市场分割理论认为预期理论的假设条件在现实中是不成立的，因此也不存在预期形成的收益曲线；事实上整个金融市场是被不同期限的债券所分割开来的，而且不同期限的债券之间完全不能替代。

　　该理论认为，由于存在法律、偏好或其他因素的限制，投资者和债券的发行者都不能无成本地实现资金在不同期限的证券之间的自由转移。因此，证券市场并不是一个统一的无差别的市场，而是分别存在着短期市场、中期市场和长期市场。不同市场上的利率分别各市场的供给和需求决定。当长期债券供给曲线与需求曲线的交点高于短期债券供给曲线与需求曲线的交点时，债券的收益率曲线向上倾斜；相反，当长期债券供给曲线与需求曲线的交点低于短期债券供给曲线与需求曲线的交点时，债券的收益率曲线向下倾斜。

市场分割理论的分割

市场分割理论认为出于一些法规、规章制度以及市场参与者目标等因素，借款人和贷款人会把他们的交易局限于一个特定的期限。这些因素归结起来主要表现在以下三方面：

　　1、由于政府的某些规章制度，借款人和贷款人被要求只能交易特定期限的金融产品。

　　2、市场参与者为了规避风险而将投资者局限于某一期限的金融产品。

　　3、按照西蒙的观点和行为金融理论，投资者并不追求最优解，而是追求满意解。

　　在以上三个因素中，政府的规章制度具有最强的约束力，受该因素制约的借款人和贷款人将不会进行期限债券之间的任何替代，这样将形成强式的市场分割。规避风险因素居于次席，出于该因素而形成的市场分割属于半强式的。第三个因素和其他因素对借款人和贷款人的约束力较差，从而形成一种弱式市场分割。

　　市场分割理论把金融市场分为短期、中期和长期三大市场：

　　1、短期市场的主要参与者是商业银行、非金融机构和货币市场基金等，它们更关注的是本金确定性(或者说安全性)而不是收入的确定性。

　　2、长期市场的参与者主要是那些债务期限结构比较长的机构，比如人寿保险公司、养老基金等，这些机构具有较强的风险回避态度，更注重收入的确定性。

　　3、虽然短期市场和长期市场的投资者的投资动机和目标各有差异，但是两个市场的参与者都同样受到法律、规章制度的制约以及降低风险的压力而集中于不同期限的市场，因此，这两个市场形成的市场分割基本上是强式的，参与者通常不会进行投资期限之间的转换。相比之下，中期市场的参与者身份就比较复杂，没有一群占主导地位的“忠实”的参与者，从而其功能是弱式的。

　　分割的具体原因：

　　1、投资者可能对某种证券具有特殊偏好或投资习惯。

　　2、投资者不能掌握足够知识，只能对某些证券感兴趣。

　　3、不同借款人也只对某些证券感兴趣，因为其资金的使用性质决定了他们只会对某些期限的证券感兴趣。

　　4、机构投资者的负债结构决定了他们在长期和短期证券之间的选择。

　　5、缺少易于在国内市场上销售的统一的债务工具

市场分割理论的现实意义

市场分割理论具有一定的现实意义，一些研究成果表明，市场分割对期限结构具有一定程度的影响。基德沃尔和科沃奇(Kidwell and Koch，1983)经过对美国市政债券和联邦政府国债的比较研究发现，市政债券市场(Municipal Bond Market)以到期日为口径存在明显的市场分割现象，换句话说，地方政府发行的短期和长期债券之间很难相互替代。

　　另外统计资料也显示：近年来，美国联邦储备体系可以在不完全影响长期利率的情况下调整短期利率。然而市场分割理论仍然受到了很多指责，批评主要来自于纯粹预期理论和流动性升水理论的支持者，他们认为金融市场的运行效率要远远高于市场分割理论所形成的市场效率，投资者事实上更愿意在不同期限的市场之间转移资金，而不是固守于某一细分市场停滞不前。现实中，大约80％ 的期限结构变动体现着利率水平的改变，这也预示着当收益率变化时，不同期限的债券基本上趋向于同步变化，因此市场分割的作用是有限的。

什么是流动性升水理论

流动性升水理论又称作流动性升水或流动性报酬理论，简称LP理论。该理论在纯预期理论的基础上，充分考虑了投资于债券的风险，认为长期债券的流动性低于短期债券，这是由于“持有长期债券确定是有风险的，而这一风险会随着债券到期日增长而增加。”

