964. This in turn led to an eventual sharing of technologies and the villains had to be even more exaggerated in power to challenge Tracy in an escalating series of stories that completely abandoned the urban crime drama roots of the strip. Finally, Junior actually marries the Moon Maid, and they produced two children! In the 1970s, Gould even less successfully tried to modernize Tracy by giving him a longer hair style and mustache, adding a supposedly "hip" sidekick, Groovy Grove. More successful was the decades-long substory of the Plenty family, a group of goofy redneck yokels headed by former villains, Bob Oscar "B.O." Plenty and Gravel Gertie. The family provided a humorous counterpoint to Tracy's adventures. Their daughter, Sparkle Plenty, first gave the strip an infant character, and later a pretty young adolescent girl character, since, unlike most comic strip children, she was allowed to grow up. Another successful addition was that of Lizz the Policewoman as one of Tracy's sidekicks. She proved be to an active and formidable female character in a manner that was groundbreaking for comic strips of that era. However, the later stories were often shackled with a stubborn grousing condemnation of the rights of the accused which often involved Tracy being frustrated by criminals because of legal technicalities and prosleytizing about it. The fact that newspaper comics were sharply reduced in space for each feature during that time also negatively affected Gould's storytelling abilities as he failed to adjust. The strip often included a frame devoted to "Crimestoppers' Textbook", a series of handy illustrated hints for the amateur crimefighter; for instance, when attempting to memorize the face of an evildoer for later identification purposes, make sure to note the size and shape of the ears and earlobes.

Later years

earlobe Gould retired from the strip in 1977 and Dick Tracy was taken over by Max Allan Collins and longtime Gould assistant Rick Fletcher. Collins reversed some of Gould's science-fiction changes by having the character Moon Maid killed off in 1978, as well as doing away with other Gould creations of the 1960s and 1970s, and generally taking a less cynical and simplistic view of the justice system. Rick Fletcher died in 1983 and was succeeded by Dick Locher, who had assisted Gould on the strip in the late 1950s, early 1960s. In 1992, Tribune writer and columnist Mike Kilian took over the writing; Kilian died on October 27 2005. In 1995, the strip was one of twenty included in the Comic Strip Classics series of commemorative postage stamps.

Other media depictions

Early films

The popularity and success of the Dick Tracy’s comic strip spread to radio and to movie serials. Ralph Byrd first played Dick Tracy in a movie of the same name in 1937. Byrd’s career continued through a series of B-grade Tracy movies. The best known of the films is Dick Tracy Meets Gruesome, with the title's villain played by Boris Karloff.

Television

The strip also had limited exposure on television with a short-lived live action series and two animated ones. In the first cartoon, Mel Blanc voiced several characters including a junior detective named Go-Go Gomez which was essentially a human version of his famous fast mouse Speedy Gonzales. Tracy would simply sit back and let Gomez and his other subordinate flatfoots mop up crooks like Pruneface, Itchy, Mumbles, Flattop, Cheater Gunsmoke, B.B. Eyes, and Tracy's other idiosyncratic villains. The show has not been seen in years because of its slightly racist undertones and use of ethnic stereotypes and accents, but recently resurfaced on pay-view digital cable channels. The second exposure to television was a feature in Archie's TV Funnies, produced by Filmation, which adhered more closely to the comic strip. There was also an unsuccessful television pilot from the producers of the live action Batman television series.

1990 film & Video Games

In 1990, Warren Beatty revived some interest in the character with his film Dick Tracy. Beatty directed the movie and starred as Tracy. In the film, Beatty was after a surreal comic strip-inspired world with only primary colors, restyled automobiles, and extensive makeup treatments for Tracy's famed villains. However, some people complained that the weak storyline was slightly overwhelmed by the casting which included stars such as Beatty, Madonna, Al Pacino, Dustin Hoffman, and Dick van Dyke as well as many other notable cameos. Madonna's soundtrack album I'm Breathless: Music from and Inspired by Dick Tracy spawned two top-ten hits including "Vogue" and "Hanky Panky". Broadway composer Stephen Sondheim wrote several songs for the film, including "Sooner or Later (I Always Get My Man)" which won the Academy Award for Best Song. There were also several other unreleased Madonna songs that were recorded for the film but not used at all. In August 1990, Bandai America, Inc. made Dick Tracy into an NES game loosley based on Beatty's film. It was also released in 1991 on the Game Boy. Sega also made a Dick Tracy video game for the Sega Genesis and Master System in 1991 as a side-scrolling arcade action adventure game.

