Integration and derivatives
Integrate univariate and multivariate functions.
Integrant is substituted in case of improper integration bounds.
Derivatives (finite differences method):
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
(cubature f lower upper)
(cubature f lower upper options)
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
- integrantlower
- seq of lower boundsupper
- seq of upper boundsOptions:
: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: falseFunction 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.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.
(derivative f)
(derivative f n)
(derivative f n options)
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 optionsCreate 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
(extrapolate g)
(extrapolate g options)
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 integerRichardson 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
(f' f)
First central derivative with order of accuracy 2.
First central derivative with order of accuracy 2.
(f'' f)
Second central derivative with order of accuracy 2.
Second central derivative with order of accuracy 2.
(f''' f)
Third central derivative with order of accuracy 2.
Third central derivative with order of accuracy 2.
(fx->gx+h f)
Convert f(x) to g(x,h)=f(x+h)
Convert f(x) to g(x,h)=f(x+h)
(fx->gx-h f)
Convert f(x) to g(x,h)=f(x-h)
Convert f(x) to g(x,h)=f(x-h)
(gradient f)
(gradient f options)
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
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`
(hessian f)
(hessian f options)
Creates function returning Hessian matrix for mulitvariate function f
and given :h
step (default: 5.0e-3
).
Creates function returning Hessian matrix for mulitvariate function `f` and given `:h` step (default: `5.0e-3`).
(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
- integrantlower
- lower boundupper
- upper boundOptions:
: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.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.
(vegas f lower upper)
(vegas f lower upper options)
VEGAS+ - Monte Carlo integration of multivariate function, n>1 dimensions.
Improper integrals with infinite bounds are handled by a substitution.
Arguments:
f
- integrantlower
- seq of lower boundsupper
- seq of upper boundsAdditional 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)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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