Connection With Differential Equations
The problem of evaluating the integral
can be reduced to an initial value problem for an ordinary differential equation. If the above integral is denoted by I(b), then the function I satisfies
Methods developed for ordinary differential equations, such as Runge–Kutta methods, can be applied to the restated problem and thus be used to evaluate the integral. For instance, the standard fourth-order Runge–Kutta method applied to the differential equation yields Simpson's rule from above.
The differential equation I ' (x) = ƒ(x) has a special form: the right-hand side contains only the dependent variable (here x) and not the independent variable (here I). This simplifies the theory and algorithms considerably. The problem of evaluating integrals is thus best studied in its own right.
Read more about this topic: Numerical Integration (quadrature)
Famous quotes containing the words connection with, connection and/or differential:
“We should always remember that the work of art is invariably the creation of a new world, so that the first thing we should do is to study that new world as closely as possible, approaching it as something brand new, having no obvious connection with the worlds we already know. When this new world has been closely studied, then and only then let us examine its links with other worlds, other branches of knowledge.”
—Vladimir Nabokov (1899–1977)
“Children of the same family, the same blood, with the same first associations and habits, have some means of enjoyment in their power, which no subsequent connections can supply; and it must be by a long and unnatural estrangement, by a divorce which no subsequent connection can justify, if such precious remains of the earliest attachments are ever entirely outlived.”
—Jane Austen (1775–1817)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)