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 ·