Search results
Results From The WOW.Com Content Network
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann . One very common application is in numerical integration , i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule .
Generally speaking, Riemann solvers are specific methods for computing the numerical flux across a discontinuity in the Riemann problem. [1] They form an important part of high-resolution schemes; typically the right and left states for the Riemann problem are calculated using some form of nonlinear reconstruction, such as a flux limiter or a WENO method, and then used as the input for the ...
Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be integrable (or more specifically Riemann-integrable). The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.
Discrete integral calculus is the study of the definitions, properties, and applications of the Riemann sums. The process of finding the value of a sum is called integration . In technical language, integral calculus studies a certain linear operator .
The Poisson summation formula is also useful to bound the errors obtained when an integral is approximated by a (Riemann) sum. Consider an approximation of S ( 0 ) = ∫ − ∞ ∞ d x s ( x ) {\textstyle S(0)=\int _{-\infty }^{\infty }dx\,s(x)} as δ ∑ n = − ∞ ∞ s ( n δ ) {\textstyle \delta \sum _{n=-\infty }^{\infty }s(n\delta ...
Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral: ∑ x < n ≤ y a n ϕ ( n ) = A ( y ) ϕ ( y ) − A ( x ) ϕ ( x ) − ∫ x y A ( u ) d ϕ ( u ) . {\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x ...
Riemann multiple integral; Riemann invariant; Riemann mapping theorem. Measurable Riemann mapping theorem; Riemann problem; Riemann solver; Riemann sphere; Riemann–Hilbert correspondence; Riemann–Hilbert problem; Riemann–Lebesgue lemma; Riemann–Liouville integral; Riemann–Roch theorem. Arithmetic Riemann–Roch theorem; Riemann–Roch ...
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite ...