GQRUL

Computes a Gauss, Gauss-Radau, or Gauss-Lobatto quadrature rule with various classical weight functions.

Required Arguments

N — Number of quadrature points. (Input)

QX — Array of length N containing quadrature points. (Output)

QW — Array of length N containing quadrature weights. (Output)

Optional Arguments

IWEIGH — Index of the weight function. (Input)

Default: IWEIGH = 1.

Default: IWEIGH = 1.

IWEIGH | wt(x) | Interval | Name |
---|---|---|---|

1 | 1 | (-1,1) | Legendre |

2 | (-1,1) | Chebyshev 1st kind | |

3 | (-1,1) | Chebyshev 2nd kind | |

4 | (-∞,∞) | Hermite | |

5 | (-1,1) | Jacobi | |

6 | (0,∞) | Generalized Laguerre | |

7 | (-∞,∞) | Hyperbolic Cosine |

ALPHA — Parameter used in the weight function with some values of IWEIGH, otherwise it is ignored. (Input)

Default: ALPHA = 2.0.

Default: ALPHA = 2.0.

BETAW — Parameter used in the weight function with some values of IWEIGH, otherwise it is ignored. (Input)

Default: BETAW = 2.0.

Default: BETAW = 2.0.

NFIX — Number of fixed quadrature points. (Input)

NFIX = 0, 1 or 2. For the usual Gauss quadrature rules, NFIX = 0.

Default: NFIX = 0.

NFIX = 0, 1 or 2. For the usual Gauss quadrature rules, NFIX = 0.

Default: NFIX = 0.

QXFIX — Array of length NFIX (ignored if NFIX = 0) containing the preset quadrature point(s). (Input)

FORTRAN 90 Interface

Generic: CALL GQRUL (N, QX, QW [, …])

Specific: The specific interface names are S_GQRUL and D_GQRUL.

FORTRAN 77 Interface

Single: CALL GQRUL (N, IWEIGH, ALPHA, BETAW, NFIX, QXFIX, QX, QW)

Double: The double precision name is DGQRUL.

Description

The routine GQRUL produces the points and weights for the Gauss, Gauss-Radau, or Gauss-Lobatto quadrature formulas for some of the most popular weights. In fact, it is slightly more general than this suggests because the extra one or two points that may be specified do not have to lie at the endpoints of the interval. This routine is a modification of the subroutine GAUSSQUADRULE (Golub and Welsch 1969).

In the simple case when NFIX = 0, the routine returns points in x = QX and weights in w = QW so that

for all functions f that are polynomials of degree less than 2N.

If NFIX = 1, then one of the above xi equals the first component of QXFIX. Similarly, if NFIX = 2, then two of the components of x will equal the first two components of QXFIX. In general, the accuracy of the above quadrature formula degrades when NFIX increases. The quadrature rule will integrate all functions f that are polynomials of degree less than 2N − NFIX.

Comments

1. Workspace may be explicitly provided, if desired, by use of G2RUL/DG2RUL. The reference is

CALL G2RUL (N, IWEIGH, ALPHA, BETAW, NFIX, QXFIX, QX, QW, WK)

The additional argument is

WK — Work array of length N.

2. If IWEIGH specifies the weight WT(X) and the interval (a, b), then approximately

3. Gaussian quadrature is always the method of choice when the function F(X) behaves like a polynomial. Gaussian quadrature is also useful on infinite intervals (with appropriate weight functions), because other techniques often fail.

4. The weight function 1/cosh(X) behaves like a polynomial near zero and like e|X| far from zero.

Examples

Example 1

In this example, we obtain the classical Gauss-Legendre quadrature formula, which is accurate for polynomials of degree less than 2N, and apply this when N = 6 to the function x8 on the interval [−1, 1]. This quadrature rule is accurate for polynomials of degree less than 12.

USE GQRUL_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER N

PARAMETER (N=6)

INTEGER I, NOUT

REAL ANSWER, QW(N), QX(N), SUM

! Get output unit number

CALL UMACH (2, NOUT)

!

! Get points and weights from GQRUL

CALL GQRUL (N, QX, QW)

! Write results from GQRUL

WRITE (NOUT,99998) (I,QX(I),I,QW(I),I=1,N)

99998 FORMAT (6(6X,'QX(',I1,') = ',F8.4,7X,'QW(',I1,') = ',F8.5,/))

! Evaluate the integral from these

! points and weights

SUM = 0.0

DO 10 I=1, N

SUM = SUM + QX(I)**8*QW(I)

10 CONTINUE

ANSWER = SUM

WRITE (NOUT,99999) ANSWER

99999 FORMAT (/, ' The quadrature result making use of these ', &

'points and weights is ', 1PE10.4, '.')

END

Output

QX(1) = -0.9325 QW(1) = 0.17132

QX(2) = -0.6612 QW(2) = 0.36076

QX(3) = -0.2386 QW(3) = 0.46791

QX(4) = 0.2386 QW(4) = 0.46791

QX(5) = 0.6612 QW(5) = 0.36076

QX(6) = 0.9325 QW(6) = 0.17132

The quadrature result making use of these points and weights is 2.2222E-01.

Example 2

We modify Example 1 by requiring that both endpoints be included in the quadrature formulas and again apply the new formulas to the function x8 on the interval [−1, 1]. This quadrature rule is accurate for polynomials of degree less than 10.

USE GQRUL_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER N

PARAMETER (N=6)

INTEGER I, IWEIGH, NFIX, NOUT

REAL ALPHA, ANSWER, BETAW, QW(N), QX(N), QXFIX(2), SUM

! Get output unit number

CALL UMACH (2, NOUT)

!

IWEIGH = 1

ALPHA = 0.0

BETAW = 0.0

NFIX = 2

QXFIX(1) = -1.0

QXFIX(2) = 1.0

! Get points and weights from GQRUL

CALL GQRUL (N, QX, QW, ALPHA=ALPHA, BETAW=BETAW, NFIX=NFIX, &

QXFIX=QXFIX)

! Write results from GQRUL

WRITE (NOUT,99998) (I,QX(I),I,QW(I),I=1,N)

99998 FORMAT (6(6X,'QX(',I1,') = ',F8.4,7X,'QW(',I1,') = ',F8.5,/))

! Evaluate the integral from these

! points and weights

SUM = 0.0

DO 10 I=1, N

SUM = SUM + QX(I)**8*QW(I)

10 CONTINUE

ANSWER = SUM

WRITE (NOUT,99999) ANSWER

99999 FORMAT (/, ' The quadrature result making use of these ', &

'points and weights is ', 1PE10.4, '.')

END

Output

QX(1) = -1.0000 QW(1) = 0.06667

QX(2) = -0.7651 QW(2) = 0.37847

QX(3) = -0.2852 QW(3) = 0.55486

QX(4) = 0.2852 QW(4) = 0.55486

QX(5) = 0.7651 QW(5) = 0.37847

QX(6) = 1.0000 QW(6) = 0.06667

The quadrature result making use of these points and weights is 2.2222E-01.