Plot summary.

Clarissa Dalloway goes around London in the morning, getting ready to host a party that evening. The nice day reminds her of her youth spent in the countryside in Bourton and makes her wonder about her choice of husband; she married the reliable Richard Dalloway instead of the enigmatic and demanding Peter Walsh, and she "had not the option" to be with a close female friend, Sally Seton. Peter reintroduces these conflicts by paying a visit that morning.  Septimus Warren Smith, a First World War veteran suffering from deferred traumatic stress, spends his day in the park with his Italian-born wife Lucrezia, where Peter Walsh observes them. Septimus is visited by frequent and indecipherable hallucinations, mostly concerning his dear friend Evans who died in the war. Later that day, after he is prescribed involuntary commitment to a psychiatric hospital, he commits suicide by jumping out of a window.  Clarissa's party in the evening is a slow success. It is attended by most of the characters she has met throughout the book, including people from her past. She hears about Septimus' suicide at the party and gradually comes to admire this stranger's act, which she considers an effort to preserve the purity of his happiness.  Sir William Bradshaw is a famous psychiatrist to whom Septimus' physician, Dr

Characters.

Holmes, refers Septimus. Bradshaw notes that Septimus has had a complete nervous breakdown and suggests spending time in the country as a cure.  Clarissa Dalloway is the 51-year-old protagonist of the novel. She is Richard's wife and Elizabeth's mother, and, while reminiscing about her past, spends the day organising a party that will be held that night. She is self-conscious about her role in London high society.  Elizabeth Dalloway is Clarissa and Richard's 17-year-old daughter. She is said to look "oriental" and has great composure. Compared to her mother, she takes great pleasure in politics and modern history, hoping to be either a doctor or farmer in the future. She would rather spend time in the country with her father than at her mother's party.  Richard Dalloway is Clarissa's practical, "simple" husband, who feels disconnected from his wife. He is immersed in his work in government.  Miss Doris Kilman, originally "Kiehlman", is Elizabeth's schoolmistress for history and is a born-again Christian. She has a degree in history and during the Great War was dismissed from her teaching job because "Miss Dolby thought she would be happier with people who shared her views about the Germans". She has a German ancestry and wears an unattractive mackintosh coat because she is uninterested in dressing to please others. She dislikes Clarissa intensely but loves to spend time with Elizabeth.  Sally Seton is a love interest of Clarissa's, with whom she shared a kiss, who is now married to Lord Rosseter and has five boys. Sally had a strained relationship with her family and spent substantial time with Clarissa's family in her youth. She once could be described as feisty as well as a youthful ragamuffin, although she has become more conventional with age.  Lucrezia "Rezia" Smith is Septimus' Italian wife. She is burdened by his mental illness and believes she is judged because of it. During most of the novel she is homesick for her family and country, which she left to marry Septimus after the Armistice.  Septimus Warren Smith is a World War I veteran who suffers from "shell shock" and hallucinations of his deceased friend, Evans. Educated and decorated in the war, he is detached from society and believes himself to be unable to feel. He is married to Lucrezia, from whom he has grown distant.  Peter Walsh is an old friend of Clarissa's who has failed at most of his ventures in life. In the past, Clarissa rejected his marriage proposal. Now he has returned to England from India and is one of Clarissa's party guests. He plans to marry Daisy, a married woman in India, and has returned to try to arrange a divorce for his current wife.  Hugh Whitbread is a pompous friend of Clarissa's, who holds an unspecified position in the British Royal household. Like Clarissa, he places great importance on his place in society. Although he believes he is an essential member of the British aristocracy, Lady Bruton, Clarissa, Richard, and Peter find him obnoxious.

Style.

In "Mrs Dalloway", all of the action, aside from the flashbacks, takes place on a day in "the middle of June" of 1923. It is an example of stream of consciousness storytelling: every scene closely tracks the momentary thoughts of a particular character. Woolf blurs the distinction between direct and indirect speech throughout the novel, freely alternating her mode of narration between omniscient description, indirect interior monologue, and soliloquy. The narration follows at least twenty characters in this way, but the bulk of the novel is spent with Clarissa Dalloway and Septimus Smith.  Woolf laid out some of her literary goals with the characters of "Mrs Dalloway" while still working on the novel. A year before its publication, she gave a talk at Cambridge University called "Character in Fiction", revised and retitled later that year as "Mr. Bennett and Mrs. Brown".

Comparisons with Joyce's "Ulysses".

Because of structural and stylistic similarities, "Mrs Dalloway" is commonly thought to be a response to James Joyce's "Ulysses", a text that is often considered one of the greatest novels of the twentieth century (though Woolf herself, writing in 1928, denied any deliberate "method" to the book, saying instead that the structure came about "without any conscious direction"). In her essay "Modern Fiction", Woolf praised "Ulysses", saying of the scene in the cemetery, "on a first reading at any rate, it is difficult not to acclaim a masterpiece." At the same time, Woolf's personal writings throughout her reading of "Ulysses" are abundant in criticisms. While in the initial reading process, she recorded the following response to the aforementioned passages,  I... have been amused, stimulated, charmed interested by the first 2 or 3 chapters—to the end of the Cemetery scene; & then puzzled, bored, irritated, & disillusioned as by a queasy undergraduate scratching his pimples. And Tom, great Tom, thinks this on a par with "War & Peace"! An illiterate, underbred book it seems to me: the book of a self-taught working man, & we all know how distressing they are, how egotistic, insistent, raw, striking, & ultimately nauseating. When one can have cooked flesh, why have the raw? But I think if you are anaemic, as Tom is, there is glory in blood. Being fairly normal myself I am soon ready for the classics again. I may revise this later. I do not compromise my critical sagacity. I plant a stick in the ground to mark page 200  Woolf's distaste for Joyce's work only solidified after she completed reading it. She summed up her thoughts on the work as a whole:  I finished "Ulysses", & think it is a mis-fire. Genius it has I think; but of the inferior water. The book is diffuse. It is brackish. It is pretentious. It is underbred, not only in the obvious sense, but in the literary sense. A first rate writer, I mean, respects writing too much to be tricky; startling; doing stunts. I’m reminded all the time of some callow board schoolboy, say like Henry Lamb, full of wits & powers, but so self-conscious and egotistical that he loses his head, becomes extravagant, mannered, uproarious, ill at ease, makes kindly people feel sorry for him, & stern ones merely annoyed; & one hopes he’ll grow out of it; but as Joyce is 40 this scarcely seems likely. I have not read it carefully; & only once; & it is very obscure; so no doubt I have scamped the virtue of it more than is fair. I feel that myriads of tiny bullets pepper one & spatter one; but one does not get one deadly wound straight in the face—as from Tolstoy, for instance; but it is entirely absurd to compare him with Tolstoy  The Hogarth Press, run by her and her husband Leonard, had to turn down the chance to publish the novel in 1919 because of the obscenity law in England, as well as the practical issues regarding publishing such a substantial text.

Themes.

The novel has two main narrative lines involving two separate characters (Clarissa Dalloway and Septimus Smith); within each narrative there is a particular time and place in the past that the main characters keep returning to in their minds. For Clarissa, the "continuous present" (Gertrude Stein's phrase) of her charmed youth at Bourton keeps intruding into her thoughts on this day in London. For Septimus, the "continuous present" of his time as a soldier during the "Great War" keeps intruding, especially in the form of Evans, his fallen comrade.

Time and secular living.

Time plays an integral role in the theme of faith and doubt in Mrs Dalloway. The overwhelming presence of the passing of time and the impending fate of death for each of the characters is felt throughout the novel. As Big Ben arches over the city of London and rings for each half-hour, characters can’t help but stop and notice the loss of life to time in regular intervals throughout the development of the story. Experiencing the vicious war, the notion of death constantly floats in Septimus' mind as he continues to see his friend Evans talking of such things. A constant stream of consciousness from the characters, especially Clarissa, can serve as a distraction from this passing of time and ultimate march towards death but each character has a constant reminder of the inevitability of these facts. However evident time and death may be throughout the novel, only a day passes over the course of the entire story, as in Joyce's "Ulysses".  Constant connections to memories and overarching ideas made by simple things passing through the character’s mind also demonstrate the meaning in every detail and the appreciation that can be drawn as a result. Clarissa even feels that her job (by throwing her parties) is to offer “the gift” of connections to the inhabitants of London. Herein lies Woolf’s underlying message. Woolf’s writing style crosses the boundaries of the past, present and future, emphasising her idea of time as a constant flow, connected only by some force (or divinity) within each person. There lies an evident contrast between the constant passing of time signalled by Big Ben and the random crossing of time in Woolf’s writing. Although it seems random, it only demonstrates the infinite number of possibilities that the world can offer once connected by the individuality of each person inside.

