Previous: Introduction to QUADPACK, Up: Integration [Contents][Index]

### 19.4 Functions and Variables for QUADPACK

Function: quad_qag
`quad_qag` (f(x), x, a, b, key, [epsrel, epsabs, limit])
`quad_qag` (f, x, a, b, key, [epsrel, epsabs, limit])

  Integration of a general function over a finite interval. `quad_qag` implements a simple globally adaptive integrator using the strategy of Aind (Piessens, 1973). The caller may choose among 6 pairs of Gauss-Kronrod quadrature formulae for the rule evaluation component. The high-degree rules are suitable for strongly oscillating integrands.

  `quad_qag` computes the integral

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

  The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b. key is the integrator to be used and should be an integer between 1 and 6, inclusive. The value of key selects the order of the Gauss-Kronrod integration rule. High-order rules are suitable for strongly oscillating integrands.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The numerical integration is done adaptively by subdividing the integration region into sub-intervals until the desired accuracy is achieved.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qag` returns a list of four elements:

  - an approximation to the integral,
  - the estimated absolute error of the approximation,
  - the number integrand evaluations,
  - an error code.

  The error code (fourth element of the return value) can have the values:

  `0`

    if no problems were encountered;

  `1`

    if too many sub-intervals were done;

  `2`

    if excessive roundoff error is detected;

  `3`

    if extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qag (x^(1/2)*log(1/x), x, 0, 1, 3, 'epsrel=5d-8);
      (%o1)    [.4444444444492108, 3.1700968502883E-9, 961, 0]

      (%i2) integrate (x^(1/2)*log(1/x), x, 0, 1);
                                      4
      (%o2)                           -
                                      9

  Categories:Numerical methods ·Package quadpack ·

Function: quad_qags
`quad_qags` (f(x), x, a, b, [epsrel, epsabs, limit])
`quad_qags` (f, x, a, b, [epsrel, epsabs, limit])

  Integration of a general function over a finite interval. `quad_qags` implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

  `quad_qags` computes the integral

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

  The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qags` returns a list of four elements:

  - an approximation to the integral,
  - the estimated absolute error of the approximation,
  - the number integrand evaluations,
  - an error code.

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `4`

    failed to converge

  `5`

    integral is probably divergent or slowly convergent

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qags (x^(1/2)*log(1/x), x, 0, 1, 'epsrel=1d-10);
      (%o1)   [.4444444444444448, 1.11022302462516E-15, 315, 0]

  Note that `quad_qags` is more accurate and efficient than `quad_qag` for this integrand.

  Categories:Numerical methods ·Package quadpack ·

Function: quad_qagi
`quad_qagi` (f(x), x, a, b, [epsrel, epsabs, limit])
`quad_qagi` (f, x, a, b, [epsrel, epsabs, limit])

  Integration of a general function over an infinite or semi-infinite interval. The interval is mapped onto a finite interval and then the same strategy as in `quad_qags` is applied.

  `quad_qagi` evaluates one of the following integrals

  $$ \int_a^\infty f(x) \, dx $$ $$ \int_\infty^a f(x) \, dx $$ $$ \int_{-\infty}^\infty f(x) \, dx $$

  using the Quadpack QAGI routine. The function to be integrated is f(x), with dependent variable x, and the function is to be integrated over an infinite range.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  One of the limits of integration must be infinity. If not, then `quad_qagi` will just return the noun form.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qagi` returns a list of four elements:

  - an approximation to the integral,
  - the estimated absolute error of the approximation,
  - the number integrand evaluations,
  - an error code.

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `4`

    failed to converge

  `5`

    integral is probably divergent or slowly convergent

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qagi (x^2*exp(-4*x), x, 0, inf, 'epsrel=1d-8);
      (%o1)        [0.03125, 2.95916102995002E-11, 105, 0]

      (%i2) integrate (x^2*exp(-4*x), x, 0, inf);
                                     1
      (%o2)                          --
                                     32

  Categories:Numerical methods ·Package quadpack ·

Function: quad_qawc
`quad_qawc` (f(x), x, c, a, b, [epsrel, epsabs, limit])
`quad_qawc` (f, x, c, a, b, [epsrel, epsabs, limit])

  Computes the Cauchy principal value of f(x)/(x - c) over a finite interval. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point x = c.

  `quad_qawc` computes the Cauchy principal value of

  $$ \int_{a}^{b}{{{f\left(x\right)}\over{x-c}}\>dx} $$

  using the Quadpack QAWC routine. The function to be integrated is f(x)/(x-c), with dependent variable x, and the function is to be integrated over the interval a to b.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qawc` returns a list of four elements:

  - an approximation to the integral,
  - the estimated absolute error of the approximation,
  - the number integrand evaluations,
  - an error code.