　　最早由希克斯(1946)提出和阐发，并经科塞尔(Kessil，1965)做了进一步补充。

　　该理论认为金融市场是有风险的，投资者持有长期证券将比持有短期证券更具风险，如果不向长期证券持有人进行补偿，则投资者将偏好于短期证券，这样可以降低风险和增加流动性。但从借贷的角度来看，借款人却又偏好长期借款以保证他们有一稳定的资金来源，这就导致了对不同期限证券的供给和需求形式上的不平衡——投机者或许希望抵消这种不平衡。因此，该理论断言投资者和投机者都是风险厌恶的，必须支付一份流动性升水才能使他们愿意持有长期证券。

　　希克斯认为随着期限的延长，投资风险也随之增加，因此流动性升水是时间的递增函数，因此有:。与纯粹预期理论的“完全替代”前提不同，流动性升水理论的关键假设是不同期限的债券可以替代，这意味着预期仍然可以影响不同期限债券的预期回报率；不过这些债券并非完全的替代品，投资者对短期债券和长期债券的偏好不同的事实说明流动性升水也会影响预期回报率。

流动性升水理论的质疑和争论

　　有些学者对升水部分提出质疑，因为现代资产组合理论认为证券收益中含有风险升水，而流动性升水与投资者的流动性偏好有关，因此他们主张债券价格变化的风险也会影响流动性升水。恩格尔和讷格(Engle and Ng，1993)就是其中的代表。他们研究了美国1964年到1989年的数据，得出与希克斯和科塞尔不同的结论：

　　1、如果预期未来短期利率与当前短期利率相同，则收益曲线的形状完全取决于升水部分，这样当波动率较低时，收益曲线将变得弯曲，而当波动率较高时，收益曲线将向上倾斜。

　　2、如果预期短期利率下降，则收益曲线仍然会向上倾斜，前提是短期利率预期下降幅度不大，同时波动率比较高。

　　3、当波动率较低时，预期部分在决定收益曲线形状时将起主要作用，这样，当预期短期利率下降(上升)时，收益曲线将向下(向上)倾斜。

　　流动性升水理论对远期利率的预测能力得到了大多数学者的认同，但是对流动性升水的性质却始终无法形成一致的意见。争论主要表现在两个方面：

　　1、流动性升水是否始终大于零？

　　莫顿(Merton，1973)、龙(Long，1974)、布利登(Breeden，1979)和考克斯等人(Cox et al，1985)认为流动性升水是随时间而变化的，但并不是时间递增函数，从而隐含了升水可能为负的可能性。法马和布里斯(1987)通过研究发现，流动性升水与经济周期有关，当经济处于繁荣时期时，预期的流动性升水将大于零；而当经济陷入萧条时，流动性升水则更可能为负。

　　2、流动性升水的大小到底是多少？

　　一些学者认为流动性升水大约在0.54％ ～1.56％ 的幅度内变动；另外一些学者认为即使对于较长的期限，流动性升水也不会超过0.5％；甚至一部分学者的研究成果表明流动性升水不仅不会随着时间而增加，相反会随着期限的延长而降低。

纯粹预期理论概述

　　纯粹预期理论认为“只有”预期的未来短期利率，决定收益率曲线的形状；当预期未来短期利率上升时，会有上升的收益率曲线，反之，收益率曲线呈下降态势。

　　纯粹预期理论是最早产生的一种期限结构理论，其最早提出者可追溯到费雪(Fisher，1896)，不过他本人并没有作深人研究，该理论的主要发展是希克斯(Hicks，1946)和卢茨(Lutz，1940)作出的，此后马尔基尔(Malkiel，1966)、梅塞尔曼(Meiselman，1962)和罗尔(Roll，1970、1971)进一步发展了该理论，提出了新的见解。

　　纯粹预期理论的主要观点是：可观察的长期利率是预期的不可观察的短期利率的平均值。需要注意的是，这里的短期利率和长期利率并没有严格的界限，两者的划分完全出于经济主体的需要和判断，但是为了方便起见，人们大多还是以一年为界来加以区分。

　　纯粹预期理论在投资者预期与利率变动之间建立了一种对应关系，预期改变了，收益曲线的形状也会随之发生变化，这样预期理论就为利率预测提供了一种基本框架，有利于利率预测模型的进一步发展。一些实证研究为纯粹预期理论在利率预测方面的重要作用提供了事实支持。

　　法马(Fama，1984b)发现利差(Spread，长期利率和短期利率之间的差异)对未来短期利率(1个月)的变化具有一定的正面预测功能；曼昆和米龙(Mankiw and Miron，1986)研究了美国1890年至1979年的国库券利率和贷款利率，认为1890年至1914年美联储成立以前，预期理论可以很好地解释收益曲线的变动，而自1915年美联储成立以后的数据则无法支持预期理论。坎普贝尔和希勒(Campbell and ShiHer，1984)、法马和布里斯(Fama and Bliss，1987)发现中期和长期的利差能在一定程度上预测未来短期利率的变化，但是他们同时也拒绝了纯粹预期理论的假设条件。希勒(1981)更是给出了具有说服力的事实来证明预期理论，他发现利差不仅对5年期债券的超额回报率具有很强的预测功能，而且他的分析数据本身也无法拒绝预期理论。哈杜威利斯(Hardouvelis，1988)利用20世纪70—80年代的国库券利率数据，发现远期利率具有关于未来即期利率走势的更多信息，远期利率因此具有很强的预测能力，即使在1979年10月到1982年10月美联储改变货币政策中介目标期间，远期利率的预测能力不但没有减弱，而且还得到了进一步的增强。这种现象表明在利率的预测能力和中央银行钉住利率目标的程度之间没有必然联系。