Mental illness.

Septimus, as the shell-shocked war hero, operates as a pointed criticism of the treatment of mental illness and depression. Woolf criticises medical discourse through Septimus' decline and suicide; his doctors make snap judgments about his condition, talk to him mainly through his wife, and dismiss his urgent confessions before he can make them. Rezia remarks that Septimus "was not ill. Dr Holmes said there was nothing the matter with him."  Woolf goes beyond commenting on the treatment of mental illness. Using the characters of Clarissa and Rezia, she makes the argument that people can only interpret Septimus' shell shock according to their cultural norms. Throughout the course of the novel Clarissa does not meet Septimus. Clarissa's reality is vastly different from that of Septimus; his presence in London is unknown to Clarissa until his death becomes the subject of idle chatter at her party. By never having these characters meet, Woolf is suggesting that mental illness can be contained to the individuals who suffer from it without others, who remain unaffected, ever having to witness it. This allows Woolf to weave her criticism of the treatment of the mentally ill with her larger argument, which is the criticism of society's class structure. Her use of Septimus as the stereotypically traumatised veteran is her way of showing that there were still reminders of the First World War in London in 1923. These ripples affect Mrs. Dalloway and readers spanning generations. Shell shock, or post traumatic stress disorder, is an important addition to the early 20th century canon of post-war British literature.  There are similarities in Septimus' condition to Woolf's struggles with bipolar disorder. Both hallucinate that birds sing in Greek, and Woolf once attempted to throw herself out of a window as Septimus does. Woolf had also been treated for her condition at various asylums, from which her antipathy towards doctors developed. Woolf committed suicide by drowning, sixteen years after the publication of "Mrs Dalloway".  Woolf's original plan for her novel called for Clarissa to kill herself during her party. In this original version, Septimus (whom Woolf called Mrs. Dalloway's "double") did not appear at all.

Existential issues.

When Peter Walsh sees a girl in the street and stalks her for half an hour, he notes that his relationship to the girl was "made up, as one makes up the better part of life." By focusing on characters' thoughts and perceptions, Woolf emphasises the significance of private thoughts on existential crisis rather than concrete events in a person's life. Most of the plot in "Mrs Dalloway" consists of realisations that the characters subjectively make.  Fueled by her bout of ill health, Clarissa Dalloway is emphasised as a woman who appreciates life. Her love of party-throwing comes from a desire to bring people together and create happy moments. Her charm, according to Peter Walsh who loves her, is a sense of "joie de vivre", always summarised by the sentence: "There she was." She interprets Septimus Smith's death as an act of embracing life and her mood remains light, even though she hears about it in the midst of the party.

Feminism.

As a commentary on inter-war society, Clarissa's character highlights the role of women as the proverbial "Angel in the House" and embodies sexual and economic repression and the narcissism of bourgeois women who have never known the hunger and insecurity of working women. She keeps up with and even embraces the social expectations of the wife of a patrician politician, but she is still able to express herself and find distinction in the parties she throws.  Her old friend Sally Seton, whom Clarissa admires dearly, is remembered as a great independent woman – she smoked cigars, once ran down a corridor naked to fetch her sponge-bag, and made bold, unladylike statements to get a reaction from people. When Clarissa meets her in the present day, Sally turns out to be a perfect housewife, having accepted her lot as a rich woman ("Yes, I have ten thousand a year"-whether before the tax was paid, or after, she couldn't remember...), married, and given birth to five sons.

Bisexuality.

Clarissa Dalloway is strongly attracted to Sally Seton at Bourton. Thirty-four years later, Clarissa still considers the kiss they shared to be the happiest moment of her life. She feels about Sally "as men feel," but she does not recognise these feelings as signs of bisexuality.  Similarly, Septimus is haunted by the image of his dear friend Evans. Evans, his commanding officer, is described as being "undemonstrative in the company of women." The narrator describes Septimus and Evans behaving together like "two dogs playing on a hearth-rug" who, inseparable, "had to be together, share with each other, fight with each other, quarrel with each other..." Jean E. Kennard notes that the word "share" could easily be read in a Forsteran manner, perhaps as in Forster's "Maurice", which shows the word's use in this period to describe homosexual relations. Kennard is one to note Septimus' "increasing revulsion at the idea of heterosexual sex," abstaining from sex with Rezia and feeling that "the business of copulation was filth to him before the end."

Film adaptation.

Dutch film director Marleen Gorris made a film version of "Mrs Dalloway" in 1997. It was adapted from Woolf's novel by British actress Eileen Atkins and starred Vanessa Redgrave in the title role. The cast included Natascha McElhone, Lena Headey, Rupert Graves, Michael Kitchen, Alan Cox, Sarah Badel, and Katie Carr. A related 2002 film, based on Michael Cunningham's novel of the same name, is "The Hours", starring Meryl Streep as New York editor Clarissa, Julianne Moore as Los Angeles housewife Laura, and Nicole Kidman as Virginia Woolf. "The Hours" is about a single day in the lives of three women of different generations who are affected by "Mrs Dalloway": Woolf is writing it, Laura is reading it, and Clarissa is living it out.

Other appearances.

Mrs Dalloway also appears in Virginia Woolf's first novel, "The Voyage Out", as well as five of her short stories, in which she hosts dinner parties to which the main subject of the narrative is invited:  The stories (except for "The Introduction") all appear in the 1944 collection "A Haunted House and Other Short Stories", and in the 1973 collection "Mrs Dalloway's Party".

Mrs Dalloway (published on 14 May 1925) is a novel by Virginia Woolf that details a day in the life of Clarissa Dalloway, a fictional woman in post–war. It is one of Woolf's best-known novels.

Paní Dallowayová ("Mrs Dalloway") je impresionistický román britské autorky Virginie Woolfové (1882–1941), na němž pracovala v letech 1923–1925. Je považován za její vrcholné a rovněž nejznámější dílo, které inspirovalo také amerického spisovatele Michaela Cunninghama k napsání světoznámého románu "Hodiny" (jak měl znít i původní název "Paní Dallowayové"), podle kterého byl natočen i stejnojmenný oscarový film. V češtině vyšla kniha v překladu Vlasty Dvořáčkové a později Kateřiny Hilské.


Introduction.

The FME(C)A is a design tool used to systematically analyze postulated component failures and identify the resultant effects on system operations. The analysis is sometimes characterized as consisting of two sub-analyses, the first being the failure modes and effects analysis (FMEA), and the second, the criticality analysis (CA). Successful development of an FMEA requires that the analyst include all significant failure modes for each contributing element or part in the system. FMEAs can be performed at the system, subsystem, assembly, subassembly or part level. The FMECA should be a living document during development of a hardware design. It should be scheduled and completed concurrently with the design. If completed in a timely manner, the FMECA can help guide design decisions. The usefulness of the FMECA as a design tool and in the decision-making process is dependent on the effectiveness and timeliness with which design problems are identified. Timeliness is probably the most important consideration. In the extreme case, the FMECA would be of little value to the design decision process if the analysis is performed after the hardware is built. While the FMECA identifies all part failure modes, its primary benefit is the early identification of all critical and catastrophic subsystem or system failure modes so they can be eliminated or minimized through design modification at the earliest point in the development effort; therefore, the FMECA should be performed at the system level as soon as preliminary design information is available and extended to the lower levels as the detail design progresses.  Remark: For more complete scenario modelling another type of Reliability analysis may be considered, for example fault tree analysis (FTA); a "deductive" (backward logic) failure analysis that may handle multiple failures within the item and/or external to the item including maintenance and logistics. It starts at higher functional / system level. An FTA may use the basic failure mode FMEA records or an effect summary as one of its inputs (the basic events). Interface hazard analysis, human error analysis and others may be added for completion in scenario modelling.

Functional Failure mode and effects analysis.

