stdlib_quadrature:local:
:depth: 2
trapz - integrate sampled values using trapezoidal ruleExperimental
Returns the trapezoidal rule integral of an array y representing discrete samples of a function. The integral is computed assuming either equidistant abscissas with spacing dx or arbitrary abscissas x.
result = trapz (y, x)
result = trapz (y, dx)
y: Shall be a rank-one array of type real.
x: Shall be a rank-one array of type real having the same kind and size as y.
dx: Shall be a scalar of type real having the same kind as y.
The result is a scalar of type real having the same kind as y.
If the size of y is zero or one, the result is zero.
```{literalinclude} ../../example/quadrature/example_trapz.f90 :language: fortran
## `trapz_weights` - trapezoidal rule weights for given abscissas
### Status
Experimental
### Description
Given an array of abscissas `x`, computes the array of weights `w` such that if `y` represented function values tabulated at `x`, then `sum(w*y)` produces a trapezoidal rule approximation to the integral.
### Syntax
`result = ` `trapz_weights` `(x)`
### Arguments
`x`: Shall be a rank-one array of type `real`.
### Return value
The result is a `real` array with the same size and kind as `x`.
If the size of `x` is one, then the sole element of the result is zero.
### Example
```{literalinclude} ../../example/quadrature/example_trapz_weights.f90
:language: fortran
simps - integrate sampled values using Simpson’s ruleExperimental
Returns the Simpson’s rule integral of an array y representing discrete samples of a function. The integral is computed assuming either equidistant abscissas with spacing dx or arbitrary abscissas x.
Simpson’s ordinary (“1/3”) rule is used for odd-length arrays. For even-length arrays, Simpson’s 3/8 rule is also utilized in a way that depends on the value of even. If even is negative (positive), the 3/8 rule is used at the beginning (end) of the array. If even is zero or not present, the result is as if the 3/8 rule were first used at the beginning of the array, then at the end of the array, and these two results were averaged.
result = simps (y, x [, even])
result = simps (y, dx [, even])
y: Shall be a rank-one array of type real.
x: Shall be a rank-one array of type real having the same kind and size as y.
dx: Shall be a scalar of type real having the same kind as y.
even: (Optional) Shall be a default-kind integer.
The result is a scalar of type real having the same kind as y.
If the size of y is zero or one, the result is zero.
If the size of y is two, the result is the same as if trapz had been called instead.
```{literalinclude} ../../example/quadrature/example_simps.f90 :language: fortran
## `simps_weights` - Simpson's rule weights for given abscissas
### Status
Experimental
### Description
Given an array of abscissas `x`, computes the array of weights `w` such that if `y` represented function values tabulated at `x`, then `sum(w*y)` produces a Simpson's rule approximation to the integral.
Simpson's ordinary ("1/3") rule is used for odd-length arrays. For even-length arrays, Simpson's 3/8 rule is also utilized in a way that depends on the value of `even`. If `even` is negative (positive), the 3/8 rule is used at the beginning (end) of the array and the 1/3 rule used elsewhere. If `even` is zero or not present, the result is as if the 3/8 rule were first used at the beginning of the array, then at the end of the array, and then these two results were averaged.
### Syntax
`result = ` `simps_weights` `(x [, even])`
### Arguments
`x`: Shall be a rank-one array of type `real`.
`even`: (Optional) Shall be a default-kind `integer`.
### Return value
The result is a `real` array with the same size and kind as `x`.
If the size of `x` is one, then the sole element of the result is zero.
If the size of `x` is two, then the result is the same as if `trapz_weights` had been called instead.
### Example
```{literalinclude} ../../example/quadrature/example_simps_weights.f90
:language: fortran
gauss_legendre - Gauss-Legendre quadrature (a.k.a. Gaussian quadrature) nodes and weightsExperimental
Computes Gauss-Legendre quadrature (also known as simply Gaussian quadrature) nodes and weights,
for any N (number of nodes).
Using the nodes x and weights w, you can compute the integral of some function f as follows:
integral = sum(f(x) * w).
Only double precision is supported - if lower precision is required, you must do the appropriate conversion yourself. Accuracy has been validated up to N=64 by comparing computed results to tablulated values known to be accurate to machine precision (maximum difference from those values is 2 epsilon).
subroutine gauss_legendre ` (x, w[, interval])`
x: Shall be a rank-one array of type real(real64). It is an output argument, representing the quadrature nodes.
w: Shall be a rank-one array of type real(real64), with the same dimension as x.
It is an output argument, representing the quadrature weights.
interval: (Optional) Shall be a two-element array of type real(real64).
If present, the nodes and weigts are calculated for integration from interval(1) to interval(2).
If not specified, the default integral is -1 to 1.
```{literalinclude} ../../example/quadrature/example_gauss_legendre.f90 :language: fortran
## `gauss_legendre_lobatto` - Gauss-Legendre-Lobatto quadrature nodes and weights
### Status
Experimental
### Description
Computes Gauss-Legendre-Lobatto quadrature nodes and weights,
for any `N` (number of nodes).
Using the nodes `x` and weights `w`, you can compute the integral of some function `f` as follows:
`integral = sum(f(x) * w)`.
Only double precision is supported - if lower precision is required, you must do the appropriate conversion yourself.
Accuracy has been validated up to N=64 by comparing computed results to tablulated values known to be accurate to machine precision
(maximum difference from those values is 2 epsilon).
### Syntax
`subroutine ` `gauss_legendre_lobatto` ` (x, w[, interval])`
### Arguments
`x`: Shall be a rank-one array of type `real(real64)`. It is an *output* argument, representing the quadrature nodes.
`w`: Shall be a rank-one array of type `real(real64)`, with the same dimension as `x`.
It is an *output* argument, representing the quadrature weights.
`interval`: (Optional) Shall be a two-element array of type `real(real64)`.
If present, the nodes and weigts are calculated for integration from `interval(1)` to `interval(2)`.
If not specified, the default integral is -1 to 1.
### Example
```{literalinclude} ../../example/quadrature/example_gauss_legendre_lobatto.f90
:language: fortran