　　根据纯粹预期理论，预期是决定未来利率水平的唯一因素，只要能获得未来利率预期的足够信息和证据，就可以判断收益曲线的形状。为此，很多经济学家着手研究各种类型的形成机制，以期为纯粹预期理论寻找现实的支持，这些研究成果包括梅塞尔曼的错误学习机制、莫迪利安尼和萨奇(Modi—gliaIli and Sutch，1966)的过去利率分布滞后、莫迪利安尼和希勒(1973)、法马(1976)的通货膨胀分布滞后以及凯恩、马尔基尔(K／lne and Malkiel，1967)和弗里德曼(Friedman，1979)的专家预期模型等，他们的共同结论是：预期是期限结构决定中的一个重要因素，但其他因素显然也发挥着重要作用。他们主张放弃预期理论的纯粹形式，而用预期和流动性升水来共同解释收益曲线向上倾斜的普遍形式。

　　然而纯粹预期理论也受到了很多批评，因为它无法对金融市场上“收益曲线通常向上倾斜”的事实作出解释。最早提出质疑的是麦考利，他于1938年指出“预期理论的结论几乎与现实经验相反”。此外，坎普贝尔和希勒(1984)、法马(1984a、b)、法马和布里斯(1987)、琼斯和罗利(Jones and Roley，1983)、曼昆和萨莫斯(Summers，1984)、希勒以及坎普贝尔和斯科恩霍兹(Schoenholtz，1983)等人考察了美国战后以来的收益率数据，结果证明1年期以下的美国国库券收益率并不遵循纯粹预期理论，真实数据与理性预期假设具有很大冲突，因此主张拒绝纯粹预期理论。

纯粹预期理论

　　纯预期理论基于下述假设：

　　1、投资者希望持有债券期间收益最大。

　　2、投资者对特定期限无特殊偏好，他们认为各种期限都是可以完全替代的。

　　3、买卖债券没有交易成本，一旦投资者察觉到收益率差异即可变换期限。

　　4、绝大多数投资者都可以对未来利率形成预期，并根据这些预期指导投资行为。

　　学者们认为，纯预期理论作为一种精巧的理论，可以较好地解释用收益率曲线表示的利率期限结构在不同时期变动的原因，但它最大的缺陷是“忽视了投资于债券或类似于票据上的内在风险”。为了克服这个缺点，人们又提出了流动性升水理论加以解释。

From Wikipedia, the free encyclopedia

"The Breakers", a gilded-age Victorian mansion in Newport, Rhode Island.

In American history, the Gilded Age refers to substantial growth in population in the United States and extravagant displays of wealth and excess of America's upper-class during the post-Civil War and post-Reconstruction era, in the late 19th century (1865-1901). The wealth polarization derived primarily from industrial and population expansion. The businessmen of the Second Industrial Revolution created industrial towns and cities in the Northeast with new factories, and contributed to the creation of an ethnically diverse industrial working class which produced the wealth owned by rising super-rich industrialists and financiers such as Cornelius Vanderbilt, John D. Rockefeller, Andrew Carnegie, Henry Flagler, and J.P. Morgan. Their critics called them "robber barons", referring to their use of overpowering and sometimes unethical financial manipulations. There was a small, growing labor union movement, led in part by Samuel Gompers, who created the American Federation of Labor (AFL), founded in 1886. It featured very close contests between the Republicans and Democrats, with occasional third parties. Nearly all the eligible men were political partisans and voter turnout often exceeded 90% in some states.

This period also witnessed the creation of a modern industrial economy. A national transportation and communication network was created, the corporation became the dominant form of business organization, and a managerial revolution transformed business operations. By the beginning of the twentieth century, per capita income and industrial production in the United States exceeded that of any other country except Britain. Long hours and hazardous working conditions led many workers to attempt to form labor unions despite strong opposition from industrialists and the courts.

The wealth of the period is highlighted by the American upper class's opulent self-indulgence, but also the rise of the American philanthropy (Andrew Carnegie called it the "Gospel of Wealth") that endowed thousands of colleges, hospitals, museums, academies, schools, opera houses, public libraries, symphony orchestras, and charities. The Beaux-Arts architectural idiom of the era clothed public buildings in Neo-Renaissance architecture.