The analysis should always be started by listing the functions that the design needs to fulfill. Functions are the starting point of a well done FMEA, and using functions as baseline provides the best yield of an FMEA. After all, a design is only one possible solution to perform functions that need to be fulfilled. This way an FMEA can be done on concept designs as well as detail designs, on hardware as well as software, and no matter how complex the design.  When performing an FMECA, interfacing hardware (or software) is first considered to be operating within specification. After that it can be extended by consequently using one of the 5 possible failure modes of one function of the interfacing hardware as a cause of failure for the design element under review. This gives the opportunity to make the design robust for function failure elsewhere in the system.  In addition, each part failure postulated is considered to be the only failure in the system (i.e., it is a single failure analysis). In addition to the FMEAs done on systems to evaluate the impact lower level failures have on system operation, several other FMEAs are done. Special attention is paid to interfaces between systems and in fact at all functional interfaces. The purpose of these FMEAs is to assure that irreversible physical and/or functional damage is not propagated across the interface as a result of failures in one of the interfacing units. These analyses are done to the piece part level for the circuits that directly interface with the other units. The FMEA can be accomplished without a CA, but a CA requires that the FMEA has previously identified system level critical failures. When both steps are done, the total process is called an FMECA.

Ground rules.

The ground rules of each FMEA include a set of project selected procedures; the assumptions on which the analysis is based; the hardware that has been included and excluded from the analysis and the rationale for the exclusions. The ground rules also describe the indenture level of the analysis (i.e. the level in the hierarchy of the part to the sub-system, sub-system to the system, etc.), the basic hardware status, and the criteria for system and mission success. Every effort should be made to define all ground rules before the FMEA begins; however, the ground rules may be expanded and clarified as the analysis proceeds. A typical set of ground rules (assumptions) follows:

Benefits.

Major benefits derived from a properly implemented FMECA effort are as follows:  From the above list, early identifications of SFPS, input to the troubleshooting procedure and locating of performance monitoring / fault detection devices are probably the most important benefits of the FMECA. In addition, the FMECA procedures are straightforward and allow orderly evaluation of the design.

History.

Procedures for conducting FMECA were described in US Armed Forces Military Procedures document MIL-P-1629 (1949); revised in 1980 as MIL-STD-1629A. By the early 1960s, contractors for the U.S. National Aeronautics and Space Administration (NASA) were using variations of FMECA or FMEA under a variety of names. NASA programs using FMEA variants included Apollo, Viking, Voyager, Magellan, Galileo, and Skylab. The civil aviation industry was an early adopter of FMEA, with the Society for Automotive Engineers (SAE, an organization covering aviation and other transportation beyond just automotive, despite its name) publishing ARP926 in 1967. After two revisions, Aerospace Recommended Practice ARP926 has been replaced by ARP4761, which is now broadly used in civil aviation.  During the 1970s, use of FMEA and related techniques spread to other industries. In 1971 NASA prepared a report for the U.S. Geological Survey recommending the use of FMEA in assessment of offshore petroleum exploration. A 1973 U.S. Environmental Protection Agency report described the application of FMEA to wastewater treatment plants. FMEA as application for HACCP on the Apollo Space Program moved into the food industry in general.  The automotive industry began to use FMEA by the mid 1970s. The Ford Motor Company introduced FMEA to the automotive industry for safety and regulatory consideration after the Pinto affair. Ford applied the same approach to processes (PFMEA) to consider potential process induced failures prior to launching production. In 1993 the Automotive Industry Action Group (AIAG) first published an FMEA standard for the automotive industry. It is now in its fourth edition. The SAE first published related standard J1739 in 1994. This standard is also now in its fourth edition.  Although initially developed by the military, FMEA methodology is now extensively used in a variety of industries including semiconductor processing, food service, plastics, software, and healthcare. Toyota has taken this one step further with its Design Review Based on Failure Mode (DRBFM) approach. The method is now supported by the American Society for Quality which provides detailed guides on applying the method. The standard Failure Modes and Effects Analysis (FMEA) and Failure Modes, Effects and Criticality Analysis (FMECA) procedures identify the product failure mechanisms, but may not model them without specialized software. This limits their applicability to provide a meaningful input to critical procedures such as virtual qualification, root cause analysis, accelerated test programs, and to remaining life assessment. To overcome the shortcomings of FMEA and FMECA a Failure Modes, Mechanisms and Effect Analysis (FMMEA) has often been used.

Basic terms.

The following covers some basic FMEA terminology.

Example of FMEA worksheet.

Probability (P).

It is necessary to look at the cause of a failure mode and the likelihood of occurrence. This can be done by analysis, calculations / FEM, looking at similar items or processes and the failure modes that have been documented for them in the past. A failure cause is looked upon as a design weakness. All the potential causes for a failure mode should be identified and documented. This should be in technical terms. Examples of causes are: Human errors in handling, Manufacturing induced faults, Fatigue, Creep, Abrasive wear, erroneous algorithms, excessive voltage or improper operating conditions or use (depending on the used ground rules). A failure mode may given a "Probability Ranking" with a defined number of levels.  For a piece part FMEA, quantitative probability may be calculated from the results of a reliability prediction analysis and the failure mode ratios from a failure mode distribution catalog, such as RAC FMD-97. This method allows a quantitative FTA to use the FMEA results to verify that undesired events meet acceptable levels of risk.

Severity (S).

Determine the Severity for the worst-case scenario adverse end effect (state). It is convenient to write these effects down in terms of what the user might see or experience in terms of functional failures. Examples of these end effects are: full loss of function x, degraded performance, functions in reversed mode, too late functioning, erratic functioning, etc. Each end effect is given a Severity number (S) from, say, I (no effect) to V (catastrophic), based on cost and/or loss of life or quality of life. These numbers prioritize the failure modes (together with probability and detectability). Below a typical classification is given. Other classifications are possible. See also hazard analysis.

Detection (D).

The means or method by which a failure is detected, isolated by operator and/or maintainer and the time it may take. This is important for maintainability control (availability of the system) and it is especially important for multiple failure scenarios. This may involve dormant failure "modes" (e.g. No direct system effect, while a redundant system / item automatically takes over or when the failure only is problematic during specific mission or system states) or latent failures (e.g. deterioration failure "mechanisms", like a metal growing crack, but not a critical length). It should be made clear how the failure mode or cause can be discovered by an operator under normal system operation or if it can be discovered by the maintenance crew by some diagnostic action or automatic built in system test. A dormancy and/or latency period may be entered.

Dormancy or Latency Period.

The average time that a failure mode may be undetected may be entered if known. For example:

Indication.

If the undetected failure allows the system to remain in a "safe" / working state, a second failure situation should be explored to determine whether or not an indication will be evident to all "operators" and what corrective action they may or should take.  Indications to the operator should be described as follows:  PERFORM DETECTION COVERAGE ANALYSIS FOR TEST PROCESSES AND MONITORING (From ARP4761 Standard):  This type of analysis is useful to determine how effective various test processes are at the detection of latent and dormant faults. The method used to accomplish this involves an examination of the applicable failure modes to determine whether or not their effects are detected, and to determine the percentage of failure rate applicable to the failure modes which are detected. The possibility that the detection means may itself fail latently should be accounted for in the coverage analysis as a limiting factor (i.e., coverage cannot be more reliable than the detection means availability). Inclusion of the detection coverage in the FMEA can lead to each individual failure that would have been one effect category now being a separate effect category due to the detection coverage possibilities. Another way to include detection coverage is for the FTA to conservatively assume that no holes in coverage due to latent failure in the detection method affect detection of all failures assigned to the failure effect category of concern. The FMEA can be revised if necessary for those cases where this conservative assumption does not allow the top event probability requirements to be met.  After these three basic steps the Risk level may be provided.

Risk level (P×S) and (D).

Risk is the combination of End Effect Probability And Severity where probability and severity includes the effect on non-detectability (dormancy time). This may influence the end effect probability of failure or the worst case effect Severity. The exact calculation may not be easy in all cases, such as those where multiple scenarios (with multiple events) are possible and detectability / dormancy plays a crucial role (as for redundant systems). In that case Fault Tree Analysis and/or Event Trees may be needed to determine exact probability and risk levels.  Preliminary Risk levels can be selected based on a Risk Matrix like shown below, based on Mil. Std. 882. The higher the Risk level, the more justification and mitigation is needed to provide evidence and lower the risk to an acceptable level. High risk should be indicated to higher level management, who are responsible for final decision-making.

