# Quadrature A structure that approximates the definite integral of a function over a finite interval. ``` struct Quadrature ``` ## Overview The following code illustrates using a Quadrature structure to calculate the area under a curve, defined by `y = sqrt(radius * radius - pow(x - radius, 2))`: ```swift let quadrature = Quadrature(integrator: .qags(maxIntervals: 10), absoluteTolerance: 1.0e-8, relativeTolerance: 1.0e-2) let result = quadrature.integrate(over: 0.0 ... 25.0) { x in let radius: Double = 12.5 return sqrt(radius * radius - pow(x - radius, 2)) } switch result { case .success(let integralResult, let estimatedAbsoluteError): print("quadrature success:", integralResult, estimatedAbsoluteError) case .failure(let error): print("quadrature error:", error.errorDescription) } ``` Alternatively, you can integrate over a function that uses vectors for its source and destination. For example: ```swift func vectorExp(x: UnsafeBufferPointer, y: UnsafeMutableBufferPointer) { let radius: Double = 12.5 for i in 0 ..< x.count { y[i] = sqrt(radius * radius - pow(x[i] - radius, 2)) } } let vRresult = quadrature.integrate(over: 0.0 ... diameter, integrand: vectorExp) ``` ## Topics ### Initializers [`init(integrator:absoluteTolerance:relativeTolerance:)`](/documentation/Accelerate/Quadrature/init(integrator:absoluteTolerance:relativeTolerance:)) Initializes and returns a quadrature instance. ### Instance Properties [`absoluteTolerance`](/documentation/Accelerate/Quadrature/absoluteTolerance) The requested absolute tolerance on the result. [`relativeTolerance`](/documentation/Accelerate/Quadrature/relativeTolerance) The requested relative tolerance on the result. ### Instance Methods [`integrate(over:integrand:)`](/documentation/Accelerate/Quadrature/integrate(over:integrand:)-2d2y2) Performs the integration over the supplied scalar function. [`integrate(over:integrand:)`](/documentation/Accelerate/Quadrature/integrate(over:integrand:)-56h8b) Performs the integration over the supplied vector function. ### Structures [`Quadrature.QAGPointsPerInterval`](/documentation/Accelerate/Quadrature/QAGPointsPerInterval) Constants that specify the number of points per interval for the globally adaptive integrator. ### Enumerations [`Quadrature.Error`](/documentation/Accelerate/Quadrature/Error) Errors thrown by the Quadrature structure. [`Quadrature.Integrator`](/documentation/Accelerate/Quadrature/Integrator) Constants that define different integrators. ### Type Aliases [`quadrature_function_array`](/documentation/Accelerate/quadrature_function_array) [`quadrature_integrate_function`](/documentation/Accelerate/quadrature_integrate_function) [`quadrature_integrate_options`](/documentation/Accelerate/quadrature_integrate_options) --- Copyright © 2026 Apple Inc. All rights reserved. | [Terms of Use](https://www.apple.com/legal/internet-services/terms/site.html) | [Privacy Policy](https://www.apple.com/privacy/privacy-policy)