I propose the following cleanup of the article Rayleigh–Ritz method. -- jftsang 05:28, 20 April 2014 (UTC)
In mathematics, the Rayleigh–Ritz method is a method for finding an approximation to the smallest eigenvalue of a Sturm-Liouville operator. It is named after Walther Ritz and Lord Rayleigh.
The Rayleigh-Ritz method is a direct variational method, in which the minimum of a functional defined on a normed linear space is approximated by a linear combination of elements from that space.
Suppose we are given the eigenvalue equation
where is the Sturm-Liouville operator
with appropriate boundary conditions. We seek the smallest eigenvalue of . Consider the functionals
and
It can be shown that the problem of finding the smallest eigenvalue of is equivalent to the variational problem of minimising , subject to , which in turn is equivalent to minimising . Here is a Lagrange multiplier, the possible values of which being also the eigenvalues of . We can minimise by considering a trial function , which satisfies the boundary conditions and . We make an ansatz about the form of and minimise amongst functions of this form. For example, might be taken to be a truncated Taylor series; and we conduct the minimisation of by choosing the coefficients of the series properly. The method is effective if the form that we guess for can approximate the lowest-eigenvalue eigenfunction well.
Consider the eigenvalue problem
subject to as . This is the eigenvalue problem for the quantum harmonic oscillator, in suitable units. We seek approximations to the lowest eigenvalue and eigenfunction. Here , and
The Rayleigh-Ritz method is widely used in applied mathematics and mechanical engineering for the calculation of the natural vibration frequency of a structure in the second or higher order, as it can give a useful approximation even when the true solution may be intractable.
Typically in mechanical engineering it is used for finding the approximate real resonant frequencies of multi degree of freedom systems, such as spring mass systems or flywheels on a shaft with varying cross section. It is an extension of Rayleigh's method. It can also be used for finding buckling loads and post-buckling behaviour for columns.
The Rayleigh-Ritz method is also widely used in quantum chemistry.
Category:Numerical differential equations
Category:Dynamical systems
The first direct contact between the English and the Chinese was on 27 June 1637, when four heavily armed ships under Captain John Wendell arrived at Macau in an attempt to open trade between England and China. They were not backed by the East India Company, but rather by a private group led by Sir William Courteen, including King Charles I's personal interest of £10,000. They were opposed by the Portuguese authorities in Macau (who were required to do so under their agreements with China) and quickly infuriated the Ming authorities. Later in the summer they easily captured one of the Bogue forts, and spent several weeks engaged in low-level fighting and smuggling. After being forced to seek Portuguese help in the release of three hostages, they left the Pearl River on 27 December. It is unclear whether they returned home. [1] [2] [3]
The Ming dynasty was conquered by the Manchu-ruled Qing dynasty in 1644. In 1706, England and Scotland signed the Treaty of Union and became the Kingdom of Great Britain. Later, the Acts of Union 1800 would establish the United Kingdom of Great Britain and Ireland.
In 1685, Michael Shen Fu-Tsung visited England and met James II. [4] A century later, George Macartney, 1st Earl Macartney led the Macartney Embassy to Beijing in 1793.
The countries traded throughout the 18th century, especially as China gradually loosened restrictions. During this time, Britain imported much tea and other goods from China, against whom it built up a large trade deficit. To correct the deficit, Britain started exporting opium to China in 1817. In the 1820s, British merchants turned Lintin Island in the Pearl River estuary into a centre of opium trade. [5] [6] Opium use became very prevalent in China and the balance of trade was reversed. The Qing, concerned at both the trade deficit and the social ills caused by opium, began to restrict the import of opium. Britain declared the First Opium War in 1839, after opium carried by British traders was seized and destroyed by a Chinese official.
In 1841, the Convention of Chuenpee was signed. The convention was intended to end the war with Hong Kong Island ceded to Britain, but it was never ratified. The war lasted until the signing of the Treaty of Nanking on 29 August 1842. Under the treaty, Hong Kong Island was ceded the British, and China opened five treaty ports to international trade. The Treaty of Nanking was supplemented by the Treaty of the Bogue (October 1843), which granted extraterritoriality to British subjects in China and most favoured nation status to Britain.
Further British and French demands for the legalisation of opium and the opening of ports were rejected by the Qing, and in 1856 the Second Opium War broke out. The war had a brief interlude in 1858, in the June of which the Treaty of Tientsin was signed by Lord Elgin. The treaty would open more ports to trade, grant more privileges to foreigners in China, and legalise opium in China, but it was at first not ratified by China. Fighting resumed until October 1860, when the victorious British and French troops sacked and destroyed Old Summer Palace. China ratified the Treaty of Tientsin and signed the Convention of Peking, ceding Kowloon Peninsula to Britain.