Recent events

Media outlets recently reported that there is a legal battle being waged over just who owns the rights to the Dick Tracy character. Warren Beatty has announced plans to make a sequel to his 1990 movie. At the same time, television producers have announced plans for a new Dick Tracy TV series. Both sides claim that they are the legal owners of the rights to Dick Tracy. A lawsuit is pending. Although the comic strip's public profile has diminished since the 1990 Beatty film, it is still run in several newspapers. Apart from that, it is a common allusion in North America for unusual-looking criminals often to be described as resembling the strip's grotesque villains, while the lead character's wrist communicator is a typical example used when the possibility of an actual communication device being developed along the lines of something from science fiction is raised.

Dick Tracy on film and televison


- Dick Tracy (1937, serial, 15 episodes, starring Ralph Byrd)
- Dick Tracy (1937, feature version of the above serial, starring Ralph Byrd)
- Dick Tracy's G-Men (1939, serial, 15 episodes, starring Ralph Byrd)
- Dick Tracy (1945, film starring Morgan Conway)
- Dick Tracy vs. Cueball (1946, film starring Morgan Conway)
- Dick Tracy's Dilemma (1947, film starring Ralph Byrd)
- Dick Tracy Meets Gruesome (1947, film starring Ralph Byrd)
- The Dick Tracy Show (1961, animated television series starring Everett Sloane)
- Dick Tracy (1967, television pilot starring Ray McDonnell)
- Archie's T.V. Funnies, Dick Tracy episode, 1971
- Dick Tracy (1990, film starring Warren Beatty)

Trivia


- In Al Capp's Li'l Abner, there is a comic-in-the-comic detective called Fearless Fosdick, clearly modeled on Dick Tracy.
- Many of the comic characters were based on local citizens of Woodstock, Illinois where Chester Gould wrote the majority of the strip. However, Gould modeled many characters after close associates like his publisher, Joseph Patterson, as Big Frost and even himself, as Pear-Shape.

See also


- List of recurring characters in Dick Tracy

External links


- [http://www.comicspage.com/dicktracy/ Dick Tracy comic strip]
- [http://www.toonopedia.com/tracy.htm Don Markstein's Toonopedia - Dick Tracy]
- [http://www.marklansdown.com/pinbacks/pages/strip-dicktracy.html Dick Tracy Pinbacks]
- [http://www.ivideoblast.com/Member/MemberMain.cfm/u/guest/cid/Movie.html iVideoBlast: Dick Tracy on your iPod] Category:Dick Tracy Tracy, Dick Tracy, Dick Category:Comic strips Category:Dieselpunk

Mandrake the Magician

Mandrake the Magician is a U.S. comic strip created in 1934 by Lee Falk (also creator of The Phantom) and mainly appearing in syndication in newspapers. Falk soon gave the job of drawing the comic strip to artist Phil Davis, whilst continuing to write the storylines. Davis worked on the strip until his death in 1964; Falk then recruited current artist Fred Fredericks. On Falk's death in 1999, Fredericks took over writing the strip as well. 1999. Art by Fred Fredericks.]] Mandrake was an illusionist whose work was based on an impossibly fast hypnotic technique. As the narrator informed us: "Mandrake gestured hypnotically" and the subject or subjects of this hypnosis would suddenly see a cane transformed in a bouquet of flowers or a cat into a tiger. Mandrake also worked against crooks and other bad guys in his spare time. In their case, he would gesture hypnotically and they would see their guns change into snakes or red hot irons. Leon Mandrake, the stage magician, who was known for his top hat, pencil line moustache and scarlet-lined cape, bears a strong resemblance to the comic strip character. In fact, Leon Mandrake had been performing for well over ten years before Lee Falk introduced the comic strip character and today most people would acknowledge the striking resemblance between them. Many diverse sources assert that the comic character was drawn to resemble Leon. Davis did meet Leon Mandrake, they became good friends and corresponded for years afterwards.

