Quadrature Rules of Gauss-Jacobi Type

**QUADRATURE_RULES_JACOBI**
is a dataset directory which
contains examples of quadrature rules of Gauss-Jacobi type.

The Gauss-Jacobi quadrature rule is designed
to approximate integrals on the interval [-1,1], with a weight function of the
form **(1-x) ^{ALPHA} * (1+x)^{BETA}**. ALPHA and BETA are
real parameters that must be greater than -1.

Gauss-Jacobi quadrature assumes that the integrand we are considering has a form like:

Integral ( -1 <= x <= +1 ) (1-x)^alpha (1+x)^beta f(x) dx

The *standard Gauss-Jacobi quadrature rule * is used as follows:

Integral ( -1 <= x <= +1 ) (1-x)^alpha (1+x)^beta f(x) dxis to be approximated by

Sum ( 1 <= i <= order ) w(i) * f(x(i))

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the endpoints of the region.

We consider a Gauss-Jacobi quadrature rule of order 4 with ALPHA = 0.5 and BETA = 1.5.

Here is the text of the "W" file storing the weights of such a rule:

```
0.1018214503045086
0.4757517664488109
0.6787436549282700
0.3144794551129494
```

Here is the text of the "X" file storing the abscissas of such a rule:

```
-0.6827529985532060
-0.1614690409023143
0.4056256275378191
0.8385964119177013
```

Here is the text of the "R" file storing the lower and upper limits of the region:

```
-1.0
+1.0
```

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

INT_EXACTNESS_JACOBI, a C++ program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

JACOBI_POLYNOMIAL, a C++ library which evaluates the Jacobi polynomial and associated functions.

JACOBI_RULE, a C++ program which can compute and print a Gauss-Jacobi quadrature rule.

Gauss-Jacobi Rule, Order 4, ALPHA = 1.0, BETA = 0.0:

- jac_o4_a1.0_b0.0_x.txt, the abscissas for the order 4 rule.
- jac_o4_a1.0_b0.0_w.txt, the weights.
- jac_o4_a1.0_b0.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 2.0, BETA = 0.0:

- jac_o4_a2.0_b0.0_x.txt, the abscissas for the order 4 rule.
- jac_o4_a2.0_b0.0_w.txt, the weights.
- jac_o4_a2.0_b0.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 3.0, BETA = 0.0:

- jac_o4_a3.0_b0.0_x.txt, the abscissas for the order 4 rule.
- jac_o4_a3.0_b0.0_w.txt, the weights.
- jac_o4_a3.0_b0.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 1, ALPHA = 0.5, BETA = 1.5:

- jac_o1_a0.5_b1.5_x.txt, the abscissas for the order 1 rule.
- jac_o1_a0.5_b1.5_w.txt, the weights.
- jac_o1_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 2, ALPHA = 0.5, BETA = 1.5:

- jac_o2_a0.5_b1.5_x.txt, the abscissas for the order 2 rule.
- jac_o2_a0.5_b1.5_w.txt, the weights.
- jac_o2_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 0.5, BETA = 1.5:

- jac_o4_a0.5_b1.5_x.txt, the abscissas for the order 4 rule.
- jac_o4_a0.5_b1.5_w.txt, the weights.
- jac_o4_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 8, ALPHA = 0.5, BETA = 1.5:

- jac_o8_a0.5_b1.5_x.txt, the abscissas for the order 8 rule.
- jac_o8_a0.5_b1.5_w.txt, the weights.
- jac_o8_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 16, ALPHA = 0.5, BETA = 1.5:

- jac_o16_a0.5_b1.5_x.txt, the abscissas for the order 16 rule.
- jac_o16_a0.5_b1.5_w.txt, the weights.
- jac_o16_a0.5_b1.5_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 0.0, BETA = 1.0:

- jac_o4_a0.0_b1.0_x.txt, the abscissas for the order 4 rule.
- jac_o4_a0.0_b1.0_w.txt, the weights.
- jac_o4_a0.0_b1.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 0.0, BETA = 2.0:

- jac_o4_a0.0_b2.0_x.txt, the abscissas for the order 4 rule.
- jac_o4_a0.0_b2.0_w.txt, the weights.
- jac_o4_a0.0_b2.0_r.txt, the range of the integration region.

Gauss-Jacobi Rule, Order 4, ALPHA = 0.0, BETA = 3.0:

- jac_o4_a0.0_b3.0_x.txt, the abscissas for the order 4 rule.
- jac_o4_a0.0_b3.0_w.txt, the weights.
- jac_o4_a0.0_b3.0_r.txt, the range of the integration region.

You can go up one level to the DATASETS page.