# `stdlib_quadrature` ```{contents} :local: :depth: 2 ``` ## `trapz` - integrate sampled values using trapezoidal rule ### Status Experimental ### Description 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`. ### Syntax `result = ` `trapz` `(y, x)` `result = ` `trapz` `(y, dx)` ### Arguments `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`. ### Return value 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. ### Example ```{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 rule ### Status Experimental ### Description 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. ### Syntax `result = ` `simps` `(y, x [, even])` `result = ` `simps` `(y, dx [, even])` ### Arguments `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`. ### Return value 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. ### Example ```{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 weights ### Status Experimental ### Description 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). ### Syntax `subroutine ` `gauss_legendre` ` (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.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 ```