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qawc (2^(-5)*((x-1)^2+4^(-5))^(-1), x, 2, 0, 5,
                       'epsrel=1d-7);
      (%o1)    [- 3.130120337415925, 1.306830140249558E-8, 495, 0]

      (%i2) integrate (2^(-alpha)*(((x-1)^2 + 4^(-alpha))*(x-2))^(-1),
            x, 0, 5);
      Principal Value
                             alpha
              alpha       9 4                 9
             4      log(------------- + -------------)
                            alpha           alpha
                        64 4      + 4   64 4      + 4
      (%o2) (-----------------------------------------
                              alpha
                           2 4      + 2

             3 alpha                       3 alpha
             -------                       -------
                2            alpha/2          2          alpha/2
          2 4        atan(4 4       )   2 4        atan(4       )   alpha
        - --------------------------- - -------------------------)/2
                    alpha                        alpha
                 2 4      + 2                 2 4      + 2

      (%i3) ev (%, alpha=5, numer);
      (%o3)                    - 3.130120337415917

  Categories:Numerical methods ·Package quadpack ·

Function: quad_qawf
`quad_qawf` (f(x), x, a, omega, trig, [epsabs, limit, maxp1, limlst])
`quad_qawf` (f, x, a, omega, trig, [epsabs, limit, maxp1, limlst])

  Calculates a Fourier cosine or Fourier sine transform on a semi-infinite interval using the Quadpack QAWF function. The same approach as in `quad_qawo` is applied on successive finite intervals, and convergence acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the series of the integral contributions.

  `quad_qawf` computes the integral

  $$ \int_a^\infty f(x) \, w(x) \, dx $$

  The weight function w is selected by trig:

  `cos`
    \(w(x) = \cos\omega x\)
  `sin`
    \(w(x) = \sin\omega x\)

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsabs`

    Desired absolute error of approximation. Default is 1d-10.

  `limit`

    Size of internal work array. (limit - limlst)/2 is the maximum number of subintervals to use. Default is 200.

  `maxp1`

    Maximum number of Chebyshev moments. Must be greater than 0. Default is 100.

  `limlst`

    Upper bound on the number of cycles. Must be greater than or equal to 3. Default is 10.

  `quad_qawf` returns a list of four elements:

  - an approximation to the integral,
  - the estimated absolute error of the approximation,
  - the number integrand evaluations,
  - an error code.

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qawf (exp(-x^2), x, 0, 1, 'cos, 'epsabs=1d-9);
      (%o1)   [.6901942235215714, 2.84846300257552E-11, 215, 0]

      (%i2) integrate (exp(-x^2)*cos(x), x, 0, inf);
                                - 1/4
                              %e      sqrt(%pi)
      (%o2)                   -----------------
                                      2

      (%i3) ev (%, numer);
      (%o3)                   .6901942235215714

  Categories:Numerical methods ·Package quadpack ·

Function: quad_qawo
`quad_qawo` (f(x), x, a, b, omega, trig, [epsrel, epsabs, limit, maxp1, limlst])
`quad_qawo` (f, x, a, b, omega, trig, [epsrel, epsabs, limit, maxp1, limlst])

  Integration of \(\cos(\omega x) f(x)\) or \(\sin(\omega x)\) over a finite interval, where \(\omega\) is a constant. The rule evaluation component is based on the modified Clenshaw-Curtis technique. `quad_qawo` applies adaptive subdivision with extrapolation, similar to `quad_qags`.

  `quad_qawo` computes the integral using the Quadpack QAWO routine:

  $$ \int_a^b f(x) \, w(x) \, dx $$

  The weight function w is selected by trig:

  `cos`
    \(w(x) = \cos\omega x\)
  `sin`
    \(w(x) = \sin\omega x\)

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit/2 is the maximum number of subintervals to use. Default is 200.

  `maxp1`

    Maximum number of Chebyshev moments. Must be greater than 0. Default is 100.

  `limlst`

    Upper bound on the number of cycles. Must be greater than or equal to 3. Default is 10.

  `quad_qawo` returns a list of four elements:

  - an approximation to the integral,
  - the estimated absolute error of the approximation,
  - the number integrand evaluations,
  - an error code.