This period overlaps with the nadir of American race relations, during which African Americans lost many of the civil rights obtained during the Reconstruction period. Increased racist violence, as well as exile of African Americans from the Southern states to the Midwest, started as soon as 1879. The end of the Gilded Age coincided with the Panic of 1893, a deep depression. The depression lasted until 1897 and marked a major political realignment in the election of 1896. After that came the Progressive Era.

The term "Gilded Age" was coined by Mark Twain and Charles Dudley Warner in their 1873 book, The Gilded Age: A Tale of Today.

[edit] Industrial and technological advances

The Gilded Age was rooted in industrialization, especially heavy industry like factories, railroads and coal mining. During the Gilded Age, American manufacturing production surpassed the combined total of Great Britain, Germany, and France. Railroad mileage tripled between 1860 and 1880, and tripled again by 1920, opening new areas to commercial farming, creating a truly national marketplace and inspiring a boom in coal mining and steel production. The voracious appetite for capital of the great trunk railroads facilitated the consolidation of the nation's financial market in Wall Street. By 1900, the process of economic concentration had extended into most branches of industry—a few large corporations, called "trusts", dominated in steel, oil, sugar, meatpacking, and the manufacture of agriculture machinery. Other major components of this infrastructure were the new methods for fabricating steel: the Bessemer and the Siemens steel making processes. The first billion-dollar corporation was United States Steel, formed by financier J. P. Morgan in 1901, who purchased and consolidated steel firms built by Andrew Carnegie and others.

Increased mechanization of industry is a major mark of the Gilded Age's search for cheaper ways to create more product. Frederick Winslow Taylor observed that worker efficiency in steel could be improved through the use of machines to make fewer motions in less time. His redesign increased the speed of factory machines and the productivity of factories while undercutting the need for skilled labor. This mechanization made some factories an assemblage of unskilled laborers performing simple and repetitive tasks under the direction of skilled foremen and engineers. Machine shops grew rapidly, and they comprised highly skilled workers and engineers. Both the number of unskilled and skilled workers increased, as their wage rates grew. Engineering colleges were established to feed the enormous demand for expertise. Railroads invented complex bureaucratic systems, using middle managers, and set up explicit career tracks. They hired young men at age 18-21 and promoted them internally until a man reached the status of locomotive engineer, conductor or station agent at age 40 or so. Career tracks were invented for skilled blue collar jobs and for white collar managers, starting in railroads and expanding into finance, manufacturing and trade. Together with rapid growth of small business, a new middle class was rapidly growing, especially in northern cities.

The United States became a world leader in applied technology. From 1860 to 1890, 500,000 patents were issued for new inventions—over ten times the number issued in the previous seventy years. George Westinghouse invented air brakes for trains (making them both safer and faster). Theodore Vail established the American Telephone & Telegraph Company. Thomas A. Edison invented a remarkable number of electrical devices, as well as the integrated power plant capable of lighting multiple buildings simultaneously; he founded General Electric corporation. Oil became an important resource, beginning with the Pennsylvania oil fields. Kerosene replaced whale oil and candles for lighting. John D. Rockefeller created Standard Oil Company to consolidate the industry.

[edit] Influential figures

Gilded age philanthropy: a table d'hôte dinner menu from March 4, 1893 for a meal in honor of symphony conductor Walter Johannes Damrosch.

Andrew Carnegie, John D. Rockefeller, and "Commodore" Cornelius Vanderbilt were amongst the most influential industrialists during the Gilded Age. Carnegie was born into a poor Scottish family; at age 18 he became an assistant to railroad superintendent Thomas A. Scott in Pittsburgh. In 1870, Carnegie erected his first blast furnace. Both Carnegie and Rockefeller gave away most of their wealth in large scale philanthropy. Carnegie created the Carnegie Institute of Technology (now part of Carnegie Mellon University) to upgrade craftsmen into trained engineers and scientists. Carnegie built hundreds of public libraries and several major research centers and foundations. Rockefeller retired from the oil business in 1897 and devoted the next 40 years of his life to giving away most of his money using systematic philanthropy, especially in the areas of education, medicine and race relations. "Commodore" C. Vanderbilt started out as a poor Staten Island farmer boy, then quickly through his sharp wit and lethal business policies built an enormous fortune in steamships and railroading to become the wealthiest man in the world in his day. His descendants and heirs would become famous for their ability to both increase and spend their wealth, building gigantic and lavish mansions and dominating Gilded Age high society.

[edit] Labor unions

[edit] The rise of unions

As American men, women, and children toiled away in the factories of the 1880s, many of them dreamed of one day living the Horatio Alger myth of moving from "rags to riches." While some Americans enjoyed unparalleled success as compared to their counterparts, many Americans still struggled to make ends meet. Workers began to organize labor unions to take responsibility for their own improvement. Modern labor unions emerged during the Civil War era. One of the earlier attempts at a national union was the National Labor Union, formed in Baltimore in 1866. The Knights of Labor, founded in 1869, had success in the late 1880s but then collapsed. The American Federation of Labor (AFL), a coalition of trades unions, became dominant in the 1890s, under Samuel Gompers. and the Industrial Workers of the World (IWW) emerged, encouraging all workers to unite.

[edit] The great strikes

While labor unions became increasingly popular in the late 1800s, their successes were also tempered by some crushing defeats. One of the earliest strikes following the Civil War was the Great Railroad Strike of 1877. It lasted for 45 days; it was finally ended when President Rutherford B. Hayes sent in federal troops. The number of deaths and the amount of damage caused by the strike scared many Americans. In 1886, the Knights of Labor were hit by two devastating events that eventually brought about an end to their union. The Great Southwest Railroad Strike of 1886 and the Haymarket Square Riot ended both internal and external support for the Knights. The Homestead Strike led by workers at the Homestead Steel Plant owned by Andrew Carnegie and Henry Clay Frick was an organized attempt at trying to secure better wages and working conditions than was ended with an assassination attempt on Frick. The final major strike of the late 1800s was the Pullman Strike which began at the company town of Pullman, Illinois. Owned by George Pullman, the town was the home to all the workers at the Pullman Palace Car Company. When Pullman refused to meet with workers to discuss intolerable conditions brought on by the Panic of 1893, a national rail strike was orchestrated by the American Railway Union led by Eugene V. Debs. This time the strike was halted by a court injunction and Debs was arrested by federal troops sent in by President Grover Cleveland. The Supreme Court upheld government action with their decision in the 1895 case In re Debs. With the Court decision coupled with the use of the Sherman Antitrust Act, labor unions were unable to regain much power until the 1930s.

[edit] Politics

A Group of Vultures Waiting for the Storm to "Blow Over"--"Let Us Prey." Cartoon of New York's Boss Tweed and other Tammany Hall figures, drawn in 1871 by Thomas Nast and published in Harper's Weekly.

Americans' sense of civic virtue was shocked by the scandals associated with the Reconstruction era: corrupt state governments, massive fraud in cities controlled by political machines, political payoffs to secure government contracts (especially the Crédit Mobilier of America scandal regarding the financing of the transcontinental railroad), and widespread evidence of government corruption during the Ulysses S. Grant Administration. This corruption divided the Republican party into two different factions, The Stalwarts led by Roscoe Conkling and the Half-Breeds led by James G. Blaine. Accordingly there were widespread calls for reform, such as Civil Service Reform led by the Bourbon Democrats and followed by the Republican Mugwumps, especially Samuel J. Tilden and Grover Cleveland. There was a sense that government intervention in the economy inevitably led to favoritism, bribery, kickbacks, inefficiency, waste, and corruption. The Bourbon Democrats led the call for a free market, low tariffs, low taxes, less spending and, in general, a Laissez-Faire (hands-off) government. They specifically denounced imperialism and overseas expansion. Many business and professional people supported this approach, although—to encourage rapid growth of industry and protect America's high wages against the low wage system in Europe—most Republicans advocated a high protective tariff. Labor activists and agrarians expressed the same spirit but focused their attacks on monopolies and railroads as unfair to the little man. Many Republicans also complained that high tariffs, for instance on British steel, benefited industrialists like Carnegie more than his employees who even at the time were regarded by many as being pitifully exploited.

In politics, the two parties engaged in very elaborate get-out-the vote campaigns that succeeded in pushing turnout to 80%, 90%, and even higher. It was financed by the "spoils system" whereby the winning party distributed most local, state and national government jobs, and many government contracts, to its loyal supporters. Large cities were dominated by political machines, in which constituents supported a candidate in exchange for anticipated patronage—favors back from the government, once that candidate was elected—and candidates were selected based on their willingness to play along. The best known example of a political machine from this time period is Tammany Hall in New York City, led by Boss Tweed. Presidential elections between the two major parties (the Republicans and Democrats), were closely contested, and Congress was marked by political stalemate. Mudslinging became an increasingly popular way of gaining advantage at the polls, and Republicans employed an election tactic known as "waving the bloody shirt". Candidates, especially when combating corruption charges, would remind voters that the Republican Party had saved the nation in the Civil War. During the 1870s, voters were repeatedly reminded that the Democrats had been responsible for the bloody upheaval, an appeal that attracted many Union veterans to the Republican camp. The Republicans consistently carried the North in presidential elections. The South, on the other hand, became the Solid South, nearly always voting Democratic. The political humiliations of Reconstruction were still fresh in many minds. Conversely, the Democrats invoked images of the "lost cause" and the glorious "stars and bars" in much the same way Republicans "waved the bloody shirt." The corruption of the Republican organization led to the defection of a group of reformers called the Mugwumps that supported Democrat Grover Cleveland in 1884. This victory gave Democrats control of the presidency for the first time since the Civil War (not counting the ascension of Andrew Johnson who was technically elected as part of the Union Party).

