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23.1 Functions of One Variable

Octave supports five different adaptive quadrature algorithms for computing the integral of a function f over the interval from a to b. These are

Numerical integration based on Gaussian quadrature.

Numerical integration using an adaptive vectorized Simpson’s rule.

Numerical integration using an adaptive Lobatto rule.

Numerical integration using an adaptive Gauss-Konrod rule.

Numerical integration using adaptive Clenshaw-Curtis rules.

In addition, the following functions are also provided:

integral

A compatibility wrapper function that will choose between quadv and quadgk depending on the integrand and options chosen.

trapz, cumtrapz

Numerical integration of data using the trapezoidal method.

The best quadrature algorithm to use depends on the integrand. If you have empirical data, rather than a function, the choice is trapz or cumtrapz. If you are uncertain about the characteristics of the integrand, quadcc will be the most robust as it can handle discontinuities, singularities, oscillatory functions, and infinite intervals. When the integrand is smooth quadgk may be the fastest of the algorithms.

FunctionCharacteristics
quadgkMedium accuracy (1e-6 – 1e-9) with smooth integrands.
Handles oscillatory functions and infinite bounds
quadccLow to High accuracy with nonsmooth/smooth integrands
Handles oscillatory functions, singularities, and infinite bounds

Here is an example of using quad to integrate the function

f(x) = x * sin (1/x) * sqrt (abs (1 - x))

from x = 0 to x = 3.

This is a fairly difficult integration (plot the function over the range of integration to see why).

The first step is to define the function:

function y = f (x)
y = x .* sin (1./x) .* sqrt (abs (1 - x));
endfunction

Note the use of the ‘dot’ forms of the operators. This is not necessary for the quad integrator, but is required by the other integrators. In any case, it makes it much easier to generate a set of points for plotting because it is possible to call the function with a vector argument to produce a vector result.

The second step is to call quad with the limits of integration:

[q, ier, nfun, err] = quad ("f", 0, 3)
⇒ 1.9819
⇒ 1
⇒ 5061
⇒ 1.1522e-07

Although quad returns a nonzero value for ier, the result is reasonably accurate (to see why, examine what happens to the result if you move the lower bound to 0.1, then 0.01, then 0.001, etc.).

The function "f" can be the string name of a function, a function handle, or an inline function. These options make it quite easy to do integration without having to fully define a function in an m-file. For example:

# Verify integral (x^3) = x^4/4
f = inline ("x.^3");
⇒ 0.25000

# Verify gamma function = (n-1)! for n = 4
f = @(x) x.^3 .* exp (-x);
⇒ 6.0000
q = quad (f, a, b)
q = quad (f, a, b, tol)
q = quad (f, a, b, tol, sing)
[q, ier, nfun, err] = quad (…)

Numerically evaluate the integral of f from a to b using Fortran routines from QUADPACK.

f is a function handle, inline function, or a string containing the name of the function to evaluate. The function must have the form y = f (x) where y and x are scalars.

a and b are the lower and upper limits of integration. Either or both may be infinite.

The optional argument tol is a vector that specifies the desired accuracy of the result. The first element of the vector is the desired absolute tolerance, and the second element is the desired relative tolerance. To choose a relative test only, set the absolute tolerance to zero. To choose an absolute test only, set the relative tolerance to zero. Both tolerances default to sqrt (eps) or approximately 1.5e-8.

The optional argument sing is a vector of values at which the integrand is known to be singular.

The result of the integration is returned in q.

ier contains an integer error code (0 indicates a successful integration).

nfun indicates the number of function evaluations that were made.

err contains an estimate of the error in the solution.

Note: because quad is written in Fortran it cannot be called recursively. This prevents its use in integrating over more than one variable by routines dblquad and triplequad.

Query or set options for the function quad.

When called with no arguments, the names of all available options and their current values are displayed.

Given one argument, return the value of the option opt.

When called with two arguments, quad_options sets the option opt to value val.

Options include

"absolute tolerance"

Absolute tolerance; may be zero for pure relative error test.

"relative tolerance"

Non-negative relative tolerance. If the absolute tolerance is zero, the relative tolerance must be greater than or equal to max (50*eps, 0.5e-28).

"single precision absolute tolerance"

Absolute tolerance for single precision; may be zero for pure relative error test.