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qawo (x^(-1/2)*exp(-2^(-2)*x), x, 1d-8, 20*2^2, 1, cos);
      (%o1)     [1.376043389877692, 4.72710759424899E-11, 765, 0]

      (%i2) rectform (integrate (x^(-1/2)*exp(-2^(-alpha)*x) * cos(x),
            x, 0, inf));
                         alpha/2 - 1/2            2 alpha
              sqrt(%pi) 2              sqrt(sqrt(2        + 1) + 1)
      (%o2)   -----------------------------------------------------
                                     2 alpha
                               sqrt(2        + 1)

      (%i3) ev (%, alpha=2, numer);
      (%o3)                     1.376043390090716

  Categories:Numerical methods ·Package quadpack ·

Function: quad_qaws
`quad_qaws` (f(x), x, a, b, alpha, beta, wfun, [epsrel, epsabs, limit])
`quad_qaws` (f, x, a, b, alpha, beta, wfun, [epsrel, epsabs, limit])

  Integration of w(x) f(x) over a finite interval, where w(x) is a certain algebraic or logarithmic function. A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain the endpoints of the interval of integration.

  `quad_qaws` computes the integral using the Quadpack QAWS routine:

  $$ \int_a^b f(x) \, w(x) \, dx $$

  The weight function w is selected by wfun:

  `1`
    \(w(x) = (x - a)^\alpha (b - x)^\beta\)
  `2`
    \(w(x) = (x - a)^\alpha (b - x)^\beta \log(x - a)\)
  `3`
    \(w(x) = (x - a)^\alpha (b - x)^\beta \log(b - x)\)
  `4`
    \(w(x) = (x - a)^\alpha (b - x)^\beta \log(x - a) \log(b - x)\)

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limitis the maximum number of subintervals to use. Default is 200.

  `quad_qaws` returns a list of four elements:

  - an approximation to the integral,
  - the estimated absolute error of the approximation,
  - the number integrand evaluations,
  - an error code.

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qaws (1/(x+1+2^(-4)), x, -1, 1, -0.5, -0.5, 1,
                       'epsabs=1d-9);
      (%o1)     [8.750097361672832, 1.24321522715422E-10, 170, 0]

      (%i2) integrate ((1-x*x)^(-1/2)/(x+1+2^(-alpha)), x, -1, 1);
             alpha
      Is  4 2      - 1  positive, negative, or zero?

      pos;
                                alpha         alpha
                         2 %pi 2      sqrt(2 2      + 1)
      (%o2)              -------------------------------
                                     alpha
                                  4 2      + 2

      (%i3) ev (%, alpha=4, numer);
      (%o3)                     8.750097361672829

  Categories:Numerical methods ·Package quadpack ·

Function: quad_qagp
`quad_qagp` (f(x), x, a, b, points, [epsrel, epsabs, limit])
`quad_qagp` (f, x, a, b, points, [epsrel, epsabs, limit])

  Integration of a general function over a finite interval. `quad_qagp` implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

  `quad_qagp` computes the integral

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

  The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b.

  The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

  To help the integrator, the user must supply a list of points where the integrand is singular or discontinuous.

  The keyword arguments are optional and may be specified in any order. They all take the form `key=val`. The keyword arguments are:

  `epsrel`

    Desired relative error of approximation. Default is 1d-8.

  `epsabs`

    Desired absolute error of approximation. Default is 0.

  `limit`

    Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

  `quad_qagp` returns a list of four elements:

  - an approximation to the integral,
  - the estimated absolute error of the approximation,
  - the number integrand evaluations,
  - an error code.

  The error code (fourth element of the return value) can have the values:

  `0`

    no problems were encountered;

  `1`

    too many sub-intervals were done;

  `2`

    excessive roundoff error is detected;

  `3`

    extremely bad integrand behavior occurs;

  `4`

    failed to converge

  `5`

    integral is probably divergent or slowly convergent

  `6`

    if the input is invalid.

  Examples:

      (%i1) quad_qagp(x^3*log(abs((x^2-1)*(x^2-2))),x,0,3,[1,sqrt(2)]);
      (%o1)   [52.74074838347143, 2.6247632689546663e-7, 1029, 0]

      (%i2) quad_qags(x^3*log(abs((x^2-1)*(x^2-2))), x, 0, 3);
      (%o2)   [52.74074847951494, 4.088443219529836e-7, 1869, 0]

  The integrand has singularities at `1` and `sqrt(2)` so we supply these points to `quad_qagp`. We also note that `quad_qagp` is more accurate and more efficient that `quad_qags`.

  Categories:Numerical methods ·Package quadpack ·

Function: quad_control (parameter, [value])

  Control error handling for quadpack. The parameter should be one of the following symbols:

  `current_error`

    The current error number

  `control`

    Controls if messages are printed or not. If it is set to zero or less, messages are suppressed.

  `max_message`

    The maximum number of times any message is to be printed.

  If value is not given, then the current value of the parameter is returned. If value is given, the value of parameter is set to the given value.

  Categories:Numerical methods ·Package quadpack ·

Previous: Introduction to QUADPACK, Up: Integration [Contents][Index]