Numeric.Tools.Integration

Funtions for numerical integration. quadRomberg or quadSimpson are reasonable choices in most cases. For non-smooth function they converge poorly and quadTrapezoid should be used then.

For example this code intergrates exponent from 0 to 1:

>>> let res = quadRomberg defQuad (0, 1) exp

>>> quadRes res     -- Integration result
Just 1.718281828459045

>>> quadPrecEst res -- Estimate of precision
2.5844957590474064e-16

>>> quadNIter res   -- Number of iterations performed
6

Integration parameters and results

data QuadParam

Integration parameters for numerical routines. Note that each additional iteration doubles number of function evaluation required to compute integral.

Number of iterations is capped at 30.

Constructors

QuadParam

Fields

quadPrecision :: Double

Relative precision of answer

quadMaxIter :: Int

Maximum number of iterations

Instances

Eq QuadParam

Data QuadParam

Show QuadParam

Typeable QuadParam

defQuad :: QuadParam

Default parameters for integration functions

• Maximum number of iterations = 20
• Precision is 10⁻⁹

data QuadRes

Result of numeric integration.

Constructors

QuadRes

Fields

quadRes :: Maybe Double

Integraion result

quadBestEst :: Double

Best estimate of integral

quadPrecEst :: Double

Rough estimate of attained precision

quadNIter :: Int

Number of iterations

Instances

Eq QuadRes

Data QuadRes

Show QuadRes

Typeable QuadRes

Integration functions

quadTrapezoid

Arguments

:: QuadParam

Parameters

-> (Double, Double)

Integration limits

-> (Double -> Double)

Function to integrate

-> QuadRes

Integration of using trapezoids. This is robust algorithm and place and useful for not very smooth. But it is very slow. It hundreds times slower then quadRomberg if function is sufficiently smooth.

quadSimpson

Arguments

:: QuadParam

Parameters

-> (Double, Double)

Integration limits

-> (Double -> Double)

Function to integrate

-> QuadRes

Integration using Simpson rule. It should be more efficient than quadTrapezoid if function being integrated have finite fourth derivative.

quadRomberg

Arguments

:: QuadParam

Parameters

-> (Double, Double)

Integration limits

-> (Double -> Double)

Function to integrate

-> QuadRes

Integration using Romberg rule. For sufficiently smooth functions (e.g. analytic) it's a fastest of three.