"single precision relative tolerance"

Non-negative relative tolerance for single precision. If the absolute tolerance is zero, the relative tolerance must be greater than or equal to max (50*eps, 0.5e-28).

q = quadv (f, a, b)
q = quadv (f, a, b, tol)
q = quadv (f, a, b, tol, trace)
q = quadv (f, a, b, tol, trace, p1, p2, …)

Numerically evaluate the integral of f from a to b using an adaptive Simpson’s rule.

f is a function handle, inline function, or string containing the name of the function to evaluate. quadv is a vectorized version of quad and the function defined by f must accept a scalar or vector as input and return a scalar, vector, or array as output.

a and b are the lower and upper limits of integration. Both limits must be finite.

The optional argument tol defines the absolute tolerance used to stop the adaptation procedure. The default value is 1e-6.

The algorithm used by quadv involves recursively subdividing the integration interval and applying Simpson’s rule on each subinterval. If trace is true then after computing each of these partial integrals display: (1) the total number of function evaluations, (2) the left end of the subinterval, (3) the length of the subinterval, (4) the approximation of the integral over the subinterval.

Additional arguments p1, etc., are passed directly to the function f. To use default values for tol and trace, one may pass empty matrices ([]).

The result of the integration is returned in q.

The optional output nfun indicates the total number of function evaluations performed.

q = quadl (f, a, b)
q = quadl (f, a, b, tol)
q = quadl (f, a, b, tol, trace)
q = quadl (f, a, b, tol, trace, p1, p2, …)

Numerically evaluate the integral of f from a to b using an adaptive Lobatto rule.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be vectorized and return a vector of output values when given a vector of input values.

a and b are the lower and upper limits of integration. Both limits must be finite.

The optional argument tol defines the absolute tolerance with which to perform the integration. The default value is 1e-6.

The algorithm used by quadl involves recursively subdividing the integration interval. If trace is defined then for each subinterval display: (1) the total number of function evaluations, (2) the left end of the subinterval, (3) the length of the subinterval, (4) the approximation of the integral over the subinterval.

Additional arguments p1, etc., are passed directly to the function f. To use default values for tol and trace, one may pass empty matrices ([]).

The result of the integration is returned in q.

The optional output nfun indicates the total number of function evaluations performed.

Reference: W. Gander and W. Gautschi, Adaptive Quadrature - Revisited, BIT Vol. 40, No. 1, March 2000, pp. 84–101. https://www.inf.ethz.ch/personal/gander/

q = quadgk (f, a, b)
q = quadgk (f, a, b, abstol)
q = quadgk (f, a, b, abstol, trace)
q = quadgk (f, a, b, prop, val, …)

Numerically evaluate the integral of f from a to b using adaptive Gauss-Konrod quadrature.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be vectorized and return a vector of output values when given a vector of input values.

a and b are the lower and upper limits of integration. Either or both limits may be infinite or contain weak end singularities. Variable transformation will be used to treat any infinite intervals and weaken the singularities. For example:

quadgk (@(x) 1 ./ (sqrt (x) .* (x + 1)), 0, Inf)

Note that the formulation of the integrand uses the element-by-element operator ./ and all user functions to quadgk should do the same.

The optional argument tol defines the absolute tolerance used to stop the integration procedure. The default value is 1e-10 (1e-5 for single).

The algorithm used by quadgk involves subdividing the integration interval and evaluating each subinterval. If trace is true then after computing each of these partial integrals display: (1) the number of subintervals at this step, (2) the current estimate of the error err, (3) the current estimate for the integral q.

Alternatively, properties of quadgk can be passed to the function as pairs "prop", val. Valid properties are

AbsTol

Define the absolute error tolerance for the quadrature. The default absolute tolerance is 1e-10 (1e-5 for single).

RelTol

Define the relative error tolerance for the quadrature. The default relative tolerance is 1e-6 (1e-4 for single).

MaxIntervalCount

quadgk initially subdivides the interval on which to perform the quadrature into 10 intervals. Subintervals that have an unacceptable error are subdivided and re-evaluated. If the number of subintervals exceeds 650 subintervals at any point then a poor convergence is signaled and the current estimate of the integral is returned. The property "MaxIntervalCount" can be used to alter the number of subintervals that can exist before exiting.

WayPoints

Discontinuities in the first derivative of the function to integrate can be flagged with the "WayPoints" property. This forces the ends of a subinterval to fall on the breakpoints of the function and can result in significantly improved estimation of the error in the integral, faster computation, or both. For example,

quadgk (@(x) abs (1 - x.^2), 0, 2, "Waypoints", 1)

signals the breakpoint in the integrand at x = 1.