Other Characters

Supporting characters

Lothar is Mandrake's best friend and crime-fighting companion. In the comic strip, Mandrake first met Lothar during his travels in Africa. Lothar was then "Prince of the Seven Nations", a mighty federation of jungle tribes, but passed on the chance to become king and instead followed Mandrake on his world travels, fighting crime and villains from all over the world (and the rest of the universe as well). He is often referred to as the strongest man in the world. Perhaps one of the first black crime-fighting heroes ever to appear in comics, Lothar made his first appearance alongside Mandrake in the very first daily strip 1934. Mandrake and Lothar are generally recognized as the world's first interracial team of crime-fighters. In the beginning, Lothar was little more than Mandrake's servant. He spoke poor English, wore a fez, short pants and a leopard skin. His muscles far exceeded his mental abilities. But despite that, Lothar was still recognized early on as Mandrake's equal by the readers. When artist Fred Fredericks took over in 1965 (after original artist Phil Davis had died), Lothar was modernized; he began to speak correct English and his clothing changed, although he still often wears shirts with leopard skin patterns. Narda is (just as Lothar) of royal blood; she is Princess of the European nation Cockaigne (today ruled by her brother Segrid). She made her first appearance in the second Mandrake story. Although they fell in love at first sight they didn't marry until 1997, but then it was an extravagant triple wedding ceremony at Mandrake's home of Xanadu, in Narda's home country Cockaigne, and Mandrake's father Theron's College of Magic (Collegium Magikos) in the Himalayas. Theron is Mandrake's father, although this was a secret to Mandrake for a long time. Mandrake's mother died shortly after she gave birth to Mandrake and his twin brother Derek. Theron is the headmaster of the College of Magic (Collegium Magikos) somewhere in the Himalayas. Theron is hundreds of years old, probably due to the energy of the powerful Mind Crystal he is the guardian of. Hojo is Mandrake's chef at his home of Xanadu. However, he is also the secret Chief of the international crime-fighting organization Inter-Intel, in addition to being a superb martial arts expert. As such, he has used Mandrake's help with many cases. Hojo's assistant at Inter-Intel is Jed. The Police Chief is named Bradley but mostly just called "Chief", and has been aided by Mandrake on several occasions. He created the "S.S.D." (Silly Stuff Dept.) for absurd and unbelievable cases that only Mandrake could solve. He has a son, Chris. Magnon is without a doubt Mandrake's most powerful friend. He is the emperor of a million planets but even he has sometimes needed Mandrake's help. Magnon and his wife Carola have a daughter, Nardraka, who is named after Mandrake and Narda and is their godchild. Lenore is Mandrake's younger half-sister. She is a world-renowned explorer. Karma is Lothar's girlfriend, an African princess, who works as a model.

Villains

The Cobra is Mandrake's most evil and dangerous foe. He made his first appearance in the very first Mandrake story. In a story from 1937, the Cobra was apparently defeated; however, he returned with a vengeance in a story from 1965. This time hiding his face (scarred and disfigured in battle with Mandrake) behind a menacing silver mask. The Cobra's main goal is to get his hands on one of the two powerful Crystal Cubes that are guarded by Mandrake and his father Theron. Mandrake then learned that The Cobra's true identity was Luciphor, Theron's oldest son - thus making The Cobra Mandrake's half-brother. In later years, the Cobra has abandoned his silver mask as his face has been reconstructed through surgery. The Cobra is sometimes accompanied by his assistant Ud. Derek is Mandrake's twin brother, and although similar to his brother in appearance, totally different when it comes to morals and ethics. Derek is only after money and women and gladly uses his magical powers (that almost rival Mandrake's) to achieve his own personal goals. Mandrake has many times tried to remove Derek's knowledge of magic through mental battles, although it has always been only temporary solutions. Derek has a son, Eric (mother unknown) who so far has shown no signs of following in his father's footsteps. The Clay Camel, real name Saki, is a master of disguises. He is known to be able to mimic anyone and change his appearance in seconds. His name comes from the symbol he leaves at the scenes of his crimes, a small camel made of clay. The Brass Monkey, daughter of The Clay Camel with a similar talent for disguises. Aleena the Enchantress is a former friend of Mandrake's from the College of Magic, a much-married spoilt temptress, who now prefers to use her magic powers for her own benefit. This includes trying to win Mandrake's heart, but when that doesn't succeed, she gladly tries to bring him trouble instead. 8 is an old and very powerful crime organization with roots to medieval times. Long thought to be dead and no more than a myth, Mandrake discovered that the organization is very much alive. They are known to often incorporate the number 8 in their crimes or leave the number 8 as a mark. They are organized like an octopus with eight arms (headquarters) spread out all over the world, and one head (the grim and mysterious leader Octon, only shown as a menacing image on a computer screen). Over the years, Mandrake has succeeded in destroying their headquarters one by one. In one of the mandrake stories the Octon of 8 is revealed as Cobra. Ekardnam is Mandrake's "evil twin", who exists on the other side of the mirror from Mandrake. Like his world (where the government is run by the "Private of the Armies", and generals do menial work like running the elevators), Ekardnam is an exact opposite, i.e., evil and treacherous, and uses his "evil eye" powers to work his magic. He was seen when Narda was drawn into the mirror, an adventure that may have been only a bad dream.