Timing.

The FMEA should be updated whenever:

Limitations.

While FMEA identifies important hazards in a system, its results may not be comprehensive and the approach has limitations. In the healthcare context, FMEA and other risk assessment methods, including SWIFT (Structured What If Technique) and retrospective approaches, have been found to have limited validity when used in isolation. Challenges around scoping and organisational boundaries appear to be a major factor in this lack of validity.  If used as a top-down tool, FMEA may only identify major failure modes in a system. Fault tree analysis (FTA) is better suited for "top-down" analysis. When used as a "bottom-up" tool FMEA can augment or complement FTA and identify many more causes and failure modes resulting in top-level symptoms. It is not able to discover complex failure modes involving multiple failures within a subsystem, or to report expected failure intervals of particular failure modes up to the upper level subsystem or system.  Additionally, the multiplication of the severity, occurrence and detection rankings may result in rank reversals, where a less serious failure mode receives a higher RPN than a more serious failure mode. The reason for this is that the rankings are ordinal scale numbers, and multiplication is not defined for ordinal numbers. The ordinal rankings only say that one ranking is better or worse than another, but not by how much. For instance, a ranking of "2" may not be twice as severe as a ranking of "1", or an "8" may not be twice as severe as a "4", but multiplication treats them as though they are. See Level of measurement for further discussion. Various solutions to this problems have been proposed, e.g., the use of fuzzy logic as an alternative to classic RPN model.  The FMEA worksheet is hard to produce, hard to understand and read, as well as hard to maintain. The use of neural network techniques to cluster and visualise failure modes were suggested starting from the 2010. An alternative approach is to combine the traditional FMEA table with set of bow-tie diagrams. The diagrams provide a visualisation of the chains of cause and effect, while the FMEA table provides the detailed information about specific events.

Failure mode and effects analysis (FMEA; often written with "failure modes" in plural) is the process of reviewing as many components, assemblies, and subsystems as possible to identify potential failure modes in a system and their causes and effects. For each component, the failure modes and their resulting effects on the rest of the system are recorded in a specific FMEA worksheet. There are numerous variations of such worksheets. An FMEA can be a qualitative analysis, but may be put on a quantitative basis when mathematical failure rate models are combined with a statistical failure mode ratio database. It was one of the first highly structured, systematic techniques for failure analysis. It was developed by reliability engineers in the late 1950s to study problems that might arise from malfunctions of military systems. An FMEA is often the first step of a system reliability study.

FMEA ("Failure Mode and Effects Analysis", analýza možného výskytu a vlivu vad) je analytická metoda, jejímž cílem je identifikovat místa možného vzniku vad ve výrobě.


Life.

Bavo was born near Liège, to a Frankish noble family that gave him the name Allowin. His father was Pippin of Landen, the Mayor of the Palace of Austrasia, and his mother Itta of Metz.  A wild, young aristocrat of Brabant, he contracted a beneficial marriage, and had a daughter. He was a soldier who led an undisciplined and disorderly life. Shortly after the death of his wife, Bavo decided to reform after hearing a sermon preached by Saint Amand on the emptiness of material things. On returning to his house he distributed his wealth to the poor, and then received the tonsure from Amand.  For some time thereafter, Bavo joined Amand in the latter's missionary travels throughout France and Flanders. On one occasion, Bavo met a man whom he had sold into slavery years before. Wishing to atone for his earlier deed, Bavo had the man lead him by a chain to the town jail. Bavo built an abbey on his grounds and became a monk. He distributed his belongings to the poor and lived as a recluse, first in a hollow tree and later in a cell in the forest by the Abbey.  He died at the Abbey in Ghent, in what is today Belgium.

Veneration.

Bavo is the patron saint of Ghent, Zellik, and Lauwe in Belgium, and Haarlem in the Netherlands. His feast in the Catholic Church and the Orthodox Church is October 1.  He is most often shown in Christian art as a knight with a sword and falcon. The most popular scene is the moment of his conversion, which has many stories attached to it. Because he is so often shown with a falcon, he came to be considered the patron saint of falconry. In medieval Ghent, taxes were paid on Bavo's feast day, and it is for this reason he is often shown holding a purse or money bag.  According to Rodulfus Glaber, the city of Bamberg is named after him, with Bamberg meaning "Mount of Bavo".

Legacy.

Several churches are dedicated to him, including:  His picture is also part of the Coat of Arms of the Antwerp suburb Wilrijk.  Rembrandt painted a "Saint Bavo", dated between 1662 and 1665.

Saint Bavo of Ghent (also known as Bavon, Allowin, Bavonius, and Baaf) (622–659) was a Roman Catholic and Eastern Orthodox saint. He was the son of Pepin of Landen and the brother of saints Begga and Gertrude of Nivelles.

Svatý Bavo, známý též jako "Allowin, Bavonius, Baaf" († kolem roku 655), je světec 7. století uctívaný ve Flandrech a především v Gentu. Dochovaly se čtyři texty o jeho životě, většinou legendy; nejstarší pochází patrně z doby "opata Eginarda", zemřelého v roce 844.


Author's note.

I had almost completed writing Ranger’s Apprentice Book 6: The Siege of Macindaw when I realized I had neglected an important period in Will and Halt’s lives. Books 1-4 deal with Will’s early years as an apprentice. Books 5 and 6 describe his first mission as a graduate Ranger. But there was a gap between them. I realized that as Will approached the moment when he would become a Ranger in his own right, he would be troubled by the worry that he wasn’t ready for the responsibility that this would entail. In addition, I wondered about Halt. As a graduate Ranger, Will would move on to his own fief. How would Halt deal with the loss of his cheerful, lively influence—one that had become so much a part of his day-to-day existence?  These questions were too important for me to ignore. So this book goes back a little in time, to the period before Books 5 and 6. It’s set a few months prior to the moment when Will is due to graduate. Halt is making his own plans to fill the empty space that Will’s imminent reassignment will leave in his life. Will is beginning to worry that he’s not ready to move away from the comforting shelter of Halt’s presence. Other characters, too, are chafing at the changes that have taken place in their lives. Principal among these is Erak, Oberjarl of the Skandians. And it’s Erak’s restlessness and dissatisfaction that precipitate the chain of events described in this book.  -John Flanagan

Plot summary.

Skandian Oberjarl Erak Starfollower becomes tired of paperwork. He decides to go on one last raid to the desert country of Arrida. His raiding party walks into an ambush and is captured; Erak's crew are eventually released in order to obtain a ransom for Erak, who is left behind. Meanwhile, at Castle Redmont, senior Ranger Halt and diplomat Lady Pauline are getting married. During the wedding after-party, Svengal, Erak's first mate, appears. In a small meeting with Ranger Will and the knight Horace, he reveals Erak's kidnapping, and also tells them that Erak thinks he was betrayed.  Princess Cassandra begs her father, King Duncan to supply the money for Erak's release. Duncan agrees, but is unable to go himself as he is in talks with the Hibernian kings. Cassandra eventually volunteers herself, much to Duncan's chagrin. Will, Cassandra, Halt, Svengal, Horace, the ranger Gilan as well as thirty of Erak's men go to deliver the ransom. Once in Arrida, they learn from Arridi leader Seley el'then that Erak had been sent to Mararoc, a fort in the desert. The party, guided by Seley el'then and his men, head to Mararoc.  During the journey, a sandstorm causes Will to lose his horse Tug. In an attempt to find him, Will gets lost and loses consciousness. A group of nomads called the Bedullin find and restores him, revealing that they have also found Tug. Will manages to reclaim Tug after winning a riding race against Hassan, who had found Tug and claimed him as his own.  Meanwhile, the others continue their journey to Mararoc. On the way they discover the corpses of the people escorting Erak. Seley el'then deduces that they were attacked by the Tualaghi, a nomadic tribe. Gilan scouts and discovers the Tualaghis, however, the Tualaghi also discover the Arridi party. The Tualaghis attack, though the Arridi group wins the first battle. However, the leader of the Tualaghi, Yusal, negotiates with Seley el'then, who is forced to surrender as their group is running out of water and cannot fight a sustained battle against Yusal. A Bedullin scout relays this information to the Bedullin, and the nomads offer to help rescue Will's friends. On the way, they find and rescue Seley el'then's soldiers, who had been betrayed and left to die by Yusal after surrendering.  The Tualaghis come into a desert town and locks the party in a storehouse where they had also put Erak. Toshak, a Skandian, reveals himself and boasts of betraying Erak. Meanwhile, Will, the Bedullins, and the Arridi devise a plan to attack the town. Erak and the other prisoners are scheduled for execution. When the execution were scheduled they built a platform for the execution. As Halt is pulled forward to be executed, Will shoots the executioner. Mayhem ensues, with Arridi and Bedullins mixed in with the crowd revealing themselves and fighting the Tualaghi. The captured prisoners join in the fight, and turn the tide against the Tualaghi. Erak kills Toshak, while Cassandra wounds Yusal, who is captured.