Trace

If logically true quadgk prints information on the convergence of the quadrature at each iteration.

If any of a, b, or waypoints is complex then the quadrature is treated as a contour integral along a piecewise continuous path defined by the above. In this case the integral is assumed to have no edge singularities. For example,

quadgk (@(z) log (z), 1+1i, 1+1i, "WayPoints",
[1-1i, -1,-1i, -1+1i])

integrates log (z) along the square defined by [1+1i, 1-1i, -1-1i, -1+1i].

The result of the integration is returned in q.

err is an approximate bound on the error in the integral abs (q - I), where I is the exact value of the integral.

Reference: L.F. Shampine, "Vectorized adaptive quadrature in MATLAB", Journal of Computational and Applied Mathematics, pp. 131–140, Vol 211, Issue 2, Feb 2008.

q = quadcc (f, a, b)
q = quadcc (f, a, b, tol)
q = quadcc (f, a, b, tol, sing)
[q, err, nr_points] = quadcc (…)

Numerically evaluate the integral of f from a to b using doubly-adaptive Clenshaw-Curtis quadrature.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be vectorized and must return a vector of output values if given a vector of input values. For example,

f = @(x) x .* sin (1./x) .* sqrt (abs (1 - x));

which uses the element-by-element “dot” form for all operators.

a and b are the lower and upper limits of integration. Either or both limits may be infinite. quadcc handles an infinite limit by substituting the variable of integration with x = tan (pi/2*u).

The optional argument tol is a 1- or 2-element vector that specifies the desired accuracy of the result. The first element of the vector is the desired absolute tolerance, and the second element is the desired relative tolerance. To choose a relative test only, set the absolute tolerance to zero. To choose an absolute test only, set the relative tolerance to zero. The default absolute tolerance is 1e-10 (1e-5 for single), and the default relative tolerance is 1e-6 (1e-4 for single).

The optional argument sing contains a list of points where the integrand has known singularities, or discontinuities in any of its derivatives, inside the integration interval. For the example above, which has a discontinuity at x=1, the call to quadcc would be as follows

int = quadcc (f, a, b, [], [ 1 ]);

The result of the integration is returned in q.

err is an estimate of the absolute integration error.

nr_points is the number of points at which the integrand was evaluated.

If the adaptive integration did not converge, the value of err will be larger than the requested tolerance. Therefore, it is recommended to verify this value for difficult integrands.

quadcc is capable of dealing with non-numeric values of the integrand such as NaN or Inf. If the integral diverges, and quadcc detects this, then a warning is issued and Inf or -Inf is returned.

Note: quadcc is a general purpose quadrature algorithm and, as such, may be less efficient for a smooth or otherwise well-behaved integrand than other methods such as quadgk.

The algorithm uses Clenshaw-Curtis quadrature rules of increasing degree in each interval and bisects the interval if either the function does not appear to be smooth or a rule of maximum degree has been reached. The error estimate is computed from the L2-norm of the difference between two successive interpolations of the integrand over the nodes of the respective quadrature rules.

Implementation Note: For Octave versions ≤ 4.2, quadcc accepted a single tolerance argument which specified the relative tolerance. For versions 4.4 and 5, quadcc will issue a warning when called with a single tolerance argument indicating that the meaning of this input has changed from relative tolerance to absolute tolerance. The warning ID for this message is "Octave:quadcc:RelTol-conversion". The warning may be disabled with warning ("off", "Octave:quadcc:RelTol-conversion").

Reference: P. Gonnet, Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants, ACM Transactions on Mathematical Software, Vol. 37, Issue 3, Article No. 3, 2010.

q = integral (f, a, b)
q = integral (f, a, b, prop, val, …)

Numerically evaluate the integral of f from a to b using adaptive quadrature.

integral is a wrapper for quadcc (general scalar integrands), quadgk (integrals with specified integration paths), and quadv (array-valued integrands) that is intended to provide MATLAB compatibility. More control of the numerical integration may be achievable by calling the various quadrature functions directly.

f is a function handle, inline function, or string containing the name of the function to evaluate. The function f must be vectorized and return a vector of output values when given a vector of input values.

a and b are the lower and upper limits of integration. Either or both limits may be infinite or contain weak end singularities. If either or both limits are complex, integral will perform a straight line path integral. Alternatively, a complex domain path can be specified using the "Waypoints" option (see below).