Comic books

octopus.]] Mandrake had a prominent role in the Magic Comics and the Big Little Books of the 30s and 40s. Dell Comics published a Mandrake the Magician issue in their comic series with various main characters. The Mandrake issue was #752 and featured original stories by unknown creators. In 1966-67 King Comics published 10 issues of a Mandrake the Magician comic magazine. Most of the stories were remakes of newspaper strip stories, and featured art by Andre LeBlanc, Ray Bailey and others. Mandrake stories also ran as back-up features in other King titles. Italian publisher Fratelli Spada produced a considerable amount of original Mandrake comic book stories in the 60s and 70s. A few of these were even published in the American Mandrake comic book mentioned above. Marvel released a Mandrake mini-series in 1995, written by Mike W. Barr and with painted art by Rob Ortaleza. However, only two of three planned issues were published. Mandrake has also enjoyed great success in comic books all over the world, for example Britain, Australia, Brazil, India, France, Spain, Italy, Germany, Norway, Denmark, Finland and Sweden (although in the case of the Nordic countries, most often as a back up feature in the Phantom comic books).

In other media

Phantom In 1939, Columbia produced a 12-part serial based on the Mandrake strip. It starred Warren Hull as Mandrake and Al Kikume as Lothar. The serial is widely available on DVD. The Mandrake Radio Serial aired over the Mutual radio network from November 11, 1940 until February 06, 1942. Originally it was a three day a week serial, and from 1941 a five day a week serial. Raymond Edward Johnson starred as Mandrake and Juano Hernandez as Lothar. NBC made a pilot for a Mandrake TV series in 1954, but no other episodes were made. Coe Norton starred as Mandrake and Woody Strode as Lothar. An unauthorized Mandrake movie was produced in Turkey in 1967. It was called Mandrake Killing'e karsi, directed by Oksal Pekmezoglu and starred Güven Erte as Mandrake. In 1979 a TV movie based on the strip was made, starring Anthony Herrera as Mandrake and Ji-Tu Cumbuka as Lothar. A musical named Mandrake the Magician and the Enchantress was produced during the late 1970s at the Lenox Arts Festival in Massachusetts. The music was by George Quincy, book by Lee Falk and Thayer Burch, and lyrics by Thayer Burch. In the cartoon Defenders of the Earth, Mandrake the Magician teams up with fellow King Features adventurers Flash Gordon and The Phantom. Mandrake's friend Lothar also has a prominent role. Peter Renaday was the voice of Mandrake and Buster Jones the voice of Lothar. A software company called MandrakeSoft started in France in 1998. Their biggest product was the GNU/Linux distribution Mandrake Linux. In February 2004 MandrakeSoft lost a court case against Hearst Corporation, owners of King Features Syndicate. Hearst contends that MandrakeSoft is infringing upon King Features' trademarked character Mandrake the Magician. The word Mandrake is not unique to the King Features character, and MandrakeSoft appealed the decision. The origin of MandrakeSoft's name however becomes apparent when you consider the Linux-tool Lothar which was developed in a project sponsored by MandrakeSoft. Lothar is also the longtime friend of Mandrake the Magician. The use of a magician's top hat and magic wand in images and logos also show that MandrakeSoft wanted to associate the name with a magician, and not (for example) the mandrake root. Consequently, the company and its product have changed names to Mandriva and Mandriva Linux, respectively.