In accordance with the treaties, a British Legation opened in Beijing (Peking) on 26 March 1861. In the following few years consulates opened throughout the Empire, including Hankou ( Wuhan), Takao ( Kaohsiung), Tamsui (near Taipei), Shanghai and Xiamen.
The United Kingdom and the anti-Communist Nationalist Chinese government were allies during World War II. Britain sought stability in China after the war to protect its more than £300 million in investments, much more than from the United States. It agreed in the Moscow Agreement of 1945 to not interfere in Chinese affairs but sympathized with the Nationalists, who until 1947 were winning the Chinese Civil War against the Communist Party of China. By August 1948, however, the Communists' victories caused the British government to begin preparing for a Communist takeover of the country. It kept open consulates in Communist-controlled areas and rejected the Nationalists' requests that British citizens assist in the defense of Shanghai. By December, the government concluded that although British property in China would likely be nationalized, British traders would benefit in the long run from a stable, industrializing Communist China. Retaining Hong Kong was especially important; although the Communists promised to not interfere with its rule, Britain reinforced the Hong Kong Garrison during 1949. When the victorious Communist government declared on 1 October 1949 that it would exchange diplomats with any country that ended relations with the Nationalists, Britain—after discussions with other Commonwealth members and European countries—formally recognised the People's Republic of China in January 1950. [12]
Suppose the symmetric matrix M0 has known eigenvalues and eigenvectors: in particular, suppose that
with x0≠0, and real λ. Assume that this is the only eigenvector (up to scalar multiplication) of M with this eigenvalue. Now consider a second matrix
where ε is small. For sufficiently small ε, there will be an eigenvalue λ and an eigenvector x of M that 'corresponds' to λ0 and x0.
We write the new eigenvalue and eigenvector as a perturbation series in ε:
Then the equation
becomes
Multiplying this out and comparing the powers of ε lets us find the corrections iteratively. For example, comparing the O(ε) gives
Multiply both sides on the left by x0T. (If M were not symmetric, then this would not necessarily be the left eigenvector of M.) The left hand side vanishes, and so
But x0≠0. Therefore
english.peopledaily.com.cn
was invoked but never defined (see the
help page).chinadaily.com.cn
was invoked but never defined (see the
help page).I propose the following cleanup of the article Rayleigh–Ritz method. -- jftsang 05:28, 20 April 2014 (UTC)
In mathematics, the Rayleigh–Ritz method is a method for finding an approximation to the smallest eigenvalue of a Sturm-Liouville operator. It is named after Walther Ritz and Lord Rayleigh.
The Rayleigh-Ritz method is a direct variational method, in which the minimum of a functional defined on a normed linear space is approximated by a linear combination of elements from that space.
Suppose we are given the eigenvalue equation
where is the Sturm-Liouville operator
with appropriate boundary conditions. We seek the smallest eigenvalue of . Consider the functionals
and
It can be shown that the problem of finding the smallest eigenvalue of is equivalent to the variational problem of minimising , subject to , which in turn is equivalent to minimising . Here is a Lagrange multiplier, the possible values of which being also the eigenvalues of . We can minimise by considering a trial function , which satisfies the boundary conditions and . We make an ansatz about the form of and minimise amongst functions of this form. For example, might be taken to be a truncated Taylor series; and we conduct the minimisation of by choosing the coefficients of the series properly. The method is effective if the form that we guess for can approximate the lowest-eigenvalue eigenfunction well.
Consider the eigenvalue problem
subject to as . This is the eigenvalue problem for the quantum harmonic oscillator, in suitable units. We seek approximations to the lowest eigenvalue and eigenfunction. Here , and
The Rayleigh-Ritz method is widely used in applied mathematics and mechanical engineering for the calculation of the natural vibration frequency of a structure in the second or higher order, as it can give a useful approximation even when the true solution may be intractable.
Typically in mechanical engineering it is used for finding the approximate real resonant frequencies of multi degree of freedom systems, such as spring mass systems or flywheels on a shaft with varying cross section. It is an extension of Rayleigh's method. It can also be used for finding buckling loads and post-buckling behaviour for columns.
The Rayleigh-Ritz method is also widely used in quantum chemistry.
