AdaptiveGaussKronrod (f,a,b)Find the integral of f over the interval [a,b] using an adaptive algorithm using Gauss-Kronrod rule G7 K15.
It will subdivide adaptively until the relative error is less than
AdaptiveGaussKronrodRelativeTolerance
or the absolute error is within
AdaptiveGaussKronrodAbsoluteTolerance.
The subinterval with the largest error is subdivided into two until we get a small enough error or until we hit
AdaptiveGaussKronrodMaxIterations iterations.
If an estimate within the given range is not achieved within the iteration limit, then null is returned and error is
printed.
See Wikipedia for more information.
Version 1.0.28 onwards.
AdaptiveGaussKronrod (f,a,b,abstol,reltol)Find the integral of f over the interval [a,b] using an adaptive algorithm using Gauss-Kronrod rule G7 K15.
It will subdivide adaptively until the relative error is less than abstol
or the absolute error is within reltol.
The subinterval with the largest error is subdivided into two until we get a small enough error or until we hit
AdaptiveGaussKronrodMaxIterations iterations.
If an estimate within the given range is not achieved within the iteration limit, then null is returned and error is
printed.
This function is useful if different precision than the defaults is needed and one does not want to change global parameters. For example, if less precision is needed and speed is paramount. Otherwise just use AdaptiveGaussKronrod or NumericalIntegral
See Wikipedia for more information.
Version 1.0.28 onwards.
CompositeSimpsonsRule (f,a,b,n)
Integration of f by Composite 1/3 Simpson's Rule
on the interval [a,b] with n subintervals with error of
max(f'''')*h^4*(b-a)/180, note that n should be even.
If the given n is odd, then 1 is added to make it even.
It is the 1/3 variant of the rule that is used, that is, if the x0,x1,x2,...,xn are the points, then the rule is ((b-a)/n) * (f(x0) + 4*f(x1) + 2*f(x2) + ... + 4*f(x(n-1)) + f(xn)).
The n argument is optional. If it is not given
the value of NumericalIntegralSteps is used, which is by default 1000.
See Wikipedia or Planetmath for more information.
CompositeSimpsonsRule (f,len)Integration using Composite Simpson's rule of a function given by a vector of values f given at equal subintervals. The integration interval is taken to be of length len, that is,
if the interval is [a,b], then len should be b-a. The vector f should have at least 3 values (representing 2 subintervals). Normally the 1/3 rule is used. If there is an odd number of subintervals, the 3/8 rule is used on the last 3 subintervals.
See Wikipedia or Planetmath for more information.
Version 1.0.28 onwards.
CompositeSimpsonsRuleTolerance (f,a,b,FourthDerivativeBound,Tolerance)
Integration of f by Composite 1/3 Simpson's Rule on the interval [a,b] with the number of steps calculated by the fourth derivative bound and the desired tolerance.
See Wikipedia or Planetmath for more information.
Derivative (f,x0)
Tente de calculer la dérivée en essayant d'abord une approche symbolique puis numérique.
See Wikipedia for more information.
EvenPeriodicExtension (f,L)
Return a function that is the even periodic extension of
f with half period L. That
is a function defined on the interval [0,L]
extended to be even on [-L,L] and then
extended to be periodic with period 2*L.
Consultez aussi OddPeriodicExtension et PeriodicExtension.
Version 1.0.7 onwards.
FourierSeriesFunction (a,b,L)
Return a function that is a Fourier series with the
coefficients given by the vectors a (sines) and
b (cosines). Note that a@(1) is
the constant coefficient! That is, a@(n) refers to
the term cos(x*(n-1)*pi/L), while
b@(n) refers to the term
sin(x*n*pi/L). Either a
or b can be null.
GaussKronrodRule (f,a,b)A single shot Gauss-Kronrod rule G7 K15 over the interval [a,b]. It returns a vector where the first element is
the approximate integral and the second is the approximate error obtained by subtracting the G7 and K15 approximates.
This is already quite good, but often it is better to call it from within the
AdaptiveGaussKronrod function,
which is the default for
NumericalIntegral.
See Wikipedia for more information.
Version 1.0.28 onwards.
InfiniteProduct (fonc,début,inc)
Essaie de calculer un produit infini pour une fonction à un seul argument.
InfiniteProduct2 (fonc,param,début,inc)
Essaie de calculer un produit infini pour une fonction à double arguments avec func(arg,n).
InfiniteSum (fonc,début,inc)
Essaie de calculer une somme infinie pour une fonction à un seul argument.
InfiniteSum2 (fonc,param,début,inc)
Essaie de calculer une somme infinie pour une fonction à double arguments avec func(arg,n).
IsContinuous (f,x0)
Essaie de voir si une fonction à valeur réelle est continue en x0 en calculant la limite en ce point.
IsDifferentiable (f,x0)
Teste de différentiabilité en approchant les limites gauche et droite et en les comparant.
LeftHandRule (f,a,b,n)Integration by left hand rule on the interval [a,b] with n subintervals.
The n argument is optional. If it is not given the value of NumericalIntegralSteps is used, which is by default 1000.
Version 1.0.28 onwards.
LeftLimit (f,x0)
Calcule la limite gauche d'une fonction à valeurs réelles en x0.
Limit (f,x0)
Calcule la limite d'une fonction à valeur réelle en x0. Essaie de calculer les deux limites à gauche et à droite.
MidpointRule (f,a,b,n)
Integration by midpoint rule on the interval [a,b] with n subintervals.
The n argument is optional. If it is not given the value of NumericalIntegralSteps is used, which is by default 1000.
The n is optional for version 1.0.28 onwards.
NumericalDerivative (f,x0)
Alias : NDerivative
Essaie de calculer la dérivée par méthode numérique.
See Wikipedia for more information.
NumericalFourierSeriesCoefficients (f,L,N)
Return a vector of vectors [a,b]
where a are the cosine coefficients and
b are the sine coefficients of
the Fourier series of
f with half-period L (that is defined
on [-L,L] and extended periodically) with coefficients
up to Nth harmonic computed numerically. The coefficients are
computed by numerical integration using
NumericalIntegral.
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
NumericalFourierSeriesFunction (f,L,N)
Return a function that is the Fourier series of
f with half-period L (that is defined
on [-L,L] and extended periodically) with coefficients
up to Nth harmonic computed numerically. This is the
trigonometric real series composed of sines and cosines. The coefficients are
computed by numerical integration using
NumericalIntegral.
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
NumericalFourierCosineSeriesCoefficients (f,L,N)
Return a vector of coefficients of
the cosine Fourier series of
f with half-period L. That is,
we take f defined on [0,L]
take the even periodic extension and compute the Fourier series, which
only has cosine terms. The series is computed up to the
Nth harmonic. The coefficients are
computed by numerical integration using
NumericalIntegral.
Note that a@(1) is
the constant coefficient! That is, a@(n) refers to
the term cos(x*(n-1)*pi/L).
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
NumericalFourierCosineSeriesFunction (f,L,N)
Return a function that is the cosine Fourier series of
f with half-period L. That is,
we take f defined on [0,L]
take the even periodic extension and compute the Fourier series, which
only has cosine terms. The series is computed up to the
Nth harmonic. The coefficients are
computed by numerical integration using
NumericalIntegral.
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
NumericalFourierSineSeriesCoefficients (f,L,N)
Return a vector of coefficients of
the sine Fourier series of
f with half-period L. That is,
we take f defined on [0,L]
take the odd periodic extension and compute the Fourier series, which
only has sine terms. The series is computed up to the
Nth harmonic. The coefficients are
computed by numerical integration using
NumericalIntegral.
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
NumericalFourierSineSeriesFunction (f,L,N)
Return a function that is the sine Fourier series of
f with half-period L. That is,
we take f defined on [0,L]
take the odd periodic extension and compute the Fourier series, which
only has sine terms. The series is computed up to the
Nth harmonic. The coefficients are
computed by numerical integration using
NumericalIntegral.
See Wikipedia or Mathworld for more information.
Version 1.0.7 onwards.
NumericalIntegral (f,a,b)
Integration by rule set in NumericalIntegralFunction of f from a to b.
By default NumericalIntegralFunction is the AdaptiveGaussKronrod,
which implements an adaptive algorithm based on the
Gauss-Kronrod G7 K15 rule. It is possible that null is returned if the algorithm cannot find
an approximation within tolerance in a tunable maximum number of iterations.
Gauss-Kronrod is the default algorithm since version 1.0.28 onwards.
NumericalLeftDerivative (f,x0)
Essaie de calculer la dérivée à gauche par méthode numérique.
NumericalLimitAtInfinity (_f,step_fun,tolerance,successive_for_success,N)
Essaie de calculer la limite de f (step_fun(i)) lorsque i va de 1 à N.
NumericalRightDerivative (f,x0)
Essaie de calculer la dérivée à droite par méthode numérique.
OddPeriodicExtension (f,L)
Return a function that is the odd periodic extension of
f with half period L. That
is a function defined on the interval [0,L]
extended to be odd on [-L,L] and then
extended to be periodic with period 2*L.
Consultez aussi EvenPeriodicExtension et PeriodicExtension.
Version 1.0.7 onwards.
OneSidedFivePointFormula (f,x0,h)
Calcule la dérivée d'un côté en utilisant une formule à cinq points.
OneSidedThreePointFormula (f,x0,h)
Calcule la dérivée d'un côté en utilisant une formule à trois points.
PeriodicExtension (f,a,b)
Return a function that is the periodic extension of
f defined on the interval [a,b]
and has period b-a.
Consultez également OddPeriodicExtension et EvenPeriodicExtension.
Version 1.0.7 onwards.
RightHandRule (f,a,b,n)Integration by right hand rule on the interval [a,b] with n subintervals.
The n argument is optional. If it is not given the value of NumericalIntegralSteps is used, which is by default 1000.
Version 1.0.28 onwards.
RightLimit (f,x0)
Calculate the right limit of a real-valued function at x0.
TrapezoidRule (f,a,b,n)Integration by trapezoid rule on the interval [a,b] with n subintervals.
The n argument is optional. If it is not given the value of NumericalIntegralSteps is used, which is by default 1000.
Version 1.0.28 onwards.
TwoSidedFivePointFormula (f,x0,h)
Calcule la dérivée des deux côtés en utilisant une formule à cinq points.
TwoSidedThreePointFormula (f,x0,h)
Calcule la dérivée des deux côtés en utilisant une formule à trois points.