Reprints


- Dragon Lady Press reprinted a 1937 Mandrake daily story in Classic Adventure Strips #1.
- Pacific Comics Club reprinted two Mandrake daily stories from 1938 Feature Books #18 and #23.
- Nostalgia Press published a hardback book reprinting two 1938 daily stories.
- Pioneer Comics reprinted a large number of Mandrake stories in comic book form.
- Comics Revue has reprinted several Mandrake daily stories.
- JAL Publications has reprinted several Mandrake stories

External links


- [http://www.kingfeatures.com/features/comics/mandrake/about.htm Mandrake the Magician at King Features]
- [http://www.toonopedia.com/mandrake.htm Toonopedia entry]
- [http://pjb.hopto.org/ Xanadu] - fan site about Mandrake the Magician
- [http://www.deepwoods.cjb.net/ The 21st Phantom's Deep Woods] - Phantom fan site that provides access to current and archived online Mandrake comic strips. Category:Comic strips Category:Comics characters Category:Fictional magicians

Category:Publishers

Category:Publishing Category:Media companies

Category:Comics

Comics are an art form using a series of static images in fixed sequence. Comic strips are often printed in newspapers, while comic books are generally published as individual documents. Webcomics are published on the World Wide Web. Category:Cartooning Category:Humor Category:Mass media Category:Art media . ko:분류:만화

Rate of convergence

In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may e.g. even make the difference between needing ten or a million iterations.

The definition of rate of convergence

Suppose that the sequence converges to the number ξ. We say that this sequence converges linearly to ξ, if : \lim_k \frac = \mu \mbox 0 < \mu < 1. \quad\quad (1) The number μ is called the rate of convergence. If (1) holds with μ = 0, then the sequence is said to converge superlinearly. One says that the sequence converges sublinearly if it converges, but (1) does not hold for any μ < 1. The next definition is used to distinguish superlinear rates of convergence. We say that the sequence converges with order q for q > 1 to ξ if : \lim_k \frac = \mu \mbox \mu > 0. \quad\quad (2) In particular, convergence with order 2 is called quadratic convergence, and convergence with order 3 is called cubic convergence.

The extended definition of rate of convergence

The drawback of the above definitions (1) and (2) is that these do not catch some sequences which still converge reasonably fast, but whose "speed" is variable, such as the sequence below. Therefore, the definition of rate of convergence is sometimes extended as follows. Under the new definition, the sequence converges with at least order q if there exists a sequence such that : |x_k - \xi| \le \varepsilon_k \quad\mbox k, and the sequence converges to zero with order q according to the above "simple" definition.

Examples

Consider the following sequences: : a_0 = 1 ,\, a_1 = \frac12 ,\, a_2 = \frac14 ,\, a_3 = \frac18 ,\, a_4 = \frac1 ,\, a_5 = \frac1 ,\, \ldots ,\, a_k = \frac1 ,\, \ldots : b_0 = 1 ,\, b_1 = 1 ,\, b_2 = \frac14 ,\, b_3 = \frac14 ,\, b_4 = \frac1 ,\, b_5 = \frac1 ,\, \ldots ,\, b_k = \frac1 ,\, \ldots : c_0 = 1 ,\, c_1 = \frac12 ,\, c_2 = \frac14 ,\, c_3 = \frac1 ,\, c_4 = \frac1 ,\, \ldots ,\, c_k = \frac1 ,\, \ldots : d_0 = 1 ,\, d_1 = \frac12 ,\, d_2 = \frac13 ,\, d_3 = \frac14 ,\, d_4 = \frac15 ,\, d_5 = \frac16 ,\, \ldots ,\, d_k = \frac1 ,\, \ldots The sequence converges linearly to 0 with rate 1/2. More generally, the sequence Cμk converges linearly with rate μ if |μ| < 1. The sequence also converges linearly to 0 with rate 1/2 under the extended definition, but not under the simple definition. The sequence converges superlinearly. In fact, it is quadratically convergent. Finally, the sequence converges sublinearly.

References

The simple definition is used in
- Michelle Schatzman (2002), Numerical analysis: a mathematical introduction, Clarendon Press, Oxford. ISBN 0-19-850279-6. The extended definition is used in
- Kendell A. Atkinson (1988), An introduction to numerical analysis (2nd ed.), John Wiley and Sons. ISBN 0-471-50023-2.
- Walter Gautschi (1997), Numerical analysis: an introduction, Birkhäuser, Boston.
- Endre Süli and David Mayers (2003), An introduction to numerical analysis, Cambridge University Press. ISBN 0-521-00794-1. Category:Mathematical analysis

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