Erak's Ransom is the seventh novel in the continuing "Ranger's Apprentice" series by Australian author John Flanagan. The book was released in Australia on 1 November 2007 and in the United States on 5 January 2010. It is set between book four in the series, "Oakleaf Bearers" and book five, "The Sorcerer of the North".

Výkupné za Eraka (v anglickém originále "Erak’s Ransom") je pátá kniha ze souboru Hraničářův učeň od australského spisovatele Johna Flanagana. V australské i americké verzi je tato kniha umístěna na místo sedmé po knihách Čaroděj na severu a Obléhání Macindawu. Příběhově se odehrává mezi 4. dílem (Nositelé dubového listu) a 6. dílem (Čaroděj na severu).


Technical description and variation.

The wingspan is 28–33 mm. Forewing yellow, thickly freckled with orange; veins finely ferruginous; lines ferruginous brown, the median thick; all running more or less parallel to each other and to termen; the inner oblique outwards from costa to subcostal vein and again from vein 1 to inner margin where it touches the median; stigmata of the ground colour, with brown rings; the orbicular large, round; both with brown centres; submarginal line formed of disconnected pale lunules edged inwardly with darker; hindwing pale yellow;- "aurantiago" Tutt is a darker form, deeper orange, and sometimes darkened by grey dusting, occurring in Britain: "incolorata" equally meriting a distinctive name is a rare form, ab. "incolorata" ab. nov. [Warren] in which the ground colour is pure pale ochreous, without any orange freckling, the veins and lines faintly brownish, the stigmata all but obsolete; the fringe pale; hindwing white; I have seen only 1 example, a female, certainly British, but without exact locality; the form "subflava" Ev., from Denmark, the Baltic, St. Fetersburg, and the Ural Mts.; recorded also from Asia Minor, has quite a different appearance, the 3 lines, inner, outer, and submarginal being accompanied by brown bands of uniform width, the stigmata marked with brown, and the hindwing with brown veins and terminal border.

Biology.

The moth flies from August to October depending on the location.  Larva grey brown with a pink tinge; dorsal and subdorsal lines whitish; between them a black spot surrounded with white tubercles on each segment; spiracular stripe yellowish white, broad. The larvae feed on the leaf-buds and later the leaves of "Tilia" species, including "Tilia cordata" and "Tilia platyphylla".

Tiliacea citrago, the orange sallow, is a species of moth of the family Noctuidae. It is found in Europe as far east as the Caucasus Mountains and the Urals.

Zlatokřídlec lipový ("Tiliacea citrago") je noční motýl z čeledi můrovitých, který dosahuje rozpětí křídel 28 až 34 mm. Obývá západopalearktickou oblast, což odpovídá téměř celé Evropě vyjma Iberského poloostrova, jižní Itálie, Řecka a větší části Skandinávie.


History.

The device appears to have been invented in the city of Languedoc of southern France one year before its pro-ported invention in England, where it was destroyed by state cloth inspectors of the rent-seeking Ancien Regime.

Operation.

In a typical frame loom, as used previous to the invention of the flying shuttle, the operator sat with the newly woven cloth before him or her, using treadles or some other mechanism to raise and lower the heddles, which opened the shed in the warp threads. The operator then had to reach forward while holding the shuttle in one hand and pass this through the shed; the shuttle carried a bobbin for the weft. The shuttle then had to be caught in the other hand, the shed closed, and the beater pulled forward to push the weft into place. This action (called a "pick") required regularly bending forward over the fabric; more importantly, the coordination between the throwing and catching of the shuttle required multiple operators if the width of the fabric exceeded that which could be reasonably reached across (typically or less).  The flying shuttle employs a board, called the "race," which runs, side to side, along the front of the beater, forming a track on which the shuttle runs. The lower threads of the shed rest on the track and the shuttle slides over them. At each end of the race, there is a box which catches the shuttle at the end of its journey, and which contains a mechanism for propelling the shuttle on its return trip. The shuttle itself has some subtle differences from the older form. The ends of the shuttle are bullet-shaped and metal-capped, and the shuttle generally has rollers to reduce friction. The weft thread is made to exit from the end rather than the side, and the thread is stored on a pirn (a long, conical, one-ended, non-turning bobbin) to allow it to feed more easily. Finally, the flying shuttle is generally somewhat heavier, so as to have sufficient momentum to carry it all the way through the shed.

Social effects.

The increase in production due to the flying shuttle exceeded the capacity of the spinning industry of the day, and prompted development of powered spinning machines, beginning with the spinning jenny and the waterframe, and culminating in the spinning mule, which could produce strong, fine thread in the quantities needed. These innovations transformed the textile industry in Great Britain. All were attacked as threats to the livelihood of spinners and weavers, and Kay's patent was largely ignored. It is often incorrectly written that Kay was attacked and fled to France, but in fact he simply moved there to attempt to rent out his looms, a business model that had failed him in England.  The flying shuttle produced a new source of injuries to the weaving process; if deflected from its path, it could be shot clear of the machine, potentially striking workers. Turn-of-the-century injury reports abound with instances in which eyes were lost or other injuries sustained and, in several instances (for example, an extended exchange in 1901), the British House of Commons was moved to take up the issue of installing guards and other contrivances to reduce these injuries.

Obsolescence.

The flying shuttle dominated commercial weaving through the middle of the twentieth century. By that time, other systems had begun to supplant it. The heavy shuttle was noisy and energy-inefficient (since the energy used to throw it was largely lost in the catching); also, its inertia limited the speed of the loom. Projectile and rapier looms eliminated the need to take the bobbin/pirn of thread through the shed; later, air- and water-jet looms reduced the weight of moving parts further. Flying shuttle looms are still used for some purposes, and old models remain in use.

The flying shuttle was one of the key developments in the industrialization of weaving during the early Industrial Revolution. It allowed a single weaver to weave much wider fabrics, and it could be mechanized, allowing for automatic machine looms. The flying shuttle, which was patented by John Kay (1704–c. 1779) in 1733, greatly sped up the previous hand process and halved the labour force. Where a broad-cloth loom previously required a weaver on each side, it could now be worked by a single operator. Until this point the textile industry had required four spinners to service one weaver. Kay's innovation, in wide use by the 1750s, greatly increased this disparity.

Létající člunek byl jedním z klíčových momentů ve vývoji tkaní, který pomohl v rozvoji průmyslové revoluce. Byl patentován Johnem Kayem (1704–1764) v roce 1733. Byl ovládán prohozním zařízením k jehož obsluze stačil jediný člověk. Až do vynálezu létajícího člunku tkalci prohazovali útek navinutý na cívce v primitivním člunku ručně a tak byla šířka tkaniny limitována délkou jejich rukou.


Life and career.

