Rock musicRock is a broad genre of popular music that originated as "rock and roll" in the United States in the late 1940s and early 1950s, developing into a range of different styles from the mid-1960s, particularly in the U.S. and the United Kingdom. It has its roots in 1940s and 1950s rock and roll, a style that drew directly from the blues and rhythm and blues genres of African-American music and from country music. Rock also drew strongly from genres such as electric blues and folk, and incorporated influences from jazz, classical, and other musical styles.
Stiff equationIn mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.
Folk rockFolk rock is a genre of rock music with heavy influences from English folk and American folk music. Combining the elements of folk and rock music, it arose in the United States, Canada, and the United Kingdom in the mid-1960s. In the U.S., folk rock emerged from the folk music revival. Performers such as Bob Dylan and the Byrds—several of whose members had earlier played in folk ensembles—attempted to blend the sounds of rock with their pre-existing folk repertoire, adopting the use of electric instrumentation and drums in a way previously discouraged in the U.
Chebyshev polynomialsThe Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind are defined by Similarly, the Chebyshev polynomials of the second kind are defined by That these expressions define polynomials in may not be obvious at first sight, but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly.
Numerical methods for ordinary differential equationsNumerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.
Euler methodIn mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).
Runge–Kutta methodsIn numerical analysis, the Runge–Kutta methods (ˈrʊŋəˈkʊtɑː ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method".
Electronics in rock musicThe use of electronic music technology in rock music coincided with the practical availability of electronic musical instruments and the genre's emergence as a distinct style. Rock music has been highly dependent on technological developments, particularly the invention and refinement of the synthesizer, the development of the MIDI digital format and computer technology.
Explicit and implicit methodsExplicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.
Progressive rockProgressive rock (shortened as prog rock or simply prog) is a broad genre of rock music that primarily developed in the United Kingdom and United States through the mid- to late 1960s, peaking in the early 1970s. Initially termed "progressive pop", the style was an emergence of psychedelic bands who abandoned standard pop traditions in favour of instrumentation and compositional techniques more frequently associated with jazz, folk, or classical music.
