Math 2.0

Numerical analysis Definition

Numerical integration

Builds on Definite integralIntegrable function

Definition

Let a<ba < b be real numbers and let FF be a class of integrable functions f ⁣:[a,b]→ℝf \colon [a, b] \to \R, with the definite integral I(f)=∫abf(x) dxI(f) = \int_a^b f(x)\, dx. A numerical integration method on FF is a rule that assigns to each f∈Ff \in F a number Q(f)Q(f). The number Q(f)Q(f) is computed from finitely many values f(x1),…,f(xn)f(x_1), \dots, f(x_n) at points xi∈[a,b]x_i \in [a, b], and possibly from finitely many values of derivatives of ff. Its error at ff is E(f)=I(f)−Q(f)E(f) = I(f) - Q(f). Numerical integration is the design and analysis of such methods, so that ∣E(f)∣\abs{E(f)} is small for the functions of interest.

Remarks

The same definition applies to integrals over regions of ℝd\R^d or with a weight function; in several dimensions one often speaks of cubature. Example: the midpoint method Q(f)=(b−a)f(a+b2)Q(f) = (b - a) f\big(\tfrac{a + b}{2}\big) on continuous functions uses one value; for f(x)=x2f(x) = x^2 on [0,1][0, 1] it gives 1/41/4, with error 1/3−1/4=1/121/3 - 1/4 = 1/12. Non-example: Q(f)=F(b)−F(a)Q(f) = F(b) - F(a) with an antiderivative FF is exact, but it is not a numerical integration method in this sense, because it needs an antiderivative, not finitely many values of ff. The points xix_i may also be chosen adaptively, depending on earlier values, as in adaptive quadrature, or at random, as in Monte Carlo integration.

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Integral of a non-negative function, concept, no evidence levelIntegral of a non-neg…conceptMeasurable function, concept, no evidence levelMeasurable functionconceptMeasure space, concept, no evidence levelMeasure spaceconceptPartition of an interval, definition, no evidence levelPartition of an inter…definitionIntegrable function, definition, no evidence levelIntegrable functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionRiemann integral, definition, no evidence levelRiemann integraldefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionNumerical integration, definition, no evidence levelNumerical integrationdefinition
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  1. Integral of a non-negative function concept
  2. Measurable function concept
  3. Measure space concept
  4. Partition of an interval definition
  5. Integrable function definition
  6. Mesh of a partition definition
  7. Riemann sum definition
  8. Riemann integrable function definition
  9. Riemann integral definition
  10. Definite integral definition
  11. Numerical integration definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Numerical_integration
About this page
TagA915CSRFEF
URIm2://A915CSRFEF
Kinddefinition
WritingW0 stub
MSC65D30, 65D32, 41A55
DefinesNumerical integration
UsesDefinite integral, Integrable function

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