Overall, Republican and Democratic political platforms remained remarkably constant during the years before 1900. The negativity and ambiguity of politics began a shift in the press to yellow journalism, in which sensationalism and sentimental stories took as prominent a role as factual news.

[edit] Immigration

Prior to the Gilded Age, the time commonly referred to as the old immigration saw the first real boom of new arrivals to the United States. During the Gilded Age, approximately 10 million immigrants came to the United States in what is known as the new immigration, many in search of religious freedom and greater prosperity. The population surge in major U.S. cities as a result of immigration gave cities an even stronger impact on government, attracting power-hungry politicians and entrepreneurs. Pressuring voters or falsifying ballots was commonplace for politicians, who often sought power only to exploit their constituents. To accommodate the influx of people into the U.S., the federal government built Ellis Island in 1892 near the Statue of Liberty. After 1892, a short physical examination was given; those with contagious diseases were not admitted. Few immigrants went to the poverty-stricken South.

[edit] Chinese immigrants

The construction of the Central Pacific railroad in California and Nevada was handled largely by Chinese laborers. In the 1870 census there were 58 Chinese men and 4 women in the entire country; these numbers grew to 100,000 men and 40,000 women in the 1880 census.^[1] Labor unions such as the American Federation of Labor strongly opposed the presence of Chinese labor, by reason of both economic competition and race. Immigrants from China were not allowed to become citizens until 1950; however, their children born in the U.S. were full citizens.

Congress banned further Chinese immigration through the Chinese Exclusion Act in 1882; the act prohibited Chinese laborers from entering the United States, but some students and businessmen were allowed in. Subsequent to the act, the Chinese population declined to only 37,000 in 1940. Although many returned to China (a greater proportion than most other immigrant groups), most of them stayed in the United States. Chinese people were unwelcome in many areas, so they resettled in the "Chinatown" districts of large cities.

[edit] Urban life

Society itself underwent significant changes in the period following the Civil War. One of the most significant changes came in the further urbanization of the northern cities. As a result of increasing demand for factory workers as well as mass immigration from Europe, the population of cities began to swell. Major American cities such as New York and Chicago even saw populations grow in excess of one million people. These rapid changes in cities brought about modern architectural and transportation features. Louis Sullivan became a noted architect using steel frames to construct skyscrapers for the first time while pioneering the idea of "form follows function". One of his earliest works was the Wainwright Building in St. Louis, Missouri. Elisha Otis’ introduction of safety measures on elevators also helped building reach newer heights. American cities also expanded underground. In order to reduce congestion on the streets, major cities such as New York began to construct underground railroads, the forerunners to modern subways. As immigration increased in cities, poverty rose as well. New immigrants were forced to live in the poorest urban areas including the Five Points and Hell’s Kitchen in Manhattan. These areas were quickly overridden with crime gangs such as the Five Points Gang and the Bowery Boys rose to prominence. Families were forced into miserable living conditions in the so-called “dumbbell tenements” where multiple families would share rooms designed for single occupants. It was not until Danish photographer Jacob Riis published his photographic essay How the Other Half Lives that real attention was called to the unsafe and unsanitary conditions of tenement living.

[edit] Social changes

During the Gilded Age, many new social movements took hold in the United States. Many women abolitionists who were disappointed that the Fifteenth Amendment did not extend voting rights to them remained active in politics, this time focusing on issues important to them. Reviving the temperance movement from the Second Great Awakening, many women joined the Women’s Christian Temperance Union (WCTU) in an attempt to bring morality back to America. Other women took up the issue of women’s suffrage which had laid dormant since the Seneca Falls Convention With leaders like Susan B. Anthony the National American Woman Suffrage Association (NAWSA) was formed in order to secure the right of women to vote. Science also played an important part in social though as the work of Charles Darwin became popular. Following in Darwin’ s idea of natural selection, English philosopher Herbert Spencer proposed the idea of social Darwinism. This new concept justified the stratification of the wealthy and poor and coined the term “survival of the fittest.” Joining Spencer was Yale University professor William Graham Sumner whose book What Social Classes Owe to Each Other argued that assistance to the poor actually weakens their ability to survive in society. Sumner argued for a laissez faire and free market economy. Not everyone agreed with the social Darwinists and soon a whole movement to help the poor arose. Henry George proposed a “single tax” in his book Progress and Poverty. The tax would be leveled on the rich and poor alike, with the excess money collected used to equalize wealth and level out society. cIn Chicago, noted attorney Clarence Darrow made vocal arguments that poverty and not biology created crime. Wisconsin-born author Thorstein Veblen argued in his book The Theory of the Leisure Class that the “conspicuous consumption and conspicuous leisure" of the wealth had become the basis of social status in America. In Looking Backward author Edward Bellamy writes of a future America set in the year 2000 in which a socialist paradise has been established. The works of authors such as George and Bellamy became popular and soon clubs were created across America to discuss their ideas although these organizations rarely made any real social change. The Third Great Awakening which began before the Civil War returned and made a significant change in religious attitudes toward social progress. Followers of the new Awakening promoted the idea of the Social Gospel which gave rise to organizations such as the YMCA; Salvation Army, and settlement houses such as Hull House founded by Jane Addams in Chicago.

In computational mathematics, an iterative method attempts to solve a problem (for example an equation or system of equations) by finding successive approximations to the solution starting from an initial guess. This approach is in contrast to direct methods, which attempt to solve the problem by a finite sequence of operations, and, in the absence of rounding errors, would deliver an exact solution (like solving a linear system of equations Ax = b by Gaussian elimination). Iterative methods are usually the only choice for nonlinear equations. However, iterative methods are often useful even for linear problems involving a large number of variables (sometimes of the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.

[edit] Attractive fixed points

If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x₁ in the basin of attraction of x, and let x_n+1 = f(x_n) for n ≥ 1, and the sequence {x_n}_n ≥ 1 will converge to the solution x. If the function f is continuously differentiable, a sufficient condition for convergence is that the spectral radius of the derivative is strictly bounded by one in a neighborhood of the fixed point. If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist.

[edit] Linear systems

In the case of a system of linear equations, the two main classes of iterative methods are the stationary iterative methods, and the more general Krylov subspace methods.

[edit] Stationary iterative methods

Stationary iterative methods solve a linear system with an operator approximating the original one; and based on a measurement of the error (the residual), form a correction equation for which this process is repeated. While these methods are simple to derive, implement, and analyse, convergence is only guaranteed for a limited class of matrices. Examples of stationary iterative methods are the Jacobi method and the Gauss–Seidel method.

[edit] Krylov subspace methods

Krylov subspace methods form an orthogonal basis of the sequence of successive matrix powers times the initial residual (the Krylov sequence). The approximations to the solution are then formed by minimizing the residual over the subspace formed. The prototypical method in this class is the conjugate gradient method (CG). Other methods are the generalized minimal residual method (GMRES) and the biconjugate gradient method (BiCG).

[edit] Convergence

Since these methods form a basis, it is evident that the method converges in N iterations, where N is the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practice N can be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of the spectrum of the operator.

[edit] Preconditioners

The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such as GMRES (alternatively, preconditioned Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area.

[edit] History

Probably the first iterative method for solving a linear system appeared in a letter of Gauss to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest.

The theory of stationary iterative methods was solidly established with the work of D.M. Young starting in the 1950s. The Conjugate Gradient method was also invented in the 1950s, with independent developments by Cornelius Lanczos, Magnus Hestenes and Eduard Stiefel, but its nature and applicability were misunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for partial differential equations, especially the elliptic type.

Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or the approximate gradient) of the function at the current point. If instead one takes steps proportional to the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.

Gradient descent is also known as steepest descent, or the method of steepest descent. When known as the latter, gradient descent should not be confused with the method of steepest descent for approximating integrals.

[edit] Description

Gradient descent is based on the observation that if the real-valued function is defined and differentiable in a neighborhood of a point , then decreases fastest if one goes from in the direction of the negative gradient of F at , . It follows that, if

for γ > 0 a small enough number, then . With this observation in mind, one starts with a guess for a local minimum of F, and considers the sequence such that

We have

so hopefully the sequence converges to the desired local minimum. Note that the value of the step size γ is allowed to change at every iteration.

This process is illustrated in the picture to the right. Here F is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of F is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the function F is minimal.

[edit] Examples

Gradient descent has problems with pathological functions such as the Rosenbrock function shown here. The Rosenbrock function has a narrow curved valley which contains the minimum. The bottom of the valley is very flat. Because of the curved flat valley the optimization is zig-zagging slowly with small stepsizes towards the minimum.

The gradient ascent method applied to :

[edit] Comments

Gradient descent works in spaces of any number of dimensions, even in infinite-dimensional ones. In the latter case the search space is typically a function space, and one calculates the Gâteaux derivative of the functional to be minimized to determine the descent direction.