When Led Zeppelin was formed in 1968, Deborah was six years old. While living on the family's Old Hyde estate, Deborah and John's son Jason started playing and recording their own songs. At the age of 17, at Robert Plant's house in a nearby village, she recorded her first demos. With some encouragement and advice from mentor Plant, Bonham sent out the songs anonymously and soon landed her first record deal with the major European label Carerre Records in 1985, for whom she recorded the critically acclaimed album "For You and the Moon". The album reached the top 5 in the "NME" chart, was DJ Simon Bates record of the week on BBC Radio 1 and sold well in Europe, being voted "Record of the Year" by Musik Mart magazine, in Germany.  A period of recording for the Japanese Sam Corporation followed, with press and radio promotion including interviews for music publication "Burrn!" and J-Wave FM Radio. After this, English label RPM records released the CD single, "Perfect World" in 1995. Bonham spent 1996 organising a new band, together with new songs for an album and playing a series of shows such as headlining the Belfast Festival and a sell out show at the Bottom Line, in Shepherd's Bush. The beginning of 1997 saw the first of many trips to the United States. She performed live on Los Angeles's Rockline syndicated radio show to three million listeners throughout North America. This was followed by a special appearance with Jason Bonham and his band at the Whisky a Go Go in Los Angeles. Her band also opened for Jason Bonham Band's shows in Los Angeles House of Blues with guests Slash and Terry Reid, and subsequent other House of Blues appearances. Bonham also played at festivals including Glastonbury and Fairport's Cropredy Convention.  In 2004, she recorded her second album on the Track label and toured in support of it. Neither of these first two albums were released in the United States. In June 2008, her third album "Duchess" was issued on ATCO by Rhino Records and released in the United States.  Bonham has also toured and performed with Van Halen, Alannah Myles, Tim Rose, Uli Jon Roth, Humble Pie, Donovan, Lonnie Donegan, Jools Holland, Foreigner and Paul Rodgers.  Side One Side Two

Deborah Bonham (born 7 February 1962) is an English rock and blues vocalist and the sister of John Bonham, the late drummer for the band Led Zeppelin. Born in Redditch, Worcestershire, England, she lived with her father in The Old Hyde farm, Cutnall Green, Worcestershire (the location where John Bonham's fantasy sequence was filmed for "The Song Remains the Same"). While living there, she started playing and recording music with her nephew Jason Bonham who has played drums on her two most recent studio releases.

Deborah Bonham (* 7. února 1962, Redditch, Worcestershire, Anglie) je anglická rocková a bluesová zpěvačka, sestra Johna Bonhama, který se proslavil jako bubeník legendární skupiny Led Zeppelin.


History.

On 7 November 1856, Dr. Joseph Girard and Urs Schild, a 28-year-old school teacher, founded the ébauche factory "Dr. Girard & Schild“. Eight years later the enterprise set out the mutual rights and obligations of employees and employers in writing. Urs Schild became a National Councilor in 1882. Six years later, he died at the age of 58. After the death of Urs Schild, his son Max took over the company and soon made a business trip to the United States. He returned to introduce machinery to the craftsmen. Max Schild was ahead of his time and his ideas were unpopular. Discouraged, he left the company and handed power over to his brother Theodore, with whom Eterna entered a prosperous 20th century.  In the 1900s, wristwatches were just starting to become fashionable. Schild Fréres, as the company was then known, started to produce ladies’ wristwatches adapted from small pocket watches. In 1905, the company changed its name to Eterna. The company continued to be at the leading edge of watch development, and in 1908 it patented the first alarm wristwatch. The watch went into production in 1914 and was launched at the Swiss National Exhibition in Bern that year.  By 1932, Eterna had set up a subsidiary company, ETA SA, to make movements for itself and other Swiss watch companies. This same year, Theodore retired and handed over the control of the company to his nephew Rudolf Schild. Theodore remained on the board of directors after retirement until his death in 1950.  Eterna produced many innovations in their history: the smallest production wristwatch with a Baguette movement in 1930, an eight-day alarm watch in the 1930s and their first automatic watch in 1938.  In 1948, Eterna advanced self-winding watch technology with the development of the Eterna-matic automatic movement. The use of five strategically placed ball bearings made the movement very efficient and significantly reduced friction and resistance on the oscillating weight that wound the mainspring. This reduced the wear and tear on internal parts, increasing the watch’s accuracy and useful life. The new watch became popular soon, is probably the most famous Eterna watch and may be one of Eterna’s greatest designs. Its popularity was such that in 1948 Eterna adopted the image of five balls as its corporate logo.  After 1982, Eterna was sold several times. By 1995, it was owned by FAP Beteiligungs GmbH. In 1999, Eterna produced a range of watches marketed under the Porsche Design label. Porsche Design sold Eterna to International Volant Ltd, a subsidiary of China Haidian on 30 June 2012. China Haidian was renamed in 2014 to Citychamp Watch & Jewellery Group Limited.  In the 2000s Eterna resumed manufacturing watches and creating the Calibre 6036 in 2004. The ultra-thin automatic 3030 was produced for the 150th anniversary in 2006. In 2009, a half-century after the Eterna-Matic, the "Spherodrive" once again highlighted the significance of tiny ball-bearings in mechanical watchmaking. The company remains active in wristwatch design.

Eterna is a Swiss luxury watch company founded in Grenchen, Canton Solothurn on 7 November 1856 by Josef Girard and Urs Schild. The company is now owned by Hong Kong-based Citychamp Watch & Jewellery Group Limited, an investment holding company formerly known as China Haidian Holdings.

Hodinářská společnost Eterna SA byla založena 7. listopadu 1856 ve Švýcarsku Dr. Josefem Girandem a Ursem Shildem jako společnost "Rohwerke Girard & Schild" v Grenchen, kanton Solothurn. Společnost se specializovala na výrobu kapesních hodinek s alarmem. V roce 1906 byla společnost přejmenována pod vedením U. Schildena na společnost Eterna. Tato společnost působila samostatně až do roku 1908, kdy sfůzovala s největším výrobcem hodinek ve Švýcarsku, společností ETA SA. V roce 1995 byla značka Eterna koupena německou společností Porsche Design.


History.

In 1904, after years of successful Mississippi River-based lumber and mill operations with Frederick Denkmann and others, Frederick Weyerhäuser moved west to fresh timber areas and founded the Weyerhäuser Timber Company. Fifteen partners and 900,000 acres (3,600 km2) of Washington timberland were involved in the founding, and the land was purchased from James J. Hill of the Great Northern Railway. In 1929, the company built what was then the world's largest sawmill in Longview, Washington. Weyerhäuser's pulp mill in Longview, which began production in 1931, sustained the company financially during the Great Depression. In 1959, the company eliminated the word "Timber" from its name to better reflect its operations. In 1965, Weyerhäuser built its first bleached kraft pulp mill in Canada. Weyerhäuser implemented its High Yield Forestry Plan in 1967 which drew upon 30 years of forestry research and field experience. It called for the planting of seedlings within one year of a harvest, soil fertilization, thinning, rehabilitation of brushlands, and, eventually, genetic improvement of trees. In 1975 the company bought the 3,200 acres of land of the Northwest Landing and developed the town of DuPont using a New Urbanism model.  Weyerhäuser consolidated its core businesses in the late 1990s and ended its services in mortgage banking, personal care products, financial services, and information systems consulting. Weyerhäuser also expanded into South America, Australia, and Asia. In 1999, Weyerhäuser purchased MacMillan Bloedel Limited, a large Canadian forestry company. Then in 2002 after a protracted hostile buyout, the company acquired Willamette Industries, Inc. of Portland, Oregon. On August 23, 2006, Weyerhäuser announced a deal which spun off its fine paper business to be combined with Domtar, a $3.3 billion cash and stock deal leaving Weyerhauser stock holders with 55 percent ownership of the new Domtar company.  In March 2008, Weyerhäuser Company announced the sale of its Containerboard Packaging and Recycling business to International Paper for $6 billion in cash, subject to post closing adjustments. The transaction included nine containerboard mills, 72 packaging locations, 10 specialty-packaging plants, four kraft bag and sack locations and 19 recycling facilities. The transaction affected approximately 14,300 employees. The deal closed on August 4, 2008.  Weyerhäuser converted into a real estate investment trust to avoid all federal income taxes when it filed its 2010 tax return.  In 2013, Weyerhäuser purchased Longview Timber for $2.65 billion including debt from Brookfield Asset Management. The acquisition added 645,000 acres of timberland to Weyerhaeuser's holdings in Oregon and Washington.  In 2014, Weyerhäeuser spun off its home building unit to TRI Pointe Homes in a $2.8 billion transaction. The company also announced its intention to sell its Federal Way headquarters and relocate to Seattle's Pioneer Square in 2016. The sale and move were completed in 2016.  On November 8, 2015, it was announced that Weyerhäuser would buy Plum Creek Timber for $8.4 billion, forming the largest private owner of timberland in the United States. The transaction closed on February 19, 2016. At the time of the merger the combined companies own about 13 million acres of timberlands.  In 2018, it won its case in the Supreme Court regarding whether private land can be classified as critical habitat if the land is not currently suitable as habitat for the protected species.

