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quadrature |
[TOC]
Experimental
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 = [[stdlib_quadrature(module):trapz(interface)]] (y, x)
result = [[stdlib_quadrature(module):trapz(interface)]] (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.
{!example/quadrature/example_trapz.f90!}Experimental
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.
result = [[stdlib_quadrature(module):trapz_weights(interface)]] (x)
x: Shall be a rank-one array of type real.
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/quadrature/example_trapz_weights.f90!}Experimental
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 = [[stdlib_quadrature(module):simps(interface)]] (y, x [, even])
result = [[stdlib_quadrature(module):simps(interface)]] (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.
{!example/quadrature/example_simps.f90!}Experimental
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.
result = [[stdlib_quadrature(module):simps_weights(interface)]] (x [, even])
x: Shall be a rank-one array of type real.
even: (Optional) Shall be a default-kind integer.
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/quadrature/example_simps_weights.f90!}Experimental
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 [[stdlib_quadrature(module):gauss_legendre(interface)]] (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.
{!example/quadrature/example_gauss_legendre.f90!}Experimental
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).
subroutine [[stdlib_quadrature(module):gauss_legendre_lobatto(interface)]] (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.
{!example/quadrature/example_gauss_legendre_lobatto.f90!}