Two weaknesses of gradient descent are:

1. The algorithm can take many iterations to converge towards a local minimum, if the curvature in different directions is very different.
2. Finding the optimal γ per step can be time-consuming. Conversely, using a fixed γ can yield poor results. Methods based on Newton's method and inversion of the Hessian using conjugate gradient techniques are often a better alternative.

A more powerful algorithm is given by the BFGS method which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated line search algorithm, to find the "best" value of γ.

Gradient descent is in fact Euler's method for solving ordinary differential equations applied to a gradient flow. As the goal is to find the minimum, not the flow line, the error in finite methods is less significant.

[edit] A computational example

The gradient descent algorithm is applied to find a local minimum of the function f(x)=x⁴-3x³+2 , with derivative f'(x)=4x³-9x². Here is an implementation in the C programming language.

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
 
int main ()
{
	// From calculation, we expect that the local minimum occurs at x=9/4
	// The algorithm starts at x=6
 
	double xOld = 0;
	double xNew = 6;
	double eps = 0.01; // step size
	double precision = 0.00001;
	while (fabs(xNew - xOld) > precision)
	{
             xOld = xNew;
             xNew = xNew - eps*(4*xNew*xNew*xNew-9*xNew*xNew);
	}
 
	printf ("Local minimum occurs at %lg\n", xNew);
 
}

With this precision, the algorithm converges to a local minimum of 2.24996 in 70 iterations.

A more robust implementation of the algorithm would also check whether the function value indeed decreases at every iteration and would make the step size smaller otherwise. One can also use an adaptive step size which may make the algorithm converge faster.

In mathematics, the steepest descent method or saddle-point approximation is a method used to approximate integrals of the form

where f(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b could possibly be infinite. The technique is also often referred to as Laplace's method, which in fact concerns the special case of real-valued functions f admitting a maximum at a real point.

The idea of Laplace's method

Assume that the function f(x) has a unique global maximum at x₀. Then, the value f(x₀) will be larger than other values f(x). If we multiply this function by a large number M, the gap between Mf(x₀) and Mf(x) will only increase, and then it will grow exponentially for the function

As such, significant contributions to the integral of this function will come only from points x in a neighborhood of x₀, which can then be estimated.

General theory of Laplace's method

To state and prove the method, we need several assumptions. We will assume that x₀ is not an endpoint of the interval of integration, that the values f(x) cannot be very close to f(x₀) unless x is close to x₀, and that f'''(x₀) < 0.

We can expand f(x) around x₀ by Taylor's theorem,

Since f has a global maximum at x₀, and since x₀ is not an endpoint, it is a stationary point, the derivative of f vanishes at x₀. Therefore, the function f(x) may be approximated to quadratic order

for x close to x₀ (recall that the second derivative is negative at the global maximum f(x₀)). The assumptions made ensure the accuracy of the approximation

where the integral is taken in a neighborhood of x₀. This latter integral is a Gaussian integral if the limits of integration go from −∞ to +∞ (which can be assumed so because the exponential decays very fast away from x₀), and thus it can be calculated. We find

A generalization of this method and extension to arbitrary precision is provided by Fog (2008).

Steepest descent

In extensions of Laplace's method, complex analysis, and in particular Cauchy's integral formula, is used to find a contour of steepest descent for an (asymptotically with large M) equivalent integral, expressed as a line integral. In particular, if no point x₀ where the derivative of f vanishes exists on the real line, it may be necessary to deform the integration contour to an optimal one, where the above analysis will be possible. Again the main idea is to reduce, at least asymptotically, the calculation of the given integral to that of a simpler integral that can be explicitly evaluated. See the book of Erdelyi (1956) for a simple discussion (where the method is termed steepest descents).

Further generalizations

An extension of the steepest descent method is the so-called nonlinear stationary phase/steepest descent method. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann-Hilbert factorization problems.

Given a contour C in the complex sphere, a function f defined on that contour and a special point, say infinity, one seeks a function M holomorphic away from the contour C, with prescribed jump across C, and with a given normalization at infinity. If f and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution.

An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann-Hilbert problem to that of a simpler, explicitly solvable, Riemann-Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.

The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou.

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.

Complex integrals

For complex integrals in the form:

with t >> 1, we make the substitution t = iu and the change of variable s = c + ix to get the Laplace bilateral transform:

We then split g(c+ix) in its real and complex part, after which we recover u = t / i. This is useful for inverse Laplace transforms, the Perron formula and complex integration.

Example: Stirling's approximation

Laplace's method can be used to derive Stirling's approximation

for a large integer N.

From the definition of the Gamma function, we have

Now we change variables, letting

so that

Plug these values back in to obtain

This integral has the form necessary for Laplace's method with

which is twice-differentiable:

The maximum of f(z) lies at z₀=1, and the second derivative of f(z) has at this point the value -1. Therefore, we obtain

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