Gaussian Quadrature - Change of Interval

Change of Interval

An integral over must be changed into an integral over before applying the Gaussian quadrature rule. This change of interval can be done in the following way:


\int_a^b f(x)\,dx = \frac{b-a}{2} \int_{-1}^1 f\left(\frac{b-a}{2}z
+ \frac{a+b}{2}\right)\,dz.

After applying the Gaussian quadrature rule, the following approximation is:


\int_a^b f(x)\,dx \approx \frac{b-a}{2} \sum_{i=1}^n w_i f\left(\frac{b-a}{2}z_i + \frac{a+b}{2}\right).

Read more about this topic:  Gaussian Quadrature

Famous quotes containing the words change and/or interval:

    The more specific idea of evolution now reached is—a change from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, accompanying the dissipation of motion and integration of matter.
    Herbert Spencer (1820–1903)

    I was interested to see how a pioneer lived on this side of the country. His life is in some respects more adventurous than that of his brother in the West; for he contends with winter as well as the wilderness, and there is a greater interval of time at least between him and the army which is to follow. Here immigration is a tide which may ebb when it has swept away the pines; there it is not a tide, but an inundation, and roads and other improvements come steadily rushing after.
    Henry David Thoreau (1817–1862)