Operations.

The company's operations are divided into three major business segments:

Corporate governance.

Devin Stockfish is the CEO and president of Weyerhaeuser Company. The Weyerhaeuser board of directors consists of: Mark Emmert, Sara Grootwassink Lewis, Rick Holley, Nicole Piasecki, Marc Racicot, Lawrence Selzer, D. Michael Steuert, Devin Stockfish, Kim Williams and Charles Williamson.

Weyerhaeuser Company () is an American timberland company which owns nearly 12.4 million acres of timberlands in the U.S. and manages an additional 14.0 million acres timberlands under long-term licenses in Canada. The company also manufactures wood products. Weyerhaeuser is a real estate investment trust.

Weyerhaeuser je společnost, která je jedním z největších výrobců papíru a produktů ze dřeva na světě. Je to největší soukromý vlastník jehličnatých lesů na světě a druhý největší soukromý vlastník lesů v USA, po společnosti Plum Creek Timber. Společnost zaměstnává přibližně 15 tisíc lidí ve třinácti zemích, kam patří mimo jiné i Spojené státy, Kanada, Austrálie, Nový Zéland a Francie.


Overview.

Floating-point numbers.

A number representation specifies some way of encoding a number, usually as a string of digits.  There are several mechanisms by which strings of digits can represent numbers. In common mathematical notation, the digit string can be of any length, and the location of the radix point is indicated by placing an explicit "point" character (dot or comma) there. If the radix point is not specified, then the string implicitly represents an integer and the unstated radix point would be off the right-hand end of the string, next to the least significant digit. In fixed-point systems, a position in the string is specified for the radix point. So a fixed-point scheme might be to use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345.  In scientific notation, the given number is scaled by a

Alternatives to floating-point numbers.

The floating-point representation is by far the most common

History.

In 1914, Leonardo Torres y Quevedo designed an electro-mechanical version of Charles Babbage's Analytical Engine, and included floating-point arithmetic. In 1938, Konrad Zuse of Berlin completed the Z1, the first binary, programmable mechanical computer; it uses a 24-bit binary floating-point number representation with a 7-bit signed exponent, a 17-bit significand (including one implicit bit), and a sign bit. The more reliable relay-based Z3, completed in 1941, has representations for both positive and negative infinities; in particular, it implements defined operations with infinity, such as formula_11, and it stops

Range of floating-point numbers.

A floating-point number consists of two fixed-point components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floating-point range linearly depends on the significand range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number.  On a typical computer system, a "double-precision" (64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2 =

IEEE 754: floating point in modern computers.

The IEEE standardized the computer representation for binary floating-point numbers in IEEE 754 (a.k.a. IEC 60559) in 1985. This first standard is followed by almost all modern machines. It was revised in 2008. IBM mainframes support IBM's own hexadecimal floating point format and IEEE 754-2008 decimal floating point in addition to the IEEE 754 binary format. The Cray T90 series had an IEEE version, but the SV1 still uses Cray floating-point format.  The standard provides for many closely related formats, differing in only a few details. Five of these formats are called "basic formats" and others are termed "extended formats"; three of these are especially widely used in computer hardware and languages:  Increasing the precision of the floating point representation generally reduces the amount of accumulated round-off error caused by intermediate calculations. Less common IEEE formats include: Any integer with absolute value less than 2 can be exactly represented in the single precision format, and any integer with absolute value less than 2 can be exactly represented in the double precision format. Furthermore, a wide range of powers of 2 times such a number can be represented. These properties are sometimes used for purely integer data, to get 53-bit integers on platforms that have double precision floats but only 32-bit integers.  The standard specifies some special values, and their representation: positive infinity (+∞), negative infinity (−∞), a negative zero (−0) distinct from ordinary ("positive") zero, and "not a number" values (NaNs).  Comparison of floating-point numbers, as defined by the IEEE standard, is a bit different from usual integer comparison. Negative and positive zero compare equal, and every NaN compares unequal to every value, including itself. All values except NaN are strictly smaller than +∞ and strictly greater than −∞. Finite floating-point numbers are ordered in the same way as their values (in the set of real numbers).

Internal representation.

Floating-point numbers are typically packed into a computer datum as the sign bit, the exponent field, and the significand or mantissa, from left to right. For the IEEE 754 binary formats (basic and extended) which have extant hardware implementations, they are apportioned as follows:  While the exponent can be positive or negative, in binary formats it is stored as an unsigned number that has a fixed "bias" added to it. Values of all 0s in this field are reserved for the zeros and subnormal numbers; values of all 1s are reserved for the infinities

Special values.

Signed zero.

In the IEEE 754 standard, zero is signed, meaning that there exist both a "positive zero" (+0) and a "negative zero" (−0). In most run-time environments, positive zero is usually printed as "0" and the negative zero as "-0". The two values behave as equal in numerical comparisons, but some operations return different results for +0 and

Subnormal numbers.

Subnormal values fill the underflow gap with values where the absolute distance between them is the same as for adjacent values just outside the underflow gap. This is an improvement

Infinities.

The infinities of the extended real number line can be represented in IEEE floating-point datatypes, just like ordinary floating-point values like 1, 1.5, etc. They are not error values in any way, though they are often (but not always, as it depends

NaNs.

IEEE 754 specifies a special value called "Not a Number" (NaN) to be returned as the result of certain "invalid" operations, such as 0/0, ∞×0, or sqrt(−1). In general, NaNs will be propagated i.e. most operations involving a NaN will result in a NaN, although functions that would give some defined result for any given floating-point value will do so for NaNs as well, e.g. NaN ^ 0 =

IEEE 754 design rationale.

It is a common misconception that the more esoteric features of the IEEE 754 standard discussed here, such as extended formats, NaN, infinities, subnormals etc., are only of interest to numerical analysts, or for advanced numerical applications; in fact the opposite is true: these features are designed to give safe robust defaults for numerically unsophisticated programmers, in addition to supporting sophisticated numerical libraries by experts. The key designer of IEEE 754, William Kahan notes that it is incorrect to "... [deem] features of IEEE Standard 754 for Binary Floating-Point Arithmetic that...[are] not appreciated to be features usable by none but numerical experts. The facts are quite the opposite. In 1977 those features were designed into the Intel 8087 to serve the widest possible market... Error-analysis tells us how to design floating-point arithmetic, like IEEE Standard 754, moderately tolerant of well-meaning ignorance among programmers".

Other notable floating-point formats.

In addition to the widely used IEEE 754 standard

Representable numbers, conversion and rounding.

By their nature, all numbers expressed in floating-point format are rational numbers with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base-10, or a terminating binary expansion in base-2). Irrational numbers, such as π or √2, or non-terminating rational numbers, must be approximated. The number of digits (or bits) of precision also limits the set of rational numbers that can be represented exactly. For example, the decimal number 123456789 cannot be exactly represented if only eight decimal digits of precision are available (would be rounded to 123456790 or 123456780 where the rightmost digit 0 is not explicitly stored).  When a number is represented in some format (such as a character string) which is not a native floating-point representation supported in a computer implementation, then it will require a conversion before it can be used in that implementation. If the number can be represented exactly in the floating-point format then the conversion is exact. If there is not an exact representation then the conversion requires a choice of which floating-point number to use to represent the original value. The representation chosen will have a different value from the original, and the value thus adjusted is called the "rounded value".  Whether or not a rational number has a terminating expansion depends on the base. For example, in base-10 the number 1/2 has a terminating expansion (0.5) while the number 1/3 does not (0.333...). In base-2 only rationals with denominators that are powers of 2 (such as 1/2 or 3/16) are terminating. Any rational with a denominator that has a prime factor other than 2 will have an infinite binary expansion. This means that numbers which appear to be short and exact when written in decimal format may need to be approximated when converted to binary floating-point. For example, the decimal number 0.1 is not representable in binary floating-point of any finite precision; the exact binary representation would have a "1100" sequence continuing endlessly: where, as previously, "s" is the significand and "e" is the exponent.  When rounded to 24 bits this becomes which is actually 0.100000001490116119384765625 in decimal. As a further example, the real number π, represented in binary as an infinite sequence of bits is but is when approximated by rounding to a precision of 24 bits.  In binary single-precision floating-point, this is represented as "s" = 1.10010010000111111011011 with "e" = 1. This has a decimal value of whereas a more accurate approximation of the true value of π is  The result of rounding differs from the true value by about 0.03 parts per million, and matches the decimal representation of π in the first 7 digits. The difference is the discretization error and is limited by the machine epsilon.  The arithmetical difference between two consecutive representable floating-point numbers which have the same exponent is called a unit in the last place (ULP). For example, if there is no representable number lying between the representable numbers 1.45a70c22 and 1.45a70c24, the ULP is 2×16, or 2. For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2 or about 10 in single precision, and exactly 2 or about 10 in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.