Category:Numerical differential equations
Category:Dynamical systems
The first direct contact between the English and the Chinese was on 27 June 1637, when four heavily armed ships under Captain John Wendell arrived at Macau in an attempt to open trade between England and China. They were not backed by the East India Company, but rather by a private group led by Sir William Courteen, including King Charles I's personal interest of £10,000. They were opposed by the Portuguese authorities in Macau (who were required to do so under their agreements with China) and quickly infuriated the Ming authorities. Later in the summer they easily captured one of the Bogue forts, and spent several weeks engaged in low-level fighting and smuggling. After being forced to seek Portuguese help in the release of three hostages, they left the Pearl River on 27 December. It is unclear whether they returned home. [1] [2] [3]
The Ming dynasty was conquered by the Manchu-ruled Qing dynasty in 1644. In 1706, England and Scotland signed the Treaty of Union and became the Kingdom of Great Britain. Later, the Acts of Union 1800 would establish the United Kingdom of Great Britain and Ireland.
In 1685, Michael Shen Fu-Tsung visited England and met James II. [4] A century later, George Macartney, 1st Earl Macartney led the Macartney Embassy to Beijing in 1793.
The countries traded throughout the 18th century, especially as China gradually loosened restrictions. During this time, Britain imported much tea and other goods from China, against whom it built up a large trade deficit. To correct the deficit, Britain started exporting opium to China in 1817. In the 1820s, British merchants turned Lintin Island in the Pearl River estuary into a centre of opium trade. [5] [6] Opium use became very prevalent in China and the balance of trade was reversed. The Qing, concerned at both the trade deficit and the social ills caused by opium, began to restrict the import of opium. Britain declared the First Opium War in 1839, after opium carried by British traders was seized and destroyed by a Chinese official.
In 1841, the Convention of Chuenpee was signed. The convention was intended to end the war with Hong Kong Island ceded to Britain, but it was never ratified. The war lasted until the signing of the Treaty of Nanking on 29 August 1842. Under the treaty, Hong Kong Island was ceded the British, and China opened five treaty ports to international trade. The Treaty of Nanking was supplemented by the Treaty of the Bogue (October 1843), which granted extraterritoriality to British subjects in China and most favoured nation status to Britain.
Further British and French demands for the legalisation of opium and the opening of ports were rejected by the Qing, and in 1856 the Second Opium War broke out. The war had a brief interlude in 1858, in the June of which the Treaty of Tientsin was signed by Lord Elgin. The treaty would open more ports to trade, grant more privileges to foreigners in China, and legalise opium in China, but it was at first not ratified by China. Fighting resumed until October 1860, when the victorious British and French troops sacked and destroyed Old Summer Palace. China ratified the Treaty of Tientsin and signed the Convention of Peking, ceding Kowloon Peninsula to Britain.
In accordance with the treaties, a British Legation opened in Beijing (Peking) on 26 March 1861. In the following few years consulates opened throughout the Empire, including Hankou ( Wuhan), Takao ( Kaohsiung), Tamsui (near Taipei), Shanghai and Xiamen.
The United Kingdom and the anti-Communist Nationalist Chinese government were allies during World War II. Britain sought stability in China after the war to protect its more than £300 million in investments, much more than from the United States. It agreed in the Moscow Agreement of 1945 to not interfere in Chinese affairs but sympathized with the Nationalists, who until 1947 were winning the Chinese Civil War against the Communist Party of China. By August 1948, however, the Communists' victories caused the British government to begin preparing for a Communist takeover of the country. It kept open consulates in Communist-controlled areas and rejected the Nationalists' requests that British citizens assist in the defense of Shanghai. By December, the government concluded that although British property in China would likely be nationalized, British traders would benefit in the long run from a stable, industrializing Communist China. Retaining Hong Kong was especially important; although the Communists promised to not interfere with its rule, Britain reinforced the Hong Kong Garrison during 1949. When the victorious Communist government declared on 1 October 1949 that it would exchange diplomats with any country that ended relations with the Nationalists, Britain—after discussions with other Commonwealth members and European countries—formally recognised the People's Republic of China in January 1950. [12]
Suppose the symmetric matrix M0 has known eigenvalues and eigenvectors: in particular, suppose that
with x0≠0, and real λ. Assume that this is the only eigenvector (up to scalar multiplication) of M with this eigenvalue. Now consider a second matrix
where ε is small. For sufficiently small ε, there will be an eigenvalue λ and an eigenvector x of M that 'corresponds' to λ0 and x0.
We write the new eigenvalue and eigenvector as a perturbation series in ε:
Then the equation
becomes
Multiplying this out and comparing the powers of ε lets us find the corrections iteratively. For example, comparing the O(ε) gives
Multiply both sides on the left by x0T. (If M were not symmetric, then this would not necessarily be the left eigenvector of M.) The left hand side vanishes, and so
But x0≠0. Therefore
english.peopledaily.com.cn
was invoked but never defined (see the
help page).chinadaily.com.cn
was invoked but never defined (see the
help page).