Additional optional parameters can be specified using "property", value pairs. Valid properties are:

Waypoints

Specifies points to be used in defining subintervals of the quadrature algorithm, or if a, b, or waypoints are complex then the quadrature is calculated as a contour integral along a piecewise continuous path. For more detail see quadgk.

ArrayValued

integral expects f to return a scalar value unless arrayvalued is specified as true. This option will cause integral to perform the integration over the entire array and return q with the same dimensions as returned by f. For more detail see quadv.

AbsTol

Define the absolute error tolerance for the quadrature. The default absolute tolerance is 1e-10 (1e-5 for single).

RelTol

Define the relative error tolerance for the quadrature. The default relative tolerance is 1e-6 (1e-4 for single).

Adaptive quadrature is used to minimize the estimate of error until the following is satisfied:

error <= max (AbsTol, RelTol*|q|).

Known MATLAB incompatibilities:

1. If tolerances are left unspecified, and any integration limits or waypoints are of type single, then Octave’s integral functions automatically reduce the default absolute and relative error tolerances as specified above. If tighter tolerances are desired they must be specified. MATLAB leaves the tighter tolerances appropriate for double inputs in place regardless of the class of the integration limits.
2. As a consequence of using quadcc, quadgk, and quadv, certain option combinations are not supported. Currently, "ArrayValued" cannot be combined with "RelTol" or "Waypoints".

Sometimes one does not have the function, but only the raw (x, y) points from which to perform an integration. This can occur when collecting data in an experiment. The trapz function can integrate these values as shown in the following example where "data" has been collected on the cosine function over the range [0, pi/2).

x = 0:0.1:pi/2;  # Uniformly spaced points
y = cos (x);
trapz (x, y)
⇒ 0.99666

The answer is reasonably close to the exact value of 1. Ordinary quadrature is sensitive to the characteristics of the integrand. Empirical integration depends not just on the integrand, but also on the particular points chosen to represent the function. Repeating the example above with the sine function over the range [0, pi/2) produces far inferior results.

x = 0:0.1:pi/2;  # Uniformly spaced points
y = sin (x);
trapz (x, y)
⇒ 0.92849

However, a slightly different choice of data points can change the result significantly. The same integration, with the same number of points, but spaced differently produces a more accurate answer.

x = linspace (0, pi/2, 16);  # Uniformly spaced, but including endpoint
y = sin (x);
trapz (x, y)
⇒ 0.99909

In general there may be no way of knowing the best distribution of points ahead of time. Or the points may come from an experiment where there is no freedom to select the best distribution. In any case, one must remain aware of this issue when using trapz.

q = trapz (y)
q = trapz (x, y)
q = trapz (…, dim)

Numerically evaluate the integral of points y using the trapezoidal method.

trapz (y) computes the integral of y along the first non-singleton dimension. When the argument x is omitted an equally spaced x vector with unit spacing (1) is assumed. trapz (x, y) evaluates the integral with respect to the spacing in x and the values in y. This is useful if the points in y have been sampled unevenly.

If the optional dim argument is given, operate along this dimension.

Application Note: If x is not specified then unit spacing will be used. To scale the integral to the correct value you must multiply by the actual spacing value (deltaX). As an example, the integral of x^3 over the range [0, 1] is x^4/4 or 0.25. The following code uses trapz to calculate the integral in three different ways.

x = 0:0.1:1;
y = x.^3;
## No scaling
q = trapz (y)
⇒ q = 2.5250
## Approximation to integral by scaling
q * 0.1
⇒ 0.25250
## Same result by specifying x
trapz (x, y)
⇒ 0.25250

q = cumtrapz (y)
q = cumtrapz (x, y)
q = cumtrapz (…, dim)

Cumulative numerical integration of points y using the trapezoidal method.

cumtrapz (y) computes the cumulative integral of y along the first non-singleton dimension. Where trapz reports only the overall integral sum, cumtrapz reports the current partial sum value at each point of y.

When the argument x is omitted an equally spaced x vector with unit spacing (1) is assumed. cumtrapz (x, y) evaluates the integral with respect to the spacing in x and the values in y. This is useful if the points in y have been sampled unevenly.

If the optional dim argument is given, operate along this dimension.

Application Note: If x is not specified then unit spacing will be used. To scale the integral to the correct value you must multiply by the actual spacing value (deltaX).