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fastmath.calculus

Integration and derivatives

Integrate univariate and multivariate functions.

  • VEGAS / VEGAS+ - Monte Carlo integration of multivariate function
  • h-Cubature - h-adaptive integration of multivariate function
  • Guass-Kronrod and Gauss-Legendre - quadrature integration of univariate functions
  • Romberg, Simpson, MidPoint and Trapezoid

Integrant is substituted in case of improper integration bounds.

Derivatives (finite differences method):

  • derivatives of any degree and any order of accuracy
  • gradient and hessian for multivariate functions

Categories

    Other vars: ->CubatureBox ->QuadGKSegment cubature derivative extrapolate f' f'' f''' fx->gx+h fx->gx-h gradient hessian integrate vegas

    cubature

    (cubature f lower upper)(cubature f lower upper {:keys [rel abs max-evals max-iters initdiv info?], :or {rel 1.0E-7, abs 1.0E-7, max-evals Integer/MAX_VALUE, max-iters 64, initdiv 2, info? false}})

    Cubature - h-adaptive integration of multivariate function, n>1 dimensions.

    Algorithm uses Genz Malik method.

    In each iteration a box with biggest error is subdivided and reevaluated.

    Improper integrals with infinite bounds are handled by a substitution.

    Arguments:

    • f - integrant
    • lower - seq of lower bounds
    • upper - seq of upper bounds

    Options:

    • :initvid - initial subdivision per dimension, default: 2.
    • :max-evals - maximum number of evaluations, default: max integer value.
    • :max-iters - maximum number of iterations, default: 64.
    • :rel - relative error, 1.0e-7
    • :abs - absolute error, 1.0e-7
    • :info? - return full information about integration, default: false

    Function returns a map containing (if info? is true, returns result otherwise):

    • :result - integration value
    • :error - integration error
    • :iterations - number of iterations
    • :evaluations - number of evaluations
    • :subdivisions - final number of boxes
    • :fail? - set to :max-evals or :max-iters when one of the limits has been reached without the convergence.
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    derivative

    (derivative f)(derivative f n)(derivative f n {:keys [acc h method extrapolate?], :or {acc 2, h 0.0, method :central}})

    Create nth derivative of f using finite difference method for given accuracy :acc and step :h.

    Returns function.

    Arguments:

    • n - derivative
    • :acc - order of accuracy (default: 2)
    • :h - step, (default: 0.0, automatic)
    • :method - :central (default), :forward or :backward
    • :extrapolate? - creates extrapolated derivative if set to true or a map with extrapolate function options
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    extrapolate

    (extrapolate g)(extrapolate g {:keys [contract power init-h rel abs max-evals tol], :or {contract 0.5, power 1.0, init-h 0.5, abs m/MACHINE-EPSILON, rel (m/sqrt (m/ulp init-h)), max-evals Integer/MAX_VALUE, tol 2.0}})

    Richardson extrapolation for given function g=g(x,h). Returns extrapolated function f(x).

    Options:

    • :contract - shrinkage factor, default=1/2
    • :power - set to 2.0 for even functions around x0, default 1.0
    • :init-h - initial step h, default=1/2
    • :abs - absolute error, default: machine epsilon
    • :rel - relative error, default: ulp for init-h
    • :tol - tolerance for error, default: 2.0
    • :max-evals - maximum evaluations, default: maximum integer
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    f'

    (f' f)

    First central derivative with order of accuracy 2.

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    f''

    (f'' f)

    Second central derivative with order of accuracy 2.

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    f'''

    (f''' f)

    Third central derivative with order of accuracy 2.

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    fx->gx+h

    (fx->gx+h f)

    Convert f(x) to g(x,h)=f(x+h)

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    fx->gx-h

    (fx->gx-h f)

    Convert f(x) to g(x,h)=f(x-h)

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    gradient

    (gradient f)(gradient f {:keys [h acc], :or {h 1.0E-6, acc 2}})

    Create first partial derivatives of multivariate function for given accuracy :acc and step :h.

    Returns function.

    Options:

    • :acc - order of accuracy, 2 (default) or 4.
    • :h - step, default 1.0e-6
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    hessian

    (hessian f)(hessian f {:keys [h], :or {h 0.005}})

    Creates function returning Hessian matrix for mulitvariate function f and given :h step (default: 5.0e-3).