Floating-point arithmetic operations.

For ease of presentation and understanding, decimal radix with 7 digit precision will be used in the examples, as in the IEEE 754 "decimal32" format. The fundamental principles are the same in any radix or precision, except that normalization is optional (it does not affect the numerical value of the result). Here, "s" denotes the significand and "e" denotes the exponent.

Addition and subtraction.

A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number is shifted right by three digits, and one then proceeds with the usual addition method:  In detail:  This is the true result, the exact sum of the operands. It will be rounded to seven digits and then normalized if necessary. The final result is  The lowest three digits of the second operand (654) are essentially lost. This is round-off error. In extreme cases, the sum of two non-zero numbers may be equal to one of them:  In the above conceptual examples it would appear that a large number of

Multiplication and division.

To multiply, the significands are multiplied while the exponents are added, and the result is rounded and normalized.  Similarly, division is accomplished by subtracting the divisor's exponent from the dividend's exponent, and dividing the dividend's significand by

Dealing with exceptional cases.

Floating-point computation in a computer can run into three kinds of problems:  Prior to the IEEE standard, such conditions usually caused the program to terminate, or triggered some kind of trap that the programmer might be able to catch. How this worked was system-dependent, meaning that floating-point programs were not portable. (The term "exception" as used in IEEE 754 is a general term meaning an exceptional condition, which is not necessarily an error, and is a different usage to that typically defined in programming languages such as a C++ or Java, in which an "exception" is an alternative flow of control, closer to what is termed a "trap" in IEEE 754 terminology).  Here, the required default method of handling exceptions according to IEEE 754 is discussed (the IEEE 754 optional trapping and other "alternate exception handling" modes are not discussed). Arithmetic exceptions are (by default) required to be recorded in "sticky" status flag bits. That they are "sticky" means that they are not reset by the next (arithmetic) operation, but stay set until explicitly reset. The use of "sticky" flags thus allows for testing of exceptional conditions to be delayed until after a full floating-point expression or subroutine: without them exceptional conditions that could not be otherwise ignored would require explicit testing immediately after every floating-point operation. By default, an operation always returns a result according to specification without interrupting computation. For instance, 1/0 returns +∞, while

Accuracy problems.

The fact that floating-point numbers cannot precisely represent all real numbers, and that floating-point operations cannot precisely represent true arithmetic operations, leads to many surprising situations. This is related to the finite precision with which computers generally represent numbers.  For example, the non-representability of 0.1 and 0.01 (in binary) means that the result of attempting to square 0.1 is neither 0.01 nor the representable number closest to it. In 24-bit (single precision) representation, 0.1 (decimal) was given previously as "e" = −4; "s" = 110011001100110011001101, which is Squaring this number gives Squaring it with single-precision floating-point hardware (with rounding) gives But the representable number closest to 0.01 is  Also, the non-representability of π (and π/2) means that an attempted computation of tan(π/2) will not yield a result of infinity, nor will it even overflow. It is simply not possible for standard floating-point hardware to attempt to compute tan(π/2), because π/2 cannot be represented exactly. This computation in C:  /* Enough digits to be sure we get the correct approximation. */ double pi = 3.1415926535897932384626433832795; double z = tan(pi/2.0);  will give a result of 16331239353195370.0. In single precision (using the tanf function), the result will be −22877332.0.  By the same token, an attempted computation of sin(π) will not yield zero. The result will be (approximately) 0.1225 in double precision, or −0.8742 in single precision.  While floating-point addition and multiplication are both commutative ("a" + "b" = "b" + "a" and "a" × "b" = "b" × "a"), they are not necessarily associative. That is, ("a" + "b") + "c" is not necessarily equal to "a" + ("b" + "c"). Using 7-digit significand decimal arithmetic:  They are also not necessarily distributive. That is, ("a" + "b") × "c" may not be the same as "a" × "c" + "b" × "c":  In addition to loss of significance, inability to represent numbers such as π and 0.1 exactly, and other slight inaccuracies, the following phenomena may occur:

Machine precision and backward error analysis.

"Machine precision" is a quantity that characterizes the accuracy of a floating-point system, and is used in backward error analysis of floating-point algorithms. It is also known as unit roundoff or "machine epsilon". Usually denoted Ε, its value depends on the particular rounding being used.  With rounding to zero, whereas rounding to nearest,  This is important since it bounds the "relative error" in representing any non-zero real number x within the normalized range of a floating-point system:  Backward error analysis, the theory of which was developed and popularized by James H. Wilkinson, can be used to establish that an algorithm implementing a numerical function is numerically stable. The basic approach is to show that although the

Minimizing the effect of accuracy problems.

Although, as noted previously, individual arithmetic operations of IEEE 754 are guaranteed accurate to within half a ULP, more complicated formulae can suffer from larger errors due to round-off. The loss of accuracy can be substantial if a problem or its data are ill-conditioned, meaning that the correct result is hypersensitive to tiny perturbations in its data. However, even functions that are well-conditioned can suffer from large loss of accuracy if an algorithm numerically unstable for that data is used: apparently equivalent formulations of expressions in a programming language can differ markedly in their numerical stability. One approach to remove the risk of such loss of accuracy is the design and analysis of numerically stable algorithms, which is an aim of the branch of mathematics known as numerical analysis. Another approach that can protect against the risk of numerical instabilities is the computation of intermediate (scratch) values in an algorithm at a higher precision than the final result requires, which can remove, or reduce by orders of magnitude, such risk: IEEE 754 quadruple precision and extended precision are designed for this purpose when computing at double precision.  For example, the following algorithm is a direct implementation to compute the function A(x) = (x−1) / (exp(x−1) − 1) which is well-conditioned at 1.0, however it can be shown to be numerically unstable and lose up to half the significant digits carried by the arithmetic when computed near 1.0.  double A(double X)  If, however, intermediate computations are all performed in extended precision (e.g. by setting line [1] to C99 long double), then up to full precision in the final double result can be maintained. Alternatively, a numerical analysis of the algorithm reveals that if the following non-obvious change to line [2] is made:  then the algorithm becomes numerically stable and can compute to full double precision.  To maintain the properties of such carefully constructed numerically stable programs, careful handling by the compiler is required. Certain "optimizations" that compilers might make (for example, reordering operations) can work against the goals of well-behaved software. There is some controversy about the failings of compilers and language designs in this area: C99 is an example of a language where such optimizations are carefully specified to maintain numerical precision. See the external references at the bottom of this article.  A detailed treatment of the techniques for writing high-quality floating-point software is beyond the scope of this article, and the reader is referred to, and the other references at the bottom of this article. Kahan suggests several rules of thumb that can substantially decrease by orders of magnitude the risk of numerical anomalies, in addition to, or in lieu of, a more careful numerical analysis. These include: as noted above, computing all expressions and intermediate results in the highest precision supported in hardware (a common rule of thumb is to carry twice the precision of the desired result i.e. compute in double precision for a final single precision result, or in double extended or quad precision for up to double precision results); and rounding

In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between range and precision. For this reason, floating-point computation is often found in systems which include very small and very large real numbers, which require fast processing times. A number is, in general, represented approximately to a fixed number of significant digits (the significand) and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form: where significand is an integer, base is an integer greater than or equal to two, and exponent is also an integer. For example:

Ve výpočetní technice se pohyblivou řádovou čárkou nebo plovoucí řádovou čárkou rozumí způsob reprezentace čísel, která by byla moc malá nebo velká pro vyjádření v pevné řádové čárce. Čísla jsou obecně uložena jako určité množství platných číslic vynásobený exponentem. Základem exponentu bývá většinou 2, 10 nebo 16 (což odpovídá dvojkové, desítkové a šestnáctkové soustavě). Čísla, která mohou být v pohyblivé řádové čárce vyjádřena přesně, jsou ve tvaru: