FunctionIntegrator.chebyshev_quadratureMethod
chebyshev_quadrature(f::Function, N::Number, k::Integer, a::Number, b::Number)

uses Chebyshev-Gauss quadrature to approximate the integral:

$\displaystyle\int_a^b f(x) dx$.

$N$ is the number of nodes (or grid points) used. Whilst the type mentioned in the function definition is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer.

$k$ determines the kind of the Chebyshev polynomial used in the quadrature.

• $k=1$ uses Chebyshev polynomials of the first kind $T_n(x)$.
• $k=2$ uses Chebyshev polynomials of the second kind $U_n(x)$.
• $k=3$ uses Chebyshev polynomials of the third kind $V_n(x)$.
• $k=4$ uses Chebyshev polynomials of the fourth kind $W_n(x)$.
FunctionIntegrator.hermite_quadratureFunction
hermite_quadrature(f::Function, N::Number, k::Integer=1)

when $k=2$ approximates the integral:

$\displaystyle \int_{-\infty}^{\infty} e^{-x^2} f(x) dx$

and otherwise approximates the integral:

$\displaystyle \int_{-\infty}^{\infty} f(x) dx$

in both instances it uses Gauss-Hermite quadrature to make this approximation.

$N$ is the number of nodes (or grid points) used. Whilst the type mentioned in the function definition is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer.

FunctionIntegrator.jacobi_quadratureMethod
jacobi_quadrature(f::Function, N::Number, α::Number, β::Number, a::Number, b::Number)

$\displaystyle \int_a^b f(x) dx.$

The remaining inputs of this function are:

• $N$ is the number of nodes (or grid points) used. Whilst the type mentioned in the function definition is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer.
• $α$ and $β$, which are parameters of the weighting function.
FunctionIntegrator.laguerre_quadratureFunction
laguerre_quadrature(f::Function, N::Number, k::Integer=1)

approximates the integral:

$\displaystyle \int_0^{\infty} f(x) dx$

or if $k$ is set to something other than 1:

$\displaystyle \int_0^{\infty} e^{-x} f(x) dx$

$N$ is the number of nodes (or grid points) used. Whilst the type mentioned in the function definition is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer.

FunctionIntegrator.legendre_quadratureMethod
legendre_quadrature(f::Function, N::Number, a::Number, b::Number)

$\displaystyle \int_a^b f(x) dx$.

$N$ is the number of nodes (or grid points) used. Whilst the type mentioned in the function definition is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer.

FunctionIntegrator.lobatto_quadratureMethod
lobatto_quadrature(f::Function, N::Number, a::Number, b::Number)

$\displaystyle \int_a^b f(x) dx.$

$N$ is the number of nodes (or grid points) used. Whilst the type mentioned in the function definition is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer.

FunctionIntegrator.radau_quadratureMethod
radau_quadrature(f::Function, N::Number, a::Number, b::Number)

$\displaystyle \int_a^b f(x) dx.$

$N$ is the number of nodes (or grid points) used. Whilst the type mentioned in the function definition is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer.

FunctionIntegrator.rectangle_rule_leftMethod
rectangle_rule_left(f::Function, N::Number, a::Number, b::Number)

numerically approximates:

$\displaystyle\int_a^b f(x) dx$

using the left rectangle rule with $N$ gridpoint values. Whilst the type mentioned in the function definition for $N$ is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'); an error message will be printed if it is not a positive integer.

FunctionIntegrator.rectangle_rule_midpointMethod
rectangle_rule_midpoint(f::Function, N::Number, a::Number, b::Number)

numerically approximates:

$\displaystyle\int_a^b f(x) dx$

using the midpoint rectangle rule with $N$ gridpoint values. Whilst the type mentioned in the function definition for $N$ is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'); an error message will be printed if it is not a positive integer.

FunctionIntegrator.rectangle_rule_rightMethod
rectangle_rule_right(f::Function, N::Number, a::Number, b::Number)

numerically approximates:

$\displaystyle\int_a^b f(x) dx$

using the right rectangle rule with $N$ gridpoint values. Whilst the type mentioned in the function definition for $N$ is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'); an error message will be printed if it is not a positive integer.

FunctionIntegrator.simpsons38_ruleMethod
simpsons38_rule(f::Function, N::Number, a::Number, b::Number)

uses Simpson's 3/8 rule to approximate:

$\displaystyle\int_a^b f(x) dx$.

$N+1$ steps are used, if endpoints are included. Whilst the type mentioned in the function definition for $N$ is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer. In order to apply Simpson's 3/8 rule, $N$ must be divisible by 3.

FunctionIntegrator.simpsons_ruleMethod
simpsons_rule(f::Function, N::Number, a::Number, b::Number)

uses Simpson's rule to approximate:

$\displaystyle\int_a^b f(x) dx$.

$N+1$ steps are used, if endpoints are included. Whilst the type mentioned in the function definition for $N$ is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive even integer.

FunctionIntegrator.trapezoidal_ruleMethod
trapezoidal_rule(f::Function, N::Number, a::Number, b::Number)

uses the trapezoidal rule to approximate:

$\displaystyle\int_a^b f(x) dx$.

$N+1$ steps are used, if endpoints are included. Whilst the type mentioned in the function definition for $N$ is just 'Number' (as opposed to 'Integer'), this is just so that scientific notation can be used to define it (as scientific notation gives the type 'Float64'), an error message will be printed if it is not a positive integer.