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    integrate

    (integrate f)(integrate f lower upper)(integrate f lower upper {:keys [rel abs max-iters min-iters max-evals info? integrator integration-points], :or {rel BaseAbstractUnivariateIntegrator/DEFAULT_RELATIVE_ACCURACY, abs BaseAbstractUnivariateIntegrator/DEFAULT_ABSOLUTE_ACCURACY, min-iters BaseAbstractUnivariateIntegrator/DEFAULT_MIN_ITERATIONS_COUNT, max-evals Integer/MAX_VALUE, integration-points 7, integrator :gauss-kronrod, info? false}, :as options})

    Univariate integration.

    Improper integrals with infinite bounds are handled by a substitution.

    Arguments:

    • f - integrant
    • lower - lower bound
    • upper - upper bound

    Options:

    • :integrator - integration algorithm, one of: :romberg, :trapezoid, :midpoint, :simpson, :gauss-legendre and :gauss-kronrod (default).
    • :min-iters - minimum number of iterations (default: 3), not used in :gauss-kronrod
    • :max-iters - maximum number of iterations (default: 32 or 64)
    • :max-evals - maximum number of evaluations, (default: maximum integer)
    • :rel - relative error
    • :abs - absolute error
    • :integration-points - number of integration (quadrature) points for :gauss-legendre and :gauss-kronrod, default 7
    • :initdiv - initial number of subdivisions for :gauss-kronrod, default: 1
    • :info? - return full information about integration, default: false

    :gauss-kronrod is h-adaptive implementation

    Function returns a map containing (if info? is true, returns result otherwise):

    • :result - integration value
    • :error - integration error (:gauss-kronrod only)
    • :iterations - number of iterations
    • :evaluations - number of evaluations
    • :subdivisions - final number of boxes (:gauss-kronrod only)
    • :fail? - set to :max-evals or :max-iters when one of the limits has been reached without the convergence.
    view source

    vegas

    (vegas f lower upper)(vegas f lower upper {:keys [max-iters rel abs nevals alpha beta warmup info? record-data?], :or {max-iters 10, rel 5.0E-4, abs 5.0E-4, nevals 10000, beta 0.75, warmup 0, info? false, record-data? false}, :as options})

    VEGAS+ - Monte Carlo integration of multivariate function, n>1 dimensions.

    Improper integrals with infinite bounds are handled by a substitution.

    Arguments:

    • f - integrant
    • lower - seq of lower bounds
    • upper - seq of upper bounds

    Additional options:

    • :max-iters - maximum number of iterations, default: 10
    • :nevals - number of evaluations per iteration, default: 10000
    • :nintervals - number of grid intervals per dimension (default: 1000)
    • :nstrats - number of stratifications per dimension (calculated)
    • :warmup - number of warmup iterations (results are used to train stratification and grid spacings, default: 0
    • :alpha - grid refinement parameter, 0.5 slow (default for vegas+), 1.5 moderate/fast (defatult for vegas)
    • :beta - stratification damping parameter for startification adaptation, default: 0.75
    • :rel - relative accuracy, default: 5.0e-4
    • :abs - absolute accuracy, default: 5.0e-4
    • :random-sequence - random sequence used for generating samples: :uniform (default), low-discrepancy sequences: :r2, :sobol and :halton.
    • :jitter - jittering factor for low-discrepancy random sequence, default: 0.75
    • :info? - return full information about integration, default: false
    • :record-data? - stores samples, number of strata, x and dx, default: false (requires, :info? to be set to true)

    For original VEGAS algorithm set :nstrats to 1.

    :nstrats can be also a list, then each dimension is divided independently according to a given number. If list is lower then number of dimensions, then it’s cycled.

    Function returns a map with following keys (if info? is true, returns result otherwise):

    • :result - value of integral
    • :iterations - number of iterations (excluding warmup)
    • :sd - standard deviation of results
    • :nintervals - actual grid size
    • :nstrats - number of stratitfications per dimension
    • :nhcubes - number of hypercubes
    • :evaluations - number of function calls
    • :chi2-avg - average of chi2
    • :dof - degrees of freedom
    • :Q - goodness of fit indicator, 1 - very good, <0.25 very poor
    • :data - recorded data (if available)
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